diff --git a/frontiers/determinant_moment_20261025/LINEAGE_VALIDATION.md b/frontiers/determinant_moment_20261025/LINEAGE_VALIDATION.md index 5a0c5d20e..aff89fa91 100644 --- a/frontiers/determinant_moment_20261025/LINEAGE_VALIDATION.md +++ b/frontiers/determinant_moment_20261025/LINEAGE_VALIDATION.md @@ -1,5 +1,22 @@ # Declared-author lineage validation +## 25 October Gaussian research successor contributor disclosure + +Root author: OpenAI/Codex /root. All scientific proposal/code/API helpers in this +successor use the same OpenAI model family: /root/uniform_jet_moment_sources, +/root/normalizer_palm_research and its scratch-only gaussian_reference_probe, +/root/regression_kernel and its gaussian_api helper, +/root/dm2_landed_composition, /root/palm_measure_transfer and its scratch-only +density_api helper. Source contracts assign their mathematical scope. +Their compiler checks, mathematical controls, kernel evidence and technical +reading receive zero provider-distinct scientific-review credit. An attempted +bridge helper was not created and is not represented as a contributor/reviewer. +Historical source authors/reviews remain at their original immutable identities; +neither inherited code nor historical scientific acceptance is reauthored here. + +The text below describes the unchanged historical validator. This disclosure +does not create an accepted alignment record or change its distinctness rules. + Scientific effect: **NONE**. This is an engineering successor to the validator, not an alignment acceptance, new proof, or scientific-status authority. Implementer: OpenAI / GPT-6 Astra Pro, ChatGPT session diff --git a/frontiers/determinant_moment_20261025/README.md b/frontiers/determinant_moment_20261025/README.md index 0b9eae248..397baff49 100644 --- a/frontiers/determinant_moment_20261025/README.md +++ b/frontiers/determinant_moment_20261025/README.md @@ -1,39 +1,40 @@ -# Exact endpoint-Hessian determinant second moment +# Gaussian endpoint moments, canonical conditioning and normalizer research -P02-LM-002: for the source's exact nine-jet endpoint convention and 0 (egm_HM r j).det) := by + simp only [egm_HM, Matrix.det_fin_two] + fun_prop + have hs : Continuous (fun j : Fin 6 → ℝ => (egm_HS r j).det) := by + simp only [egm_HS, Matrix.det_fin_two] + fun_prop + have htype : MeasurableSet {j : Fin 6 → ℝ | + dm2Negative (egm_HM r j) ∧ (egm_HS r j).det < 0} := by + simp only [egm_HM, dm2_negative_iff] + apply MeasurableSet.inter + · apply MeasurableSet.inter + · have h : Continuous (fun j : Fin 6 → ℝ => r * j 0) := by fun_prop + exact (isOpen_lt h continuous_const).measurableSet + · have h : Continuous (fun j : Fin 6 → ℝ => + (r * j 0) * j 2 - (r * j 1) ^ 2) := by fun_prop + exact (isOpen_lt continuous_const h).measurableSet + · exact (isOpen_lt hs continuous_const).measurableSet + change Measurable (fun j => if dm2Negative (egm_HM r j) ∧ (egm_HS r j).det < 0 + then |(egm_HM r j).det * (egm_HS r j).det| else 0) + exact Measurable.ite htype (hm.measurable.mul hs.measurable).abs measurable_const + +private theorem egm_coordinate_bound (j : Fin 6 → ℝ) (i : Fin 6) : + |j i| ≤ gm8Radius j := by + have hsum : |j i| ≤ ∑ k, |j k| := + Finset.single_le_sum (fun k _ => abs_nonneg (j k)) (Finset.mem_univ i) + unfold gm8Radius + linarith + +private theorem egm_factor_bound (r a b c R : ℝ) (hr : 0 ≤ r) (hr1 : r ≤ 1) + (hR : 0 ≤ R) (ha : |a| ≤ R) (hb : |b| ≤ R) (hc : |c| ≤ R) : + |a * c - r * b ^ 2| ≤ 2 * R ^ 2 := by + have hac : |a * c| ≤ R ^ 2 := by + rw [abs_mul, pow_two] + exact mul_le_mul ha hc (abs_nonneg c) hR + have hbb : b ^ 2 ≤ R ^ 2 := by nlinarith [sq_abs b, abs_nonneg b] + have hrbb : |r * b ^ 2| ≤ R ^ 2 := by + rw [abs_of_nonneg (mul_nonneg hr (sq_nonneg b))] + exact (mul_le_of_le_one_left (sq_nonneg b) hr1).trans hbb + exact (abs_sub _ _).trans (by linarith) + +private theorem egm_untyped (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) (hr1 : r ≤ 1) : + ((egm_HM r j).det * (egm_HS r j).det) ^ 2 ≤ + 16 * r ^ 4 * gm8Radius j ^ 8 := by + let R := gm8Radius j + let dM := j 0 * j 2 - r * j 1 ^ 2 + let dS := j 3 * j 5 - r * j 4 ^ 2 + have hR : 0 ≤ R := by dsimp [R, gm8Radius]; positivity + have hM : |dM| ≤ 2 * R ^ 2 := + egm_factor_bound r _ _ _ R hr.le hr1 hR + (egm_coordinate_bound j 0) (egm_coordinate_bound j 1) (egm_coordinate_bound j 2) + have hS : |dS| ≤ 2 * R ^ 2 := + egm_factor_bound r _ _ _ R hr.le hr1 hR + (egm_coordinate_bound j 3) (egm_coordinate_bound j 4) (egm_coordinate_bound j 5) + have hprod : |dM * dS| ≤ (2 * R ^ 2) ^ 2 := by + rw [abs_mul, pow_two] + exact mul_le_mul hM hS (abs_nonneg _) (by positivity) + have hsq : (dM * dS) ^ 2 ≤ ((2 * R ^ 2) ^ 2) ^ 2 := by + simpa only [sq_abs] using + (sq_le_sq₀ (abs_nonneg (dM * dS)) (sq_nonneg (2 * R ^ 2))).2 hprod + have hdet : ((egm_HM r j).det * (egm_HS r j).det) ^ 2 = + r ^ 4 * (dM * dS) ^ 2 := by + simp [egm_HM, egm_HS, Matrix.det_fin_two, dM, dS] + ring + calc + _ = r ^ 4 * (dM * dS) ^ 2 := hdet + _ ≤ r ^ 4 * ((2 * R ^ 2) ^ 2) ^ 2 := + mul_le_mul_of_nonneg_left hsq (by positivity) + _ = 16 * r ^ 4 * gm8Radius j ^ 8 := by dsimp [R]; ring + +/-- Exact endpoint determinant domination in the actual normalized six-vector. +The type indicators can only reduce the nonnegative determinant product. -/ +theorem egm_pointwise (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) (hr1 : r ≤ 1) : + egm_weight r j ^ 2 ≤ 16 * r ^ 4 * gm8Radius j ^ 8 := by + classical + unfold egm_weight + split_ifs + · simpa only [sq_abs] using egm_untyped r j hr hr1 + · simpa only [zero_pow (by decide : (2 : ℕ) ≠ 0)] using + (show 0 ≤ 16 * r ^ 4 * gm8Radius j ^ 8 by positivity) + +/-- Genuine Gaussian marginal laws of the actual normalized endpoint vector +supply both integrability and its determinant second moment. The derivative-map, +conditioning law and uniform parameter bounds remain explicit model inputs. -/ +theorem egm_gaussian_moment {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsProbabilityMeasure μ] (J : Ω → Fin 6 → ℝ) + (hJ : Measurable J) (m : Fin 6 → ℝ) (v : Fin 6 → ℝ≥0) + (B V r : ℝ) (hB : 0 ≤ B) (hV : 0 ≤ V) (hr : 0 < r) (hr1 : r ≤ 1) + (hmean : ∀ i, |m i| ≤ B) (hvar : ∀ i, (v i : ℝ) ≤ V) + (hlaw : ∀ i, HasLaw (fun x => J x i) (gaussianReal (m i) (v i)) μ) : + Integrable (fun x => egm_weight r (J x) ^ 2) μ ∧ + (∫ x, egm_weight r (J x) ^ 2 ∂μ) ≤ 16 * gm8FiniteM8 6 B V * r ^ 4 := by + obtain ⟨h8, hbound⟩ := gm8_finite_integrable_bound 6 μ J hJ m v B V hB hV + hmean hvar hlaw + have hmajor : Integrable (fun x => 16 * r ^ 4 * gm8Radius (J x) ^ 8) μ := + h8.const_mul _ + have hmeas : AEStronglyMeasurable (fun x => egm_weight r (J x) ^ 2) μ := + (((egm_weight_measurable r).comp hJ).pow_const 2).aestronglyMeasurable + have hi : Integrable (fun x => egm_weight r (J x) ^ 2) μ := + hmajor.mono' hmeas (Filter.Eventually.of_forall (fun x => by + rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] + exact egm_pointwise r (J x) hr hr1)) + refine ⟨hi, ?_⟩ + calc + _ ≤ ∫ x, 16 * r ^ 4 * gm8Radius (J x) ^ 8 ∂μ := + integral_mono hi hmajor (fun x => egm_pointwise r (J x) hr hr1) + _ = 16 * r ^ 4 * (∫ x, gm8Radius (J x) ^ 8 ∂μ) := integral_const_mul _ _ + _ ≤ 16 * r ^ 4 * gm8FiniteM8 6 B V := + mul_le_mul_of_nonneg_left hbound (by positivity) + _ = 16 * gm8FiniteM8 6 B V * r ^ 4 := by ring + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/EndpointNormalizerBridge.lean b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/EndpointNormalizerBridge.lean new file mode 100644 index 000000000..73d861b7e --- /dev/null +++ b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/EndpointNormalizerBridge.lean @@ -0,0 +1,191 @@ +import ResearchFormalCoreR1.EndpointGaussianMoments +import ResearchFormalCoreR1.NormalizerFloor + +/-! +# Literal endpoint weight and continuous normalized weight + +The actual endpoint vector has order `(aM,bM,cM,aS,bS,cS)`, while the +continuous normalizer uses `(aM,aS,bM,bS,cM,cS)`. The permutation below +and the identity hold pointwise, including all determinant/type boundaries. +See ../SOURCE_ENDPOINT_NORMALIZER_BRIDGE.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory Filter +open scoped Topology NNReal + +set_option autoImplicit false + +/-- Convert the actual six endpoint entries to the continuous weight's order. -/ +def enb_permutation (j : Fin 6 → ℝ) : Fin 6 → ℝ := + ![j 0, j 3, j 1, j 4, j 2, j 5] + +theorem enb_permutation_continuous : Continuous enb_permutation := by + unfold enb_permutation + fun_prop + +private theorem enb_maximum_factor (r a b c : ℝ) (hr : 0 < r) : + max (max (-a) 0 * max (-c) 0 - r * b ^ 2) 0 = + if a < 0 ∧ 0 < a * c - r * b ^ 2 then a * c - r * b ^ 2 else 0 := by + by_cases h : a < 0 ∧ 0 < a * c - r * b ^ 2 + · have hc : c < 0 := by + by_contra hn + have hac : a * c ≤ 0 := mul_nonpos_of_nonpos_of_nonneg h.1.le (le_of_not_gt hn) + have hrb : 0 ≤ r * b ^ 2 := mul_nonneg hr.le (sq_nonneg b) + linarith [h.2] + rw [ite_eq_left h, max_eq_left (neg_nonneg.mpr h.1.le), + max_eq_left (neg_nonneg.mpr hc.le)] + have heq : -a * -c - r * b ^ 2 = a * c - r * b ^ 2 := by ring + rw [heq, max_eq_left h.2.le] + · rw [ite_eq_right h] + by_cases ha : a < 0 + · have hd : a * c - r * b ^ 2 ≤ 0 := + le_of_not_gt (fun hp => h ⟨ha, hp⟩) + by_cases hc : c ≤ 0 + · rw [max_eq_left (neg_nonneg.mpr ha.le), max_eq_left (neg_nonneg.mpr hc)] + have heq : -a * -c - r * b ^ 2 = a * c - r * b ^ 2 := by ring + rw [heq, max_eq_right hd] + · rw [max_eq_right (by linarith : -c ≤ 0)] + simpa using (max_eq_right (neg_nonpos.mpr (mul_nonneg hr.le (sq_nonneg b)))) + · rw [max_eq_right (by linarith : -a ≤ 0)] + simpa using (max_eq_right (neg_nonpos.mpr (mul_nonneg hr.le (sq_nonneg b)))) + +/-- Strict negative definiteness in actual endpoint coordinates, with no +assumed sign of the other diagonal entry or omission of singular boundaries. -/ +theorem enb_maximum_iff (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) : + dm2Negative (egm_HM r j) ↔ j 0 < 0 ∧ 0 < j 0 * j 2 - r * j 1 ^ 2 := by + rw [egm_HM, dm2_negative_iff] + have heq : r * j 0 * j 2 - (r * j 1) ^ 2 = + r * (j 0 * j 2 - r * j 1 ^ 2) := by ring + have hmul (x : ℝ) : r * x < 0 ↔ x < 0 := by + constructor <;> intro hx <;> nlinarith + rw [heq, hmul, mul_pos_iff_of_pos_left hr] + +/-- The literal strict saddle condition is the sign of its factored determinant. -/ +theorem enb_saddle_iff (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) : + (egm_HS r j).det < 0 ↔ j 3 * j 5 - r * j 4 ^ 2 < 0 := by + have heq : (egm_HS r j).det = r * (j 3 * j 5 - r * j 4 ^ 2) := by + simp [egm_HS, Matrix.det_fin_two] + ring + rw [heq] + constructor <;> intro h <;> nlinarith + +/-- Exact bridge from the literal typed physical endpoint weight to the +continuous expression. All excluded types and all zero boundaries give zero. -/ +theorem enb_weight_identity (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) : + egm_weight r j = r ^ 2 * gn_normalizedWeight r (enb_permutation j) := by + classical + have hM := enb_maximum_iff r j hr + have hS := enb_saddle_iff r j hr + have hdetM : (egm_HM r j).det = r * (j 0 * j 2 - r * j 1 ^ 2) := by + simp [egm_HM, Matrix.det_fin_two] + ring + have hdetS : (egm_HS r j).det = r * (j 3 * j 5 - r * j 4 ^ 2) := by + simp [egm_HS, Matrix.det_fin_two] + ring + unfold egm_weight gn_normalizedWeight + change (if dm2Negative (egm_HM r j) ∧ (egm_HS r j).det < 0 then + |(egm_HM r j).det * (egm_HS r j).det| else 0) = + r ^ 2 * (max (max (-j 0) 0 * max (-j 2) 0 - r * j 1 ^ 2) 0 * + max (r * j 4 ^ 2 - j 3 * j 5) 0) + rw [enb_maximum_factor r (j 0) (j 1) (j 2) hr] + by_cases hmax : j 0 < 0 ∧ 0 < j 0 * j 2 - r * j 1 ^ 2 + · rw [ite_eq_left hmax] + by_cases hsad : j 3 * j 5 - r * j 4 ^ 2 < 0 + · rw [ite_eq_left ⟨hM.mpr hmax, hS.mpr hsad⟩, hdetM, hdetS] + rw [max_eq_left (by linarith : 0 ≤ r * j 4 ^ 2 - j 3 * j 5)] + have hp : r * (j 0 * j 2 - r * j 1 ^ 2) * + (r * (j 3 * j 5 - r * j 4 ^ 2)) ≤ 0 := + mul_nonpos_of_nonneg_of_nonpos (mul_nonneg hr.le hmax.2.le) + (mul_nonpos_of_nonneg_of_nonpos hr.le hsad.le) + rw [abs_of_nonpos hp] + ring + · rw [ite_eq_right (fun h => hsad (hS.mp h.2)), + max_eq_right (by linarith : r * j 4 ^ 2 - j 3 * j 5 ≤ 0)] + ring + · rw [ite_eq_right hmax, ite_eq_right (fun h => hmax (hM.mp h.1))] + ring + +/-- The physical and continuous normalizers are the same integral up to the +exact radius factor, for any common random six-vector and any measure. -/ +theorem enb_integral_identity {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (J : Ω → Fin 6 → ℝ) (r : ℝ) (hr : 0 < r) : + (∫ ω, egm_weight r (J ω) ∂P) = + r ^ 2 * ∫ ω, gn_normalizedWeight r (enb_permutation (J ω)) ∂P := by + calc + _ = ∫ ω, r ^ 2 * gn_normalizedWeight r (enb_permutation (J ω)) ∂P := + integral_congr_ae (Eventually.of_forall (fun ω => enb_weight_identity r (J ω) hr)) + _ = _ := integral_const_mul _ _ + +/-- Positive radius makes the exact scaling nonzero, so either weight is +integrable precisely when the other is. -/ +theorem enb_integrable_iff {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (J : Ω → Fin 6 → ℝ) (r : ℝ) (hr : 0 < r) : + Integrable (fun ω => egm_weight r (J ω)) P ↔ + Integrable (fun ω => gn_normalizedWeight r (enb_permutation (J ω))) P := by + have heq : (fun ω => egm_weight r (J ω)) = + (fun ω => r ^ 2 * gn_normalizedWeight r (enb_permutation (J ω))) := + funext (fun ω => enb_weight_identity r (J ω) hr) + rw [heq, integrable_const_mul_iff (isUnit_iff_ne_zero.mpr (pow_ne_zero 2 hr.ne'))] + +/-- On a finite measure, the actual endpoint second-moment theorem supplies +the first-moment hypotheses needed by the normalizer join. -/ +theorem enb_integrable_of_integrable_sq {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) [IsFiniteMeasure P] + (J : Ω → Fin 6 → ℝ) (hJ : Measurable J) (r : ℝ) (hr : 0 < r) + (hsq : Integrable (fun ω => egm_weight r (J ω) ^ 2) P) : + Integrable (fun ω => egm_weight r (J ω)) P ∧ + Integrable (fun ω => gn_normalizedWeight r (enb_permutation (J ω))) P := by + have hmajor : Integrable (fun ω => 1 + egm_weight r (J ω) ^ 2) P := + (integrable_const 1).add hsq + have hphysical : Integrable (fun ω => egm_weight r (J ω)) P := by + apply hmajor.mono' (((egm_weight_measurable r).comp hJ).aestronglyMeasurable) + filter_upwards [] with ω + rw [Real.norm_eq_abs] + change |egm_weight r (J ω)| ≤ 1 + egm_weight r (J ω) ^ 2 + nlinarith [sq_nonneg (|egm_weight r (J ω)| - 1/2), sq_abs (egm_weight r (J ω))] + exact ⟨hphysical, (enb_integrable_iff P J r hr).mp hphysical⟩ + +/-- The proved common-coupling normalizer floor transfers to the literal +physical endpoint weight, with its own actual coordinate order. Current +physical integrability and the conditioned-field convergence remain inputs. -/ +theorem enb_normalizer_floor_of_common_coupling + {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) [IsProbabilityMeasure P] + (J : ℝ → Ω → Fin 6 → ℝ) (hJ : ∀ r, Measurable (J r)) + (A Q : Ω → ℝ) (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) + (hQ : HasLaw Q (gaussianReal m v) P) + (hconv : ∀ᵐ ω ∂P, Tendsto (fun r => J r ω) (𝓝[>] (0 : ℝ)) + (𝓝 ![-1,-A ω/2,Q ω,1,A ω/2,Q ω])) + (hint : ∀ᶠ r in 𝓝[>] (0 : ℝ), + Integrable (fun ω => egm_weight r (J r ω)) P) : + ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → + c * r ^ 2 ≤ ∫ ω, egm_weight r (J r ω) ∂P := by + let X : ℝ → Ω → Fin 6 → ℝ := fun r ω => enb_permutation (J r ω) + have hX (r : ℝ) : Measurable (X r) := + enb_permutation_continuous.measurable.comp (hJ r) + have hXconv : ∀ᵐ ω ∂P, Tendsto (fun r => X r ω) (𝓝[>] (0 : ℝ)) + (𝓝 ![-1,1,-A ω/2,A ω/2,Q ω,Q ω]) := by + filter_upwards [hconv] with ω hω + have hp := (enb_permutation_continuous.tendsto + ![-1,-A ω/2,Q ω,1,A ω/2,Q ω]).comp hω + have hcontact : enb_permutation ![-1,-A ω/2,Q ω,1,A ω/2,Q ω] = + ![-1,1,-A ω/2,A ω/2,Q ω,Q ω] := rfl + simpa only [Function.comp_def, hcontact, X] using hp + have hXint : ∀ᶠ r in 𝓝[>] (0 : ℝ), + Integrable (fun ω => gn_normalizedWeight r (X r ω)) P := by + filter_upwards [hint, self_mem_nhdsWithin] with r hri hr + exact (enb_integrable_iff P (J r) r hr).mp hri + obtain ⟨c, hc, δ, hδ, hfloor⟩ := + gn_normalizer_floor_of_common_coupling P X hX A Q m v hv hQ hXconv hXint + refine ⟨c, hc, δ, hδ, ?_⟩ + intro r hr hrδ + have hbound := hfloor r hr hrδ + calc + c * r ^ 2 ≤ ∫ ω, r ^ 2 * gn_normalizedWeight r (X r ω) ∂P := hbound + _ = ∫ ω, egm_weight r (J r ω) ∂P := + integral_congr_ae (Eventually.of_forall (fun ω => (enb_weight_identity r (J r ω) hr).symm)) + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianCoupling.lean b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianCoupling.lean new file mode 100644 index 000000000..143c6253a --- /dev/null +++ b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianCoupling.lean @@ -0,0 +1,65 @@ +import Mathlib.Probability.Distributions.Gaussian.Multivariate +import Mathlib.Probability.HasLaw +import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric + +/-! +# One continuous standard-Gaussian coupling of a finite Gaussian family + +PSD square-root continuity allows singular covariance and arbitrary varying +means. The affine maps have the literal pinned multivariateGaussian laws. +Actual source endpoint means/covariances and their limits remain application +joins; see ../SOURCE_GAUSSIAN_COUPLING.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter Matrix WithLp +open scoped Topology ENNReal NNReal MatrixOrder Matrix.Norms.L2Operator +set_option autoImplicit false + +private theorem gc_matrix_eval_tendsto {ι R : Type*} [Fintype ι] [DecidableEq ι] + {u : Filter R} {S : R → Matrix ι ι ℝ} {S0 : Matrix ι ι ℝ} + (hS : Tendsto S u (𝓝 S0)) (z : EuclideanSpace ℝ ι) : + Tendsto (fun r => Matrix.toEuclideanCLM (𝕜 := ℝ) (S r) z) u + (𝓝 (Matrix.toEuclideanCLM (𝕜 := ℝ) S0 z)) := by + have hcoord (i j : ι) : Tendsto (fun r => S r i j) u (𝓝 (S0 i j)) := + ((continuous_apply j).tendsto (S0 i)).comp + (((continuous_apply i).tendsto S0).comp hS) + have hvec : Tendsto (fun r => (S r) *ᵥ ofLp z) u (𝓝 (S0 *ᵥ ofLp z)) := by + apply tendsto_pi_nhds.mpr + intro i + exact tendsto_finsetSum Finset.univ (fun j _ => + (hcoord i j).mul tendsto_const_nhds) + change Tendsto (fun r => toLp 2 ((S r) *ᵥ ofLp z)) u + (𝓝 (toLp 2 (S0 *ᵥ ofLp z))) + exact ((PiLp.continuous_toLp 2 (fun _ : ι => ℝ)).tendsto _).comp hvec + +/-- The canonical affine vector on one common standard Gaussian space. -/ +noncomputable def gc_coupledVector {ι : Type*} [Fintype ι] [DecidableEq ι] + (m : EuclideanSpace ℝ ι) (S : Matrix ι ι ℝ) (z : EuclideanSpace ℝ ι) : + EuclideanSpace ℝ ι := m + Matrix.toEuclideanCLM (𝕜 := ℝ) (CFC.sqrt S) z + +/-- Exact marginal law; this is the pinned multivariateGaussian definition. -/ +theorem gc_coupledVector_hasLaw {ι : Type*} [Fintype ι] [DecidableEq ι] + (m : EuclideanSpace ℝ ι) (S : Matrix ι ι ℝ) : + HasLaw (gc_coupledVector m S) (multivariateGaussian m S) + (stdGaussian (EuclideanSpace ℝ ι)) where + aemeasurable := by unfold gc_coupledVector; fun_prop + map_eq := rfl + +/-- Every common Gaussian coordinate converges when means and PSD covariances +converge. No invertibility, fixed rank, or square-root Lipschitz estimate. -/ +theorem gc_coupledVector_tendsto {ι κ : Type*} [Fintype ι] [DecidableEq ι] + (u : Filter κ) (m : κ → EuclideanSpace ℝ ι) (S : κ → Matrix ι ι ℝ) + (m0 : EuclideanSpace ℝ ι) (S0 : Matrix ι ι ℝ) + (hm : Tendsto m u (𝓝 m0)) (hS : Tendsto S u (𝓝 S0)) + (hPSD : ∀ᶠ t in u, (S t).PosSemidef) (hPSD0 : S0.PosSemidef) + (z : EuclideanSpace ℝ ι) : + Tendsto (fun t => gc_coupledVector (m t) (S t) z) u + (𝓝 (gc_coupledVector m0 S0 z)) := by + have hwithin : Tendsto S u (𝓝[{B : Matrix ι ι ℝ | 0 ≤ B}] S0) := + tendsto_nhdsWithin_iff.mpr ⟨hS, hPSD.mono (fun t ht => ht.nonneg)⟩ + have hsqrt : Tendsto (fun t => CFC.sqrt (S t)) u (𝓝 (CFC.sqrt S0)) := + (CFC.continuousOn_sqrt S0 hPSD0.nonneg).tendsto.comp hwithin + have heval := gc_matrix_eval_tendsto hsqrt z + exact hm.add heval +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianJetMoments.lean b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianJetMoments.lean new file mode 100644 index 000000000..52498daf0 --- /dev/null +++ b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianJetMoments.lean @@ -0,0 +1,124 @@ +import ResearchFormalCoreR1.DeterminantMoment +import ResearchFormalCoreR1.GaussianScalarMoments +import Mathlib.Probability.HasLaw +import Mathlib.Algebra.Order.Chebyshev + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory +open scoped BigOperators NNReal + +set_option autoImplicit false + +/-- Radius for a finite vector, including its constant one. -/ +def gm8Radius {n : ℕ} (j : Fin n → ℝ) : ℝ := 1 + ∑ i, |j i| + +/-- One finite-dimensional Gaussian radius moment constant. -/ +noncomputable def gm8FiniteM8 (n : ℕ) (B V : ℝ) : ℝ := + ((n+1 : ℕ) : ℝ)^7 * (1 + (n : ℝ)*128*(B^8 + V^4*gm8C8)) + +theorem gm8FiniteM8_nonneg (n : ℕ) (B V : ℝ) : 0 ≤ gm8FiniteM8 n B V := by + unfold gm8FiniteM8 + have hC := gm8_standard_nonneg + positivity + +/-- The source's exact fixed nine-coordinate constant. -/ +noncomputable def gm8M8 (B V : ℝ) : ℝ := + (10 : ℝ)^7 * (1 + 9*128*(B^8 + V^4*gm8C8)) + +theorem gm8M8_nonneg (B V : ℝ) : 0 ≤ gm8M8 B V := by + simpa [gm8M8, gm8FiniteM8] using gm8FiniteM8_nonneg 9 B V + +theorem gm8_finite_radius_power_bound (n : ℕ) (j : Fin n → ℝ) : + gm8Radius j ^ 8 ≤ ((n+1 : ℕ) : ℝ)^7 * (1 + ∑ i, |j i|^8) := by + let f : Fin (n+1) → ℝ := Fin.cases 1 (fun i : Fin n => |j i|) + have h := pow_sum_le_card_mul_sum_pow (s := Finset.univ) (f := f) + (fun i _ => by + refine Fin.cases ?_ (fun k => ?_) i + · norm_num [f] + · exact abs_nonneg (j k)) 7 + simpa [f, gm8Radius, Fin.sum_univ_succ] using h + +theorem gm8_finite_integrable_bound (n : ℕ) {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsProbabilityMeasure μ] (J : Ω → Fin n → ℝ) + (hJ : Measurable J) (m : Fin n → ℝ) (v : Fin n → ℝ≥0) + (B V : ℝ) (hB : 0 ≤ B) (hV : 0 ≤ V) + (hmean : ∀ i, |m i| ≤ B) (hvar : ∀ i, (v i : ℝ) ≤ V) + (hlaw : ∀ i, HasLaw (fun x => J x i) (gaussianReal (m i) (v i)) μ) : + Integrable (fun x => gm8Radius (J x)^8) μ ∧ + (∫ x, gm8Radius (J x)^8 ∂μ) ≤ gm8FiniteM8 n B V := by + have hi (i : Fin n) : Integrable (fun x => |J x i|^8) μ := by + exact (hlaw i).integrable_comp + (gm8_scalar_integrable_bound (m i) B V (v i) hB hV (hmean i) (hvar i)).1 + have hsum : Integrable (fun x => ∑ i, |J x i|^8) μ := + integrable_finsetSum _ (fun i _ => hi i) + have hmajor : Integrable (fun x => ((n+1 : ℕ) : ℝ)^7*(1+∑ i, |J x i|^8)) μ := + ((integrable_const 1).add hsum).const_mul _ + have hc : Continuous (gm8Radius (n := n)) := by unfold gm8Radius; fun_prop + have hmeas : AEStronglyMeasurable (fun x => gm8Radius (J x)^8) μ := + ((hc.measurable.comp hJ).pow_const 8).aestronglyMeasurable + have hR : Integrable (fun x => gm8Radius (J x)^8) μ := + hmajor.mono' hmeas (Filter.Eventually.of_forall (fun x => by + rw [Real.norm_eq_abs, abs_of_nonneg (pow_nonneg (by unfold gm8Radius; positivity) 8)] + exact gm8_finite_radius_power_bound n (J x))) + have hcoord (i : Fin n) : + (∫ x, |J x i|^8 ∂μ) ≤ 128*(B^8+V^4*gm8C8) := by + have heq := (hlaw i).integral_comp + (show AEStronglyMeasurable (fun y : ℝ => |y|^8) (gaussianReal (m i) (v i)) by fun_prop) + exact heq.trans_le (gm8_scalar_integrable_bound (m i) B V (v i) hB hV + (hmean i) (hvar i)).2 + refine ⟨hR, ?_⟩ + calc + _ ≤ ∫ x, ((n+1 : ℕ) : ℝ)^7*(1+∑ i, |J x i|^8) ∂μ := + integral_mono hR hmajor (fun x => gm8_finite_radius_power_bound n (J x)) + _ = ((n+1 : ℕ) : ℝ)^7*(1+∑ i, ∫ x, |J x i|^8 ∂μ) := by + rw [integral_const_mul, integral_add (integrable_const _) hsum, + integral_finsetSum _ (fun i _ => hi i)] + simp + _ ≤ ((n+1 : ℕ) : ℝ)^7*(1+∑ _i : Fin n, 128*(B^8+V^4*gm8C8)) := by + apply mul_le_mul_of_nonneg_left _ (by positivity) + exact add_le_add le_rfl (Finset.sum_le_sum (fun i _ => hcoord i)) + _ = gm8FiniteM8 n B V := by simp [gm8FiniteM8, mul_assoc] + +/-- The exact DM2 radius is the nine-dimensional specialization. -/ +theorem gm8_radius_power_bound (j : DM2Jet) : + dm2Radius j ^ 8 ≤ (10 : ℝ)^7 * (1 + ∑ i, |j i|^8) := by + simpa [gm8Radius, dm2Radius] using gm8_finite_radius_power_bound 9 j + +/-- The nine-jet Gaussian marginal moment interface, with no independence or +moment conclusion assumed. -/ +theorem gm8_jet_integrable_bound {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsProbabilityMeasure μ] (J : Ω → DM2Jet) + (hJ : Measurable J) (m : Fin 9 → ℝ) (v : Fin 9 → ℝ≥0) + (B V : ℝ) (hB : 0 ≤ B) (hV : 0 ≤ V) + (hmean : ∀ i, |m i| ≤ B) (hvar : ∀ i, (v i : ℝ) ≤ V) + (hlaw : ∀ i, HasLaw (fun x => J x i) (gaussianReal (m i) (v i)) μ) : + Integrable (fun x => dm2Radius (J x)^8) μ ∧ + (∫ x, dm2Radius (J x)^8 ∂μ) ≤ gm8M8 B V := by + simpa [gm8Radius, dm2Radius, gm8M8, gm8FiniteM8] using + gm8_finite_integrable_bound 9 μ J hJ m v B V hB hV hmean hvar hlaw + +/-- The same B,V give one determinant second-moment coefficient throughout +0 J r x i) (gaussianReal (m r i) (v r i)) (μ r)) : + 0 ≤ 16*gm8M8 B V ∧ ∀ r, 0 < r → r ≤ r0 → + Integrable (fun x => dm2Weight r (J r x)^2) (μ r) ∧ + (∫ x, dm2Weight r (J r x)^2 ∂μ r) ≤ (16*gm8M8 B V)*r^4 := by + refine ⟨mul_nonneg (by norm_num) (gm8M8_nonneg B V), ?_⟩ + intro r hr hr0 + have : IsProbabilityMeasure (μ r) := hprob r hr hr0 + obtain ⟨h8, hbound⟩ := gm8_jet_integrable_bound (μ r) (J r) (hJ r hr hr0) + (m r) (v r) B V hB hV (hmean r hr hr0) (hvar r hr hr0) (hlaw r hr hr0) + exact dm2_moment (μ r) (J r) (hJ r hr hr0) r (gm8M8 B V) hr (hr0.trans hr01) + h8 hbound + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianNormalizer.lean b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianNormalizer.lean new file mode 100644 index 000000000..a9657cfb8 --- /dev/null +++ b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianNormalizer.lean @@ -0,0 +1,110 @@ +import Mathlib.Probability.Distributions.Gaussian.Real +import Mathlib.MeasureTheory.Integral.Bochner.Basic +import Mathlib.MeasureTheory.Measure.Lebesgue.Basic +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum +import Mathlib.Tactic.FunProp +import Mathlib.Tactic.Positivity + +/-! +# Gaussian reference positivity and an endpoint-box weight floor + +The literal continuous normalized endpoint weight is from LM006 stability, +sections 1–2. These components do not construct the conditioned field law, +its endpoint convergence, a positive box mass, or the full normalizer floor. +See ../SOURCE_GAUSSIAN_NORMALIZER.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory +open scoped ENNReal NNReal + +set_option autoImplicit false + +/-- The six coordinates are aM,aS,bM,bS,cM,cS. For physical positive radius, +the source identifies this continuous expression with its typed W_r/r². -/ +noncomputable def gn_normalizedWeight (r : ℝ) (v : Fin 6 → ℝ) : ℝ := + max (max (-v 0) 0 * max (-v 4) 0 - r * (v 2)^2) 0 * + max (r * (v 3)^2 - v 1 * v 5) 0 + +/-- The contact negative-curvature reference has an integrable, strictly +positive squared negative part, with arbitrary mean and nonzero variance. -/ +theorem gn_gaussian_reference_integrable_pos (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) : + Integrable (fun q : ℝ => (max (-q) 0)^2) (gaussianReal m v) ∧ + 0 < ∫ q : ℝ, (max (-q) 0)^2 ∂gaussianReal m v := by + have hsq : Integrable (fun q : ℝ => q^2) (gaussianReal m v) := by + simpa using (memLp_id_gaussianReal (μ := m) (v := v) 2).integrable_sq + have hint : Integrable (fun q : ℝ => (max (-q) 0)^2) (gaussianReal m v) := by + apply hsq.mono' (by fun_prop) + filter_upwards [] with q + rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] + rcases le_total 0 q with hq | hq + · rw [max_eq_right (neg_nonpos.mpr hq)] + simpa using sq_nonneg q + · rw [max_eq_left (neg_nonneg.mpr hq)] + nlinarith + have hsupp : Function.support (fun q : ℝ => (max (-q) 0)^2) = Set.Iio 0 := by + ext q + simp only [Function.mem_support, Set.mem_Iio] + constructor + · intro h + by_contra hn + have hq : 0 ≤ q := le_of_not_gt hn + apply h + simp [max_eq_right (neg_nonpos.mpr hq)] + · intro hq + have hp : 0 < (max (-q) 0)^2 := sq_pos_of_pos (lt_max_of_lt_left (neg_pos.mpr hq)) + exact hp.ne' + have hmass : 0 < gaussianReal m v (Set.Iio 0) := by + apply pos_iff_ne_zero.mpr + intro hzero + have hvol := (gaussianReal_absolutelyContinuous' m hv) hzero + simp at hvol + refine ⟨hint, ?_⟩ + apply (integral_pos_iff_support_of_nonneg (fun q => sq_nonneg _) hint).mpr + simpa only [hsupp] using hmass + +/-- An explicit open endpoint box has a uniform positive normalized weight. +This is a pointwise component, with no claimed probability mass for the box. -/ +theorem gn_open_box_floor + (r K : ℝ) (v : Fin 6 → ℝ) + (hr : 0 ≤ r) (hK : 0 < K) (hsmall : r * K^2 ≤ 1/4) + (haM : v 0 ∈ Set.Ioo (-5/4) (-3/4)) + (haS : v 1 ∈ Set.Ioo (3/4) (5/4)) + (hbM : |v 2| < K) (_hbS : |v 3| < K) + (hcM : v 4 ∈ Set.Ioo (-2) (-1)) + (hcS : v 5 ∈ Set.Ioo (-2) (-1)) : + (3 : ℝ)/8 ≤ gn_normalizedWeight r v := by + have ha0 : 3/4 ≤ -v 0 := by linarith [haM.2] + have hc4 : 1 ≤ -v 4 := by linarith [hcM.2] + have ha1 : 3/4 ≤ v 1 := haS.1.le + have hc5 : 1 ≤ -v 5 := by linarith [hcS.2] + have hAM : max (-v 0) 0 = -v 0 := max_eq_left (by linarith) + have hCM : max (-v 4) 0 = -v 4 := max_eq_left (by linarith) + have hb2 : (v 2)^2 ≤ K^2 := by + simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg (v 2)) hK.le).2 hbM.le + have hrb : r * (v 2)^2 ≤ 1/4 := + (mul_le_mul_of_nonneg_left hb2 hr).trans hsmall + have hprodM : (3 : ℝ)/4 ≤ (-v 0) * (-v 4) := by + have h := mul_le_mul ha0 hc4 (by norm_num : (0 : ℝ) ≤ 1) (by linarith : 0 ≤ -v 0) + norm_num at h ⊢ + exact h + have hprodS : (3 : ℝ)/4 ≤ -(v 1 * v 5) := by + have h := mul_le_mul ha1 hc5 (by norm_num : (0 : ℝ) ≤ 1) (by linarith : 0 ≤ v 1) + norm_num at h + nlinarith + have hF : (1 : ℝ)/2 ≤ max (max (-v 0) 0 * max (-v 4) 0 - r * (v 2)^2) 0 := by + rw [hAM, hCM] + exact (show (1 : ℝ)/2 ≤ (-v 0) * (-v 4) - r * (v 2)^2 by linarith).trans + (le_max_left _ _) + have hG : (3 : ℝ)/4 ≤ max (r * (v 3)^2 - v 1 * v 5) 0 := by + have hp : 0 ≤ r * (v 3)^2 := mul_nonneg hr (sq_nonneg _) + exact (show (3 : ℝ)/4 ≤ r * (v 3)^2 - v 1 * v 5 by linarith).trans + (le_max_left _ _) + have h := mul_le_mul hF hG (by norm_num : (0 : ℝ) ≤ 3/4) + (le_max_right _ _) + norm_num at h + exact h + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianNormalizerLimit.lean b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianNormalizerLimit.lean new file mode 100644 index 000000000..be08ef92c --- /dev/null +++ b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianNormalizerLimit.lean @@ -0,0 +1,210 @@ +import ResearchFormalCoreR1.GaussianCoupling +import ResearchFormalCoreR1.NormalizerFloor +import ResearchFormalCoreR1.EndpointNormalizerBridge +import Mathlib.Probability.Moments.Variance +import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp + +/-! +# A full normalizer floor from finite Gaussian parameter limits + +Contact identities follow from the displayed zero variances. The common +coupling, noncentered nondegenerate scalar reference and weight integrability +are derived. No positive normalizer, coupling limit or moment bound is assumed. +Actual torus conditional means/covariances remain a separate source join. +See ../SOURCE_GAUSSIAN_NORMALIZER_LIMIT.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter Matrix WithLp +open scoped Topology ENNReal NNReal MatrixOrder Matrix.Norms.L2Operator +set_option autoImplicit false + +private theorem gnl_contact_relations (m : EuclideanSpace ℝ (Fin 6)) (S : Matrix (Fin 6) (Fin 6) ℝ) + (hS : S.PosSemidef) + (hm0 : m 0 = -1) (hm1 : m 1 = 1) + (hm23 : m 2 + m 3 = 0) (hm45 : m 4 - m 5 = 0) + (hS0 : S 0 0 = 0) (hS1 : S 1 1 = 0) + (hS23 : S 2 2 + 2*S 2 3 + S 3 3 = 0) + (hS45 : S 4 4 - 2*S 4 5 + S 5 5 = 0) : + ∀ᵐ x ∂multivariateGaussian m S, + x 0 = -1 ∧ x 1 = 1 ∧ x 2 + x 3 = 0 ∧ x 4 - x 5 = 0 := by + let P := multivariateGaussian m S + have hLp (i : Fin 6) : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => x i) 2 P := by + exact (IsGaussian.memLp_two_id (μ := P)).continuousLinearMap_comp (EuclideanSpace.proj (𝕜 := ℝ) i) + have hi (i : Fin 6) : ∫ x : EuclideanSpace ℝ (Fin 6), x i ∂P = m i := by + change ∫ x, (EuclideanSpace.proj (𝕜 := ℝ) i) x ∂P = m i + rw [ContinuousLinearMap.integral_comp_id_comm] + · simp [P] + · exact IsGaussian.integrable_id + have hvar0 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 0; P] = 0 := by + exact (variance_eval_multivariateGaussian hS 0).trans hS0 + have hvar1 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 1; P] = 0 := by + exact (variance_eval_multivariateGaussian hS 1).trans hS1 + have hvar23 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 2 + x 3; P] = 0 := by + rw [variance_fun_add (hLp 2) (hLp 3)] + simpa [P, variance_eval_multivariateGaussian hS, + covariance_eval_multivariateGaussian hS] using hS23 + have hvar45 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 4 - x 5; P] = 0 := by + rw [variance_fun_sub (hLp 4) (hLp 5)] + simpa [P, variance_eval_multivariateGaussian hS, + covariance_eval_multivariateGaussian hS] using hS45 + have hi23 : ∫ x : EuclideanSpace ℝ (Fin 6), (x 2 + x 3) ∂P = 0 := by + rw [integral_add ((hLp 2).integrable (by norm_num)) ((hLp 3).integrable (by norm_num)), hi, hi] + exact hm23 + have hi45 : ∫ x : EuclideanSpace ℝ (Fin 6), (x 4 - x 5) ∂P = 0 := by + rw [integral_sub ((hLp 4).integrable (by norm_num)) ((hLp 5).integrable (by norm_num)), hi, hi] + exact hm45 + filter_upwards [ae_eq_integral_of_variance_eq_zero (hLp 0) hvar0, + ae_eq_integral_of_variance_eq_zero (hLp 1) hvar1, + ae_eq_integral_of_variance_eq_zero ((hLp 2).add (hLp 3)) hvar23, + ae_eq_integral_of_variance_eq_zero ((hLp 4).sub (hLp 5)) hvar45] + with x hx0 hx1 hx23 hx45 + exact ⟨hx0.trans ((hi 0).trans hm0), hx1.trans ((hi 1).trans hm1), + hx23.trans hi23, hx45.trans hi45⟩ + +private lemma gnl_positivePart_memLp {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {f : Ω → ℝ} {p : ℝ≥0∞} (hf : MemLp f p P) : + MemLp (fun x => max (f x) 0) p P := by + apply hf.norm.mono' ((hf.aestronglyMeasurable.aemeasurable.sup_const 0).aestronglyMeasurable) + filter_upwards [] with x + rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg (le_max_right _ _)] + exact max_le (le_abs_self _) (abs_nonneg _) + +private lemma gnl_gaussian_weight_integrable (r : ℝ) (m : EuclideanSpace ℝ (Fin 6)) + (S : Matrix (Fin 6) (Fin 6) ℝ) (hS : S.PosSemidef) : + Integrable (fun x : EuclideanSpace ℝ (Fin 6) => gn_normalizedWeight r (ofLp x)) + (multivariateGaussian m S) := by + have : ENNReal.HolderTriple 4 4 2 := ⟨by + rw [← two_mul, show (4 : ℝ≥0∞) = 2*2 by norm_num, ENNReal.mul_inv (by simp) (by simp), + ← mul_assoc, ENNReal.mul_inv_cancel two_ne_zero ENNReal.ofNat_ne_top, one_mul]⟩ + let P := multivariateGaussian m S + have hc (i : Fin 6) : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => x i) 4 P := by + have hp := measurePreserving_eval_multivariateGaussian (μ := m) hS (i := i) + have hlaw : HasLaw (fun x : EuclideanSpace ℝ (Fin 6) => x i) + (gaussianReal (m i) (S i i).toNNReal) P := ⟨hp.measurable.aemeasurable, hp.map_eq⟩ + exact hlaw.memLp (memLp_id_gaussianReal 4) + have hpos (i : Fin 6) : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => max (-x i) 0) 4 P := + gnl_positivePart_memLp (hc i).neg + have hM : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => + max (-x 0) 0 * max (-x 4) 0 - r*(x 2)^2) 2 P := by + have h1 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => max (-x 0) 0 * max (-x 4) 0) 2 P := + (hpos 0).fun_mul (hpos 4) + have h2 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => (x 2)^2) 2 P := by + simpa only [pow_two] using (hc 2).fun_mul (hc 2) + exact h1.sub (h2.const_mul r) + have hSadd : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => + r*(x 3)^2 - x 1*x 5) 2 P := by + have h1 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => (x 3)^2) 2 P := by + simpa only [pow_two] using (hc 3).fun_mul (hc 3) + have h2 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => x 1*x 5) 2 P := + (hc 1).fun_mul (hc 5) + exact (h1.const_mul r).sub h2 + have hT : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => gn_normalizedWeight r (ofLp x)) 1 P := + (gnl_positivePart_memLp hM).fun_mul (gnl_positivePart_memLp hSadd) + exact memLp_one_iff_integrable.mp hT + + +/-- Gaussian endpoint parameter convergence and concrete contact moments supply +one positive physical normalizer coefficient at every sufficiently small radius. +The covariance may be singular; the transverse reference mean is unrestricted. -/ +theorem gnl_normalizer_floor_of_parameter_limits + (m : ℝ → EuclideanSpace ℝ (Fin 6)) (S : ℝ → Matrix (Fin 6) (Fin 6) ℝ) + (m0 : EuclideanSpace ℝ (Fin 6)) (S0 : Matrix (Fin 6) (Fin 6) ℝ) + (hm : Tendsto m (𝓝[>] (0 : ℝ)) (𝓝 m0)) + (hS : Tendsto S (𝓝[>] (0 : ℝ)) (𝓝 S0)) + (hPSD : ∀ᶠ r in 𝓝[>] (0 : ℝ), (S r).PosSemidef) (hPSD0 : S0.PosSemidef) + (hm0 : m0 0 = -1) (hm1 : m0 1 = 1) + (hm23 : m0 2 + m0 3 = 0) (hm45 : m0 4 - m0 5 = 0) + (hS0 : S0 0 0 = 0) (hS1 : S0 1 1 = 0) + (hS23 : S0 2 2 + 2*S0 2 3 + S0 3 3 = 0) + (hS45 : S0 4 4 - 2*S0 4 5 + S0 5 5 = 0) (hQpos : 0 < S0 4 4) : + ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → c*r^2 ≤ + ∫ x : EuclideanSpace ℝ (Fin 6), r^2 * gn_normalizedWeight r (ofLp x) + ∂multivariateGaussian (m r) (S r) := by + let P := stdGaussian (EuclideanSpace ℝ (Fin 6)) + let X : ℝ → EuclideanSpace ℝ (Fin 6) → Fin 6 → ℝ := + fun r z => ofLp (gc_coupledVector (m r) (S r) z) + let A : EuclideanSpace ℝ (Fin 6) → ℝ := fun z => 2 * gc_coupledVector m0 S0 z 3 + let Q : EuclideanSpace ℝ (Fin 6) → ℝ := fun z => gc_coupledVector m0 S0 z 4 + have hX (r : ℝ) : Measurable (X r) := by + dsimp [X, gc_coupledVector] + fun_prop + have hLaw0 : HasLaw (gc_coupledVector m0 S0) (multivariateGaussian m0 S0) P := + gc_coupledVector_hasLaw m0 S0 + have hcontacts : ∀ᵐ z ∂P, + gc_coupledVector m0 S0 z 0 = -1 ∧ gc_coupledVector m0 S0 z 1 = 1 ∧ + gc_coupledVector m0 S0 z 2 + gc_coupledVector m0 S0 z 3 = 0 ∧ + gc_coupledVector m0 S0 z 4 - gc_coupledVector m0 S0 z 5 = 0 := by + have hb : ∀ᵐ x ∂P.map (gc_coupledVector m0 S0), + x 0 = -1 ∧ x 1 = 1 ∧ x 2 + x 3 = 0 ∧ x 4 - x 5 = 0 := by + rw [hLaw0.map_eq] + exact gnl_contact_relations m0 S0 hPSD0 hm0 hm1 hm23 hm45 hS0 hS1 hS23 hS45 + exact ae_of_ae_map hLaw0.aemeasurable hb + have hconv : ∀ᵐ z ∂P, Tendsto (fun r => X r z) (𝓝[>] (0 : ℝ)) + (𝓝 ![-1,1,-A z/2,A z/2,Q z,Q z]) := by + filter_upwards [hcontacts] with z hz + have hshape : ofLp (gc_coupledVector m0 S0 z) = ![-1,1,-A z/2,A z/2,Q z,Q z] := by + funext i + fin_cases i <;> simp [A, Q] <;> linarith [hz.1, hz.2.1, hz.2.2.1, hz.2.2.2] + have ht := ((PiLp.continuous_ofLp 2 (fun _ : Fin 6 => ℝ)).tendsto + (gc_coupledVector m0 S0 z)).comp + (gc_coupledVector_tendsto (𝓝[>] (0 : ℝ)) m S m0 S0 hm hS hPSD hPSD0 z) + simpa only [Function.comp_def, X, hshape] using ht + have hv : (S0 4 4).toNNReal ≠ 0 := ne_of_gt (Real.toNNReal_pos.mpr hQpos) + have hQ : HasLaw Q (gaussianReal (m0 4) (S0 4 4).toNNReal) P := by + exact (measurePreserving_eval_multivariateGaussian (μ := m0) hPSD0 (i := 4)).hasLaw.fun_comp hLaw0 + have hint : ∀ᶠ r in 𝓝[>] (0 : ℝ), + Integrable (fun z => gn_normalizedWeight r (X r z)) P := by + filter_upwards [hPSD] with r hr + exact (gc_coupledVector_hasLaw (m r) (S r)).integrable_comp + (gnl_gaussian_weight_integrable r (m r) (S r) hr) + obtain ⟨c, hc, δ, hδ, hfloor⟩ := + gn_normalizer_floor_of_common_coupling P X hX A Q (m0 4) (S0 4 4).toNNReal hv hQ hconv hint + refine ⟨c, hc, δ, hδ, ?_⟩ + intro r hr hrδ + calc + c*r^2 ≤ ∫ z, r^2 * gn_normalizedWeight r (X r z) ∂P := hfloor r hr hrδ + _ = ∫ x : EuclideanSpace ℝ (Fin 6), r^2 * gn_normalizedWeight r (ofLp x) + ∂multivariateGaussian (m r) (S r) := by + simpa only [Function.comp_def, X] using + (gc_coupledVector_hasLaw (m r) (S r)).integral_comp + (f := fun x : EuclideanSpace ℝ (Fin 6) => r^2 * gn_normalizedWeight r (ofLp x)) + (by unfold gn_normalizedWeight; fun_prop) + + +/-- The same Gaussian parameter limit gives the full literal typed endpoint +normalizer, after the explicit permutation back to physical endpoint order. -/ +theorem gnl_physical_normalizer_floor_of_parameter_limits + (m : ℝ → EuclideanSpace ℝ (Fin 6)) (S : ℝ → Matrix (Fin 6) (Fin 6) ℝ) + (m0 : EuclideanSpace ℝ (Fin 6)) (S0 : Matrix (Fin 6) (Fin 6) ℝ) + (hm : Tendsto m (𝓝[>] (0 : ℝ)) (𝓝 m0)) + (hS : Tendsto S (𝓝[>] (0 : ℝ)) (𝓝 S0)) + (hPSD : ∀ᶠ r in 𝓝[>] (0 : ℝ), (S r).PosSemidef) (hPSD0 : S0.PosSemidef) + (hm0 : m0 0 = -1) (hm1 : m0 1 = 1) + (hm23 : m0 2 + m0 3 = 0) (hm45 : m0 4 - m0 5 = 0) + (hS0 : S0 0 0 = 0) (hS1 : S0 1 1 = 0) + (hS23 : S0 2 2 + 2*S0 2 3 + S0 3 3 = 0) + (hS45 : S0 4 4 - 2*S0 4 5 + S0 5 5 = 0) (hQpos : 0 < S0 4 4) : + ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → c*r^2 ≤ + ∫ x : EuclideanSpace ℝ (Fin 6), egm_weight r ![x 0,x 2,x 4,x 1,x 3,x 5] + ∂multivariateGaussian (m r) (S r) := by + obtain ⟨c, hc, δ, hδ, hfloor⟩ := gnl_normalizer_floor_of_parameter_limits + m S m0 S0 hm hS hPSD hPSD0 hm0 hm1 hm23 hm45 hS0 hS1 hS23 hS45 hQpos + refine ⟨c, hc, δ, hδ, ?_⟩ + intro r hr hrδ + calc + c*r^2 ≤ ∫ x : EuclideanSpace ℝ (Fin 6), r^2 * gn_normalizedWeight r (ofLp x) + ∂multivariateGaussian (m r) (S r) := hfloor r hr hrδ + _ = ∫ x : EuclideanSpace ℝ (Fin 6), egm_weight r ![x 0,x 2,x 4,x 1,x 3,x 5] + ∂multivariateGaussian (m r) (S r) := by + apply integral_congr_ae + filter_upwards [] with x + have hp : enb_permutation ![x 0,x 2,x 4,x 1,x 3,x 5] = ofLp x := by + funext i + fin_cases i <;> rfl + simpa only [hp] using + (enb_weight_identity r ![x 0,x 2,x 4,x 1,x 3,x 5] hr).symm + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianRegressionKernel.lean b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianRegressionKernel.lean new file mode 100644 index 000000000..face5de81 --- /dev/null +++ b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianRegressionKernel.lean @@ -0,0 +1,270 @@ +import Mathlib.Probability.HasCondDistrib +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +/-! +# Canonical Gaussian regression kernel + +For a residual law `ν`, `regressionKernel ν B mX mY x` is the literal affine pushforward +`ν.map (fun r => r + mY + B (x - mX))` at every observation `x`. The kernel is constructed +measurably from a deterministic observation and a constant residual kernel, so it has a +specified value even where the observation has probability zero. + +The Gaussian theorem derives observation/residual independence from joint Gaussianity and +zero coordinate cross-covariance. No nondegeneracy or residual covariance inverse is needed. +The direct observed/target theorem derives those residual hypotheses from the coordinate +normal equations for `B`. The construction does not identify concrete derivative observations, +regression or Schur covariance blocks, a Palm weight, or a conditional law of a literal smooth field. + +Source contract: `SOURCE_REGRESSION_KERNEL.md` in this packet. Author-side Lean verification +is separate from source alignment and the required review by another model family. +-/ + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory BigOperators + +set_option autoImplicit false + +namespace ResearchFormalCoreR1 + +/-- Copying the first coordinate and retaining an independent second coordinate. -/ +lemma regression_compProd_id_const {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] + (μ : Measure α) (ν : Measure β) [SFinite μ] [SFinite ν] : + μ ⊗ₘ (Kernel.id ×ₖ Kernel.const α ν) = + (μ.prod ν).map (fun p : α × β => (p.1, p)) := by + ext s hs + rw [Measure.compProd_apply hs, Measure.map_apply (by fun_prop) hs, + Measure.prod_apply (s := (fun p : α × β => (p.1, p)) ⁻¹' s) (hs.preimage (by fun_prop))] + simp_rw [Kernel.prod_apply, Kernel.id_apply, Kernel.const_apply, Measure.dirac_prod] + congr with x + rw [Measure.map_apply (by fun_prop) (measurable_prodMk_left hs)] + rfl + +/-- An independent residual paired with the observation has the explicit product kernel. -/ +lemma regression_pair_hasCondDistrib {Ω α β : Type*} [MeasurableSpace Ω] + [MeasurableSpace α] [MeasurableSpace β] {P : Measure Ω} [IsProbabilityMeasure P] + {X : Ω → α} {R : Ω → β} (hX : AEMeasurable X P) (hR : AEMeasurable R P) + (hXR : IndepFun X R P) : + HasCondDistrib (fun ω => (X ω, R ω)) X + (Kernel.id ×ₖ Kernel.const α (P.map R)) P := by + refine ⟨hX.prodMk (hX.prodMk hR), ?_⟩ + calc + P.map (fun ω => (X ω, (X ω, R ω))) = + (P.map (fun ω => (X ω, R ω))).map (fun p : α × β => (p.1, p)) := by + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) (hX.prodMk hR)] + rfl + _ = ((P.map X).prod (P.map R)).map (fun p : α × β => (p.1, p)) := by + rw [hXR.map_prod_eq_prod_map_map hX hR] + _ = (P.map X) ⊗ₘ (Kernel.id ×ₖ Kernel.const α (P.map R)) := + (regression_compProd_id_const (P.map X) (P.map R)).symm + + +/-- Affine reconstruction of the target from the observation and the regression residual. -/ +def regressionReconstruction {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (p : (Fin n → ℝ) × (Fin k → ℝ)) : Fin k → ℝ := + p.2 + mY + B (p.1 - mX) + +lemma regressionReconstruction_measurable {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Measurable (regressionReconstruction B mX mY) := by + unfold regressionReconstruction + fun_prop + +/-- The canonical regression law is specified at every observation, including null pins. -/ +noncomputable def regressionKernel {n k : ℕ} (ν : Measure (Fin k → ℝ)) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Kernel (Fin n → ℝ) (Fin k → ℝ) := + ((Kernel.id : Kernel (Fin n → ℝ) (Fin n → ℝ)) ×ₖ + Kernel.const (Fin n → ℝ) ν).map (regressionReconstruction B mX mY) + +instance regressionKernel_isMarkov {n k : ℕ} (ν : Measure (Fin k → ℝ)) + [IsProbabilityMeasure ν] (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) + (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + IsMarkovKernel (regressionKernel ν B mX mY) := by + unfold regressionKernel + exact Kernel.IsMarkovKernel.map _ (regressionReconstruction_measurable B mX mY) + +lemma regressionKernel_apply {n k : ℕ} (ν : Measure (Fin k → ℝ)) [SFinite ν] + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (x : Fin n → ℝ) : + regressionKernel ν B mX mY x = ν.map (fun r => r + mY + B (x - mX)) := by + rw [regressionKernel, Kernel.map_apply _ (regressionReconstruction_measurable B mX mY), + Kernel.prod_apply, Kernel.id_apply, Kernel.const_apply, Measure.dirac_prod, + Measure.map_map (regressionReconstruction_measurable B mX mY) (by fun_prop)] + rfl + +/-- Affine reconstruction from an independent residual has the canonical conditional kernel. -/ +lemma independentResidual_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {n k : ℕ} + {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hX : AEMeasurable X P) (hR : AEMeasurable R P) (hXR : IndepFun X R P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + exact (regression_pair_hasCondDistrib hX hR hXR).comp_left + (regressionReconstruction_measurable B mX mY) + +/-- Joint Gaussianity and vanishing observation/residual cross-covariance give the canonical +conditional law, without choosing an almost-everywhere version at a prescribed observation. -/ +theorem gaussianRegression_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hXR : HasGaussianLaw (fun ω => (X ω, R ω)) P) + (hzero : ∀ i j, covariance (fun ω => X ω i) (fun ω => R ω j) P = 0) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + have : IsProbabilityMeasure P := hXR.isProbabilityMeasure + exact independentResidual_hasCondDistrib hXR.fst.aemeasurable hXR.snd.aemeasurable + (hXR.indepFun_of_covariance_eval hzero) B mX mY + +/-- Kernel measurability is pointwise in the observation and does not require a Gaussian law. -/ +lemma regressionKernel_measurable {n k : ℕ} (ν : Measure (Fin k → ℝ)) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Measurable (regressionKernel ν B mX mY) := + Kernel.measurable _ + +/-- All parts of the canonical Gaussian conditional-law construction, including the literal +pushforward at every prescribed observation. Zero-dimensional and degenerate laws are allowed. -/ +theorem gaussianRegressionKernel_spec {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hXR : HasGaussianLaw (fun ω => (X ω, R ω)) P) + (hzero : ∀ i j, covariance (fun ω => X ω i) (fun ω => R ω j) P = 0) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + IsMarkovKernel (regressionKernel (P.map R) B mX mY) ∧ + Measurable (regressionKernel (P.map R) B mX mY) ∧ + (∀ x, regressionKernel (P.map R) B mX mY x = + (P.map R).map (fun r => r + mY + B (x - mX))) ∧ + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + have : IsProbabilityMeasure P := hXR.isProbabilityMeasure + exact ⟨inferInstance, regressionKernel_measurable _ B mX mY, + regressionKernel_apply _ B mX mY, gaussianRegression_hasCondDistrib hXR hzero B mX mY⟩ + +/-- Coordinates of a finite-vector continuous linear map in the standard basis. -/ +lemma regression_map_coordinate {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (v : Fin n → ℝ) (j : Fin k) : + B v j = ∑ l, B (Pi.single l 1) j * v l := by + calc + B v j = B (∑ l, v l • (Pi.single l 1 : Fin n → ℝ)) j := + congrArg (fun w => B w j) (pi_eq_sum_univ' v) + _ = ∑ l, B (Pi.single l 1) j * v l := by + simp [map_sum, mul_comm] + +/-- The covariance of a centered finite-vector linear predictor is its normal-equation sum. -/ +lemma regression_covariance_map {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {n k : ℕ} + {X : Ω → Fin n → ℝ} (hX : HasGaussianLaw X P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) + (i : Fin n) (j : Fin k) : + covariance (fun ω => X ω i) (fun ω => B (X ω - mX) j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P := by + have hcoord : (fun ω => B (X ω - mX) j) = + (fun ω => ∑ l, B (Pi.single l 1) j * (X ω l - mX l)) := by + funext ω + exact regression_map_coordinate B (X ω - mX) j + rw [hcoord, covariance_fun_sum_right] + · apply Finset.sum_congr rfl + intro l _ + rw [covariance_const_mul_right, covariance_sub_const_right (hX.eval l).integrable] + · intro l + exact ((hX.eval l).memLp_two.sub (memLp_const _)).const_mul _ + · exact (hX.eval i).memLp_two + +/-- The actual regression residual has zero observation cross-covariance by the normal equations. -/ +lemma regression_residual_zero_covariance {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (hnormal : ∀ i j, covariance (fun ω => X ω i) (fun ω => Y ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P) : + ∀ i j, covariance (fun ω => X ω i) + (fun ω => (Y ω - mY - B (X ω - mX)) j) P = 0 := by + have : IsProbabilityMeasure P := hXY.isProbabilityMeasure + intro i j + have hXi := (hXY.fst.eval i).memLp_two + have hYj := (hXY.snd.eval j).memLp_two + have hBj : MemLp (fun ω => B (X ω - mX) j) 2 P := by + have hcoord : (fun ω => B (X ω - mX) j) = + (fun ω => ∑ l, B (Pi.single l 1) j * (X ω l - mX l)) := by + funext ω + exact regression_map_coordinate B (X ω - mX) j + rw [hcoord] + exact memLp_finsetSum _ (fun l _ => + ((hXY.fst.eval l).memLp_two.sub (memLp_const _)).const_mul _) + simp only [Pi.sub_apply] + rw [covariance_fun_sub_right (Y := fun ω => Y ω j - mY j) hXi (hYj.sub (memLp_const _)) hBj, + covariance_sub_const_right (hXY.snd.eval j).integrable, + regression_covariance_map hXY.fst B mX i j, hnormal i j, sub_self] + +/-- The observation and the actual affine residual are jointly Gaussian. -/ +lemma regression_residual_jointGaussian {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {n k : ℕ} + {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasGaussianLaw (fun ω => (X ω, Y ω - mY - B (X ω - mX))) P := by + let L : ((Fin n → ℝ) × (Fin k → ℝ)) →L[ℝ] + ((Fin n → ℝ) × (Fin k → ℝ)) := + (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ)).prod + (ContinuousLinearMap.snd ℝ (Fin n → ℝ) (Fin k → ℝ) - + B.comp (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ))) + let c : (Fin n → ℝ) × (Fin k → ℝ) := (0, B mX - mY) + have hL : HasGaussianLaw (fun ω => L (X ω, Y ω)) P := hXY.map_fun L + have hshift : HasGaussianLaw (fun ω => L (X ω, Y ω) + c) P := { + aemeasurable := hL.aemeasurable.add aemeasurable_const + isGaussian_map := by + have : IsGaussian (P.map (fun ω => L (X ω, Y ω))) := hL.isGaussian_map + convert (inferInstance : IsGaussian + ((P.map (fun ω => L (X ω, Y ω))).map (fun p => p + c))) using 1 + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hL.aemeasurable] + rfl } + apply hshift.congr + filter_upwards [] with ω + apply Prod.ext + · simp [L, c] + · change Y ω - B (X ω) + (B mX - mY) = Y ω - mY - B (X ω - mX) + rw [map_sub] + abel + + + +/-- The actual target has the explicit regression conditional law when the coefficient map +satisfies the coordinate normal equations. The residual hypotheses are derived. -/ +theorem gaussianRegression_observed_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (hnormal : ∀ i j, covariance (fun ω => X ω i) (fun ω => Y ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P) : + HasCondDistrib Y X + (regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY) P := by + have hXR := regression_residual_jointGaussian hXY B mX mY + have hzero := regression_residual_zero_covariance hXY B mX mY hnormal + have hcond := gaussianRegression_hasCondDistrib hXR hzero B mX mY + convert hcond using 1 + funext ω + abel + +/-- The canonical conditional-law specification for the actual jointly Gaussian target. +The residual Gaussianity and its zero cross-covariance are conclusions of the construction. -/ +theorem gaussianRegression_observed_spec {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (hnormal : ∀ i j, covariance (fun ω => X ω i) (fun ω => Y ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P) : + IsMarkovKernel + (regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY) ∧ + Measurable + (regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY) ∧ + (∀ x, regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY x = + (P.map (fun ω => Y ω - mY - B (X ω - mX))).map + (fun r => r + mY + B (x - mX))) ∧ + HasCondDistrib Y X + (regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY) P := by + have hXR := regression_residual_jointGaussian hXY B mX mY + have hzero := regression_residual_zero_covariance hXY B mX mY hnormal + have hspec := gaussianRegressionKernel_spec hXR hzero B mX mY + exact ⟨hspec.1, hspec.2.1, hspec.2.2.1, + gaussianRegression_observed_hasCondDistrib hXY B mX mY hnormal⟩ + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianScalarMoments.lean b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianScalarMoments.lean new file mode 100644 index 000000000..ce432cb71 --- /dev/null +++ b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/GaussianScalarMoments.lean @@ -0,0 +1,99 @@ +import Mathlib.Probability.Distributions.Gaussian.Real +import Mathlib.Algebra.Order.Ring.Basic +import Mathlib.Tactic.FunProp +import Mathlib.Tactic.NormNum +import Mathlib.Tactic.Ring + +/-! Scalar eighth moments for noncentered Gaussian laws. The finite standard +Gaussian moment is proved integrable; it is not an assumed model bound. -/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory +open scoped NNReal + +set_option autoImplicit false + +noncomputable def gm8C8 : ℝ := ∫ z : ℝ, |z| ^ 8 ∂gaussianReal 0 1 + +theorem gm8_standard_integrable : + Integrable (fun z : ℝ => |z| ^ 8) (gaussianReal 0 1) := by + simpa only [id_eq, Real.norm_eq_abs] using + (memLp_id_gaussianReal' (μ := 0) (v := 1) 8 (by norm_num)).integrable_norm_pow + (by norm_num : (8 : ℕ) ≠ 0) + +theorem gm8_standard_nonneg : 0 ≤ gm8C8 := by + exact integral_nonneg (fun z => pow_nonneg (abs_nonneg z) 8) + +private theorem gm8_integrable (m : ℝ) (v : ℝ≥0) : + Integrable (fun x : ℝ => |x| ^ 8) (gaussianReal m v) := by + simpa only [id_eq, Real.norm_eq_abs] using + (memLp_id_gaussianReal' (μ := m) (v := v) 8 (by norm_num)).integrable_norm_pow + (by norm_num : (8 : ℕ) ≠ 0) + +private theorem gm8_sqrt_pow (V : ℝ) (hV : 0 ≤ V) : + (Real.sqrt V) ^ 8 = V ^ 4 := by + calc + _ = ((Real.sqrt V) ^ 2) ^ 4 := by ring + _ = V ^ 4 := by rw [Real.sq_sqrt hV] + +private theorem gm8_affine_map (m : ℝ) (v : ℝ≥0) : + (gaussianReal 0 1).map (fun z : ℝ => m + Real.sqrt (v : ℝ) * z) = + gaussianReal m v := by + calc + _ = ((gaussianReal 0 1).map (fun z : ℝ => Real.sqrt (v : ℝ) * z)).map + (fun z : ℝ => m + z) := by + rw [Measure.map_map (by fun_prop) (by fun_prop)] + rfl + _ = gaussianReal m v := by + rw [gaussianReal_map_const_mul, gaussianReal_map_const_add] + simp only [mul_zero, zero_add, mul_one] + congr 1 + apply NNReal.eq + exact Real.sq_sqrt v.property + +private theorem gm8_affine_bound (m B V : ℝ) (v : ℝ≥0) (hB : 0 ≤ B) (hV : 0 ≤ V) + (hm : |m| ≤ B) (hv : (v : ℝ) ≤ V) (z : ℝ) : + |m + Real.sqrt (v : ℝ) * z| ^ 8 ≤ 128 * (B ^ 8 + V ^ 4 * |z| ^ 8) := by + have habs : |m + Real.sqrt (v : ℝ) * z| ≤ B + Real.sqrt V * |z| := by + calc + _ ≤ |m| + |Real.sqrt (v : ℝ) * z| := abs_add_le _ _ + _ = |m| + Real.sqrt (v : ℝ) * |z| := by + rw [abs_mul, abs_of_nonneg (Real.sqrt_nonneg _)] + _ ≤ B + Real.sqrt V * |z| := + add_le_add hm (mul_le_mul_of_nonneg_right (Real.sqrt_le_sqrt hv) (abs_nonneg z)) + calc + _ ≤ (B + Real.sqrt V * |z|) ^ 8 := pow_le_pow_left₀ (abs_nonneg _) habs 8 + _ ≤ 128 * (B ^ 8 + (Real.sqrt V * |z|) ^ 8) := by + convert add_pow_le hB (mul_nonneg (Real.sqrt_nonneg V) (abs_nonneg z)) 8 using 1 + norm_num + _ = 128 * (B ^ 8 + V ^ 4 * |z| ^ 8) := by + rw [mul_pow, gm8_sqrt_pow V hV] + +/-- Bounded means and variances give a uniform eighth-moment bound, including +zero variance and nonzero means. No eighth-moment hypothesis is assumed. -/ +theorem gm8_scalar_integrable_bound (m B V : ℝ) (v : ℝ≥0) + (hB : 0 ≤ B) (hV : 0 ≤ V) (hm : |m| ≤ B) (hv : (v : ℝ) ≤ V) : + Integrable (fun x : ℝ => |x| ^ 8) (gaussianReal m v) ∧ + (∫ x : ℝ, |x| ^ 8 ∂gaussianReal m v) ≤ + 128 * (B ^ 8 + V ^ 4 * gm8C8) := by + have hi := gm8_integrable m v + have hmajor : Integrable (fun z : ℝ => 128 * (B ^ 8 + V ^ 4 * |z| ^ 8)) + (gaussianReal 0 1) := + ((integrable_const (B ^ 8)).add (gm8_standard_integrable.const_mul (V ^ 4))).const_mul 128 + have hiaff : Integrable (fun z : ℝ => |m + Real.sqrt (v : ℝ) * z| ^ 8) + (gaussianReal 0 1) := by + rw [← gm8_affine_map m v] at hi + exact (integrable_map_measure (by fun_prop) (by fun_prop)).mp hi + refine ⟨hi, ?_⟩ + calc + _ = ∫ z : ℝ, |m + Real.sqrt (v : ℝ) * z| ^ 8 ∂gaussianReal 0 1 := by + rw [← gm8_affine_map m v, integral_map (by fun_prop) (by fun_prop)] + _ ≤ ∫ z : ℝ, 128 * (B ^ 8 + V ^ 4 * |z| ^ 8) ∂gaussianReal 0 1 := + integral_mono hiaff hmajor (gm8_affine_bound m B V v hB hV hm hv) + _ = 128 * (B ^ 8 + V ^ 4 * gm8C8) := by + rw [integral_const_mul, integral_add (integrable_const _) + (gm8_standard_integrable.const_mul _), integral_const_mul] + simp only [integral_const, probReal_univ, smul_eq_mul, one_mul, gm8C8] + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/NormalizerFloor.lean b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/NormalizerFloor.lean new file mode 100644 index 000000000..3635695ef --- /dev/null +++ b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/NormalizerFloor.lean @@ -0,0 +1,120 @@ +import ResearchFormalCoreR1.GaussianNormalizer +import Mathlib.Probability.HasLaw +import Mathlib.MeasureTheory.Integral.Lebesgue.Add +import Mathlib.Topology.Instances.ENNReal.Lemmas +import Mathlib.Topology.Order.LeftRightNhds + +/-! +# A full small-radius lower normalizer from an explicit common coupling + +Nonnegative Fatou and the proved positive Gaussian reference give one lower +constant for every sufficiently small positive radius. Neither uniform +integrability nor convergence of normalizers is assumed. The literal +conditioned-field construction of the coupling remains an application join. +See ../SOURCE_GAUSSIAN_NORMALIZER.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory Filter +open scoped Topology ENNReal NNReal + +set_option autoImplicit false + +private theorem nfloor_eventually_lower_of_fatou {Ω ι : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (u : Filter ι) [IsCountablyGenerated u] [NeBot u] + (f : ι → Ω → ℝ) (g : Ω → ℝ) + (hmeas : ∀ r, Measurable (f r)) + (hnonneg : ∀ r ω, 0 ≤ f r ω) + (hg : Integrable g P) (hgpos : 0 < ∫ ω, g ω ∂P) + (hconv : ∀ᵐ ω ∂P, Tendsto (fun r => f r ω) u (𝓝 (g ω))) + (hfint : ∀ᶠ r in u, Integrable (f r) P) : + ∀ᶠ r in u, (∫ ω, g ω ∂P)/2 ≤ ∫ ω, f r ω ∂P := by + have hg0 : 0 ≤ᵐ[P] g := hconv.mono (fun ω hω => + ge_of_tendsto hω (Eventually.of_forall (fun r => hnonneg r ω))) + have hlim : (fun ω => liminf (fun r => ENNReal.ofReal (f r ω)) u) =ᵐ[P] + (fun ω => ENNReal.ofReal (g ω)) := by + filter_upwards [hconv] with ω hω + exact ((ENNReal.continuous_ofReal.tendsto (g ω)).comp hω).liminf_eq + have hfatou : ENNReal.ofReal (∫ ω, g ω ∂P) ≤ + liminf (fun r => ∫⁻ ω, ENNReal.ofReal (f r ω) ∂P) u := by + rw [ofReal_integral_eq_lintegral_ofReal hg hg0] + rw [← lintegral_congr_ae hlim] + exact lintegral_liminf_le (fun r => (hmeas r).ennreal_ofReal) + have hhalf : ENNReal.ofReal ((∫ ω, g ω ∂P)/2) < + ENNReal.ofReal (∫ ω, g ω ∂P) := by + exact (ENNReal.ofReal_lt_ofReal_iff hgpos).mpr (by linarith) + have hevent := eventually_lt_of_lt_liminf (hhalf.trans_le hfatou) + filter_upwards [hevent, hfint] with r hr hri + rw [← ofReal_integral_eq_lintegral_ofReal hri + (Eventually.of_forall (fun ω => hnonneg r ω))] at hr + exact (ENNReal.ofReal_lt_ofReal_iff'.mp hr).1.le + +private theorem nfloor_weight_continuous : + Continuous (fun p : ℝ × (Fin 6 → ℝ) => gn_normalizedWeight p.1 p.2) := by + unfold gn_normalizedWeight + fun_prop +private theorem nfloor_contact_value (a q : ℝ) : + gn_normalizedWeight 0 ![-1,1,-a/2,a/2,q,q] = (max (-q) 0)^2 := by + simp [gn_normalizedWeight, pow_two] +private theorem nfloor_weight_limit (X : ℝ → Fin 6 → ℝ) (a q : ℝ) + (hX : Tendsto X (𝓝[>] (0:ℝ)) (𝓝 ![-1,1,-a/2,a/2,q,q])) : + Tendsto (fun r => gn_normalizedWeight r (X r)) (𝓝[>] (0:ℝ)) + (𝓝 ((max (-q) 0)^2)) := by + have hr : Tendsto (fun r : ℝ => r) (𝓝[>] (0:ℝ)) (𝓝 0) := + continuousWithinAt_id + simpa only [nfloor_contact_value, Function.comp_def] using + (nfloor_weight_continuous.tendsto (0, ![-1,1,-a/2,a/2,q,q])).comp (hr.prodMk_nhds hX) +private theorem nfloor_reference {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (Q : Ω → ℝ) (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) + (hlaw : HasLaw Q (gaussianReal m v) P) : + Integrable (fun ω => (max (-Q ω) 0)^2) P ∧ + 0 < ∫ ω, (max (-Q ω) 0)^2 ∂P := by + obtain ⟨hInt,hPos⟩ := gn_gaussian_reference_integrable_pos m v hv + refine ⟨hlaw.integrable_comp hInt, ?_⟩ + have heq := hlaw.integral_comp + (show AEStronglyMeasurable (fun q : ℝ => (max (-q) 0)^2) (gaussianReal m v) by fun_prop) + exact heq.trans_gt hPos + +/-- The actual integral of the physical r²-scaled continuous endpoint weight +has a positive lower constant at every sufficiently small positive radius. +The common coupling and current integrability are explicit premises; its +positive reference integral and the lower bound are proved, not premises. -/ +theorem gn_normalizer_floor_of_common_coupling + {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) [IsProbabilityMeasure P] + (X : ℝ → Ω → Fin 6 → ℝ) (hX : ∀ r, Measurable (X r)) + (A Q : Ω → ℝ) (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) + (hQ : HasLaw Q (gaussianReal m v) P) + (hconv : ∀ᵐ ω ∂P, Tendsto (fun r => X r ω) (𝓝[>] (0 : ℝ)) + (𝓝 ![-1,1,-A ω/2,A ω/2,Q ω,Q ω])) + (hint : ∀ᶠ r in 𝓝[>] (0 : ℝ), + Integrable (fun ω => gn_normalizedWeight r (X r ω)) P) : + ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → + c * r^2 ≤ ∫ ω, r^2 * gn_normalizedWeight r (X r ω) ∂P := by + let g : Ω → ℝ := fun ω => (max (-Q ω) 0)^2 + obtain ⟨hg, hgpos⟩ := nfloor_reference P Q m v hv hQ + have hmeas (r : ℝ) : Measurable (fun ω => gn_normalizedWeight r (X r ω)) := by + have hc : Continuous (gn_normalizedWeight r) := by + unfold gn_normalizedWeight + fun_prop + exact hc.measurable.comp (hX r) + have hnonneg (r : ℝ) (ω : Ω) : 0 ≤ gn_normalizedWeight r (X r ω) := + mul_nonneg (le_max_right _ _) (le_max_right _ _) + have hweightconv : ∀ᵐ ω ∂P, + Tendsto (fun r => gn_normalizedWeight r (X r ω)) (𝓝[>] (0 : ℝ)) (𝓝 (g ω)) := by + filter_upwards [hconv] with ω hω + exact nfloor_weight_limit (fun r => X r ω) (A ω) (Q ω) hω + have hfloor : ∀ᶠ r in 𝓝[>] (0 : ℝ), + (∫ ω, g ω ∂P)/2 ≤ ∫ ω, gn_normalizedWeight r (X r ω) ∂P := + nfloor_eventually_lower_of_fatou P (𝓝[>] (0 : ℝ)) + (fun r ω => gn_normalizedWeight r (X r ω)) g hmeas hnonneg hg hgpos hweightconv hint + obtain ⟨δ, hδ, hinterval⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hfloor + refine ⟨(∫ ω, g ω ∂P)/2, by positivity, δ, hδ, ?_⟩ + intro r hr hrδ + have hbound := hinterval ⟨hr, hrδ⟩ + rw [integral_const_mul] + simpa only [mul_comm] using mul_le_mul_of_nonneg_left hbound (sq_nonneg r) + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/PalmReweighting.lean b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/PalmReweighting.lean new file mode 100644 index 000000000..f0d6abafc --- /dev/null +++ b/frontiers/determinant_moment_20261025/ResearchFormalCoreR1/PalmReweighting.lean @@ -0,0 +1,139 @@ +import Mathlib.MeasureTheory.Function.L2Space +import Mathlib.MeasureTheory.Measure.WithDensity +import Mathlib.MeasureTheory.Integral.Bochner.Set +import Mathlib.MeasureTheory.Measure.Real +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum +import Mathlib.Tactic.Positivity +import Mathlib.Tactic.FieldSimp + +/-! +# A canonical weighted probability and its square-root event bound + +This constructs the literal measure with density `W / integral W`, proves its +event formula, and applies integral Cauchy–Schwarz. The determinant second +moment and normalizer estimates cancel their exact radius powers. Identifying +this measure with a geometric Kac–Rice Palm law remains an application join. +See ../SOURCE_PALM_REWEIGHTING.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory +open scoped ENNReal + +set_option autoImplicit false + +/-- The literal normalized nonnegative real density, defined for every input. -/ +noncomputable def prwMeasure {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) : Measure Ω := + μ.withDensity (fun ω => ENNReal.ofReal (W ω / ∫ x, W x ∂μ)) + +/-- On a finite measure, a measurable square-integrable real weight is integrable. -/ +theorem prw_integrable_of_integrable_sq {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsFiniteMeasure μ] (W : Ω → ℝ) (hW : Measurable W) + (hsq : Integrable (fun ω => W ω ^ 2) μ) : Integrable W μ := by + have hmajor : Integrable (fun ω => 1 + W ω ^ 2) μ := + (integrable_const 1).add hsq + apply hmajor.mono' hW.aestronglyMeasurable + filter_upwards [] with ω + rw [Real.norm_eq_abs] + nlinarith [sq_nonneg (|W ω| - 1 / 2), sq_abs (W ω)] + +/-- The actual event mass is the normalized set integral, in extended reals. -/ +theorem prw_measure_apply {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + prwMeasure μ W E = ENNReal.ofReal ((∫ x in E, W x ∂μ) / ∫ x, W x ∂μ) := by + unfold prwMeasure + rw [withDensity_apply _ hE] + rw [← ofReal_integral_eq_lintegral_ofReal + (hW.integrableOn.div_const _) (Filter.Eventually.of_forall (fun ω => + div_nonneg (hW0 ω) hZ.le))] + rw [integral_div] + +/-- Positive finite normalizer makes the literal density a probability law. -/ +theorem prw_isProbabilityMeasure {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) : + IsProbabilityMeasure (prwMeasure μ W) := by + constructor + rw [prw_measure_apply μ W hW hW0 hZ Set.univ MeasurableSet.univ] + simp [hZ.ne'] + +/-- The event identity is proved for the constructed law, not assumed. -/ +theorem prw_measure_real {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + (prwMeasure μ W).real E = (∫ x in E, W x ∂μ) / ∫ x, W x ∂μ := by + rw [measureReal_def, prw_measure_apply μ W hW hW0 hZ E hE, + ENNReal.toReal_ofReal (div_nonneg (integral_nonneg hW0) hZ.le)] + +/-- Integral Cauchy–Schwarz against the event indicator, including null events. -/ +theorem prw_numerator_bound {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsFiniteMeasure μ] (W : Ω → ℝ) (hW : Measurable W) + (hW0 : ∀ ω, 0 ≤ W ω) (hsq : Integrable (fun ω => W ω ^ 2) μ) + (E : Set Ω) (hE : MeasurableSet E) : + (∫ x in E, W x ∂μ) ≤ + Real.sqrt (∫ x, W x ^ 2 ∂μ) * Real.sqrt (μ.real E) := by + classical + let I : Ω → ℝ := E.indicator (fun _ => 1) + have hI0 : ∀ ω, 0 ≤ I ω := by + intro ω + simp only [I, Set.indicator_apply] + split_ifs <;> norm_num + have hWLp : MemLp W (ENNReal.ofReal 2) μ := by + simpa using (memLp_two_iff_integrable_sq hW.aestronglyMeasurable).mpr hsq + have hILp : MemLp I (ENNReal.ofReal 2) μ := + (memLp_const (1 : ℝ)).indicator hE + have hCS := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two + (Filter.Eventually.of_forall hW0) (Filter.Eventually.of_forall hI0) hWLp hILp + have hleft : (fun ω => W ω * I ω) = E.indicator W := by + funext ω + by_cases hω : ω ∈ E <;> simp [I, hω] + have hright : (fun ω => I ω ^ (2 : ℝ)) = E.indicator (fun _ => (1 : ℝ)) := by + funext ω + by_cases hω : ω ∈ E <;> simp [I, hω] + have hIint : (∫ ω, I ω ^ (2 : ℝ) ∂μ) = μ.real E := by + rw [hright] + change (∫ ω, E.indicator (1 : Ω → ℝ) ω ∂μ) = μ.real E + exact integral_indicator_one hE + rw [hleft, integral_indicator hE, hIint] at hCS + simpa only [Real.rpow_two, ← Real.sqrt_eq_rpow] using hCS + +/-- The exact `r^4` second moment and `r^2` normalizer cancel in the literal +weighted probability. No event probability is assumed positive. -/ +theorem prw_square_root_event_bound {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsProbabilityMeasure μ] (W : Ω → ℝ) (hW : Measurable W) + (hW0 : ∀ ω, 0 ≤ W ω) (hsq : Integrable (fun ω => W ω ^ 2) μ) + (C c r : ℝ) (hC : 0 ≤ C) (hc : 0 < c) (hr : 0 < r) + (hmoment : (∫ x, W x ^ 2 ∂μ) ≤ C * r ^ 4) + (hfloor : c * r ^ 2 ≤ ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + IsProbabilityMeasure (prwMeasure μ W) ∧ + (prwMeasure μ W).real E ≤ (Real.sqrt C / c) * Real.sqrt (μ.real E) := by + have hZ : 0 < ∫ x, W x ∂μ := + (mul_pos hc (sq_pos_of_pos hr)).trans_le hfloor + have hW1 : Integrable W μ := prw_integrable_of_integrable_sq μ W hW hsq + refine ⟨prw_isProbabilityMeasure μ W hW1 hW0 hZ, ?_⟩ + rw [prw_measure_real μ W hW1 hW0 hZ E hE] + have hnum := prw_numerator_bound μ W hW hW0 hsq E hE + have hsqrt : Real.sqrt (∫ x, W x ^ 2 ∂μ) ≤ Real.sqrt C * r ^ 2 := by + calc + _ ≤ Real.sqrt (C * r ^ 4) := Real.sqrt_le_sqrt hmoment + _ = Real.sqrt C * r ^ 2 := by + rw [Real.sqrt_mul hC] + congr 1 + have heq : r ^ 4 = (r ^ 2) ^ 2 := by ring + rw [heq, Real.sqrt_sq (sq_nonneg r)] + calc + _ ≤ (Real.sqrt C * r ^ 2 * Real.sqrt (μ.real E)) / (c * r ^ 2) := by + apply div_le_div₀ (by positivity) + (hnum.trans (mul_le_mul_of_nonneg_right hsqrt (Real.sqrt_nonneg _))) + (mul_pos hc (sq_pos_of_pos hr)) hfloor + _ = (Real.sqrt C / c) * Real.sqrt (μ.real E) := by + field_simp [hr.ne'] + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/SCOPE.md b/frontiers/determinant_moment_20261025/SCOPE.md index 0a3e15e04..6ca8aa543 100644 --- a/frontiers/determinant_moment_20261025/SCOPE.md +++ b/frontiers/determinant_moment_20261025/SCOPE.md @@ -1,35 +1,66 @@ -# Exact DM2 formal scope +# Determinant, Gaussian conditioning and weighted-probability research scope -Author OpenAI/Codex /root; source-alignment PENDING_INDEPENDENT_REVIEW. -The complete preimplementation source binding is SOURCE_MAP.md and remains bound. +Successor to the five-statement DM2 packet landed in Math-#511. Scientific +effect NONE. All new authors/helpers are OpenAI/Codex; source alignment remains +PENDING_INDEPENDENT_REVIEW. The complete public theorem inventory and exact +source hashes are in manifest.json. The gate's elaborated-types log records +their actual hypotheses, including auxiliaries. -| Declaration | Source mapping and hypotheses | +## Mathematical coverage and concrete premises + +| Module | Result and retained premises | |---|---| -| dm2_untyped | Section 3: both exact determinant factors; arbitrary nine real jets; 00, +egm_weight r j = r^2 * gn_normalizedWeight r (permutation j). +Keep the exact strict negative-definite maximum condition and strict negative +saddle determinant. The identity must hold on all zero/type boundaries, with +no assumed negative eigenvalues, null-boundary premise or Gaussian hypothesis. +It follows by the 2x2 Sylvester criterion and positive/negative parts of the +determinants. A random-vector integral equality and the transfer of the proved +common-coupling lower floor to the literal egm_weight may be included as proved +consequences, not assumed identities. Do not replace the actual six-vector by +the source's normalized quartic nine-vector. + +An eventual integrability premise in the generic floor is permitted at this +join, but the Gaussian endpoint moment theorem supplies it in applications: +the physical weight is L2 and hence L1 on a probability space, and division by +the nonzero scalar r^2 recovers integrability of the normalized weight. Concrete +conditioned-field construction and convergence remain research obligations; +they are not claimed from this algebraic identity alone. + +Preserve existing proofs and evidence. Narrow pinned Mathlib imports only; +root alone integrates the root module, target manifest and final gate receipt. +Source review requires a different actual model family from OpenAI authors. diff --git a/frontiers/determinant_moment_20261025/SOURCE_GAUSSIAN_COUPLING.md b/frontiers/determinant_moment_20261025/SOURCE_GAUSSIAN_COUPLING.md new file mode 100644 index 000000000..985be4c5d --- /dev/null +++ b/frontiers/determinant_moment_20261025/SOURCE_GAUSSIAN_COUPLING.md @@ -0,0 +1,35 @@ +# Gaussian square-root common coupling: frozen before implementation + +25 October 2026. Author OpenAI/Codex root and +/root/normalizer_palm_research; scientific review requires another family. + +Source LM006 stability NOTE section 5, as bound in +SOURCE_GAUSSIAN_NORMALIZER.md, constructs one standard-Gaussian square-root +coupling and explicitly permits singular limiting covariance. The concrete +endpoint covariance extension is in +reviews/lm006_endpoint_extension_20261005/NOTE.md, sections 3–4, at Math-main +e75cb81e8961d178f43aa9967dddd11cd579dc8a, blob +551aa563562bc88a19e33d146259c03aa2d318f6, SHA256 +d860fd4669f168c117b67066550047bfa2fcd90d77bbc366d978b6112e9f3757. + +Frozen new GaussianCoupling.lean target: for any finite real dimension and +filter u, means m_r tending to m_0 and covariance matrices S_r tending to S_0, +with S_r positive semidefinite eventually and S_0 positive semidefinite, prove +for EVERY standard Gaussian coordinate z convergence of +m_r + toEuclideanCLM(CFC.sqrt S_r) z to +m_0 + toEuclideanCLM(CFC.sqrt S_0) z. No invertibility, positive determinant, +fixed rank, zero mean or pre-existing coupling-convergence premise is allowed. +Also prove the exact HasLaw of each such affine vector under stdGaussian as +multivariateGaussian(m_r,S_r). These are the actual pinned Mathlib definitions, +not a newly assumed representation. A measurable coupling or almost-sure +version may be exported as a consequence. CFC continuity on the PSD cone may +be used; do not assume a Lipschitz estimate at a singular covariance. + +The application still must supply actual conditional means/covariances and +their limits, contact-coordinate identities and nonzero reference variance. +This generic result constructs the common finite endpoint coupling once those +quantities are identified; it does not identify the whole conditioned field +or arbitrary field-dependent geometric events. + +Sole new production module, narrow pinned imports, preserved prior proofs and +verification. Root alone updates root/manifest. No scientific-status change. diff --git a/frontiers/determinant_moment_20261025/SOURCE_GAUSSIAN_MOMENTS.md b/frontiers/determinant_moment_20261025/SOURCE_GAUSSIAN_MOMENTS.md new file mode 100644 index 000000000..a9ad4b21c --- /dev/null +++ b/frontiers/determinant_moment_20261025/SOURCE_GAUSSIAN_MOMENTS.md @@ -0,0 +1,110 @@ +# Uniform Gaussian eighth moments: source contract before implementation + +Author OpenAI/Codex /root; scalar coauthor /root/uniform_jet_moment_sources. +Scientific effect NONE; source/correctness review must use another model family. +This new work responds to the owner's inclusion of remaining parent obligations. + +## Exact sources and mathematical role + +Math- base e75cb81e8961d178f43aa9967dddd11cd579dc8a preserves +imports/hardening_ebedb780/P02-LM-005/proof.md.export.txt, section 2 and section 6: +the exact physical midpoint nine-jet conditional law is Gaussian N(m_r,Sigma_r), +with uniformly bounded means and covariance upper bound on a small common interval; +section 6 deduces a uniform moment for R=1+sum of all nine absolute coordinates. +The radius ordering is q,a,w,z,c40,c31,c22,c13,c04, exactly as DM2. +Source export SHA256 0fe0fad5d8e5519cac0e57a42181aa53560bb38d94f23aea51b63be3f182df8b. + +Recovered primary GP-DER-044: main commit7caac254cbba5f513b2dc0afb56b78a598bc0c93, +Git blob34e38cb003a3160c154c07d293d44690a6c8b718 (catalog sources-03.md). +It fixes torus side and height b. Pins are f(-r/2,0)=b, f(r/2,0)=b-r^3/6, +with both endpoint gradients zero. Its transformed prescribed vector is +(b-r^3/12,-r^2/6,0,0,0,2), and m=C G^-1 y, Sigma=S-C G^-1 C^T. +Means are generally nonzero; the side-24 q law is not the planar N(-b,2). +No uniformity over unbounded b is supplied. We inspected the primary source; +this does not transfer its historical reviews to new Lean code. + +Landed DM2 implementation: DeterminantMoment.lean, proof commit + dcfc89366f590342b4cd30b141a22c9ae2a562bb, source SHA256 +661b73c3f21c6d2529f5e09ce426d89c01939ab27a5cc868db3802c7e66ea82d. +Its dm2_moment consumes genuine integrability and an upper bound on integral R^8. +This successor derives those premises from Gaussian coordinate laws and uniform +mean/variance bounds, instead of presupposing them. + +## Frozen mathematical statement + +Define C8=integral |z|^8 under gaussianReal 0 1. Prove its integrability and C8>=0. +For m,B,V real, v nonnegative real, B,V>=0, |m|<=B and v<=V, prove +integrability of |x|^8 under gaussianReal m v and +integral |x|^8 <=128*(B^8+V^4*C8). +All variances, including zero, are retained; no centered-law substitution. + +For a probability measure mu, measurable J:Omega->Fin9->Real, and coordinate +laws HasLaw (fun x=>J x i) (gaussianReal (m i) (v i)) mu, with common B,V as +above, prove integrability and +integral dm2Radius(J)^8 <= M8(B,V), +M8(B,V)=10^7*(1+9*128*(B^8+V^4*C8)). +There is no independence or joint-Gaussian hypothesis. This is stronger than the +source's Gaussian-vector moment deduction and preserves all its noncentered laws. + +A family theorem applies the same constants on 00. +Its contact value at v0(a,q)=(-1,1,-a/2,a/2,q,q) is (max(-q,0))². +This continuous weighted variable includes all singular/type boundaries. + +Target gn_gaussian_reference_integrable_pos: for every real m and nonnegative +real variance v with v!=0, under gaussianReal m v the actual function +q |-> (max(-q,0))² is integrable and its integral is strictly positive. No assumed +positive moment, planar reference or zero-mean substitution is permitted. + +Target gn_open_box_floor: define the literal T_r above. If r>=0, K>0, +r*K²<=1/4, aM in(-5/4,-3/4), aS in(3/4,5/4), cM,cS in(-2,-1), +and |bM|,|bS|=3/8. This is a new explicit open-box consequence +of source formula(1), not a claim that the source stated these exact constants. +Its proof uses F_r>=1/2 and G_r>=3/4, without inverse eigenvalues. + +Research application: a nondegenerate reference Gaussian Q charges(-2,-1). +Together with finite mixed coordinates, an increasing family of these boxes has +positive reference mass for some K. Under actual endpoint weak convergence, +Portmanteau gives mass>=p/2 eventually, and integration gives the full-normalizer +floor Z_r>=(3p/16)*r². The current targets prove reference positivity and the exact +pointwise weight component, not the concrete field convergence or box-mass claim. +That missing conditioned-derivative/Palm identification remains in the research +scope and must be supplied; it is not replaced by a target-assuming hypothesis. + +Implementation is one new GaussianNormalizer.lean module with narrow pinned +Mathlib imports, using the same Lean4.34.1/mathlib revision and the unchanged +kernel/axiom/control gate. Preserve all prior proof/evidence bytes and check only +new code during development. No old source review is silently rebound. This +source note does not qualify as an independent scientific review or status change. + +## Full lower-floor deduction from a common coupling: frozen before code + +New NormalizerFloor.lean target: on one probability space let X_r be measurable +six-endpoint vectors in LM006 order(aM,aS,bM,bS,cM,cS). Suppose as r->0+ they +converge almost surely to(-1,1,-A/2,A/2,Q,Q), where Q has gaussianReal(m,v) +law with v!=0 (m arbitrary), and each T_r(X_r) is genuinely integrable for +positive r sufficiently near0. Then prove existence of c,delta>0 with +integral r²*T_r(X_r) >= c*r² whenever00+ to m0,S0. Assume eventual PSD S_r and +PSD S0, allowing singular matrices. Assume only the concrete contact moment +relations + + m0_0=-1, m0_1=1, m0_2+m0_3=0, m0_4-m0_5=0, + S0_00=0, S0_11=0, + S0_22+2*S0_23+S0_33=0, + S0_44-2*S0_45+S0_55=0, and S0_44>0. + +Prove existence c,delta>0 such that for every 0=0, c>0, r>0, if integral W^2 <= C*r^4 and c*r^2 <= integral W, +prove the actual weighted event probability <= (sqrt C/c)*sqrt(mu.real E). +The numerator estimate must follow from integral Cauchy–Schwarz and the event +indicator. All r-powers must cancel in the proved inequality. An equivalent +explicit normalized set-integral bound may be an auxiliary, but the final +statement must concern the constructed probability measure, not arbitrary +symbols p,z,n supplied as premises. No nonzero event probability premise; +include empty/null events and zero C wherever compatible with the other premises. + +The already proved actual endpoint Gaussian second moment and common-coupling +normalizer floor supply the weight estimates once their concrete conditional +field inputs are identified. This generic weight construction does not by +itself identify a geometric Kac–Rice Palm law or arbitrary whole-field event. +Those actual-law/event joins remain in the original research scope. + +Use narrow pinned imports, preserve old proofs/gate/pins and reuse completed +verification. Record exact commands/exits and meaningful mathematical controls. +Root alone updates root/manifest and final evidence. Different-family source +review remains separate from proof execution and author-side technical work. diff --git a/frontiers/determinant_moment_20261025/SOURCE_REGRESSION_KERNEL.md b/frontiers/determinant_moment_20261025/SOURCE_REGRESSION_KERNEL.md new file mode 100644 index 000000000..2e9ec5127 --- /dev/null +++ b/frontiers/determinant_moment_20261025/SOURCE_REGRESSION_KERNEL.md @@ -0,0 +1,54 @@ +# Canonical Gaussian regression kernel: preimplementation source contract + +OpenAI/Codex root and coauthor /root/regression_kernel. Scientific review must +use another model family. New research remains within the owner's original goal. + +Source: GP-DER-044-v1.0 Exact-torus Gaussian regularity via confluent divided +differences, main commit7caac254cbba5f513b2dc0afb56b78a598bc0c93, Git blob +34e38cb003a3160c154c07d293d44690a6c8b718. Exact public path is catalogued by +main/docs/public-math/sources-03.md and SOURCE_GAUSSIAN_MOMENTS.md. Section4 +uses joint Gaussian observations/targets, the independent Gaussian regression +residual, mean C G^-1 y and Schur covariance S-C G^-1 C^T to define the law +at each prescribed six-pin value. These are probability-zero conditioning events. + +Generic condDistrib versions are unique only almost everywhere; substituting +an arbitrary version at the exact pin is not a valid source identification. +This target therefore constructs an explicit kernel, not just existential RCD. + +Frozen target: for finite real-vector observations X and residual R on a common +probability space, joint Gaussianity and zero cross-covariance imply independence. +Given a continuous affine reconstruction F(x,r)=r+mY+B(x-mX), define +regressionKernel(x)=Law(R).map (fun r=>F(x,r)) at EVERY x. Prove the kernel is +Markov/measurable, its pointwise application equals that literal pushforward, +and HasCondDistrib (fun omega=>F(X omega,R omega)) X regressionKernel P. +No HasCondDistrib, target probability, residual independence or Palm conclusion +may be assumed in the final Gaussian theorem; independence must follow from +joint Gaussianity and zero covariance. A generic independent-residual helper is +permitted as a proved auxiliary. All finite dimensions, including degenerate +residual Gaussian laws, are retained; no residual covariance inverse is used. + +This is the source's canonical conditional-law construction component. Concrete +application still requires jointly Gaussian derivative observations/targets, +positive pin covariance, regression/Schur block identities, correct prescribed +vector and physical jet map, and connection to the literal smooth field. Those +joins are not implied by generic conditioning, this kernel theorem or the already +landed unconditional evaluation-Gaussian constructor. The typed Palm weight and +parent geometric event identification remain separate research obligations. + +Implementation: sole new GaussianRegressionKernel.lean, narrow pinned Mathlib +imports; retain source/failed probes; no oldproof/test campaigns. Root alone +updates manifest/root after stabilization. Existing successful verification is +reused where applicable; final gate tests the changed closure and new targets. + +## Direct observed/target regression strengthening, frozen before code + +The implementation may additionally derive the residual hypotheses from a +jointly Gaussian pair(X,Y). Let B be continuous linear and satisfy the coordinate +normal equations Cov(X_i,Y_j)=sum_l B(e_l)_j*Cov(X_i,X_l). Define +R=Y-mY-B(X-mX) using these ACTUAL random vectors. Prove joint Gaussianity of +(X,R), zero cross-covariance and hence HasCondDistrib Y X for the canonical +residual kernel. Do not assume residual joint Gaussianity, residual independence, +zero residual covariance or a conditional-law conclusion in this strengthened +theorem. Normal equations may be an explicit algebraic premise; in the concrete +source they come from B=C G^-1 and pin-covariance invertibility. Identifying those +blocks and literal field derivatives remains an application obligation. diff --git a/frontiers/determinant_moment_20261025/blueprint/src/content.tex b/frontiers/determinant_moment_20261025/blueprint/src/content.tex index 9f25b5571..7cb3cee5d 100644 --- a/frontiers/determinant_moment_20261025/blueprint/src/content.tex +++ b/frontiers/determinant_moment_20261025/blueprint/src/content.tex @@ -1,5 +1,66 @@ +\lean{ResearchFormalCoreR1.ec005_fold_gap} +\lean{ResearchFormalCoreR1.ec008_factor_expansion} +\lean{ResearchFormalCoreR1.ec010_generic} +\lean{ResearchFormalCoreR1.ec010_transverse} +\lean{ResearchFormalCoreR1.ec011_scalar_cancellation} +\lean{ResearchFormalCoreR1.ec014_contact_power} +\lean{ResearchFormalCoreR1.p02_lm008_r2_cancel} +\lean{ResearchFormalCoreR1.p02_lm008_cross_multiplied} +\lean{ResearchFormalCoreR1.p02_lm008_quotient_bound} +\lean{ResearchFormalCoreR1.p02_lm009_bad_event_threshold} +\lean{ResearchFormalCoreR1.p02_lm009_power40_identity} +\lean{ResearchFormalCoreR1.p02_lm009_r4_le_r3} +\lean{ResearchFormalCoreR1.p02_lm009_palm_r4_to_r3} \lean{ResearchFormalCoreR1.dm2_untyped} \lean{ResearchFormalCoreR1.dm2_pointwise} \lean{ResearchFormalCoreR1.dm2_negative_iff} \lean{ResearchFormalCoreR1.dm2_weight_measurable} \lean{ResearchFormalCoreR1.dm2_moment} +\lean{ResearchFormalCoreR1.gm8_standard_integrable} +\lean{ResearchFormalCoreR1.gm8_standard_nonneg} +\lean{ResearchFormalCoreR1.gm8_scalar_integrable_bound} +\lean{ResearchFormalCoreR1.gm8FiniteM8_nonneg} +\lean{ResearchFormalCoreR1.gm8M8_nonneg} +\lean{ResearchFormalCoreR1.gm8_finite_radius_power_bound} +\lean{ResearchFormalCoreR1.gm8_finite_integrable_bound} +\lean{ResearchFormalCoreR1.gm8_radius_power_bound} +\lean{ResearchFormalCoreR1.gm8_jet_integrable_bound} +\lean{ResearchFormalCoreR1.gm8_family_weight_moment} +\lean{ResearchFormalCoreR1.gn_gaussian_reference_integrable_pos} +\lean{ResearchFormalCoreR1.gn_open_box_floor} +\lean{ResearchFormalCoreR1.regression_compProd_id_const} +\lean{ResearchFormalCoreR1.regression_pair_hasCondDistrib} +\lean{ResearchFormalCoreR1.regressionReconstruction_measurable} +\lean{ResearchFormalCoreR1.regressionKernel_apply} +\lean{ResearchFormalCoreR1.independentResidual_hasCondDistrib} +\lean{ResearchFormalCoreR1.gaussianRegression_hasCondDistrib} +\lean{ResearchFormalCoreR1.regressionKernel_measurable} +\lean{ResearchFormalCoreR1.gaussianRegressionKernel_spec} +\lean{ResearchFormalCoreR1.regression_map_coordinate} +\lean{ResearchFormalCoreR1.regression_covariance_map} +\lean{ResearchFormalCoreR1.regression_residual_zero_covariance} +\lean{ResearchFormalCoreR1.regression_residual_jointGaussian} +\lean{ResearchFormalCoreR1.gaussianRegression_observed_hasCondDistrib} +\lean{ResearchFormalCoreR1.gaussianRegression_observed_spec} +\lean{ResearchFormalCoreR1.egm_weight_measurable} +\lean{ResearchFormalCoreR1.egm_pointwise} +\lean{ResearchFormalCoreR1.egm_gaussian_moment} +\lean{ResearchFormalCoreR1.gn_normalizer_floor_of_common_coupling} +\lean{ResearchFormalCoreR1.enb_permutation_continuous} +\lean{ResearchFormalCoreR1.enb_maximum_iff} +\lean{ResearchFormalCoreR1.enb_saddle_iff} +\lean{ResearchFormalCoreR1.enb_weight_identity} +\lean{ResearchFormalCoreR1.enb_integral_identity} +\lean{ResearchFormalCoreR1.enb_integrable_iff} +\lean{ResearchFormalCoreR1.enb_integrable_of_integrable_sq} +\lean{ResearchFormalCoreR1.enb_normalizer_floor_of_common_coupling} +\lean{ResearchFormalCoreR1.gc_coupledVector_hasLaw} +\lean{ResearchFormalCoreR1.gc_coupledVector_tendsto} +\lean{ResearchFormalCoreR1.gnl_normalizer_floor_of_parameter_limits} +\lean{ResearchFormalCoreR1.gnl_physical_normalizer_floor_of_parameter_limits} +\lean{ResearchFormalCoreR1.prw_integrable_of_integrable_sq} +\lean{ResearchFormalCoreR1.prw_measure_apply} +\lean{ResearchFormalCoreR1.prw_isProbabilityMeasure} +\lean{ResearchFormalCoreR1.prw_measure_real} +\lean{ResearchFormalCoreR1.prw_numerator_bound} +\lean{ResearchFormalCoreR1.prw_square_root_event_bound} diff --git a/frontiers/determinant_moment_20261025/controls/egm_false_midpoint_identification.lean.txt b/frontiers/determinant_moment_20261025/controls/egm_false_midpoint_identification.lean.txt new file mode 100644 index 000000000..f11632919 --- /dev/null +++ b/frontiers/determinant_moment_20261025/controls/egm_false_midpoint_identification.lean.txt @@ -0,0 +1,8 @@ +import ResearchFormalCoreR1.EndpointGaussianMoments +open ResearchFormalCoreR1 +-- Exact endpoints of the invisible-midpoint-jet perturbation, versus the +-- quartic raw-nine-jet radius. These two charts cannot be silently identified. +example : egm_weight 1 ![-9,0,-1,9,0,-1]^2 ≤ + 16*(1 : ℝ)^4*dm2Radius ![-1,0,0,0,0,0,0,0,0]^8 := by + norm_num [egm_weight, egm_HM, egm_HS, dm2_negative_iff, + Matrix.det_fin_two, dm2Radius, Fin.sum_univ_succ] diff --git a/frontiers/determinant_moment_20261025/controls/gaussian_parent_positive.lean.txt b/frontiers/determinant_moment_20261025/controls/gaussian_parent_positive.lean.txt new file mode 100644 index 000000000..297181d4c --- /dev/null +++ b/frontiers/determinant_moment_20261025/controls/gaussian_parent_positive.lean.txt @@ -0,0 +1,26 @@ +import ResearchFormalCoreR1.GaussianJetMoments +import ResearchFormalCoreR1.GaussianRegressionKernel +import ResearchFormalCoreR1.EndpointGaussianMoments +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +open scoped NNReal +-- All nine coordinates can be the SAME Gaussian variable: independence is not used. +example : Integrable (fun z : ℝ => (1+9*|z|)^8) (gaussianReal 0 1) := by + have h := gm8_jet_integrable_bound (gaussianReal 0 1) + (fun z : ℝ => (fun _i : Fin 9 => z)) (by fun_prop) + (fun _ => 0) (fun _ => 1) 0 1 (by norm_num) (by norm_num) + (by intro i; norm_num) (by intro i; norm_num) + (by intro i; exact HasLaw.id) + simpa [dm2Radius, Finset.sum_const] using h.1 +-- Nonzero mean and zero variance are preserved, rather than centered away. +example : (∫ x : ℝ, |x|^8 ∂gaussianReal 1 0) = 1 := by + norm_num [gaussianReal_zero_var] +-- The canonical formula specifies even an observation outside the source support. +example (x : Fin 1 → ℝ) : + regressionKernel (Measure.dirac (0 : Fin 1 → ℝ)) + (ContinuousLinearMap.id ℝ (Fin 1 → ℝ)) 0 0 x = Measure.dirac x := by + rw [regressionKernel_apply, Measure.map_dirac (by fun_prop)] + simp +-- The two actual endpoint types in the six-coordinate chart are allowed. +example : dm2Negative (egm_HM 1 ![-1,0,-1,1,0,-1]) ∧ + (egm_HS 1 ![-1,0,-1,1,0,-1]).det < 0 := by + norm_num [egm_HM, egm_HS, dm2_negative_iff, Matrix.det_fin_two] diff --git a/frontiers/determinant_moment_20261025/controls/gaussian_parent_positive_successor.lean.txt b/frontiers/determinant_moment_20261025/controls/gaussian_parent_positive_successor.lean.txt new file mode 100644 index 000000000..daf6d260b --- /dev/null +++ b/frontiers/determinant_moment_20261025/controls/gaussian_parent_positive_successor.lean.txt @@ -0,0 +1,26 @@ +import ResearchFormalCoreR1.GaussianJetMoments +import ResearchFormalCoreR1.GaussianRegressionKernel +import ResearchFormalCoreR1.EndpointGaussianMoments +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +open scoped NNReal +-- All nine coordinates can be the SAME Gaussian variable: independence is not used. +example : Integrable (fun z : ℝ => (1+9*|z|)^8) (gaussianReal 0 1) := by + have h := gm8_jet_integrable_bound (gaussianReal 0 1) + (fun z : ℝ => (fun _i : Fin 9 => z)) (by fun_prop) + (fun _ => 0) (fun _ => 1) 0 1 (by norm_num) (by norm_num) + (by intro i; norm_num) (by intro i; norm_num) + (by intro i; exact HasLaw.id) + simpa [dm2Radius, Finset.sum_const] using h.1 +-- Nonzero mean and zero variance are preserved, rather than centered away. +example : (∫ x : ℝ, |x|^8 ∂gaussianReal 1 0) = 1 := by + norm_num [gaussianReal_zero_var] +-- The canonical formula specifies even an observation outside the source support. +example (x : Fin 1 → ℝ) : + regressionKernel (Measure.dirac (0 : Fin 1 → ℝ)) + (ContinuousLinearMap.id ℝ (Fin 1 → ℝ)) 0 0 x = Measure.dirac x := by + rw [regressionKernel_apply, Measure.map_dirac' (by fun_prop)] + simp +-- The two actual endpoint types in the six-coordinate chart are allowed. +example : dm2Negative (egm_HM 1 ![-1,0,-1,1,0,-1]) ∧ + (egm_HS 1 ![-1,0,-1,1,0,-1]).det < 0 := by + norm_num [egm_HM, egm_HS, dm2_negative_iff, Matrix.det_fin_two] diff --git a/frontiers/determinant_moment_20261025/controls/gm8_false_centered_substitution.lean.txt b/frontiers/determinant_moment_20261025/controls/gm8_false_centered_substitution.lean.txt new file mode 100644 index 000000000..04150e241 --- /dev/null +++ b/frontiers/determinant_moment_20261025/controls/gm8_false_centered_substitution.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1.GaussianScalarMoments +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +-- Dropping the nonzero mean term would predict zero eighth moment for N(1,0). +example : (∫ x : ℝ, |x|^8 ∂gaussianReal 1 0) ≤ 128*((0 : ℝ)^8+0^4*gm8C8) := by + norm_num [gaussianReal_zero_var] diff --git a/frontiers/determinant_moment_20261025/controls/gm8_false_omit_radius_one.lean.txt b/frontiers/determinant_moment_20261025/controls/gm8_false_omit_radius_one.lean.txt new file mode 100644 index 000000000..350cf2972 --- /dev/null +++ b/frontiers/determinant_moment_20261025/controls/gm8_false_omit_radius_one.lean.txt @@ -0,0 +1,6 @@ +import ResearchFormalCoreR1.GaussianJetMoments +open ResearchFormalCoreR1 +-- The exact radius includes one, even when every Gaussian coordinate vanishes. +example : dm2Radius (fun _i : Fin 9 => 0)^8 ≤ + (10 : ℝ)^7 * ∑ _i : Fin 9, |(0 : ℝ)|^8 := by + norm_num [dm2Radius] diff --git a/frontiers/determinant_moment_20261025/controls/gn_false_degenerate_reference_floor.lean.txt b/frontiers/determinant_moment_20261025/controls/gn_false_degenerate_reference_floor.lean.txt new file mode 100644 index 000000000..d73990e1a --- /dev/null +++ b/frontiers/determinant_moment_20261025/controls/gn_false_degenerate_reference_floor.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1.GaussianNormalizer +open MeasureTheory ProbabilityTheory +-- Strictly positive reference mass requires nonzero variance. +example : 0 < ∫ q : ℝ, (max (-q) 0)^2 ∂gaussianReal 1 0 := by + norm_num [gaussianReal_zero_var] diff --git a/frontiers/determinant_moment_20261025/controls/gn_false_no_small_radius.lean.txt b/frontiers/determinant_moment_20261025/controls/gn_false_no_small_radius.lean.txt new file mode 100644 index 000000000..3a7e99464 --- /dev/null +++ b/frontiers/determinant_moment_20261025/controls/gn_false_no_small_radius.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1.GaussianNormalizer +open ResearchFormalCoreR1 +-- The source's small r*K² budget cannot be discarded from the box floor. +example : (3 : ℝ)/8 ≤ gn_normalizedWeight 1 ![-1,1,9,0,-3/2,-3/2] := by + norm_num [gn_normalizedWeight] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/Audit.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/Audit.lean.txt new file mode 100644 index 000000000..9ed29a330 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/Audit.lean.txt @@ -0,0 +1,67 @@ +import ResearchFormalCoreR1 +#print axioms ResearchFormalCoreR1.ec005_fold_gap +#print axioms ResearchFormalCoreR1.ec008_factor_expansion +#print axioms ResearchFormalCoreR1.ec010_generic +#print axioms ResearchFormalCoreR1.ec010_transverse +#print axioms ResearchFormalCoreR1.ec011_scalar_cancellation +#print axioms ResearchFormalCoreR1.ec014_contact_power +#print axioms ResearchFormalCoreR1.p02_lm008_r2_cancel +#print axioms ResearchFormalCoreR1.p02_lm008_cross_multiplied +#print axioms ResearchFormalCoreR1.p02_lm008_quotient_bound +#print axioms ResearchFormalCoreR1.p02_lm009_bad_event_threshold +#print axioms ResearchFormalCoreR1.p02_lm009_power40_identity +#print axioms ResearchFormalCoreR1.p02_lm009_r4_le_r3 +#print axioms ResearchFormalCoreR1.p02_lm009_palm_r4_to_r3 +#print axioms ResearchFormalCoreR1.dm2_untyped +#print axioms ResearchFormalCoreR1.dm2_pointwise +#print axioms ResearchFormalCoreR1.dm2_negative_iff +#print axioms ResearchFormalCoreR1.dm2_weight_measurable +#print axioms ResearchFormalCoreR1.dm2_moment +#print axioms ResearchFormalCoreR1.gm8_standard_integrable +#print axioms ResearchFormalCoreR1.gm8_standard_nonneg +#print axioms ResearchFormalCoreR1.gm8_scalar_integrable_bound +#print axioms ResearchFormalCoreR1.gm8FiniteM8_nonneg +#print axioms ResearchFormalCoreR1.gm8M8_nonneg +#print axioms ResearchFormalCoreR1.gm8_finite_radius_power_bound +#print axioms ResearchFormalCoreR1.gm8_finite_integrable_bound +#print axioms ResearchFormalCoreR1.gm8_radius_power_bound +#print axioms ResearchFormalCoreR1.gm8_jet_integrable_bound +#print axioms ResearchFormalCoreR1.gm8_family_weight_moment +#print axioms ResearchFormalCoreR1.gn_gaussian_reference_integrable_pos +#print axioms ResearchFormalCoreR1.gn_open_box_floor +#print axioms ResearchFormalCoreR1.regression_compProd_id_const +#print axioms ResearchFormalCoreR1.regression_pair_hasCondDistrib +#print axioms ResearchFormalCoreR1.regressionReconstruction_measurable +#print axioms ResearchFormalCoreR1.regressionKernel_apply +#print axioms ResearchFormalCoreR1.independentResidual_hasCondDistrib +#print axioms ResearchFormalCoreR1.gaussianRegression_hasCondDistrib +#print axioms ResearchFormalCoreR1.regressionKernel_measurable +#print axioms ResearchFormalCoreR1.gaussianRegressionKernel_spec +#print axioms ResearchFormalCoreR1.regression_map_coordinate +#print axioms ResearchFormalCoreR1.regression_covariance_map +#print axioms ResearchFormalCoreR1.regression_residual_zero_covariance +#print axioms ResearchFormalCoreR1.regression_residual_jointGaussian +#print axioms ResearchFormalCoreR1.gaussianRegression_observed_hasCondDistrib +#print axioms ResearchFormalCoreR1.gaussianRegression_observed_spec +#print axioms ResearchFormalCoreR1.egm_weight_measurable +#print axioms ResearchFormalCoreR1.egm_pointwise +#print axioms ResearchFormalCoreR1.egm_gaussian_moment +#print axioms ResearchFormalCoreR1.gn_normalizer_floor_of_common_coupling +#print axioms ResearchFormalCoreR1.enb_permutation_continuous +#print axioms ResearchFormalCoreR1.enb_maximum_iff +#print axioms ResearchFormalCoreR1.enb_saddle_iff +#print axioms ResearchFormalCoreR1.enb_weight_identity +#print axioms ResearchFormalCoreR1.enb_integral_identity +#print axioms ResearchFormalCoreR1.enb_integrable_iff +#print axioms ResearchFormalCoreR1.enb_integrable_of_integrable_sq +#print axioms ResearchFormalCoreR1.enb_normalizer_floor_of_common_coupling +#print axioms ResearchFormalCoreR1.gc_coupledVector_hasLaw +#print axioms ResearchFormalCoreR1.gc_coupledVector_tendsto +#print axioms ResearchFormalCoreR1.gnl_normalizer_floor_of_parameter_limits +#print axioms ResearchFormalCoreR1.gnl_physical_normalizer_floor_of_parameter_limits +#print axioms ResearchFormalCoreR1.prw_integrable_of_integrable_sq +#print axioms ResearchFormalCoreR1.prw_measure_apply +#print axioms ResearchFormalCoreR1.prw_isProbabilityMeasure +#print axioms ResearchFormalCoreR1.prw_measure_real +#print axioms ResearchFormalCoreR1.prw_numerator_bound +#print axioms ResearchFormalCoreR1.prw_square_root_event_bound diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/EXPORT_IDENTITIES.json b/frontiers/determinant_moment_20261025/evidence/gaussian-final/EXPORT_IDENTITIES.json new file mode 100644 index 000000000..80a3326c0 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/EXPORT_IDENTITIES.json @@ -0,0 +1,18 @@ +{ + "checked_source_commit": "16f81b973a02783f8be83081aa7f198724cf5767", + "manifest_sha256": "0008bee4d620206e78ad72ca3d4f39006a8c495f185f556d3b2ffa60e248ccbd", + "scope_sha256": "1105cc13a0ccfe382e9582dccd4cc1d07c215733f565dcf2f75801ab7b090ee4", + "receipt_sha256": "95e502472e3d748c8b5dca3328a8644a9cb7cfc8fe40d5f9cce95a32814e5ec6", + "target_count": 66, + "kernel_gate_exit": 0, + "scientific_effect": "NONE", + "alignment": "PENDING_INDEPENDENT_REVIEW", + "evidence_reuse": "Original applicable mathematical and78Python gate-test checks reused; changed13module root checked once. Separate new actual-source modules have their own scoped receipts.", + "negative_controls": { + "custom_imported": "REJECTED_BY_AXIOM_GATE", + "false_fold": "REJECTED_BY_LEAN", + "false_power": "REJECTED_BY_LEAN", + "native": "REJECTED_BY_AXIOM_GATE", + "sorry": "REJECTED_BY_AXIOM_GATE" + } +} diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/NEW_CONTROLS.json b/frontiers/determinant_moment_20261025/evidence/gaussian-final/NEW_CONTROLS.json new file mode 100644 index 000000000..b755ae350 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/NEW_CONTROLS.json @@ -0,0 +1,99 @@ +{ + "false_controls": [ + { + "name": "gm8_false_centered_substitution", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean" + ], + "exit": 1, + "source_sha256": "96dcad5abc7a1fe24c42bf9c7544ecd6d6a81f71fe6245b9d4de07a5faffde91", + "log_sha256": "6425d08dffa942dac35c0d2c6049f7b45ea7e8ca3bd738d90d082358a6a80e6e", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean:4:76: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gm8_false_omit_radius_one", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean" + ], + "exit": 1, + "source_sha256": "e382d597a45d667a492f10127a50b54e016f8ab808f2b872d0d00acf781a8b9d", + "log_sha256": "c48135a6578a1e599f01bd5d60c41e8353fe574a3eb5a7fd6cc37e591f473b70", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean:5:46: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gn_false_degenerate_reference_floor", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean" + ], + "exit": 1, + "source_sha256": "ef3e9a137e1cb0adb1e5608af2bba29a6b475ccb56a7206e3797e3c7bf4dcdd2", + "log_sha256": "256223c08bec680444daa53eff7bf2f75a97a6046d1175b183ae599b88f98fc6", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean:4:59: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gn_false_no_small_radius", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean" + ], + "exit": 1, + "source_sha256": "c92e629433ec77f420400aaba2cdd9092d322555b0c01bd8836e87178ceca076", + "log_sha256": "aeee073a8c3898625adf42f138af830d9d75d656ddc2cac6e504341832b61958", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean:4:69: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "egm_false_midpoint_identification", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean" + ], + "exit": 1, + "source_sha256": "293faccdf06762221b99723c08acfb4130669b6b6600717a1f96c2c3aa9a4bc1", + "log_sha256": "7c68421d4fc5d388db70182f95c2a9502d5fd9523e9815ba418f1f20c766226b", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean:6:54: error: unsolved goals\n\u22a2 False\n" + } + ], + "positive_successor": { + "source_path": "controls/gaussian_parent_positive_successor.lean.txt", + "source_sha256": "c73799328c4191422c13aefc458ffe6df200da610ec8e79f666db0fbf0817e51", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gaussian_parent_positive_successor.lean" + ], + "exit": 0, + "log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855", + "previous_attempts": [ + "initial missing .olean", + "second wrong map_dirac API; original checked-source file preserved" + ], + "source_changes": "Only successor control tactic API corrected; 66 proof targets/root/manifest unchanged; final gate evidence reused." + }, + "core_proof_source_unchanged": true, + "historical_positive_control_status": "Original bound positive-control draft did not pass: missing import artifact first, wrong map_dirac tactic second. Both raw failures retained. A separately hashed successor fixes only that control API and passes. This does not change/rebind the successful66-target gate manifest." +} diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/Types.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/Types.lean.txt new file mode 100644 index 000000000..47c14c345 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/Types.lean.txt @@ -0,0 +1,68 @@ +import ResearchFormalCoreR1 +set_option pp.explicit true +#check ResearchFormalCoreR1.ec005_fold_gap +#check ResearchFormalCoreR1.ec008_factor_expansion +#check ResearchFormalCoreR1.ec010_generic +#check ResearchFormalCoreR1.ec010_transverse +#check ResearchFormalCoreR1.ec011_scalar_cancellation +#check ResearchFormalCoreR1.ec014_contact_power +#check ResearchFormalCoreR1.p02_lm008_r2_cancel +#check ResearchFormalCoreR1.p02_lm008_cross_multiplied +#check ResearchFormalCoreR1.p02_lm008_quotient_bound +#check ResearchFormalCoreR1.p02_lm009_bad_event_threshold +#check ResearchFormalCoreR1.p02_lm009_power40_identity +#check ResearchFormalCoreR1.p02_lm009_r4_le_r3 +#check ResearchFormalCoreR1.p02_lm009_palm_r4_to_r3 +#check ResearchFormalCoreR1.dm2_untyped +#check ResearchFormalCoreR1.dm2_pointwise +#check ResearchFormalCoreR1.dm2_negative_iff +#check ResearchFormalCoreR1.dm2_weight_measurable +#check ResearchFormalCoreR1.dm2_moment +#check ResearchFormalCoreR1.gm8_standard_integrable +#check ResearchFormalCoreR1.gm8_standard_nonneg +#check ResearchFormalCoreR1.gm8_scalar_integrable_bound +#check ResearchFormalCoreR1.gm8FiniteM8_nonneg +#check ResearchFormalCoreR1.gm8M8_nonneg +#check ResearchFormalCoreR1.gm8_finite_radius_power_bound +#check ResearchFormalCoreR1.gm8_finite_integrable_bound +#check ResearchFormalCoreR1.gm8_radius_power_bound +#check ResearchFormalCoreR1.gm8_jet_integrable_bound +#check ResearchFormalCoreR1.gm8_family_weight_moment +#check ResearchFormalCoreR1.gn_gaussian_reference_integrable_pos +#check ResearchFormalCoreR1.gn_open_box_floor +#check ResearchFormalCoreR1.regression_compProd_id_const +#check ResearchFormalCoreR1.regression_pair_hasCondDistrib +#check ResearchFormalCoreR1.regressionReconstruction_measurable +#check ResearchFormalCoreR1.regressionKernel_apply +#check ResearchFormalCoreR1.independentResidual_hasCondDistrib +#check ResearchFormalCoreR1.gaussianRegression_hasCondDistrib +#check ResearchFormalCoreR1.regressionKernel_measurable +#check ResearchFormalCoreR1.gaussianRegressionKernel_spec +#check ResearchFormalCoreR1.regression_map_coordinate +#check ResearchFormalCoreR1.regression_covariance_map +#check ResearchFormalCoreR1.regression_residual_zero_covariance +#check ResearchFormalCoreR1.regression_residual_jointGaussian +#check ResearchFormalCoreR1.gaussianRegression_observed_hasCondDistrib +#check ResearchFormalCoreR1.gaussianRegression_observed_spec +#check ResearchFormalCoreR1.egm_weight_measurable +#check ResearchFormalCoreR1.egm_pointwise +#check ResearchFormalCoreR1.egm_gaussian_moment +#check ResearchFormalCoreR1.gn_normalizer_floor_of_common_coupling +#check ResearchFormalCoreR1.enb_permutation_continuous +#check ResearchFormalCoreR1.enb_maximum_iff +#check ResearchFormalCoreR1.enb_saddle_iff +#check ResearchFormalCoreR1.enb_weight_identity +#check ResearchFormalCoreR1.enb_integral_identity +#check ResearchFormalCoreR1.enb_integrable_iff +#check ResearchFormalCoreR1.enb_integrable_of_integrable_sq +#check ResearchFormalCoreR1.enb_normalizer_floor_of_common_coupling +#check ResearchFormalCoreR1.gc_coupledVector_hasLaw +#check ResearchFormalCoreR1.gc_coupledVector_tendsto +#check ResearchFormalCoreR1.gnl_normalizer_floor_of_parameter_limits +#check ResearchFormalCoreR1.gnl_physical_normalizer_floor_of_parameter_limits +#check ResearchFormalCoreR1.prw_integrable_of_integrable_sq +#check ResearchFormalCoreR1.prw_measure_apply +#check ResearchFormalCoreR1.prw_isProbabilityMeasure +#check ResearchFormalCoreR1.prw_measure_real +#check ResearchFormalCoreR1.prw_numerator_bound +#check ResearchFormalCoreR1.prw_square_root_event_bound diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/axioms.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/axioms.log new file mode 100644 index 000000000..f805bd41d --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/axioms.log @@ -0,0 +1,74 @@ +'ResearchFormalCoreR1.ec005_fold_gap' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.ec008_factor_expansion' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.ec010_generic' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.ec010_transverse' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.ec011_scalar_cancellation' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.ec014_contact_power' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.p02_lm008_r2_cancel' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.p02_lm008_cross_multiplied' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.p02_lm008_quotient_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.p02_lm009_bad_event_threshold' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.p02_lm009_power40_identity' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.p02_lm009_r4_le_r3' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.p02_lm009_palm_r4_to_r3' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.dm2_untyped' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.dm2_pointwise' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.dm2_negative_iff' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.dm2_weight_measurable' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.dm2_moment' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8_standard_integrable' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8_standard_nonneg' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8_scalar_integrable_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8FiniteM8_nonneg' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8M8_nonneg' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8_finite_radius_power_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8_finite_integrable_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8_radius_power_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8_jet_integrable_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gm8_family_weight_moment' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gn_gaussian_reference_integrable_pos' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gn_open_box_floor' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.regression_compProd_id_const' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.regression_pair_hasCondDistrib' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.regressionReconstruction_measurable' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.regressionKernel_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.independentResidual_hasCondDistrib' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gaussianRegression_hasCondDistrib' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.regressionKernel_measurable' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gaussianRegressionKernel_spec' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.regression_map_coordinate' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.regression_covariance_map' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.regression_residual_zero_covariance' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.regression_residual_jointGaussian' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gaussianRegression_observed_hasCondDistrib' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'ResearchFormalCoreR1.gaussianRegression_observed_spec' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.egm_weight_measurable' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.egm_pointwise' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.egm_gaussian_moment' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gn_normalizer_floor_of_common_coupling' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.enb_permutation_continuous' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.enb_maximum_iff' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.enb_saddle_iff' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.enb_weight_identity' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.enb_integral_identity' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.enb_integrable_iff' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.enb_integrable_of_integrable_sq' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.enb_normalizer_floor_of_common_coupling' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'ResearchFormalCoreR1.gc_coupledVector_hasLaw' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gc_coupledVector_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gnl_normalizer_floor_of_parameter_limits' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'ResearchFormalCoreR1.gnl_physical_normalizer_floor_of_parameter_limits' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'ResearchFormalCoreR1.prw_integrable_of_integrable_sq' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_isProbabilityMeasure' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_real' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_numerator_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_square_root_event_bound' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/build.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/build.log new file mode 100644 index 000000000..d031c0ec5 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/build.log @@ -0,0 +1,27 @@ +✔ [3167/3194] Built ResearchFormalCoreR1.AlgebraV2 (2.6s) +⚠ [3186/3202] Built ResearchFormalCoreR1.ProbabilityCompanionsV2 (3.0s) +warning: ResearchFormalCoreR1/ProbabilityCompanionsV2.lean:27:18: Variable name `hcW` is not explicitly referenced. + +Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning: + [apply] _hcW + +Note: This linter can be disabled with `set_option linter.unusedVariables false` +warning: ResearchFormalCoreR1/ProbabilityCompanionsV2.lean:27:48: Variable name `hq` is not explicitly referenced. + +Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning: + [apply] _hq + +Note: This linter can be disabled with `set_option linter.unusedVariables false` +✔ [3189/3202] Built ResearchFormalCoreR1.GaussianScalarMoments (7.0s) +✔ [3191/3202] Built ResearchFormalCoreR1.PalmReweighting (6.5s) +✔ [3192/3202] Built ResearchFormalCoreR1.GaussianRegressionKernel (7.7s) +✔ [3193/3202] Built ResearchFormalCoreR1.GaussianNormalizer (10s) +✔ [3194/3202] Built ResearchFormalCoreR1.GaussianCoupling (5.1s) +✔ [3195/3202] Built ResearchFormalCoreR1.NormalizerFloor (3.2s) +✔ [3196/3202] Built ResearchFormalCoreR1.DeterminantMoment (14s) +✔ [3197/3202] Built ResearchFormalCoreR1.GaussianJetMoments (3.8s) +✔ [3198/3202] Built ResearchFormalCoreR1.EndpointGaussianMoments (2.8s) +✔ [3199/3202] Built ResearchFormalCoreR1.EndpointNormalizerBridge (3.9s) +✔ [3200/3202] Built ResearchFormalCoreR1.GaussianNormalizerLimit (4.8s) +✔ [3201/3202] Built ResearchFormalCoreR1 (2.4s) +Build completed successfully (3202 jobs). diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/custom_imported.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/custom_imported.lean.txt new file mode 100644 index 000000000..365006fe0 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/custom_imported.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1 +axiom hiddenPremise : False +theorem middle : False := hiddenPremise +theorem injected : False := middle +#print axioms injected diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/custom_imported.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/custom_imported.log new file mode 100644 index 000000000..52be5f132 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/custom_imported.log @@ -0,0 +1 @@ +'injected' depends on axioms: [hiddenPremise] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/elaborated-types.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/elaborated-types.log new file mode 100644 index 000000000..fb27b5974 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/elaborated-types.log @@ -0,0 +1,3450 @@ +ResearchFormalCoreR1.ec005_fold_gap (s : Real) : + @Eq Real + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (ResearchFormalCoreR1.foldPotential s s) + (ResearchFormalCoreR1.foldPotential s (@Neg.neg Real Real.instNeg s))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯)) s) + (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) + (@OfNat.ofNat Real (nat_lit 6) (@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast ⋯))) +ResearchFormalCoreR1.ec008_factor_expansion (c s : Real) : + @Eq Real + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) s + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) s + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast ⋯)) + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) c + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast ⋯)) + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) c + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) s + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + s (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) + (@OfNat.ofNat Real (nat_lit 24) (@instOfNatAtLeastTwo Real (nat_lit 24) Real.instNatCast ⋯))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) s + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast ⋯)) + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) c + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + s (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯)))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@Neg.neg Real Real.instNeg c) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + s (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯)) + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) c + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + s (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast ⋯))) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) s + (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) c + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) s + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast ⋯)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) c + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) s + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@OfNat.ofNat Real (nat_lit 24) (@instOfNatAtLeastTwo Real (nat_lit 24) Real.instNatCast ⋯))) +ResearchFormalCoreR1.ec010_generic (α K c η : Real) : + @Eq Real + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) K + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast ⋯)) α) + c) + η)) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 64) (@instOfNatAtLeastTwo Real (nat_lit 64) Real.instNatCast ⋯)) α) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) α)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) c + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) η + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) α) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) K + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) α + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) K + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast ⋯)) c) + η)) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) +ResearchFormalCoreR1.ec010_transverse (α a η : Real) : + @Eq Real + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) a + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯)) α)) + η)) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast ⋯)) α) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) α)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) η + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) α) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) a η) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) α + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) a η) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) +ResearchFormalCoreR1.ec011_scalar_cancellation (h w g : Real) : + @Eq Real + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) (@Neg.neg Real Real.instNeg h) w) g) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) w + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) h g))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) +ResearchFormalCoreR1.ec014_contact_power (r : Real) + (hr : @Ne Real r (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) : + @Eq Real + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) + (@OfNat.ofNat Real (nat_lit 6) (@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast ⋯))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) + (@OfNat.ofNat Real (nat_lit 6) (@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast ⋯))) +ResearchFormalCoreR1.p02_lm008_r2_cancel (r cW cZ q : Real) + (hr : @Ne Real r (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (hcZ : @Ne Real cZ (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) : + @Eq Real + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) cW.sqrt + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + q.sqrt) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) cZ + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) cW.sqrt cZ) q.sqrt) +ResearchFormalCoreR1.p02_lm008_cross_multiplied (n z r cW cZ q : Real) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hcW : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) cW) + (hcZ : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) cZ) + (hq : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) q) + (hz : + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) cZ + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + z) + (hn : + @LE.le Real Real.instLE n + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) cW.sqrt + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + q.sqrt)) : + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) n + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) cZ + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) cW.sqrt + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + q.sqrt) + z) +ResearchFormalCoreR1.p02_lm008_quotient_bound (n z r cW cZ q : Real) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hcW : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) cW) + (hcZ : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) cZ) + (hq : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) q) + (hzpos : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) z) + (hz : + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) cZ + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + z) + (hn : + @LE.le Real Real.instLE n + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) cW.sqrt + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + q.sqrt)) : + @LE.le Real Real.instLE + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) n z) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) cW.sqrt cZ) q.sqrt) +ResearchFormalCoreR1.p02_lm009_bad_event_threshold (r R ε : Real) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hbad : + @LT.lt Real Real.instLT ε + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) r + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) R + (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))))) : + @LT.lt Real Real.instLT + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) ε r) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) R + (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))) +ResearchFormalCoreR1.p02_lm009_power40_identity (R : Real) : + @Eq Real + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) R + (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))) + (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) R + (@OfNat.ofNat Nat (nat_lit 40) (instOfNatNat (nat_lit 40)))) +ResearchFormalCoreR1.p02_lm009_r4_le_r3 (r : Real) + (hr0 : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hr1 : @LE.le Real Real.instLE r (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) : + @LE.le Real Real.instLE + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) +ResearchFormalCoreR1.p02_lm009_palm_r4_to_r3 (p C r : Real) + (hC : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) C) + (hr0 : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hr1 : @LE.le Real Real.instLE r (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + (hp : + @LE.le Real Real.instLE p + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) C + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) : + @LE.le Real Real.instLE p + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) C + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))) +ResearchFormalCoreR1.dm2_untyped (r : Real) (j : ResearchFormalCoreR1.DM2Jet) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hr1 : @LE.le Real Real.instLE r (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) : + @LE.le Real Real.instLE + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@Matrix.det (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (instDecidableEqFin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.commRing + (ResearchFormalCoreR1.dm2HM r j)) + (@Matrix.det (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (instDecidableEqFin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.commRing + (ResearchFormalCoreR1.dm2HS r j))) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 16) (@instOfNatAtLeastTwo Real (nat_lit 16) Real.instNatCast ⋯)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Radius j) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))))) +ResearchFormalCoreR1.dm2_pointwise (r : Real) (j : ResearchFormalCoreR1.DM2Jet) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hr1 : @LE.le Real Real.instLE r (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) : + @LE.le Real Real.instLE + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Weight r j) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 16) (@instOfNatAtLeastTwo Real (nat_lit 16) Real.instNatCast ⋯)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Radius j) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))))) +ResearchFormalCoreR1.dm2_negative_iff (a b c : Real) : + Iff + (ResearchFormalCoreR1.dm2Negative + (@DFunLike.coe + (Equiv + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → + Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real) + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real)) + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → + Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real) + (fun x => + Matrix (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real) + (@EquivLike.toFunLike + (Equiv + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → + Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real) + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real)) + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → + Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real) + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real) + (@Equiv.instEquivLike + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → + Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real) + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real))) + (@Matrix.of (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real) + (@Matrix.vecCons (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real) + (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) a + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) b (@Matrix.vecEmpty Real))) + (@Matrix.vecCons (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real) + (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) b + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) c + (@Matrix.vecEmpty Real))) + (@Matrix.vecEmpty (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real)))))) + (And (@LT.lt Real Real.instLT a (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) a c) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) b + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))) +ResearchFormalCoreR1.dm2_weight_measurable (r : Real) : + @Measurable ResearchFormalCoreR1.DM2Jet Real + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))) (fun a => Real) fun a => + Real.measurableSpace) + Real.measurableSpace (ResearchFormalCoreR1.dm2Weight r) +ResearchFormalCoreR1.dm2_moment.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] (μ : @MeasureTheory.Measure Ω inst✝) + (J : Ω → ResearchFormalCoreR1.DM2Jet) + (hJ : + @Measurable Ω ResearchFormalCoreR1.DM2Jet inst✝ + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))) (fun a => Real) fun a => + Real.measurableSpace) + J) + (r M8 : Real) (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hr1 : @LE.le Real Real.instLE r (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + (h8 : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Radius (J x)) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + μ) + (hM8 : + @LE.le Real Real.instLE + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup (@NormedField.toNormedSpace Real Real.normedField) inst✝ μ + fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Radius (J x)) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + M8) : + And + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Weight r (J x)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + μ) + (@LE.le Real Real.instLE + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup (@NormedField.toNormedSpace Real Real.normedField) inst✝ μ + fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Weight r (J x)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 16) (@instOfNatAtLeastTwo Real (nat_lit 16) Real.instNatCast ⋯)) M8) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) +ResearchFormalCoreR1.gm8_standard_integrable : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Real Real.measurableSpace + (fun z => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@abs Real Real.lattice Real.instAddGroup z) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (ProbabilityTheory.gaussianReal (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@OfNat.ofNat NNReal (nat_lit 1) (@One.toOfNat1 NNReal NNReal.instOne))) +ResearchFormalCoreR1.gm8_standard_nonneg : + @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) ResearchFormalCoreR1.gm8C8 +ResearchFormalCoreR1.gm8_scalar_integrable_bound (m B V : Real) (v : NNReal) + (hB : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) B) + (hV : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) V) + (hm : @LE.le Real Real.instLE (@abs Real Real.lattice Real.instAddGroup m) B) (hv : @LE.le Real Real.instLE (↑v) V) : + And + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Real Real.measurableSpace + (fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@abs Real Real.lattice Real.instAddGroup x) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (ProbabilityTheory.gaussianReal m v)) + (@LE.le Real Real.instLE + (@MeasureTheory.integral Real Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + Real.measurableSpace (ProbabilityTheory.gaussianReal m v) fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@abs Real Real.lattice Real.instAddGroup x) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 128) (@instOfNatAtLeastTwo Real (nat_lit 128) Real.instNatCast ⋯)) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) B + (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) V + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))) + ResearchFormalCoreR1.gm8C8)))) +ResearchFormalCoreR1.gm8FiniteM8_nonneg (n : Nat) (B V : Real) : + @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (ResearchFormalCoreR1.gm8FiniteM8 n B V) +ResearchFormalCoreR1.gm8M8_nonneg (B V : Real) : + @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (ResearchFormalCoreR1.gm8M8 B V) +ResearchFormalCoreR1.gm8_finite_radius_power_bound (n : Nat) (j : Fin n → Real) : + @LE.le Real Real.instLE + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@ResearchFormalCoreR1.gm8Radius n j) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@Nat.cast Real Real.instNatCast + (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) n + (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) + (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7)))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) + (∑ i, + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@abs Real Real.lattice Real.instAddGroup (j i)) + (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))))) +ResearchFormalCoreR1.gm8_finite_integrable_bound.{u_1} (n : Nat) {Ω : Type u_1} [MeasurableSpace Ω] + (μ : @MeasureTheory.Measure Ω inst✝) [@MeasureTheory.IsProbabilityMeasure Ω inst✝ μ] (J : Ω → Fin n → Real) + (hJ : + @Measurable Ω (Fin n → Real) inst✝ (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) J) + (m : Fin n → Real) (v : Fin n → NNReal) (B V : Real) + (hB : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) B) + (hV : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) V) + (hmean : ∀ (i : Fin n), @LE.le Real Real.instLE (@abs Real Real.lattice Real.instAddGroup (m i)) B) + (hvar : ∀ (i : Fin n), @LE.le Real Real.instLE (↑(v i)) V) + (hlaw : + ∀ (i : Fin n), + @ProbabilityTheory.HasLaw Ω Real inst✝ Real.measurableSpace (fun x => J x i) + (ProbabilityTheory.gaussianReal (m i) (v i)) μ) : + And + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@ResearchFormalCoreR1.gm8Radius n (J x)) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + μ) + (@LE.le Real Real.instLE + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@ResearchFormalCoreR1.gm8Radius n (J x)) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (ResearchFormalCoreR1.gm8FiniteM8 n B V)) +ResearchFormalCoreR1.gm8_radius_power_bound (j : ResearchFormalCoreR1.DM2Jet) : + @LE.le Real Real.instLE + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Radius j) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@OfNat.ofNat Real (nat_lit 10) (@instOfNatAtLeastTwo Real (nat_lit 10) Real.instNatCast ⋯)) + (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7)))) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) + (∑ i, + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@abs Real Real.lattice Real.instAddGroup (j i)) + (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))))) +ResearchFormalCoreR1.gm8_jet_integrable_bound.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + (μ : @MeasureTheory.Measure Ω inst✝) [@MeasureTheory.IsProbabilityMeasure Ω inst✝ μ] + (J : Ω → ResearchFormalCoreR1.DM2Jet) + (hJ : + @Measurable Ω ResearchFormalCoreR1.DM2Jet inst✝ + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))) (fun a => Real) fun a => + Real.measurableSpace) + J) + (m : Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9))) → Real) + (v : Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9))) → NNReal) (B V : Real) + (hB : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) B) + (hV : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) V) + (hmean : + ∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))), + @LE.le Real Real.instLE (@abs Real Real.lattice Real.instAddGroup (m i)) B) + (hvar : ∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))), @LE.le Real Real.instLE (↑(v i)) V) + (hlaw : + ∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))), + @ProbabilityTheory.HasLaw Ω Real inst✝ Real.measurableSpace (fun x => J x i) + (ProbabilityTheory.gaussianReal (m i) (v i)) μ) : + And + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Radius (J x)) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + μ) + (@LE.le Real Real.instLE + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Radius (J x)) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))) + (ResearchFormalCoreR1.gm8M8 B V)) +ResearchFormalCoreR1.gm8_family_weight_moment.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + (μ : Real → @MeasureTheory.Measure Ω inst✝) (J : Real → Ω → ResearchFormalCoreR1.DM2Jet) + (m : Real → Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9))) → Real) + (v : Real → Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9))) → NNReal) (B V r0 : Real) + (hB : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) B) + (hV : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) V) + (hr01 : @LE.le Real Real.instLE r0 (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + (hprob : + ∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LE.le Real Real.instLE r r0 → @MeasureTheory.IsProbabilityMeasure Ω inst✝ (μ r)) + (hJ : + ∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LE.le Real Real.instLE r r0 → + @Measurable Ω ResearchFormalCoreR1.DM2Jet inst✝ + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))) (fun a => Real) + fun a => Real.measurableSpace) + (J r)) + (hmean : + ∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LE.le Real Real.instLE r r0 → + ∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))), + @LE.le Real Real.instLE (@abs Real Real.lattice Real.instAddGroup (m r i)) B) + (hvar : + ∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LE.le Real Real.instLE r r0 → + ∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))), @LE.le Real Real.instLE (↑(v r i)) V) + (hlaw : + ∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LE.le Real Real.instLE r r0 → + ∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))), + @ProbabilityTheory.HasLaw Ω Real inst✝ Real.measurableSpace (fun x => J r x i) + (ProbabilityTheory.gaussianReal (m r i) (v r i)) (μ r)) : + And + (@LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 16) (@instOfNatAtLeastTwo Real (nat_lit 16) Real.instNatCast ⋯)) + (ResearchFormalCoreR1.gm8M8 B V))) + (∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LE.le Real Real.instLE r r0 → + And + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Weight r (J r x)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (μ r)) + (@LE.le Real Real.instLE + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ (μ r) fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.dm2Weight r (J r x)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 16) (@instOfNatAtLeastTwo Real (nat_lit 16) Real.instNatCast ⋯)) + (ResearchFormalCoreR1.gm8M8 B V)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + r (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))))) +ResearchFormalCoreR1.gn_gaussian_reference_integrable_pos (m : Real) (v : NNReal) + (hv : @Ne NNReal v (@OfNat.ofNat NNReal (nat_lit 0) (@Zero.toOfNat0 NNReal NNReal.instZero))) : + And + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Real Real.measurableSpace + (fun q => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@max Real Real.instMax (@Neg.neg Real Real.instNeg q) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (ProbabilityTheory.gaussianReal m v)) + (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@MeasureTheory.integral Real Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + Real.measurableSpace (ProbabilityTheory.gaussianReal m v) fun q => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@max Real Real.instMax (@Neg.neg Real Real.instNeg q) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) +ResearchFormalCoreR1.gn_open_box_floor (r K : Real) + (v : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (hr : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hK : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) K) + (hsmall : + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) r + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) K + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) + (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast ⋯)))) + (haM : + @Membership.mem Real (Set Real) (@Set.instMembership Real) + (@Set.Ioo Real Real.instPreorder + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@Neg.neg Real Real.instNeg + (@OfNat.ofNat Real (nat_lit 5) (@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast ⋯))) + (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast ⋯))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@Neg.neg Real Real.instNeg + (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast ⋯))) + (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast ⋯)))) + (v + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0))))) + (haS : + @Membership.mem Real (Set Real) (@Set.instMembership Real) + (@Set.Ioo Real Real.instPreorder + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast ⋯)) + (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast ⋯))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@OfNat.ofNat Real (nat_lit 5) (@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast ⋯)) + (@OfNat.ofNat Real (nat_lit 4) (@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast ⋯)))) + (v + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1))))) + (hbM : + @LT.lt Real Real.instLT + (@abs Real Real.lattice Real.instAddGroup + (v + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2))))) + K) + (_hbS : + @LT.lt Real Real.instLT + (@abs Real Real.lattice Real.instAddGroup + (v + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3))))) + K) + (hcM : + @Membership.mem Real (Set Real) (@Set.instMembership Real) + (@Set.Ioo Real Real.instPreorder + (@Neg.neg Real Real.instNeg + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯))) + (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) + (v + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4))))) + (hcS : + @Membership.mem Real (Set Real) (@Set.instMembership Real) + (@Set.Ioo Real Real.instPreorder + (@Neg.neg Real Real.instNeg + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯))) + (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) + (v + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5))))) : + @LE.le Real Real.instLE + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@OfNat.ofNat Real (nat_lit 3) (@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast ⋯)) + (@OfNat.ofNat Real (nat_lit 8) (@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast ⋯))) + (ResearchFormalCoreR1.gn_normalizedWeight r v) +ResearchFormalCoreR1.regression_compProd_id_const.{u_1, u_2} {α : Type u_1} {β : Type u_2} [MeasurableSpace α] + [MeasurableSpace β] (μ : @MeasureTheory.Measure α inst✝) (ν : @MeasureTheory.Measure β inst✝¹) + [@MeasureTheory.SFinite α inst✝ μ] [@MeasureTheory.SFinite β inst✝¹ ν] : + @Eq + (@MeasureTheory.Measure (Prod α (Prod α β)) + (@Prod.instMeasurableSpace α (Prod α β) inst✝ (@Prod.instMeasurableSpace α β inst✝ inst✝¹))) + (@MeasureTheory.Measure.compProd α (Prod α β) inst✝ (@Prod.instMeasurableSpace α β inst✝ inst✝¹) μ + (@ProbabilityTheory.Kernel.prod α α inst✝ inst✝ β inst✝¹ (@ProbabilityTheory.Kernel.id α inst✝) + (@ProbabilityTheory.Kernel.const α β inst✝ inst✝¹ ν))) + (@MeasureTheory.Measure.map (Prod α β) (Prod α (Prod α β)) (@Prod.instMeasurableSpace α β inst✝ inst✝¹) + (@Prod.instMeasurableSpace α (Prod α β) inst✝ (@Prod.instMeasurableSpace α β inst✝ inst✝¹)) + (fun p => @Prod.mk α (Prod α β) (@Prod.fst α β p) p) (@MeasureTheory.Measure.prod α β inst✝ inst✝¹ μ ν)) +ResearchFormalCoreR1.regression_pair_hasCondDistrib.{u_1, u_2, u_3} {Ω : Type u_1} {α : Type u_2} {β : Type u_3} + [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace β] {P : @MeasureTheory.Measure Ω inst✝} + [@MeasureTheory.IsProbabilityMeasure Ω inst✝ P] {X : Ω → α} {R : Ω → β} (hX : @AEMeasurable Ω α inst✝¹ inst✝ X P) + (hR : @AEMeasurable Ω β inst✝² inst✝ R P) (hXR : @ProbabilityTheory.IndepFun Ω α β inst✝ inst✝¹ inst✝² X R P) : + @ProbabilityTheory.HasCondDistrib Ω α (Prod α β) inst✝ inst✝¹ (@Prod.instMeasurableSpace α β inst✝¹ inst✝²) + (fun ω => @Prod.mk α β (X ω) (R ω)) X + (@ProbabilityTheory.Kernel.prod α α inst✝¹ inst✝¹ β inst✝² (@ProbabilityTheory.Kernel.id α inst✝¹) + (@ProbabilityTheory.Kernel.const α β inst✝¹ inst✝² (@MeasureTheory.Measure.map Ω β inst✝ inst✝² R P))) + P +ResearchFormalCoreR1.regressionReconstruction_measurable {n k : Nat} + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) : + @Measurable (Prod (Fin n → Real) (Fin k → Real)) (Fin k → Real) + (@Prod.instMeasurableSpace (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (@ResearchFormalCoreR1.regressionReconstruction n k B mX mY) +ResearchFormalCoreR1.regressionKernel_apply {n k : Nat} + (ν : + @MeasureTheory.Measure (Fin k → Real) (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + [@MeasureTheory.SFinite (Fin k → Real) (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) ν] + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) (x : Fin n → Real) : + @Eq + (@MeasureTheory.Measure (Fin k → Real) (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@DFunLike.coe + (@ProbabilityTheory.Kernel (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (Fin n → Real) + (fun x => + @MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ProbabilityTheory.Kernel.instFunLike (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ResearchFormalCoreR1.regressionKernel n k ν B mX mY) x) + (@MeasureTheory.Measure.map (Fin k → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun r => + @HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) + (@HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) r mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) x mX))) + ν) +ResearchFormalCoreR1.independentResidual_hasCondDistrib.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + {P : @MeasureTheory.Measure Ω inst✝} [@MeasureTheory.IsProbabilityMeasure Ω inst✝ P] {n k : Nat} + {X : Ω → Fin n → Real} {R : Ω → Fin k → Real} + (hX : + @AEMeasurable Ω (Fin n → Real) (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) inst✝ X + P) + (hR : + @AEMeasurable Ω (Fin k → Real) (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) inst✝ R + P) + (hXR : + @ProbabilityTheory.IndepFun Ω (Fin n → Real) (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) X R P) + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) : + @ProbabilityTheory.HasCondDistrib Ω (Fin n → Real) (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun ω => + @HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) + (@HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) (R ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX))) + X + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) R P) + B mX mY) + P +ResearchFormalCoreR1.gaussianRegression_hasCondDistrib.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + {P : @MeasureTheory.Measure Ω inst✝} {n k : Nat} {X : Ω → Fin n → Real} {R : Ω → Fin k → Real} + (hXR : + @ProbabilityTheory.HasGaussianLaw Ω (Prod (Fin n → Real) (Fin k → Real)) inst✝ + (@instTopologicalSpaceProd (Fin n → Real) (Fin k → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + (@Prod.instAddCommMonoid (Fin n → Real) (Fin k → Real) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid)) + (@Prod.instModule Real (Fin n → Real) (Fin k → Real) Real.semiring + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (@Prod.instMeasurableSpace (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (fun ω => @Prod.mk (Fin n → Real) (Fin k → Real) (X ω) (R ω)) P) + (hzero : + ∀ (i : Fin n) (j : Fin k), + @Eq Real (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) (fun ω => R ω j) P) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) : + @ProbabilityTheory.HasCondDistrib Ω (Fin n → Real) (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun ω => + @HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) + (@HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) (R ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX))) + X + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) R P) + B mX mY) + P +ResearchFormalCoreR1.regressionKernel_measurable {n k : Nat} + (ν : + @MeasureTheory.Measure (Fin k → Real) (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) : + @Measurable (Fin n → Real) + (@MeasureTheory.Measure (Fin k → Real) (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasureTheory.Measure.instMeasurableSpace (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@DFunLike.coe + (@ProbabilityTheory.Kernel (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (Fin n → Real) + (fun x => + @MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ProbabilityTheory.Kernel.instFunLike (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ResearchFormalCoreR1.regressionKernel n k ν B mX mY)) +ResearchFormalCoreR1.gaussianRegressionKernel_spec.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + {P : @MeasureTheory.Measure Ω inst✝} {n k : Nat} {X : Ω → Fin n → Real} {R : Ω → Fin k → Real} + (hXR : + @ProbabilityTheory.HasGaussianLaw Ω (Prod (Fin n → Real) (Fin k → Real)) inst✝ + (@instTopologicalSpaceProd (Fin n → Real) (Fin k → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + (@Prod.instAddCommMonoid (Fin n → Real) (Fin k → Real) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid)) + (@Prod.instModule Real (Fin n → Real) (Fin k → Real) Real.semiring + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (@Prod.instMeasurableSpace (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (fun ω => @Prod.mk (Fin n → Real) (Fin k → Real) (X ω) (R ω)) P) + (hzero : + ∀ (i : Fin n) (j : Fin k), + @Eq Real (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) (fun ω => R ω j) P) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) : + And + (@ProbabilityTheory.IsMarkovKernel (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) R P) + B mX mY)) + (And + (@Measurable (Fin n → Real) + (@MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasureTheory.Measure.instMeasurableSpace (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@DFunLike.coe + (@ProbabilityTheory.Kernel (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (Fin n → Real) + (fun x => + @MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ProbabilityTheory.Kernel.instFunLike (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) R P) + B mX mY))) + (And + (∀ (x : Fin n → Real), + @Eq + (@MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@DFunLike.coe + (@ProbabilityTheory.Kernel (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (Fin n → Real) + (fun x => + @MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ProbabilityTheory.Kernel.instFunLike (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) R P) + B mX mY) + x) + (@MeasureTheory.Measure.map (Fin k → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun r => + @HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) + (@HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) r mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) x mX))) + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) R P))) + (@ProbabilityTheory.HasCondDistrib Ω (Fin n → Real) (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun ω => + @HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) + (@HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) (R ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX))) + X + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) R P) + B mX mY) + P))) +ResearchFormalCoreR1.regression_map_coordinate {n k : Nat} + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (v : Fin n → Real) (j : Fin k) : + @Eq Real + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B v j) + (∑ l, + @HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@Pi.single (Fin n) (fun a => Real) (fun i => Real.instZero) (instDecidableEqFin n) l + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + j) + (v l)) +ResearchFormalCoreR1.regression_covariance_map.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + {P : @MeasureTheory.Measure Ω inst✝} [@MeasureTheory.IsProbabilityMeasure Ω inst✝ P] {n k : Nat} + {X : Ω → Fin n → Real} + (hX : + @ProbabilityTheory.HasGaussianLaw Ω (Fin n → Real) inst✝ + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) X P) + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (i : Fin n) (j : Fin k) : + @Eq Real + (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) + (fun ω => + @DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX) + j) + P) + (∑ l, + @HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@Pi.single (Fin n) (fun a => Real) (fun i => Real.instZero) (instDecidableEqFin n) l + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + j) + (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) (fun ω => X ω l) P)) +ResearchFormalCoreR1.regression_residual_zero_covariance.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + {P : @MeasureTheory.Measure Ω inst✝} {n k : Nat} {X : Ω → Fin n → Real} {Y : Ω → Fin k → Real} + (hXY : + @ProbabilityTheory.HasGaussianLaw Ω (Prod (Fin n → Real) (Fin k → Real)) inst✝ + (@instTopologicalSpaceProd (Fin n → Real) (Fin k → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + (@Prod.instAddCommMonoid (Fin n → Real) (Fin k → Real) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid)) + (@Prod.instModule Real (Fin n → Real) (Fin k → Real) Real.semiring + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (@Prod.instMeasurableSpace (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (fun ω => @Prod.mk (Fin n → Real) (Fin k → Real) (X ω) (Y ω)) P) + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) + (hnormal : + ∀ (i : Fin n) (j : Fin k), + @Eq Real (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) (fun ω => Y ω j) P) + (∑ l, + @HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@Pi.single (Fin n) (fun a => Real) (fun i => Real.instZero) (instDecidableEqFin n) l + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + j) + (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) (fun ω => X ω l) P))) + (i : Fin n) (j : Fin k) : + @Eq Real + (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) + (fun ω => + @HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) + (@HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) (Y ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX)) + j) + P) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) +ResearchFormalCoreR1.regression_residual_jointGaussian.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + {P : @MeasureTheory.Measure Ω inst✝} {n k : Nat} {X : Ω → Fin n → Real} {Y : Ω → Fin k → Real} + (hXY : + @ProbabilityTheory.HasGaussianLaw Ω (Prod (Fin n → Real) (Fin k → Real)) inst✝ + (@instTopologicalSpaceProd (Fin n → Real) (Fin k → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + (@Prod.instAddCommMonoid (Fin n → Real) (Fin k → Real) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid)) + (@Prod.instModule Real (Fin n → Real) (Fin k → Real) Real.semiring + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (@Prod.instMeasurableSpace (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (fun ω => @Prod.mk (Fin n → Real) (Fin k → Real) (X ω) (Y ω)) P) + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) : + @ProbabilityTheory.HasGaussianLaw Ω (Prod (Fin n → Real) (Fin k → Real)) inst✝ + (@instTopologicalSpaceProd (Fin n → Real) (Fin k → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + (@Prod.instAddCommMonoid (Fin n → Real) (Fin k → Real) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid)) + (@Prod.instModule Real (Fin n → Real) (Fin k → Real) Real.semiring + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (@Prod.instMeasurableSpace (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (fun ω => + @Prod.mk (Fin n → Real) (Fin k → Real) (X ω) + (@HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) + (@HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) (Y ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX)))) + P +ResearchFormalCoreR1.gaussianRegression_observed_hasCondDistrib.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + {P : @MeasureTheory.Measure Ω inst✝} {n k : Nat} {X : Ω → Fin n → Real} {Y : Ω → Fin k → Real} + (hXY : + @ProbabilityTheory.HasGaussianLaw Ω (Prod (Fin n → Real) (Fin k → Real)) inst✝ + (@instTopologicalSpaceProd (Fin n → Real) (Fin k → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + (@Prod.instAddCommMonoid (Fin n → Real) (Fin k → Real) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid)) + (@Prod.instModule Real (Fin n → Real) (Fin k → Real) Real.semiring + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (@Prod.instMeasurableSpace (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (fun ω => @Prod.mk (Fin n → Real) (Fin k → Real) (X ω) (Y ω)) P) + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) + (hnormal : + ∀ (i : Fin n) (j : Fin k), + @Eq Real (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) (fun ω => Y ω j) P) + (∑ l, + @HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@Pi.single (Fin n) (fun a => Real) (fun i => Real.instZero) (instDecidableEqFin n) l + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + j) + (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) (fun ω => X ω l) P))) : + @ProbabilityTheory.HasCondDistrib Ω (Fin n → Real) (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) Y X + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun ω => + @HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) + (@HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) (Y ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX))) + P) + B mX mY) + P +ResearchFormalCoreR1.gaussianRegression_observed_spec.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + {P : @MeasureTheory.Measure Ω inst✝} {n k : Nat} {X : Ω → Fin n → Real} {Y : Ω → Fin k → Real} + (hXY : + @ProbabilityTheory.HasGaussianLaw Ω (Prod (Fin n → Real) (Fin k → Real)) inst✝ + (@instTopologicalSpaceProd (Fin n → Real) (Fin k → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + (@Prod.instAddCommMonoid (Fin n → Real) (Fin k → Real) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid)) + (@Prod.instModule Real (Fin n → Real) (Fin k → Real) Real.semiring + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (@Prod.instMeasurableSpace (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (fun ω => @Prod.mk (Fin n → Real) (Fin k → Real) (X ω) (Y ω)) P) + (B : + @ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (mX : Fin n → Real) (mY : Fin k → Real) + (hnormal : + ∀ (i : Fin n) (j : Fin k), + @Eq Real (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) (fun ω => Y ω j) P) + (∑ l, + @HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@Pi.single (Fin n) (fun a => Real) (fun i => Real.instZero) (instDecidableEqFin n) l + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + j) + (@ProbabilityTheory.covariance Ω inst✝ (fun ω => X ω i) (fun ω => X ω l) P))) : + And + (@ProbabilityTheory.IsMarkovKernel (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun ω => + @HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) + (@HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) (Y ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX))) + P) + B mX mY)) + (And + (@Measurable (Fin n → Real) + (@MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasureTheory.Measure.instMeasurableSpace (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@DFunLike.coe + (@ProbabilityTheory.Kernel (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (Fin n → Real) + (fun x => + @MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ProbabilityTheory.Kernel.instFunLike (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun ω => + @HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) + (@HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) (Y ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX))) + P) + B mX mY))) + (And + (∀ (x : Fin n → Real), + @Eq + (@MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@DFunLike.coe + (@ProbabilityTheory.Kernel (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (Fin n → Real) + (fun x => + @MeasureTheory.Measure (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ProbabilityTheory.Kernel.instFunLike (Fin n → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace)) + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun ω => + @HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) + (@HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) (Y ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) + mX))) + P) + B mX mY) + x) + (@MeasureTheory.Measure.map (Fin k → Real) (Fin k → Real) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun r => + @HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) + (@HAdd.hAdd (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHAdd (Fin k → Real) (@Pi.instAdd (Fin k) (fun a => Real) fun i => Real.instAdd)) r mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) x mX))) + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun ω => + @HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) + (@HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) (Y ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) + mX))) + P))) + (@ProbabilityTheory.HasCondDistrib Ω (Fin n → Real) (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin n) (fun a => Real) fun a => Real.measurableSpace) + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) Y X + (@ResearchFormalCoreR1.regressionKernel n k + (@MeasureTheory.Measure.map Ω (Fin k → Real) inst✝ + (@MeasurableSpace.pi (Fin k) (fun a => Real) fun a => Real.measurableSpace) + (fun ω => + @HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) + (@HSub.hSub (Fin k → Real) (Fin k → Real) (Fin k → Real) + (@instHSub (Fin k → Real) (@Pi.instSub (Fin k) (fun a => Real) fun i => Real.instSub)) (Y ω) mY) + (@DFunLike.coe + (@ContinuousLinearMap Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + (Fin n → Real) (fun x => Fin k → Real) + (@ContinuousLinearMap.funLike Real Real Real.semiring Real.semiring + (@RingHom.id Real (@Semiring.toNonAssocSemiring Real Real.semiring)) (Fin n → Real) + (@Pi.topologicalSpace (Fin n) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin n) (fun a => Real) fun i => Real.instAddCommMonoid) (Fin k → Real) + (@Pi.topologicalSpace (Fin k) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.addCommMonoid (Fin k) (fun a => Real) fun i => Real.instAddCommMonoid) + (@Pi.Function.module (Fin n) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring)) + (@Pi.Function.module (Fin k) Real Real Real.semiring Real.instAddCommMonoid + (@Semiring.toModule Real Real.semiring))) + B + (@HSub.hSub (Fin n → Real) (Fin n → Real) (Fin n → Real) + (@instHSub (Fin n → Real) (@Pi.instSub (Fin n) (fun a => Real) fun i => Real.instSub)) (X ω) mX))) + P) + B mX mY) + P))) +ResearchFormalCoreR1.egm_weight_measurable (r : Real) : + @Measurable (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) Real + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun a => + Real.measurableSpace) + Real.measurableSpace (ResearchFormalCoreR1.egm_weight r) +ResearchFormalCoreR1.egm_pointwise (r : Real) (j : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hr1 : @LE.le Real Real.instLE r (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) : + @LE.le Real Real.instLE + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.egm_weight r j) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 16) (@instOfNatAtLeastTwo Real (nat_lit 16) Real.instNatCast ⋯)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (@ResearchFormalCoreR1.gm8Radius (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) j) + (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))))) +ResearchFormalCoreR1.egm_gaussian_moment.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] (μ : @MeasureTheory.Measure Ω inst✝) + [@MeasureTheory.IsProbabilityMeasure Ω inst✝ μ] + (J : Ω → Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (hJ : + @Measurable Ω (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) inst✝ + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun a => + Real.measurableSpace) + J) + (m : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (v : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → NNReal) (B V r : Real) + (hB : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) B) + (hV : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) V) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hr1 : @LE.le Real Real.instLE r (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + (hmean : + ∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))), + @LE.le Real Real.instLE (@abs Real Real.lattice Real.instAddGroup (m i)) B) + (hvar : ∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))), @LE.le Real Real.instLE (↑(v i)) V) + (hlaw : + ∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))), + @ProbabilityTheory.HasLaw Ω Real inst✝ Real.measurableSpace (fun x => J x i) + (ProbabilityTheory.gaussianReal (m i) (v i)) μ) : + And + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.egm_weight r (J x)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + μ) + (@LE.le Real Real.instLE + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.egm_weight r (J x)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 16) (@instOfNatAtLeastTwo Real (nat_lit 16) Real.instNatCast ⋯)) + (ResearchFormalCoreR1.gm8FiniteM8 (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) B V)) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) +ResearchFormalCoreR1.gn_normalizer_floor_of_common_coupling.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + (P : @MeasureTheory.Measure Ω inst✝) [@MeasureTheory.IsProbabilityMeasure Ω inst✝ P] + (X : Real → Ω → Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (hX : + ∀ (r : Real), + @Measurable Ω (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) inst✝ + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun a => + Real.measurableSpace) + (X r)) + (A Q : Ω → Real) (m : Real) (v : NNReal) + (hv : @Ne NNReal v (@OfNat.ofNat NNReal (nat_lit 0) (@Zero.toOfNat0 NNReal NNReal.instZero))) + (hQ : @ProbabilityTheory.HasLaw Ω Real inst✝ Real.measurableSpace Q (ProbabilityTheory.gaussianReal m v) P) + (hconv : + @Filter.Eventually Ω + (fun ω => + @Filter.Tendsto Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) (fun r => X r ω) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) + (@nhds (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (@Pi.topologicalSpace (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) + fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) + (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@Neg.neg Real Real.instNeg (A ω)) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (A ω) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) (Q ω) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) (Q ω) + (@Matrix.vecEmpty Real))))))))) + (@MeasureTheory.ae Ω (@MeasureTheory.Measure Ω inst✝) (@MeasureTheory.Measure.instFunLike Ω inst✝) ⋯ P)) + (hint : + @Filter.Eventually Real + (fun r => + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ (fun ω => ResearchFormalCoreR1.gn_normalizedWeight r (X r ω)) P) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))) : + ∃ c, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) c) + (∃ δ, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) δ) + (∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LT.lt Real Real.instLT r δ → + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ P fun ω => + @HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (ResearchFormalCoreR1.gn_normalizedWeight r (X r ω))))) +ResearchFormalCoreR1.enb_permutation_continuous : + @Continuous (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (@Pi.topologicalSpace (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Pi.topologicalSpace (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + ResearchFormalCoreR1.enb_permutation +ResearchFormalCoreR1.enb_maximum_iff (r : Real) + (j : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) : + Iff (ResearchFormalCoreR1.dm2Negative (ResearchFormalCoreR1.egm_HM r j)) + (And + (@LT.lt Real Real.instLT + (j + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (j + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (j + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2))))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) r + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (j + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))) +ResearchFormalCoreR1.enb_saddle_iff (r : Real) + (j : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) : + Iff + (@LT.lt Real Real.instLT + (@Matrix.det (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (instDecidableEqFin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (Fin.fintype (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real Real.commRing + (ResearchFormalCoreR1.egm_HS r j)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (@LT.lt Real Real.instLT + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (j + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3)))) + (j + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5))))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) r + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (j + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) +ResearchFormalCoreR1.enb_weight_identity (r : Real) + (j : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) : + @Eq Real (ResearchFormalCoreR1.egm_weight r j) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (ResearchFormalCoreR1.gn_normalizedWeight r (ResearchFormalCoreR1.enb_permutation j))) +ResearchFormalCoreR1.enb_integral_identity.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] (P : @MeasureTheory.Measure Ω inst✝) + (J : Ω → Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) (r : Real) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) : + @Eq Real + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ P fun ω => ResearchFormalCoreR1.egm_weight r (J ω)) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ P fun ω => ResearchFormalCoreR1.gn_normalizedWeight r (ResearchFormalCoreR1.enb_permutation (J ω)))) +ResearchFormalCoreR1.enb_integrable_iff.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] (P : @MeasureTheory.Measure Ω inst✝) + (J : Ω → Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) (r : Real) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) : + Iff + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ (fun ω => ResearchFormalCoreR1.egm_weight r (J ω)) P) + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ (fun ω => ResearchFormalCoreR1.gn_normalizedWeight r (ResearchFormalCoreR1.enb_permutation (J ω))) P) +ResearchFormalCoreR1.enb_integrable_of_integrable_sq.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + (P : @MeasureTheory.Measure Ω inst✝) [@MeasureTheory.IsFiniteMeasure Ω inst✝ P] + (J : Ω → Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (hJ : + @Measurable Ω (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) inst✝ + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun a => + Real.measurableSpace) + J) + (r : Real) (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hsq : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun ω => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (ResearchFormalCoreR1.egm_weight r (J ω)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + P) : + And + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ (fun ω => ResearchFormalCoreR1.egm_weight r (J ω)) P) + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ (fun ω => ResearchFormalCoreR1.gn_normalizedWeight r (ResearchFormalCoreR1.enb_permutation (J ω))) P) +ResearchFormalCoreR1.enb_normalizer_floor_of_common_coupling.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + (P : @MeasureTheory.Measure Ω inst✝) [@MeasureTheory.IsProbabilityMeasure Ω inst✝ P] + (J : Real → Ω → Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (hJ : + ∀ (r : Real), + @Measurable Ω (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) inst✝ + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun a => + Real.measurableSpace) + (J r)) + (A Q : Ω → Real) (m : Real) (v : NNReal) + (hv : @Ne NNReal v (@OfNat.ofNat NNReal (nat_lit 0) (@Zero.toOfNat0 NNReal NNReal.instZero))) + (hQ : @ProbabilityTheory.HasLaw Ω Real inst✝ Real.measurableSpace Q (ProbabilityTheory.gaussianReal m v) P) + (hconv : + @Filter.Eventually Ω + (fun ω => + @Filter.Tendsto Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) (fun r => J r ω) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) + (@nhds (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (@Pi.topologicalSpace (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) + fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) + (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@Neg.neg Real Real.instNeg (A ω)) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) (Q ω) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) + (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) (A ω) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) (Q ω) + (@Matrix.vecEmpty Real))))))))) + (@MeasureTheory.ae Ω (@MeasureTheory.Measure Ω inst✝) (@MeasureTheory.Measure.instFunLike Ω inst✝) ⋯ P)) + (hint : + @Filter.Eventually Real + (fun r => + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ (fun ω => ResearchFormalCoreR1.egm_weight r (J r ω)) P) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))) : + ∃ c, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) c) + (∃ δ, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) δ) + (∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LT.lt Real Real.instLT r δ → + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ P fun ω => ResearchFormalCoreR1.egm_weight r (J r ω)))) +ResearchFormalCoreR1.gc_coupledVector_hasLaw.{u_1} {ι : Type u_1} [Fintype ι] [DecidableEq ι] + (m : EuclideanSpace Real ι) (S : Matrix ι ι Real) : + @ProbabilityTheory.HasLaw (EuclideanSpace Real ι) (EuclideanSpace Real ι) + (@WithLp.measurableSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : ι) → (fun x => Real) i) (@MeasurableSpace.pi ι (fun x => Real) fun a => Real.measurableSpace)) + (@WithLp.measurableSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : ι) → (fun x => Real) i) (@MeasurableSpace.pi ι (fun x => Real) fun a => Real.measurableSpace)) + (@ResearchFormalCoreR1.gc_coupledVector ι inst✝ inst✝¹ m S) + (@ProbabilityTheory.multivariateGaussian ι inst✝ inst✝¹ m S) + (@ProbabilityTheory.stdGaussian (EuclideanSpace Real ι) + (@PiLp.normedAddCommGroup + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ι (fun x => Real) fact_one_le_two_ennreal inst✝ fun i => Real.normedAddCommGroup) + (@PiLp.innerProductSpace Real Real.instRCLike ι inst✝ (fun x => Real) (fun i => Real.normedAddCommGroup) fun i => + @RCLike.toInnerProductSpaceReal Real Real.instRCLike) + ⋯ + (@WithLp.measurableSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : ι) → (fun x => Real) i) (@MeasurableSpace.pi ι (fun x => Real) fun a => Real.measurableSpace))) +ResearchFormalCoreR1.gc_coupledVector_tendsto.{u_1, u_2} {ι : Type u_1} {κ : Type u_2} [Fintype ι] [DecidableEq ι] + (u : Filter κ) (m : κ → EuclideanSpace Real ι) (S : κ → Matrix ι ι Real) (m0 : EuclideanSpace Real ι) + (S0 : Matrix ι ι Real) + (hm : + @Filter.Tendsto κ (EuclideanSpace Real ι) m u + (@nhds (EuclideanSpace Real ι) + (@PiLp.topologicalSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ι (fun x => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + m0)) + (hS : + @Filter.Tendsto κ (Matrix ι ι Real) S u + (@nhds (Matrix ι ι Real) + (@instTopologicalSpaceMatrix ι ι Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + S0)) + (hPSD : + @Filter.Eventually κ (fun t => @Matrix.PosSemidef ι Real Real.instRing Real.partialOrder instStarRingReal (S t)) u) + (hPSD0 : @Matrix.PosSemidef ι Real Real.instRing Real.partialOrder instStarRingReal S0) (z : EuclideanSpace Real ι) : + @Filter.Tendsto κ (EuclideanSpace Real ι) + (fun t => @ResearchFormalCoreR1.gc_coupledVector ι inst✝ inst✝¹ (m t) (S t) z) u + (@nhds (EuclideanSpace Real ι) + (@PiLp.topologicalSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ι (fun x => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@ResearchFormalCoreR1.gc_coupledVector ι inst✝ inst✝¹ m0 S0 z)) +ResearchFormalCoreR1.gnl_normalizer_floor_of_parameter_limits + (m : Real → EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) + (S : + Real → + Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (m0 : EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) + (S0 : + Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (hm : + @Filter.Tendsto Real (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) m + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) + (@nhds (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) + (@PiLp.topologicalSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun x => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + m0)) + (hS : + @Filter.Tendsto Real + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + S + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) + (@nhds + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (@instTopologicalSpaceMatrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + S0)) + (hPSD : + @Filter.Eventually Real + (fun r => + @Matrix.PosSemidef (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real Real.instRing + Real.partialOrder instStarRingReal (S r)) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))) + (hPSD0 : + @Matrix.PosSemidef (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real Real.instRing + Real.partialOrder instStarRingReal S0) + (hm0 : + @Eq + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (@Neg.neg + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + Real.instNeg + (@OfNat.ofNat + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (nat_lit 1) + (@One.toOfNat1 + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + Real.instOne)))) + (hm1 : + @Eq + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + (@OfNat.ofNat + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + (nat_lit 1) + (@One.toOfNat1 + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + Real.instOne))) + (hm23 : + @Eq + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (@HAdd.hAdd + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3)))) + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (@instHAdd + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + Real.instAdd) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3))))) + (@OfNat.ofNat + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (nat_lit 0) + (@Zero.toOfNat0 + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + Real.instZero))) + (hm45 : + @Eq + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (@HSub.hSub + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5)))) + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (@instHSub + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + Real.instSub) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5))))) + (@OfNat.ofNat + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (nat_lit 0) + (@Zero.toOfNat0 + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + Real.instZero))) + (hS0 : + @Eq Real + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (hS1 : + @Eq Real + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (hS23 : + @Eq Real + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯)) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3)))))) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3))))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (hS45 : + @Eq Real + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯)) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5)))))) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5))))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (hQpos : + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4))))) : + ∃ c, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) c) + (∃ δ, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) δ) + (∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LT.lt Real Real.instLT r δ → + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@MeasureTheory.integral + (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) Real + Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + (@WithLp.measurableSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (fun x => Real) fun a => Real.measurableSpace)) + (@ProbabilityTheory.multivariateGaussian + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin.fintype (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (instDecidableEqFin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (m r) (S r)) + fun x => + @HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (ResearchFormalCoreR1.gn_normalizedWeight r + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) x))))) +ResearchFormalCoreR1.gnl_physical_normalizer_floor_of_parameter_limits + (m : Real → EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) + (S : + Real → + Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (m0 : EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) + (S0 : + Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (hm : + @Filter.Tendsto Real (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) m + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) + (@nhds (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) + (@PiLp.topologicalSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun x => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + m0)) + (hS : + @Filter.Tendsto Real + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + S + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) + (@nhds + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (@instTopologicalSpaceMatrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + S0)) + (hPSD : + @Filter.Eventually Real + (fun r => + @Matrix.PosSemidef (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real Real.instRing + Real.partialOrder instStarRingReal (S r)) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))) + (hPSD0 : + @Matrix.PosSemidef (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real Real.instRing + Real.partialOrder instStarRingReal S0) + (hm0 : + @Eq + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (@Neg.neg + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + Real.instNeg + (@OfNat.ofNat + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (nat_lit 1) + (@One.toOfNat1 + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + Real.instOne)))) + (hm1 : + @Eq + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + (@OfNat.ofNat + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + (nat_lit 1) + (@One.toOfNat1 + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + Real.instOne))) + (hm23 : + @Eq + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (@HAdd.hAdd + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3)))) + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (@instHAdd + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + Real.instAdd) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3))))) + (@OfNat.ofNat + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (nat_lit 0) + (@Zero.toOfNat0 + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + Real.instZero))) + (hm45 : + @Eq + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (@HSub.hSub + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5)))) + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (@instHSub + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + Real.instSub) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) m0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5))))) + (@OfNat.ofNat + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (nat_lit 0) + (@Zero.toOfNat0 + ((fun x => Real) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + Real.instZero))) + (hS0 : + @Eq Real + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (hS1 : + @Eq Real + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 1)))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (hS23 : + @Eq Real + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯)) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3)))))) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 3))))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (hS45 : + @Eq Real + (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) + (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@OfNat.ofNat Real (nat_lit 2) (@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast ⋯)) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5)))))) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 5))))) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) + (hQpos : + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (S0 + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4))) + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 4))))) : + ∃ c, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) c) + (∃ δ, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) δ) + (∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LT.lt Real Real.instLT r δ → + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@MeasureTheory.integral + (EuclideanSpace Real (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))) Real + Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + (@WithLp.measurableSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (fun x => Real) fun a => Real.measurableSpace)) + (@ProbabilityTheory.multivariateGaussian + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin.fintype (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (instDecidableEqFin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (m r) (S r)) + fun x => + ResearchFormalCoreR1.egm_weight r + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) x + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 0) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 0)))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) x + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 2) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ (nat_lit 2)))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → (fun x => Real) i) + x + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (nat_lit 4) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ + (nat_lit 4)))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal + ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → + (fun x => Real) i) + x + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (nat_lit 1) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ + (nat_lit 1)))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal + ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → + (fun x => Real) i) + x + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (nat_lit 3) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ + (nat_lit 3)))) + (@Matrix.vecCons Real (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) + (@WithLp.ofLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal + ENNReal.instAddCommMonoidWithOne)) + PiLp.innerProductSpace._proof_1)) + ((i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) → + (fun x => Real) i) + x + (@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (nat_lit 5) + (@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) ⋯ + (nat_lit 5)))) + (@Matrix.vecEmpty Real)))))))))) +ResearchFormalCoreR1.prw_integrable_of_integrable_sq.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + (μ : @MeasureTheory.Measure Ω inst✝) [@MeasureTheory.IsFiniteMeasure Ω inst✝ μ] (W : Ω → Real) + (hW : @Measurable Ω Real inst✝ Real.measurableSpace W) + (hsq : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun ω => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) (W ω) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + μ) : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ W μ +ResearchFormalCoreR1.prw_measure_apply.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] (μ : @MeasureTheory.Measure Ω inst✝) + (W : Ω → Real) + (hW : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ W μ) + (hW0 : ∀ (ω : Ω), @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) (W ω)) + (hZ : + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => W x)) + (E : Set Ω) (hE : @MeasurableSet Ω inst✝ E) : + @Eq ENNReal + (@DFunLike.coe (@MeasureTheory.Measure Ω inst✝) (Set Ω) (fun x => ENNReal) + (@MeasureTheory.Measure.instFunLike Ω inst✝) (@ResearchFormalCoreR1.prwMeasure Ω inst✝ μ W) E) + (ENNReal.ofReal + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ (@MeasureTheory.Measure.restrict Ω inst✝ μ E) fun x => W x) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => W x))) +ResearchFormalCoreR1.prw_isProbabilityMeasure.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + (μ : @MeasureTheory.Measure Ω inst✝) (W : Ω → Real) + (hW : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ W μ) + (hW0 : ∀ (ω : Ω), @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) (W ω)) + (hZ : + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => W x)) : + @MeasureTheory.IsProbabilityMeasure Ω inst✝ (@ResearchFormalCoreR1.prwMeasure Ω inst✝ μ W) +ResearchFormalCoreR1.prw_measure_real.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] (μ : @MeasureTheory.Measure Ω inst✝) + (W : Ω → Real) + (hW : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ W μ) + (hW0 : ∀ (ω : Ω), @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) (W ω)) + (hZ : + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => W x)) + (E : Set Ω) (hE : @MeasurableSet Ω inst✝ E) : + @Eq Real (@MeasureTheory.Measure.real Ω inst✝ (@ResearchFormalCoreR1.prwMeasure Ω inst✝ μ W) E) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ (@MeasureTheory.Measure.restrict Ω inst✝ μ E) fun x => W x) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => W x)) +ResearchFormalCoreR1.prw_numerator_bound.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] (μ : @MeasureTheory.Measure Ω inst✝) + [@MeasureTheory.IsFiniteMeasure Ω inst✝ μ] (W : Ω → Real) (hW : @Measurable Ω Real inst✝ Real.measurableSpace W) + (hW0 : ∀ (ω : Ω), @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) (W ω)) + (hsq : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun ω => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) (W ω) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + μ) + (E : Set Ω) (hE : @MeasurableSet Ω inst✝ E) : + @LE.le Real Real.instLE + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ (@MeasureTheory.Measure.restrict Ω inst✝ μ E) fun x => W x) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) (W x) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))).sqrt + (@MeasureTheory.Measure.real Ω inst✝ μ E).sqrt) +ResearchFormalCoreR1.prw_square_root_event_bound.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] + (μ : @MeasureTheory.Measure Ω inst✝) [@MeasureTheory.IsProbabilityMeasure Ω inst✝ μ] (W : Ω → Real) + (hW : @Measurable Ω Real inst✝ Real.measurableSpace W) + (hW0 : ∀ (ω : Ω), @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) (W ω)) + (hsq : + @MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + Ω inst✝ + (fun ω => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) (W ω) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + μ) + (C c r : Real) (hC : @LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) C) + (hc : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) c) + (hr : @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r) + (hmoment : + @LE.le Real Real.instLE + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => + @HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) (W x) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) C + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) + (hfloor : + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c + (@HPow.hPow Real Nat Real (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@MeasureTheory.integral Ω Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + inst✝ μ fun x => W x)) + (E : Set Ω) (hE : @MeasurableSet Ω inst✝ E) : + And (@MeasureTheory.IsProbabilityMeasure Ω inst✝ (@ResearchFormalCoreR1.prwMeasure Ω inst✝ μ W)) + (@LE.le Real Real.instLE (@MeasureTheory.Measure.real Ω inst✝ (@ResearchFormalCoreR1.prwMeasure Ω inst✝ μ W) E) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) C.sqrt c) + (@MeasureTheory.Measure.real Ω inst✝ μ E).sqrt)) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_fold.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_fold.lean.txt new file mode 100644 index 000000000..775823dc0 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_fold.lean.txt @@ -0,0 +1,3 @@ +import ResearchFormalCoreR1 +example : ResearchFormalCoreR1.foldPotential 1 1 - ResearchFormalCoreR1.foldPotential 1 (-1) = (2 : Real)^3 / 7 := by + norm_num [ResearchFormalCoreR1.foldPotential] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_fold.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_fold.log new file mode 100644 index 000000000..361f876d9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_fold.log @@ -0,0 +1,2 @@ +/workspace/Math-/frontiers/determinant_moment_20261025/.lake/formal-evidence/false_fold.lean:2:115: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_power.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_power.lean.txt new file mode 100644 index 000000000..b5aa5b89f --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_power.lean.txt @@ -0,0 +1,3 @@ +import ResearchFormalCoreR1 +example : (2 : Real)^4 <= (2 : Real)^3 := by + norm_num diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_power.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_power.log new file mode 100644 index 000000000..baa89174a --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/false_power.log @@ -0,0 +1,2 @@ +/workspace/Math-/frontiers/determinant_moment_20261025/.lake/formal-evidence/false_power.lean:2:42: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/gate-terminal.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/gate-terminal.log new file mode 100644 index 000000000..eface0d3f --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/gate-terminal.log @@ -0,0 +1,421 @@ +version-preflight: exit 0 +build: exit 0 +leanchecker: exit 0 +axioms: exit 0 +elaborated-types: exit 0 +false_fold: exit 1 +false_power: exit 1 +sorry: exit 0 +custom_imported: exit 0 +native: exit 0 +version: exit 0 +{ + "alignment_status": "PENDING_INDEPENDENT_REVIEW", + "axioms": { + "ResearchFormalCoreR1.dm2_moment": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.dm2_negative_iff": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.dm2_pointwise": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.dm2_untyped": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.dm2_weight_measurable": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.ec005_fold_gap": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.ec008_factor_expansion": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.ec010_generic": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.ec010_transverse": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.ec011_scalar_cancellation": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.ec014_contact_power": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.egm_gaussian_moment": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.egm_pointwise": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.egm_weight_measurable": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.enb_integrable_iff": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.enb_integrable_of_integrable_sq": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.enb_integral_identity": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.enb_maximum_iff": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.enb_normalizer_floor_of_common_coupling": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.enb_permutation_continuous": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.enb_saddle_iff": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "ResearchFormalCoreR1.enb_weight_identity": [ 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"f8089be2e69db162c2e3b1e3062dbec69234a00b4400caee57f4ff6fc3ec3e06", + "false_fold.log": "0c0d91c62fbbdeed9ac1f8b968f9d73b701b8079089bd90c07c0af6fcf9ce686", + "false_power.log": "eccbb6ccf7bded2d8ba23f1c2b78b146564a467b848e2c431a76d1dabd053a9c", + "leanchecker.log": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855", + "native.log": "8a0b9f79f3265724af58b27461627f9d532535f8a23681361da731c9ea37c101", + "sorry.log": "afc2bcbd48dbe1be41b06656bc3cbb8eb5b2fae535b72bca51dacfe3071fb469", + "version-preflight.log": "355a9a028f81641782d75d6e6b58dec70f29d2f61238a9e271690259d9bf5680", + "version.log": "355a9a028f81641782d75d6e6b58dec70f29d2f61238a9e271690259d9bf5680" + }, + "manifest_sha256": "0008bee4d620206e78ad72ca3d4f39006a8c495f185f556d3b2ffa60e248ccbd", + "negative_controls": { + "custom_imported": "REJECTED_BY_AXIOM_GATE", + "false_fold": "REJECTED_BY_LEAN", + "false_power": "REJECTED_BY_LEAN", + "native": "REJECTED_BY_AXIOM_GATE", + "sorry": "REJECTED_BY_AXIOM_GATE" + }, + "observations": { + "final": { + "dependency_revisions": { + "Cli": "e92c9f15fdfacc8536f31cfb3b7ad26c3c8cd204", + "LeanSearchClient": "ddf04cf3949fa556442341e87d47f9f6e6074707", + "Qq": "6a489d9af5d0c47e5b259e2e8bcdfc1811b5a259", + "aesop": "355695d523e41d0554926416cba2a2b3544fbbc9", + "batteries": "f2effa3d803fda822b1f97b806c47cf2adfbcbc2", + "importGraph": "e928b72544873815af278d38681b31c0293588e3", + "mathlib": "d13f23b723b8a846827a245b89c10fc7d3f11612", + "plausible": "118aa17ee84656b8bd727fef7c458ee8c833385c", + "proofwidgets": "106ff4fafc74ef4ac99d81dbf3ab399118f497a5" + }, + "dependency_worktrees": "CLEAN", + "lean_version": "Lean (version 4.34.1, x86_64-unknown-linux-gnu, commit 5045d0056413266e57c625dcd7c365b10e377c52, Release)", + "manifest_sha256": "0008bee4d620206e78ad72ca3d4f39006a8c495f185f556d3b2ffa60e248ccbd" + }, + "preflight": { + "dependency_revisions": { + "Cli": "e92c9f15fdfacc8536f31cfb3b7ad26c3c8cd204", + "LeanSearchClient": "ddf04cf3949fa556442341e87d47f9f6e6074707", + "Qq": "6a489d9af5d0c47e5b259e2e8bcdfc1811b5a259", + "aesop": "355695d523e41d0554926416cba2a2b3544fbbc9", + "batteries": "f2effa3d803fda822b1f97b806c47cf2adfbcbc2", + "importGraph": "e928b72544873815af278d38681b31c0293588e3", + "mathlib": "d13f23b723b8a846827a245b89c10fc7d3f11612", + "plausible": "118aa17ee84656b8bd727fef7c458ee8c833385c", + "proofwidgets": "106ff4fafc74ef4ac99d81dbf3ab399118f497a5" + }, + "dependency_worktrees": "CLEAN", + "lean_version": "Lean (version 4.34.1, x86_64-unknown-linux-gnu, commit 5045d0056413266e57c625dcd7c365b10e377c52, Release)", + "manifest_sha256": "0008bee4d620206e78ad72ca3d4f39006a8c495f185f556d3b2ffa60e248ccbd" + } + }, + "repository": "d6g8k5htny-coder/Math-", + "schema_version": 1, + "scientific_effect": "NONE", + "workflow_run_attempt": null, + "workflow_run_id": null +} diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/sorry.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/sorry.lean.txt new file mode 100644 index 000000000..ace09ca23 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/sorry.lean.txt @@ -0,0 +1,3 @@ +import ResearchFormalCoreR1 +theorem injected : False := by sorry +#print axioms injected diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/sorry.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/sorry.log new file mode 100644 index 000000000..e34e95e9e --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/sorry.log @@ -0,0 +1,2 @@ +/workspace/Math-/frontiers/determinant_moment_20261025/.lake/formal-evidence/sorry.lean:2:8: warning: declaration uses `sorry` +'injected' depends on axioms: [sorryAx] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/POSITIVE_SUCCESSOR.json b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/POSITIVE_SUCCESSOR.json new file mode 100644 index 000000000..45609ceaa --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/POSITIVE_SUCCESSOR.json @@ -0,0 +1,18 @@ +{ + "source_path": "controls/gaussian_parent_positive_successor.lean.txt", + "source_sha256": "c73799328c4191422c13aefc458ffe6df200da610ec8e79f666db0fbf0817e51", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gaussian_parent_positive_successor.lean" + ], + "exit": 0, + "log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855", + "previous_attempts": [ + "initial missing .olean", + "second wrong map_dirac API; original checked-source file preserved" + ], + "source_changes": "Only successor control tactic API corrected; 66 proof targets/root/manifest unchanged; final gate evidence reused." +} diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/RESULTS.json b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/RESULTS.json new file mode 100644 index 000000000..3da9b4674 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/RESULTS.json @@ -0,0 +1,92 @@ +[ + { + "name": "gm8_false_centered_substitution", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean" + ], + "exit": 1, + "source_sha256": "96dcad5abc7a1fe24c42bf9c7544ecd6d6a81f71fe6245b9d4de07a5faffde91", + "log_sha256": "6425d08dffa942dac35c0d2c6049f7b45ea7e8ca3bd738d90d082358a6a80e6e", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean:4:76: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gm8_false_omit_radius_one", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean" + ], + "exit": 1, + "source_sha256": "e382d597a45d667a492f10127a50b54e016f8ab808f2b872d0d00acf781a8b9d", + "log_sha256": "c48135a6578a1e599f01bd5d60c41e8353fe574a3eb5a7fd6cc37e591f473b70", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean:5:46: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gn_false_degenerate_reference_floor", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean" + ], + "exit": 1, + "source_sha256": "ef3e9a137e1cb0adb1e5608af2bba29a6b475ccb56a7206e3797e3c7bf4dcdd2", + "log_sha256": "256223c08bec680444daa53eff7bf2f75a97a6046d1175b183ae599b88f98fc6", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean:4:59: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gn_false_no_small_radius", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean" + ], + "exit": 1, + "source_sha256": "c92e629433ec77f420400aaba2cdd9092d322555b0c01bd8836e87178ceca076", + "log_sha256": "aeee073a8c3898625adf42f138af830d9d75d656ddc2cac6e504341832b61958", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean:4:69: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "egm_false_midpoint_identification", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean" + ], + "exit": 1, + "source_sha256": "293faccdf06762221b99723c08acfb4130669b6b6600717a1f96c2c3aa9a4bc1", + "log_sha256": "7c68421d4fc5d388db70182f95c2a9502d5fd9523e9815ba418f1f20c766226b", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean:6:54: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gaussian_parent_positive", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean" + ], + "exit": 1, + "source_sha256": "ae2c3934d498a862e829b878222385d8d8a39c9fbbb2485df65cb04c45d63ddc", + "log_sha256": "669a75be6dc262dcf45355ae315367e86320c9309005363270d6cc189597e9f8", + "expected_success": true, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean:1:0: error: object file '/workspace/Math-/frontiers/determinant_moment_20261025/.lake/build/lib/lean/ResearchFormalCoreR1/GaussianRegressionKernel.olean' of module ResearchFormalCoreR1.GaussianRegressionKernel does not exist\n" + } +] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.command b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.command new file mode 100644 index 000000000..37fd6ee98 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.lean.txt new file mode 100644 index 000000000..f11632919 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.lean.txt @@ -0,0 +1,8 @@ +import ResearchFormalCoreR1.EndpointGaussianMoments +open ResearchFormalCoreR1 +-- Exact endpoints of the invisible-midpoint-jet perturbation, versus the +-- quartic raw-nine-jet radius. These two charts cannot be silently identified. +example : egm_weight 1 ![-9,0,-1,9,0,-1]^2 ≤ + 16*(1 : ℝ)^4*dm2Radius ![-1,0,0,0,0,0,0,0,0]^8 := by + norm_num [egm_weight, egm_HM, egm_HS, dm2_negative_iff, + Matrix.det_fin_two, dm2Radius, Fin.sum_univ_succ] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.log new file mode 100644 index 000000000..b8715a9f5 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/egm_false_midpoint_identification.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean:6:54: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-02.command b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-02.command new file mode 100644 index 000000000..bdc80c5da --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-02.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-02.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-02.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-02.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-02.log new file mode 100644 index 000000000..78435ee85 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-02.log @@ -0,0 +1,7 @@ +/workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean:21:30: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + Measure.map ?m.41 (Measure.dirac ?m.42) +in the target expression + Measure.map (fun r => r + 0 + (ContinuousLinearMap.id ℝ (Fin 1 → ℝ)) (x - 0)) (Measure.dirac 0) = Measure.dirac x + +x : Fin 1 → ℝ +⊢ Measure.map (fun r => r + 0 + (ContinuousLinearMap.id ℝ (Fin 1 → ℝ)) (x - 0)) (Measure.dirac 0) = Measure.dirac x diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-03.command b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-03.command new file mode 100644 index 000000000..5d8b229c7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-03.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gaussian_parent_positive_successor.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-03.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-03.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-03.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.command b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.command new file mode 100644 index 000000000..bdc80c5da --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.lean.txt new file mode 100644 index 000000000..297181d4c --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.lean.txt @@ -0,0 +1,26 @@ +import ResearchFormalCoreR1.GaussianJetMoments +import ResearchFormalCoreR1.GaussianRegressionKernel +import ResearchFormalCoreR1.EndpointGaussianMoments +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +open scoped NNReal +-- All nine coordinates can be the SAME Gaussian variable: independence is not used. +example : Integrable (fun z : ℝ => (1+9*|z|)^8) (gaussianReal 0 1) := by + have h := gm8_jet_integrable_bound (gaussianReal 0 1) + (fun z : ℝ => (fun _i : Fin 9 => z)) (by fun_prop) + (fun _ => 0) (fun _ => 1) 0 1 (by norm_num) (by norm_num) + (by intro i; norm_num) (by intro i; norm_num) + (by intro i; exact HasLaw.id) + simpa [dm2Radius, Finset.sum_const] using h.1 +-- Nonzero mean and zero variance are preserved, rather than centered away. +example : (∫ x : ℝ, |x|^8 ∂gaussianReal 1 0) = 1 := by + norm_num [gaussianReal_zero_var] +-- The canonical formula specifies even an observation outside the source support. +example (x : Fin 1 → ℝ) : + regressionKernel (Measure.dirac (0 : Fin 1 → ℝ)) + (ContinuousLinearMap.id ℝ (Fin 1 → ℝ)) 0 0 x = Measure.dirac x := by + rw [regressionKernel_apply, Measure.map_dirac (by fun_prop)] + simp +-- The two actual endpoint types in the six-coordinate chart are allowed. +example : dm2Negative (egm_HM 1 ![-1,0,-1,1,0,-1]) ∧ + (egm_HS 1 ![-1,0,-1,1,0,-1]).det < 0 := by + norm_num [egm_HM, egm_HS, dm2_negative_iff, Matrix.det_fin_two] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.log new file mode 100644 index 000000000..91b97ba01 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive.log @@ -0,0 +1 @@ +/workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean:1:0: error: object file '/workspace/Math-/frontiers/determinant_moment_20261025/.lake/build/lib/lean/ResearchFormalCoreR1/GaussianRegressionKernel.olean' of module ResearchFormalCoreR1.GaussianRegressionKernel does not exist diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive_successor.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive_successor.lean.txt new file mode 100644 index 000000000..daf6d260b --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gaussian_parent_positive_successor.lean.txt @@ -0,0 +1,26 @@ +import ResearchFormalCoreR1.GaussianJetMoments +import ResearchFormalCoreR1.GaussianRegressionKernel +import ResearchFormalCoreR1.EndpointGaussianMoments +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +open scoped NNReal +-- All nine coordinates can be the SAME Gaussian variable: independence is not used. +example : Integrable (fun z : ℝ => (1+9*|z|)^8) (gaussianReal 0 1) := by + have h := gm8_jet_integrable_bound (gaussianReal 0 1) + (fun z : ℝ => (fun _i : Fin 9 => z)) (by fun_prop) + (fun _ => 0) (fun _ => 1) 0 1 (by norm_num) (by norm_num) + (by intro i; norm_num) (by intro i; norm_num) + (by intro i; exact HasLaw.id) + simpa [dm2Radius, Finset.sum_const] using h.1 +-- Nonzero mean and zero variance are preserved, rather than centered away. +example : (∫ x : ℝ, |x|^8 ∂gaussianReal 1 0) = 1 := by + norm_num [gaussianReal_zero_var] +-- The canonical formula specifies even an observation outside the source support. +example (x : Fin 1 → ℝ) : + regressionKernel (Measure.dirac (0 : Fin 1 → ℝ)) + (ContinuousLinearMap.id ℝ (Fin 1 → ℝ)) 0 0 x = Measure.dirac x := by + rw [regressionKernel_apply, Measure.map_dirac' (by fun_prop)] + simp +-- The two actual endpoint types in the six-coordinate chart are allowed. +example : dm2Negative (egm_HM 1 ![-1,0,-1,1,0,-1]) ∧ + (egm_HS 1 ![-1,0,-1,1,0,-1]).det < 0 := by + norm_num [egm_HM, egm_HS, dm2_negative_iff, Matrix.det_fin_two] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.command b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.command new file mode 100644 index 000000000..2736f2ec2 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.lean.txt new file mode 100644 index 000000000..04150e241 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1.GaussianScalarMoments +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +-- Dropping the nonzero mean term would predict zero eighth moment for N(1,0). +example : (∫ x : ℝ, |x|^8 ∂gaussianReal 1 0) ≤ 128*((0 : ℝ)^8+0^4*gm8C8) := by + norm_num [gaussianReal_zero_var] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.log new file mode 100644 index 000000000..3fa387ae1 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_centered_substitution.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean:4:76: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.command b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.command new file mode 100644 index 000000000..90ab78411 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.lean.txt new file mode 100644 index 000000000..350cf2972 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.lean.txt @@ -0,0 +1,6 @@ +import ResearchFormalCoreR1.GaussianJetMoments +open ResearchFormalCoreR1 +-- The exact radius includes one, even when every Gaussian coordinate vanishes. +example : dm2Radius (fun _i : Fin 9 => 0)^8 ≤ + (10 : ℝ)^7 * ∑ _i : Fin 9, |(0 : ℝ)|^8 := by + norm_num [dm2Radius] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.log new file mode 100644 index 000000000..15931bfd5 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gm8_false_omit_radius_one.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean:5:46: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.command b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.command new file mode 100644 index 000000000..4f40a5f27 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.lean.txt new file mode 100644 index 000000000..d73990e1a --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1.GaussianNormalizer +open MeasureTheory ProbabilityTheory +-- Strictly positive reference mass requires nonzero variance. +example : 0 < ∫ q : ℝ, (max (-q) 0)^2 ∂gaussianReal 1 0 := by + norm_num [gaussianReal_zero_var] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.log new file mode 100644 index 000000000..41a6514b5 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_degenerate_reference_floor.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean:4:59: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.command b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.command new file mode 100644 index 000000000..54ee6dde6 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.lean.txt new file mode 100644 index 000000000..3a7e99464 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1.GaussianNormalizer +open ResearchFormalCoreR1 +-- The source's small r*K² budget cannot be discarded from the box floor. +example : (3 : ℝ)/8 ≤ gn_normalizedWeight 1 ![-1,1,9,0,-3/2,-3/2] := by + norm_num [gn_normalizedWeight] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.log new file mode 100644 index 000000000..aa1607266 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/supplemental-controls/gn_false_no_small_radius.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean:4:69: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/version-preflight.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/version-preflight.log new file mode 100644 index 000000000..3bdff924e --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/version-preflight.log @@ -0,0 +1 @@ +Lean (version 4.34.1, x86_64-unknown-linux-gnu, commit 5045d0056413266e57c625dcd7c365b10e377c52, Release) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-final/version.log b/frontiers/determinant_moment_20261025/evidence/gaussian-final/version.log new file mode 100644 index 000000000..3bdff924e --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-final/version.log @@ -0,0 +1 @@ +Lean (version 4.34.1, x86_64-unknown-linux-gnu, commit 5045d0056413266e57c625dcd7c365b10e377c52, Release) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/endpoint_bridge_compile_transcript.md b/frontiers/determinant_moment_20261025/evidence/gaussian-history/endpoint_bridge_compile_transcript.md new file mode 100644 index 000000000..6700d9ceb --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/endpoint_bridge_compile_transcript.md @@ -0,0 +1,35 @@ +# Reconstructed transcript of actual scoped compilation + +This file reconstructs the actual tool command and returned output after the +fact. No redirected raw log was created. This is command-history evidence, +not a replacement for the final gate's raw kernel evidence. No unchanged proof +was rerun to create this transcript. + +Working directory: +`/workspace/Math-/frontiers/determinant_moment_20261025` + +Actual command: +```sh +lake env lean -o .lake/build/lib/lean/ResearchFormalCoreR1/EndpointNormalizerBridge.olean ResearchFormalCoreR1/EndpointNormalizerBridge.lean +``` + +The actual command was launched by `exec_command`, which returned running +session ID `12809`. `write_stdin` then returned: + +```json +{"chunk_id":"7bcfd2","wall_time_seconds":0.000028968,"exit_code":0,"original_token_count":0,"output":""} +``` + +Raw process output: empty. Exit code: 0. The preceding compile attempt failed +on the new L2-to-L1 lemma because the normalized composition expression was +not syntactically exposed to `nlinarith`. The sole correction was the explicit +`change |egm_weight r (J ω)| ≤ 1 + egm_weight r (J ω) ^ 2` line. The command +above compiled the resulting final file cleanly and generated its `.olean`. + +Author-side helper lineage: OpenAI/Codex agent +`/root/dm2_landed_composition`, actual task delegated by `/root`. No helper was +spawned: a requested independent read-only strategy helper was rejected by the +thread limit before creation. This agent authored the new bridge module; its +work is not an independent scientific or provider-distinct review. + +Completed applicable verification must be reused per `/workspace/AGENTS.md`. diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-controls-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-controls-01.log new file mode 100644 index 000000000..a05c379df --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-controls-01.log @@ -0,0 +1,10 @@ +ResearchFormalCoreR1.gc_coupledVector.{u_1} {ι : Type u_1} [Fintype ι] [DecidableEq ι] (m : EuclideanSpace ℝ ι) + (S : Matrix ι ι ℝ) (z : EuclideanSpace ℝ ι) : EuclideanSpace ℝ ι +ResearchFormalCoreR1.gc_coupledVector_hasLaw.{u_1} {ι : Type u_1} [Fintype ι] [DecidableEq ι] (m : EuclideanSpace ℝ ι) + (S : Matrix ι ι ℝ) : HasLaw (gc_coupledVector m S) (multivariateGaussian m S) (stdGaussian (EuclideanSpace ℝ ι)) +ResearchFormalCoreR1.gc_coupledVector_tendsto.{u_1, u_2} {ι : Type u_1} {κ : Type u_2} [Fintype ι] [DecidableEq ι] + (u : Filter κ) (m : κ → EuclideanSpace ℝ ι) (S : κ → Matrix ι ι ℝ) (m0 : EuclideanSpace ℝ ι) (S0 : Matrix ι ι ℝ) + (hm : Tendsto m u (𝓝 m0)) (hS : Tendsto S u (𝓝 S0)) (hPSD : ∀ᶠ (t : κ) in u, (S t).PosSemidef) (hPSD0 : S0.PosSemidef) + (z : EuclideanSpace ℝ ι) : Tendsto (fun t => gc_coupledVector (m t) (S t) z) u (𝓝 (gc_coupledVector m0 S0 z)) +'ResearchFormalCoreR1.gc_coupledVector_hasLaw' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gc_coupledVector_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-controls.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-controls.lean.txt new file mode 100644 index 000000000..4d224488c --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-controls.lean.txt @@ -0,0 +1,34 @@ +import ResearchFormalCoreR1.GaussianCoupling +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter Matrix WithLp +open scoped Topology MatrixOrder Matrix.Norms.L2Operator + +-- Nonzero changing mean and rank drop to zero covariance, on the full real filter. +example (z : EuclideanSpace ℝ (Fin 1)) : + Tendsto (fun r : ℝ => gc_coupledVector (toLp 2 (fun _ : Fin 1 => 3+r)) + (Matrix.diagonal (fun _ : Fin 1 => r^2)) z) (𝓝 0) + (𝓝 (gc_coupledVector (toLp 2 (fun _ : Fin 1 => 3)) 0 z)) := by + apply gc_coupledVector_tendsto (𝓝 (0 : ℝ)) + (fun r : ℝ => toLp 2 (fun _ : Fin 1 => 3+r)) + (fun r : ℝ => Matrix.diagonal (fun _ : Fin 1 => r^2)) + (toLp 2 (fun _ : Fin 1 => 3)) 0 + · have hc : Continuous (fun r : ℝ => toLp 2 (fun _ : Fin 1 => 3+r)) := by fun_prop + simpa using hc.tendsto 0 + · have hc : Continuous (fun r : ℝ => Matrix.diagonal (fun _ : Fin 1 => r^2)) := by fun_prop + simpa using hc.tendsto 0 + · filter_upwards [] with r + exact (Matrix.posSemidef_diagonal_iff).2 (fun _ => sq_nonneg r) + · exact Matrix.PosSemidef.zero + +-- Its literal singular law is still the same declared multivariate Gaussian. +example : HasLaw (gc_coupledVector (toLp 2 (fun _ : Fin 1 => (3 : ℝ))) 0) + (multivariateGaussian (toLp 2 (fun _ : Fin 1 => (3 : ℝ))) 0) + (stdGaussian (EuclideanSpace ℝ (Fin 1))) := + gc_coupledVector_hasLaw _ _ + +#check gc_coupledVector +#check gc_coupledVector_hasLaw +#check gc_coupledVector_tendsto +#print axioms gc_coupledVector_hasLaw +#print axioms gc_coupledVector_tendsto +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-module-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-module-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-01.log new file mode 100644 index 000000000..29e70637c --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-01.log @@ -0,0 +1,45 @@ +/workspace/scratch/gaussian-coupling/coupling-probe.lean:12:30: error: Function expected at + toEuclideanCLM +but this term has type + Matrix ?m.2 ?m.2 ?m.1 ≃⋆ₐ[?m.1] EuclideanSpace ?m.1 ?m.2 →L[?m.1] EuclideanSpace ?m.1 ?m.2 + +Note: Expected a function because this term is being applied to the argument + (CFC.sqrt S) +/workspace/scratch/gaussian-coupling/coupling-probe.lean:19:12: error: Type mismatch + rfl +has type + ?m.24 = ?m.24 +but is expected to have type + Measure.map (gc_coupledVector m S) (stdGaussian (EuclideanSpace ℝ ι)) = multivariateGaussian m S +/workspace/scratch/gaussian-coupling/coupling-probe.lean:33:33: error: Function expected at + toEuclideanCLM +but this term has type + Matrix ?m.131 ?m.131 ?m.130 ≃⋆ₐ[?m.130] EuclideanSpace ?m.130 ?m.131 →L[?m.130] EuclideanSpace ?m.130 ?m.131 + +Note: Expected a function because this term is being applied to the argument + (CFC.sqrt (S t)) +/workspace/scratch/gaussian-coupling/coupling-probe.lean:34:10: error: Function expected at + toEuclideanCLM +but this term has type + Matrix ?m.139 ?m.139 ?m.138 ≃⋆ₐ[?m.138] EuclideanSpace ?m.138 ?m.139 →L[?m.138] EuclideanSpace ?m.138 ?m.139 + +Note: Expected a function because this term is being applied to the argument + (CFC.sqrt S0) +/workspace/scratch/gaussian-coupling/coupling-probe.lean:28:40: error: unsolved goals +ι : Type u_1 +κ : Type u_2 +inst✝¹ : Fintype ι +inst✝ : DecidableEq ι +u : Filter κ +m : κ → EuclideanSpace ℝ ι +S : κ → Matrix ι ι ℝ +m0 : EuclideanSpace ℝ ι +S0 : Matrix ι ι ℝ +hm : Tendsto m u (𝓝 m0) +hS : Tendsto S u (𝓝 S0) +hPSD : ∀ᶠ (t : κ) in u, (S t).PosSemidef +hPSD0 : S0.PosSemidef +z : EuclideanSpace ℝ ι +hwithin : Tendsto S u (𝓝[{B | 0 ≤ B}] S0) +hsqrt : Tendsto (fun t => CFC.sqrt (S t)) u (𝓝 (CFC.sqrt S0)) +⊢ Tendsto (fun t => gc_coupledVector (m t) (S t) z) u (𝓝 (gc_coupledVector m0 S0 z)) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-02.log new file mode 100644 index 000000000..0325e8a86 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-02.log @@ -0,0 +1,10 @@ +/workspace/scratch/gaussian-coupling/coupling-probe.lean:35:10: error: (deterministic) timeout at `isDefEq`, maximum number of heartbeats (200000) has been reached + +Note: Use `set_option maxHeartbeats ` to set the limit. + +Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command. +/workspace/scratch/gaussian-coupling/coupling-probe.lean:21:0: error: (deterministic) timeout at `whnf`, maximum number of heartbeats (200000) has been reached + +Note: Use `set_option maxHeartbeats ` to set the limit. + +Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command. diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-03.log new file mode 100644 index 000000000..f6d620b4f --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-03.log @@ -0,0 +1,10 @@ +/workspace/scratch/gaussian-coupling/coupling-probe.lean:35:10: error: (deterministic) timeout at `whnf`, maximum number of heartbeats (200000) has been reached + +Note: Use `set_option maxHeartbeats ` to set the limit. + +Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command. +/workspace/scratch/gaussian-coupling/coupling-probe.lean:21:0: error: (deterministic) timeout at `whnf`, maximum number of heartbeats (200000) has been reached + +Note: Use `set_option maxHeartbeats ` to set the limit. + +Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command. diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-04.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-04.log new file mode 100644 index 000000000..a0b7543cd --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe-04.log @@ -0,0 +1,10 @@ +/workspace/scratch/gaussian-coupling/coupling-probe.lean:36:10: error: (deterministic) timeout at `isDefEq`, maximum number of heartbeats (1000000) has been reached + +Note: Use `set_option maxHeartbeats ` to set the limit. + +Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command. +/workspace/scratch/gaussian-coupling/coupling-probe.lean:22:0: error: (deterministic) timeout at `whnf`, maximum number of heartbeats (1000000) has been reached + +Note: Use `set_option maxHeartbeats ` to set the limit. + +Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command. diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe.lean.txt new file mode 100644 index 000000000..05756081b --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/coupling-probe.lean.txt @@ -0,0 +1,39 @@ +import Mathlib.Probability.Distributions.Gaussian.Multivariate +import Mathlib.Probability.HasLaw +import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric + +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter Matrix +open scoped Topology ENNReal NNReal MatrixOrder Matrix.Norms.L2Operator +set_option autoImplicit false + +noncomputable def gc_coupledVector {ι : Type*} [Fintype ι] [DecidableEq ι] + (m : EuclideanSpace ℝ ι) (S : Matrix ι ι ℝ) (z : EuclideanSpace ℝ ι) : + EuclideanSpace ℝ ι := m + Matrix.toEuclideanCLM (𝕜 := ℝ) (CFC.sqrt S) z + +theorem gc_coupledVector_hasLaw {ι : Type*} [Fintype ι] [DecidableEq ι] + (m : EuclideanSpace ℝ ι) (S : Matrix ι ι ℝ) : + HasLaw (gc_coupledVector m S) (multivariateGaussian m S) + (stdGaussian (EuclideanSpace ℝ ι)) where + aemeasurable := by unfold gc_coupledVector; fun_prop + map_eq := rfl + +set_option maxHeartbeats 1000000 in +theorem gc_coupledVector_tendsto {ι κ : Type*} [Fintype ι] [DecidableEq ι] + (u : Filter κ) (m : κ → EuclideanSpace ℝ ι) (S : κ → Matrix ι ι ℝ) + (m0 : EuclideanSpace ℝ ι) (S0 : Matrix ι ι ℝ) + (hm : Tendsto m u (𝓝 m0)) (hS : Tendsto S u (𝓝 S0)) + (hPSD : ∀ᶠ t in u, (S t).PosSemidef) (hPSD0 : S0.PosSemidef) + (z : EuclideanSpace ℝ ι) : + Tendsto (fun t => gc_coupledVector (m t) (S t) z) u + (𝓝 (gc_coupledVector m0 S0 z)) := by + have hwithin : Tendsto S u (𝓝[{B : Matrix ι ι ℝ | 0 ≤ B}] S0) := + tendsto_nhdsWithin_iff.mpr ⟨hS, hPSD.mono (fun t ht => ht.nonneg)⟩ + have hsqrt : Tendsto (fun t => CFC.sqrt (S t)) u (𝓝 (CFC.sqrt S0)) := + (CFC.continuousOn_sqrt S0 hPSD0.nonneg).tendsto.comp hwithin + have heval : Tendsto (fun t => Matrix.toEuclideanCLM (𝕜 := ℝ) (CFC.sqrt (S t)) z) u + (𝓝 (Matrix.toEuclideanCLM (𝕜 := ℝ) (CFC.sqrt S0) z)) := by + exact ((Matrix.continuous_uncurry_toEuclideanCLM (𝕜 := ℝ) (n := ι)).tendsto (CFC.sqrt S0, z)).comp + (hsqrt.prodMk_nhds tendsto_const_nhds) + exact hm.add heval +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/evaluation-probe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/evaluation-probe-01.log new file mode 100644 index 000000000..41be8b242 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/evaluation-probe-01.log @@ -0,0 +1 @@ +/workspace/scratch/gaussian-coupling/evaluation-probe.lean:20:10: warning: `tendsto_finset_sum` has been deprecated: Use `tendsto_finsetSum` instead diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/evaluation-probe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/evaluation-probe-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/evaluation-probe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/evaluation-probe.lean.txt new file mode 100644 index 000000000..1478acb9b --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-coupling/evaluation-probe.lean.txt @@ -0,0 +1,25 @@ +import Mathlib.Probability.Distributions.Gaussian.Multivariate + +open Filter Matrix WithLp +open scoped Topology +set_option autoImplicit false + +namespace ResearchFormalCoreR1 + +theorem eval_matrix_tendsto {ι R : Type*} [Fintype ι] [DecidableEq ι] + {u : Filter R} {S : R → Matrix ι ι ℝ} {S0 : Matrix ι ι ℝ} + (hS : Tendsto S u (𝓝 S0)) (z : EuclideanSpace ℝ ι) : + Tendsto (fun r => Matrix.toEuclideanCLM (𝕜 := ℝ) (S r) z) u + (𝓝 (Matrix.toEuclideanCLM (𝕜 := ℝ) S0 z)) := by + have hcoord (i j : ι) : Tendsto (fun r => S r i j) u (𝓝 (S0 i j)) := + ((continuous_apply j).tendsto (S0 i)).comp + (((continuous_apply i).tendsto S0).comp hS) + have hvec : Tendsto (fun r => (S r) *ᵥ ofLp z) u (𝓝 (S0 *ᵥ ofLp z)) := by + apply tendsto_pi_nhds.mpr + intro i + exact tendsto_finsetSum Finset.univ (fun j _ => + (hcoord i j).mul tendsto_const_nhds) + change Tendsto (fun r => toLp 2 ((S r) *ᵥ ofLp z)) u + (𝓝 (toLp 2 (S0 *ᵥ ofLp z))) + exact ((PiLp.continuous_toLp 2 (fun _ : ι => ℝ)).tendsto _).comp hvec +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/contact-probe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/contact-probe-01.log new file mode 100644 index 000000000..2222ece96 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/contact-probe-01.log @@ -0,0 +1,8 @@ +/workspace/scratch/gaussian-normalizer-limit/contact-probe.lean:20:70: error: Application type mismatch: The argument + EuclideanSpace.proj i +has type + StrongDual ?m.257 (EuclideanSpace ?m.257 (Fin 6)) +but is expected to have type + EuclideanSpace ℝ (Fin 6) →L[?m.244] ?m.246 +in the application + MemLp.continuousLinearMap_comp IsGaussian.memLp_two_id (EuclideanSpace.proj i) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/contact-probe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/contact-probe-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/contact-probe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/contact-probe.lean.txt new file mode 100644 index 000000000..879447e8a --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/contact-probe.lean.txt @@ -0,0 +1,51 @@ +import ResearchFormalCoreR1.GaussianCoupling +import Mathlib.Probability.Moments.Variance + +open MeasureTheory ProbabilityTheory Matrix WithLp +open scoped ENNReal NNReal MatrixOrder Matrix.Norms.L2Operator +set_option autoImplicit false +namespace ResearchFormalCoreR1 + +theorem contact_relations (m : EuclideanSpace ℝ (Fin 6)) (S : Matrix (Fin 6) (Fin 6) ℝ) + (hS : S.PosSemidef) + (hm0 : m 0 = -1) (hm1 : m 1 = 1) + (hm23 : m 2 + m 3 = 0) (hm45 : m 4 - m 5 = 0) + (hS0 : S 0 0 = 0) (hS1 : S 1 1 = 0) + (hS23 : S 2 2 + 2*S 2 3 + S 3 3 = 0) + (hS45 : S 4 4 - 2*S 4 5 + S 5 5 = 0) : + ∀ᵐ x ∂multivariateGaussian m S, + x 0 = -1 ∧ x 1 = 1 ∧ x 2 + x 3 = 0 ∧ x 4 - x 5 = 0 := by + let P := multivariateGaussian m S + have hLp (i : Fin 6) : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => x i) 2 P := by + exact (IsGaussian.memLp_two_id (μ := P)).continuousLinearMap_comp (EuclideanSpace.proj (𝕜 := ℝ) i) + have hi (i : Fin 6) : ∫ x : EuclideanSpace ℝ (Fin 6), x i ∂P = m i := by + change ∫ x, (EuclideanSpace.proj (𝕜 := ℝ) i) x ∂P = m i + rw [ContinuousLinearMap.integral_comp_id_comm] + · simp [P] + · exact IsGaussian.integrable_id + have hvar0 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 0; P] = 0 := by + exact (variance_eval_multivariateGaussian hS 0).trans hS0 + have hvar1 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 1; P] = 0 := by + exact (variance_eval_multivariateGaussian hS 1).trans hS1 + have hvar23 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 2 + x 3; P] = 0 := by + rw [variance_fun_add (hLp 2) (hLp 3)] + simpa [P, variance_eval_multivariateGaussian hS, + covariance_eval_multivariateGaussian hS] using hS23 + have hvar45 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 4 - x 5; P] = 0 := by + rw [variance_fun_sub (hLp 4) (hLp 5)] + simpa [P, variance_eval_multivariateGaussian hS, + covariance_eval_multivariateGaussian hS] using hS45 + have hi23 : ∫ x : EuclideanSpace ℝ (Fin 6), (x 2 + x 3) ∂P = 0 := by + rw [integral_add ((hLp 2).integrable (by norm_num)) ((hLp 3).integrable (by norm_num)), hi, hi] + exact hm23 + have hi45 : ∫ x : EuclideanSpace ℝ (Fin 6), (x 4 - x 5) ∂P = 0 := by + rw [integral_sub ((hLp 4).integrable (by norm_num)) ((hLp 5).integrable (by norm_num)), hi, hi] + exact hm45 + filter_upwards [ae_eq_integral_of_variance_eq_zero (hLp 0) hvar0, + ae_eq_integral_of_variance_eq_zero (hLp 1) hvar1, + ae_eq_integral_of_variance_eq_zero ((hLp 2).add (hLp 3)) hvar23, + ae_eq_integral_of_variance_eq_zero ((hLp 4).sub (hLp 5)) hvar45] + with x hx0 hx1 hx23 hx45 + exact ⟨hx0.trans ((hi 0).trans hm0), hx1.trans ((hi 1).trans hm1), + hx23.trans hi23, hx45.trans hi45⟩ +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/gaussian-normalizer-limit-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/gaussian-normalizer-limit-01.log new file mode 100644 index 000000000..801791594 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/gaussian-normalizer-limit-01.log @@ -0,0 +1,44 @@ +ResearchFormalCoreR1/GaussianNormalizerLimit.lean:138:4: error: Tactic `apply` failed: could not unify the conclusion of `ae_of_ae_map hLaw0.aemeasurable` + ∀ᵐ (x : EuclideanSpace ℝ (Fin 6)) ∂P, ?m.527 (gc_coupledVector m0 S0 x) +with the goal + ∀ᵐ (z : EuclideanSpace ℝ (Fin 6)) ∂P, + (gc_coupledVector m0 S0 z).ofLp 0 = -1 ∧ + (gc_coupledVector m0 S0 z).ofLp 1 = 1 ∧ + (gc_coupledVector m0 S0 z).ofLp 2 + (gc_coupledVector m0 S0 z).ofLp 3 = 0 ∧ + (gc_coupledVector m0 S0 z).ofLp 4 - (gc_coupledVector m0 S0 z).ofLp 5 = 0 + +Note: The full type of `ae_of_ae_map hLaw0.aemeasurable` is + (∀ᵐ (y : EuclideanSpace ℝ (Fin 6)) ∂Measure.map (gc_coupledVector m0 S0) P, ?m.527 y) → + ∀ᵐ (x : EuclideanSpace ℝ (Fin 6)) ∂P, ?m.527 (gc_coupledVector m0 S0 x) + +m : ℝ → EuclideanSpace ℝ (Fin 6) +S : ℝ → Matrix (Fin 6) (Fin 6) ℝ +m0 : EuclideanSpace ℝ (Fin 6) +S0 : Matrix (Fin 6) (Fin 6) ℝ +hm : Tendsto m (𝓝[>] 0) (𝓝 m0) +hS : Tendsto S (𝓝[>] 0) (𝓝 S0) +hPSD : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (S r).PosSemidef +hPSD0 : S0.PosSemidef +hm0 : m0.ofLp 0 = -1 +hm1 : m0.ofLp 1 = 1 +hm23 : m0.ofLp 2 + m0.ofLp 3 = 0 +hm45 : m0.ofLp 4 - m0.ofLp 5 = 0 +hS0 : S0 0 0 = 0 +hS1 : S0 1 1 = 0 +hS23 : S0 2 2 + 2 * S0 2 3 + S0 3 3 = 0 +hS45 : S0 4 4 - 2 * S0 4 5 + S0 5 5 = 0 +hQpos : 0 < S0 4 4 +P : Measure (EuclideanSpace ℝ (Fin 6)) := stdGaussian (EuclideanSpace ℝ (Fin 6)) +X : ℝ → EuclideanSpace ℝ (Fin 6) → Fin 6 → ℝ := fun r z => (gc_coupledVector (m r) (S r) z).ofLp +A : EuclideanSpace ℝ (Fin 6) → ℝ := fun z => 2 * (gc_coupledVector m0 S0 z).ofLp 3 +Q : EuclideanSpace ℝ (Fin 6) → ℝ := fun z => (gc_coupledVector m0 S0 z).ofLp 4 +hX : ∀ (r : ℝ), Measurable (X r) +hLaw0 : HasLaw (gc_coupledVector m0 S0) (multivariateGaussian m0 S0) P +⊢ ∀ᵐ (z : EuclideanSpace ℝ (Fin 6)) ∂P, + (gc_coupledVector m0 S0 z).ofLp 0 = -1 ∧ + (gc_coupledVector m0 S0 z).ofLp 1 = 1 ∧ + (gc_coupledVector m0 S0 z).ofLp 2 + (gc_coupledVector m0 S0 z).ofLp 3 = 0 ∧ + (gc_coupledVector m0 S0 z).ofLp 4 - (gc_coupledVector m0 S0 z).ofLp 5 = 0 +ResearchFormalCoreR1/GaussianNormalizerLimit.lean:167:68: error: `fun_prop` was unable to prove `AEStronglyMeasurable ?m.1785 (multivariateGaussian (m r) (S r))` + +Issues: diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/gaussian-normalizer-limit-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/gaussian-normalizer-limit-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/gaussian-normalizer-limit-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/gaussian-normalizer-limit-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-01.log new file mode 100644 index 000000000..9e259b2fc --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-01.log @@ -0,0 +1,16 @@ +/workspace/scratch/gaussian-normalizer-limit/integrability-probe.lean:33:6: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ENNReal.HolderTriple 4 4 2 + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/gaussian-normalizer-limit/integrability-probe.lean:35:33: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ENNReal.HolderTriple 4 4 2 + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/gaussian-normalizer-limit/integrability-probe.lean:40:33: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ENNReal.HolderTriple 4 4 2 + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/gaussian-normalizer-limit/integrability-probe.lean:42:6: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ENNReal.HolderTriple 4 4 2 + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-02.log new file mode 100644 index 000000000..85b7d8d73 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-02.log @@ -0,0 +1,14 @@ +/workspace/scratch/gaussian-normalizer-limit/integrability-probe.lean:22:40: error: unsolved goals +r : ℝ +m : EuclideanSpace ℝ (Fin 6) +S : Matrix (Fin 6) (Fin 6) ℝ +hS : S.PosSemidef +⊢ 4⁻¹ + 4⁻¹ = 2⁻¹ +/workspace/scratch/gaussian-normalizer-limit/integrability-probe.lean:22:2: warning: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-03.log new file mode 100644 index 000000000..00ee4e941 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-03.log @@ -0,0 +1,22 @@ +/workspace/scratch/gaussian-normalizer-limit/integrability-probe.lean:22:41: error: unsolved goals +case ha +r : ℝ +m : EuclideanSpace ℝ (Fin 6) +S : Matrix (Fin 6) (Fin 6) ℝ +hS : S.PosSemidef +⊢ 2 ≠ 0 ∨ 2 ≠ ∞ + +case hb +r : ℝ +m : EuclideanSpace ℝ (Fin 6) +S : Matrix (Fin 6) (Fin 6) ℝ +hS : S.PosSemidef +⊢ 2 ≠ ∞ ∨ 2 ≠ 0 +/workspace/scratch/gaussian-normalizer-limit/integrability-probe.lean:22:2: warning: Try this: + haveI̵ + +The goal is a proposition, so `have` is preferred over `haveI`. +The difference between `have` and `haveI` is that `haveI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-04.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe-04.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe.lean.txt new file mode 100644 index 000000000..15ea3020a --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/integrability-probe.lean.txt @@ -0,0 +1,50 @@ +import ResearchFormalCoreR1.GaussianCoupling +import ResearchFormalCoreR1.GaussianNormalizer +import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp + +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter Matrix WithLp +open scoped NNReal ENNReal Topology +set_option autoImplicit false + +private lemma positivePart_memLp {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {f : Ω → ℝ} {p : ℝ≥0∞} (hf : MemLp f p P) : + MemLp (fun x => max (f x) 0) p P := by + apply hf.norm.mono' ((hf.aestronglyMeasurable.aemeasurable.sup_const 0).aestronglyMeasurable) + filter_upwards [] with x + rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg (le_max_right _ _)] + exact max_le (le_abs_self _) (abs_nonneg _) + +lemma gaussian_weight_integrable (r : ℝ) (m : EuclideanSpace ℝ (Fin 6)) + (S : Matrix (Fin 6) (Fin 6) ℝ) (hS : S.PosSemidef) : + Integrable (fun x : EuclideanSpace ℝ (Fin 6) => gn_normalizedWeight r (ofLp x)) + (multivariateGaussian m S) := by + have : ENNReal.HolderTriple 4 4 2 := ⟨by + rw [← two_mul, show (4 : ℝ≥0∞) = 2*2 by norm_num, ENNReal.mul_inv (by simp) (by simp), + ← mul_assoc, ENNReal.mul_inv_cancel two_ne_zero ENNReal.ofNat_ne_top, one_mul]⟩ + let P := multivariateGaussian m S + have hc (i : Fin 6) : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => x i) 4 P := by + have hp := measurePreserving_eval_multivariateGaussian (μ := m) hS (i := i) + have hlaw : HasLaw (fun x : EuclideanSpace ℝ (Fin 6) => x i) + (gaussianReal (m i) (S i i).toNNReal) P := ⟨hp.measurable.aemeasurable, hp.map_eq⟩ + exact hlaw.memLp (memLp_id_gaussianReal 4) + have hpos (i : Fin 6) : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => max (-x i) 0) 4 P := + positivePart_memLp (hc i).neg + have hM : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => + max (-x 0) 0 * max (-x 4) 0 - r*(x 2)^2) 2 P := by + have h1 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => max (-x 0) 0 * max (-x 4) 0) 2 P := + (hpos 0).fun_mul (hpos 4) + have h2 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => (x 2)^2) 2 P := by + simpa only [pow_two] using (hc 2).fun_mul (hc 2) + exact h1.sub (h2.const_mul r) + have hSadd : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => + r*(x 3)^2 - x 1*x 5) 2 P := by + have h1 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => (x 3)^2) 2 P := by + simpa only [pow_two] using (hc 3).fun_mul (hc 3) + have h2 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => x 1*x 5) 2 P := + (hc 1).fun_mul (hc 5) + exact (h1.const_mul r).sub h2 + have hT : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => gn_normalizedWeight r (ofLp x)) 1 P := + (positivePart_memLp hM).fun_mul (positivePart_memLp hSadd) + exact memLp_one_iff_integrable.mp hT +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/parameter-floor-controls-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/parameter-floor-controls-01.log new file mode 100644 index 000000000..24a0b6d4e --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/parameter-floor-controls-01.log @@ -0,0 +1,39 @@ +ResearchFormalCoreR1.gnl_normalizer_floor_of_parameter_limits (m : ℝ → EuclideanSpace ℝ (Fin 6)) + (S : ℝ → Matrix (Fin 6) (Fin 6) ℝ) (m0 : EuclideanSpace ℝ (Fin 6)) (S0 : Matrix (Fin 6) (Fin 6) ℝ) + (hm : Tendsto m (𝓝[>] 0) (𝓝 m0)) (hS : Tendsto S (𝓝[>] 0) (𝓝 S0)) (hPSD : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (S r).PosSemidef) + (hPSD0 : S0.PosSemidef) (hm0 : m0.ofLp 0 = -1) (hm1 : m0.ofLp 1 = 1) (hm23 : m0.ofLp 2 + m0.ofLp 3 = 0) + (hm45 : m0.ofLp 4 - m0.ofLp 5 = 0) (hS0 : S0 0 0 = 0) (hS1 : S0 1 1 = 0) (hS23 : S0 2 2 + 2 * S0 2 3 + S0 3 3 = 0) + (hS45 : S0 4 4 - 2 * S0 4 5 + S0 5 5 = 0) (hQpos : 0 < S0 4 4) : + ∃ c, + 0 < c ∧ + ∃ δ, + 0 < δ ∧ + ∀ (r : ℝ), + 0 < r → + r < δ → + c * r ^ 2 ≤ + ∫ (x : EuclideanSpace ℝ (Fin 6)), + r ^ 2 * gn_normalizedWeight r x.ofLp ∂multivariateGaussian (m r) (S r) +ResearchFormalCoreR1.gnl_physical_normalizer_floor_of_parameter_limits (m : ℝ → EuclideanSpace ℝ (Fin 6)) + (S : ℝ → Matrix (Fin 6) (Fin 6) ℝ) (m0 : EuclideanSpace ℝ (Fin 6)) (S0 : Matrix (Fin 6) (Fin 6) ℝ) + (hm : Tendsto m (𝓝[>] 0) (𝓝 m0)) (hS : Tendsto S (𝓝[>] 0) (𝓝 S0)) (hPSD : ∀ᶠ (r : ℝ) in 𝓝[>] 0, (S r).PosSemidef) + (hPSD0 : S0.PosSemidef) (hm0 : m0.ofLp 0 = -1) (hm1 : m0.ofLp 1 = 1) (hm23 : m0.ofLp 2 + m0.ofLp 3 = 0) + (hm45 : m0.ofLp 4 - m0.ofLp 5 = 0) (hS0 : S0 0 0 = 0) (hS1 : S0 1 1 = 0) (hS23 : S0 2 2 + 2 * S0 2 3 + S0 3 3 = 0) + (hS45 : S0 4 4 - 2 * S0 4 5 + S0 5 5 = 0) (hQpos : 0 < S0 4 4) : + ∃ c, + 0 < c ∧ + ∃ δ, + 0 < δ ∧ + ∀ (r : ℝ), + 0 < r → + r < δ → + c * r ^ 2 ≤ + ∫ (x : EuclideanSpace ℝ (Fin 6)), + egm_weight r + ![x.ofLp 0, x.ofLp 2, x.ofLp 4, x.ofLp 1, x.ofLp 3, x.ofLp 5] ∂multivariateGaussian (m r) (S r) +'ResearchFormalCoreR1.gnl_normalizer_floor_of_parameter_limits' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'ResearchFormalCoreR1.gnl_physical_normalizer_floor_of_parameter_limits' depends on axioms: [propext, + Classical.choice, + Quot.sound] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/parameter-floor-controls.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/parameter-floor-controls.lean.txt new file mode 100644 index 000000000..64375ec39 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer-limit/parameter-floor-controls.lean.txt @@ -0,0 +1,31 @@ +import ResearchFormalCoreR1.GaussianNormalizerLimit +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter Matrix WithLp +open scoped Topology ENNReal NNReal MatrixOrder Matrix.Norms.L2Operator + +private def controlMean : EuclideanSpace ℝ (Fin 6) := toLp 2 ![-1,1,0,0,3,3] +private def controlDirection : Fin 6 → ℝ := ![0,0,0,0,1,1] +private def controlCov : Matrix (Fin 6) (Fin 6) ℝ := vecMulVec controlDirection controlDirection + +-- Fully inhabited hypotheses with positive nonzero mean Q and singular rank-one covariance. +example : ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ ∀ r : ℝ, 0 < r → r < δ → + c*r^2 ≤ ∫ x : EuclideanSpace ℝ (Fin 6), egm_weight r ![x 0,x 2,x 4,x 1,x 3,x 5] + ∂multivariateGaussian controlMean controlCov := by + have hPSD : controlCov.PosSemidef := by + simpa [controlCov] using Matrix.posSemidef_vecMulVec_self_star controlDirection + exact gnl_physical_normalizer_floor_of_parameter_limits + (fun _r => controlMean) (fun _r => controlCov) controlMean controlCov + tendsto_const_nhds tendsto_const_nhds (Eventually.of_forall (fun _r => hPSD)) hPSD + (by norm_num [controlMean]) (by norm_num [controlMean]) + (by norm_num [controlMean]) (by norm_num [controlMean]) + (by norm_num [controlCov, controlDirection, vecMulVec]) + (by norm_num [controlCov, controlDirection, vecMulVec]) + (by norm_num [controlCov, controlDirection, vecMulVec]) + (by norm_num [controlCov, controlDirection, vecMulVec]) + (by norm_num [controlCov, controlDirection, vecMulVec]) + +#check gnl_normalizer_floor_of_parameter_limits +#check gnl_physical_normalizer_floor_of_parameter_limits +#print axioms gnl_normalizer_floor_of_parameter_limits +#print axioms gnl_physical_normalizer_floor_of_parameter_limits +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-api-before.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-api-before.lean.txt new file mode 100644 index 000000000..122131dd2 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-api-before.lean.txt @@ -0,0 +1,4 @@ +import ResearchFormalCoreR1.GaussianNormalizer +#check ResearchFormalCoreR1.gn_normalizedWeight +#check ResearchFormalCoreR1.gn_gaussian_reference_integrable_pos +#check ResearchFormalCoreR1.gn_open_box_floor diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-api-before.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-api-before.log new file mode 100644 index 000000000..7ef8f723f --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-api-before.log @@ -0,0 +1 @@ +/workspace/scratch/gaussian-normalizer/normalizer-api-before.lean:1:0: error: object file '/workspace/Math-/frontiers/determinant_moment_20261025/.lake/build/lib/lean/ResearchFormalCoreR1/GaussianNormalizer.olean' of module ResearchFormalCoreR1.GaussianNormalizer does not exist diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-compile-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-compile-01.log new file mode 100644 index 000000000..f23b13e80 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-compile-01.log @@ -0,0 +1 @@ +ResearchFormalCoreR1/GaussianNormalizer.lean:1:0: error: object file '/workspace/Math-/frontiers/determinant_moment_20261025/.lake/packages/mathlib/.lake/build/lib/lean/Mathlib/Probability/Distributions/Gaussian/Moments.olean' of module Mathlib.Probability.Distributions.Gaussian.Moments does not exist diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-compile-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-compile-02.log new file mode 100644 index 000000000..13bb93681 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-compile-02.log @@ -0,0 +1,6 @@ +ResearchFormalCoreR1/GaussianNormalizer.lean:36:23: warning: Variable name `hbS` is not explicitly referenced. + +Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning: + [apply] _hbS + +Note: This linter can be disabled with `set_option linter.unusedVariables false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-compile-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-compile-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-controls.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-controls.lean.txt new file mode 100644 index 000000000..551b9139d --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-controls.lean.txt @@ -0,0 +1,18 @@ +import ResearchFormalCoreR1.GaussianNormalizer +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory +example (a q : ℝ) : + gn_normalizedWeight 0 ![-1,1,-a/2,a/2,q,q] = (max (-q) 0)^2 := by + simp [gn_normalizedWeight, pow_two] +example : + (∫ q : ℝ, (max (-q) 0)^2 ∂gaussianReal 1 0) = 0 := by + simp [gaussianReal_zero_var] +example : + ¬ 0 < ∫ q : ℝ, (max (-q) 0)^2 ∂gaussianReal 1 0 := by + simp [gaussianReal_zero_var] +#check gn_normalizedWeight +#check gn_gaussian_reference_integrable_pos +#check gn_open_box_floor +#print axioms gn_gaussian_reference_integrable_pos +#print axioms gn_open_box_floor +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-controls.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-controls.log new file mode 100644 index 000000000..622008582 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-controls.log @@ -0,0 +1,8 @@ +ResearchFormalCoreR1.gn_normalizedWeight (r : ℝ) (v : Fin 6 → ℝ) : ℝ +ResearchFormalCoreR1.gn_gaussian_reference_integrable_pos (m : ℝ) (v : NNReal) (hv : v ≠ 0) : + Integrable (fun q => max (-q) 0 ^ 2) (gaussianReal m v) ∧ 0 < ∫ (q : ℝ), max (-q) 0 ^ 2 ∂gaussianReal m v +ResearchFormalCoreR1.gn_open_box_floor (r K : ℝ) (v : Fin 6 → ℝ) (hr : 0 ≤ r) (hK : 0 < K) (hsmall : r * K ^ 2 ≤ 1 / 4) + (haM : v 0 ∈ Set.Ioo (-5 / 4) (-3 / 4)) (haS : v 1 ∈ Set.Ioo (3 / 4) (5 / 4)) (hbM : |v 2| < K) (_hbS : |v 3| < K) + (hcM : v 4 ∈ Set.Ioo (-2) (-1)) (hcS : v 5 ∈ Set.Ioo (-2) (-1)) : 3 / 8 ≤ gn_normalizedWeight r v +'ResearchFormalCoreR1.gn_gaussian_reference_integrable_pos' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.gn_open_box_floor' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-module-build.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/normalizer-module-build.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe-01.log new file mode 100644 index 000000000..3354109a9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe-01.log @@ -0,0 +1,4 @@ +/workspace/scratch/gaussian-normalizer/reference-probe.lean:25:6: error: not a positivity goal +/workspace/scratch/gaussian-normalizer/reference-probe.lean:44:4: warning: Try `simp at hvol` instead of `simpa using hvol` + +Note: This linter can be disabled with `set_option linter.unnecessarySimpa false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe-02.log new file mode 100644 index 000000000..6db2283e4 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe-02.log @@ -0,0 +1,9 @@ +/workspace/scratch/gaussian-normalizer/reference-probe.lean:25:6: error: Type mismatch + sq_nonneg q +has type + 0 ≤ q ^ 2 +but is expected to have type + 0 ^ 2 ≤ q ^ 2 +/workspace/scratch/gaussian-normalizer/reference-probe.lean:44:4: warning: Try `simp at hvol` instead of `simpa using hvol` + +Note: This linter can be disabled with `set_option linter.unnecessarySimpa false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe.lean.txt new file mode 100644 index 000000000..da035327d --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-normalizer/reference-probe.lean.txt @@ -0,0 +1,48 @@ +import Mathlib.Probability.Distributions.Gaussian.Real +import Mathlib.MeasureTheory.Integral.Bochner.Basic +import Mathlib.MeasureTheory.Measure.Lebesgue.Basic +import Mathlib.Tactic.FunProp +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum +import Mathlib.Tactic.Positivity + +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory +open scoped ENNReal NNReal +set_option autoImplicit false + +theorem reference_integrable_pos (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) : + Integrable (fun q : ℝ => (max (-q) 0)^2) (gaussianReal m v) ∧ + 0 < ∫ q : ℝ, (max (-q) 0)^2 ∂gaussianReal m v := by + have hsq : Integrable (fun q : ℝ => q^2) (gaussianReal m v) := by + simpa using (memLp_id_gaussianReal (μ := m) (v := v) 2).integrable_sq + have hint : Integrable (fun q : ℝ => (max (-q) 0)^2) (gaussianReal m v) := by + apply hsq.mono' (by fun_prop) + filter_upwards [] with q + rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] + rcases le_total 0 q with hq | hq + · rw [max_eq_right (neg_nonpos.mpr hq)] + simpa using sq_nonneg q + · rw [max_eq_left (neg_nonneg.mpr hq)] + nlinarith + have hsupp : Function.support (fun q : ℝ => (max (-q) 0)^2) = Set.Iio 0 := by + ext q + simp only [Function.mem_support, Set.mem_Iio] + constructor + · intro h + by_contra hn + have hq : 0 ≤ q := le_of_not_gt hn + apply h + simp [max_eq_right (neg_nonpos.mpr hq)] + · intro hq + have hp : 0 < (max (-q) 0)^2 := sq_pos_of_pos (lt_max_of_lt_left (neg_pos.mpr hq)) + exact hp.ne' + have hmass : 0 < gaussianReal m v (Set.Iio 0) := by + apply pos_iff_ne_zero.mpr + intro hzero + have hvol := (gaussianReal_absolutelyContinuous' m hv) hzero + simp at hvol + refine ⟨hint, ?_⟩ + apply (integral_pos_iff_support_of_nonneg (fun q => sq_nonneg _) hint).mpr + simpa only [hsupp] using hmass +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/RESULTS.json b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/RESULTS.json new file mode 100644 index 000000000..3da9b4674 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/RESULTS.json @@ -0,0 +1,92 @@ +[ + { + "name": "gm8_false_centered_substitution", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean" + ], + "exit": 1, + "source_sha256": "96dcad5abc7a1fe24c42bf9c7544ecd6d6a81f71fe6245b9d4de07a5faffde91", + "log_sha256": "6425d08dffa942dac35c0d2c6049f7b45ea7e8ca3bd738d90d082358a6a80e6e", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean:4:76: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gm8_false_omit_radius_one", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean" + ], + "exit": 1, + "source_sha256": "e382d597a45d667a492f10127a50b54e016f8ab808f2b872d0d00acf781a8b9d", + "log_sha256": "c48135a6578a1e599f01bd5d60c41e8353fe574a3eb5a7fd6cc37e591f473b70", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean:5:46: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gn_false_degenerate_reference_floor", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean" + ], + "exit": 1, + "source_sha256": "ef3e9a137e1cb0adb1e5608af2bba29a6b475ccb56a7206e3797e3c7bf4dcdd2", + "log_sha256": "256223c08bec680444daa53eff7bf2f75a97a6046d1175b183ae599b88f98fc6", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean:4:59: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gn_false_no_small_radius", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean" + ], + "exit": 1, + "source_sha256": "c92e629433ec77f420400aaba2cdd9092d322555b0c01bd8836e87178ceca076", + "log_sha256": "aeee073a8c3898625adf42f138af830d9d75d656ddc2cac6e504341832b61958", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean:4:69: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "egm_false_midpoint_identification", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean" + ], + "exit": 1, + "source_sha256": "293faccdf06762221b99723c08acfb4130669b6b6600717a1f96c2c3aa9a4bc1", + "log_sha256": "7c68421d4fc5d388db70182f95c2a9502d5fd9523e9815ba418f1f20c766226b", + "expected_success": false, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean:6:54: error: unsolved goals\n\u22a2 False\n" + }, + { + "name": "gaussian_parent_positive", + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean" + ], + "exit": 1, + "source_sha256": "ae2c3934d498a862e829b878222385d8d8a39c9fbbb2485df65cb04c45d63ddc", + "log_sha256": "669a75be6dc262dcf45355ae315367e86320c9309005363270d6cc189597e9f8", + "expected_success": true, + "diagnostics": "/workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean:1:0: error: object file '/workspace/Math-/frontiers/determinant_moment_20261025/.lake/build/lib/lean/ResearchFormalCoreR1/GaussianRegressionKernel.olean' of module ResearchFormalCoreR1.GaussianRegressionKernel does not exist\n" + } +] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.command new file mode 100644 index 000000000..37fd6ee98 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.lean.txt new file mode 100644 index 000000000..f11632919 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.lean.txt @@ -0,0 +1,8 @@ +import ResearchFormalCoreR1.EndpointGaussianMoments +open ResearchFormalCoreR1 +-- Exact endpoints of the invisible-midpoint-jet perturbation, versus the +-- quartic raw-nine-jet radius. These two charts cannot be silently identified. +example : egm_weight 1 ![-9,0,-1,9,0,-1]^2 ≤ + 16*(1 : ℝ)^4*dm2Radius ![-1,0,0,0,0,0,0,0,0]^8 := by + norm_num [egm_weight, egm_HM, egm_HS, dm2_negative_iff, + Matrix.det_fin_two, dm2Radius, Fin.sum_univ_succ] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.log new file mode 100644 index 000000000..b8715a9f5 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/egm_false_midpoint_identification.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/egm_false_midpoint_identification.lean:6:54: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.command new file mode 100644 index 000000000..bdc80c5da --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.lean.txt new file mode 100644 index 000000000..297181d4c --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.lean.txt @@ -0,0 +1,26 @@ +import ResearchFormalCoreR1.GaussianJetMoments +import ResearchFormalCoreR1.GaussianRegressionKernel +import ResearchFormalCoreR1.EndpointGaussianMoments +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +open scoped NNReal +-- All nine coordinates can be the SAME Gaussian variable: independence is not used. +example : Integrable (fun z : ℝ => (1+9*|z|)^8) (gaussianReal 0 1) := by + have h := gm8_jet_integrable_bound (gaussianReal 0 1) + (fun z : ℝ => (fun _i : Fin 9 => z)) (by fun_prop) + (fun _ => 0) (fun _ => 1) 0 1 (by norm_num) (by norm_num) + (by intro i; norm_num) (by intro i; norm_num) + (by intro i; exact HasLaw.id) + simpa [dm2Radius, Finset.sum_const] using h.1 +-- Nonzero mean and zero variance are preserved, rather than centered away. +example : (∫ x : ℝ, |x|^8 ∂gaussianReal 1 0) = 1 := by + norm_num [gaussianReal_zero_var] +-- The canonical formula specifies even an observation outside the source support. +example (x : Fin 1 → ℝ) : + regressionKernel (Measure.dirac (0 : Fin 1 → ℝ)) + (ContinuousLinearMap.id ℝ (Fin 1 → ℝ)) 0 0 x = Measure.dirac x := by + rw [regressionKernel_apply, Measure.map_dirac (by fun_prop)] + simp +-- The two actual endpoint types in the six-coordinate chart are allowed. +example : dm2Negative (egm_HM 1 ![-1,0,-1,1,0,-1]) ∧ + (egm_HS 1 ![-1,0,-1,1,0,-1]).det < 0 := by + norm_num [egm_HM, egm_HS, dm2_negative_iff, Matrix.det_fin_two] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.log new file mode 100644 index 000000000..91b97ba01 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gaussian_parent_positive.log @@ -0,0 +1 @@ +/workspace/scratch/gaussian-parent-controls/gaussian_parent_positive.lean:1:0: error: object file '/workspace/Math-/frontiers/determinant_moment_20261025/.lake/build/lib/lean/ResearchFormalCoreR1/GaussianRegressionKernel.olean' of module ResearchFormalCoreR1.GaussianRegressionKernel does not exist diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.command new file mode 100644 index 000000000..2736f2ec2 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.lean.txt new file mode 100644 index 000000000..04150e241 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1.GaussianScalarMoments +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +-- Dropping the nonzero mean term would predict zero eighth moment for N(1,0). +example : (∫ x : ℝ, |x|^8 ∂gaussianReal 1 0) ≤ 128*((0 : ℝ)^8+0^4*gm8C8) := by + norm_num [gaussianReal_zero_var] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.log new file mode 100644 index 000000000..3fa387ae1 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_centered_substitution.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/gm8_false_centered_substitution.lean:4:76: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.command new file mode 100644 index 000000000..90ab78411 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.lean.txt new file mode 100644 index 000000000..350cf2972 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.lean.txt @@ -0,0 +1,6 @@ +import ResearchFormalCoreR1.GaussianJetMoments +open ResearchFormalCoreR1 +-- The exact radius includes one, even when every Gaussian coordinate vanishes. +example : dm2Radius (fun _i : Fin 9 => 0)^8 ≤ + (10 : ℝ)^7 * ∑ _i : Fin 9, |(0 : ℝ)|^8 := by + norm_num [dm2Radius] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.log new file mode 100644 index 000000000..15931bfd5 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gm8_false_omit_radius_one.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/gm8_false_omit_radius_one.lean:5:46: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.command new file mode 100644 index 000000000..4f40a5f27 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean.txt new file mode 100644 index 000000000..d73990e1a --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1.GaussianNormalizer +open MeasureTheory ProbabilityTheory +-- Strictly positive reference mass requires nonzero variance. +example : 0 < ∫ q : ℝ, (max (-q) 0)^2 ∂gaussianReal 1 0 := by + norm_num [gaussianReal_zero_var] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.log new file mode 100644 index 000000000..41a6514b5 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_degenerate_reference_floor.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/gn_false_degenerate_reference_floor.lean:4:59: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.command new file mode 100644 index 000000000..54ee6dde6 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.lean.txt new file mode 100644 index 000000000..3a7e99464 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.lean.txt @@ -0,0 +1,5 @@ +import ResearchFormalCoreR1.GaussianNormalizer +open ResearchFormalCoreR1 +-- The source's small r*K² budget cannot be discarded from the box floor. +example : (3 : ℝ)/8 ≤ gn_normalizedWeight 1 ![-1,1,9,0,-3/2,-3/2] := by + norm_num [gn_normalizedWeight] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.log new file mode 100644 index 000000000..aa1607266 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gaussian-parent-controls/gn_false_no_small_radius.log @@ -0,0 +1,2 @@ +/workspace/scratch/gaussian-parent-controls/gn_false_no_small_radius.lean:4:69: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/00-initial-api.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/00-initial-api.command new file mode 100644 index 000000000..4be1b4c97 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/00-initial-api.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean /workspace/scratch/gm8-endpoint/initial_api.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/00-initial-api.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/00-initial-api.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/00-initial-api.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/00-initial-api.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/00-initial-api.log new file mode 100644 index 000000000..ec827f0df --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/00-initial-api.log @@ -0,0 +1,6 @@ +/workspace/scratch/gm8-endpoint/initial_api.lean:3:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.egm_HM` +/workspace/scratch/gm8-endpoint/initial_api.lean:4:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.egm_HS` +/workspace/scratch/gm8-endpoint/initial_api.lean:5:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.egm_weight` +/workspace/scratch/gm8-endpoint/initial_api.lean:6:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.egm_weight_measurable` +/workspace/scratch/gm8-endpoint/initial_api.lean:7:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.egm_pointwise` +/workspace/scratch/gm8-endpoint/initial_api.lean:8:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.egm_gaussian_moment` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/01-first-compile.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/01-first-compile.command new file mode 100644 index 000000000..f996b187a --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/01-first-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/ResearchFormalCoreR1/EndpointGaussianMoments.olean ResearchFormalCoreR1/EndpointGaussianMoments.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/01-first-compile.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/01-first-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/01-first-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/01-first-compile.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/01-first-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/02-export-contract.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/02-export-contract.command new file mode 100644 index 000000000..8517546ea --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/02-export-contract.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/gm8-endpoint/export_contract.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/02-export-contract.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/02-export-contract.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/02-export-contract.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/02-export-contract.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/02-export-contract.log new file mode 100644 index 000000000..a3ffaf216 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/02-export-contract.log @@ -0,0 +1,11 @@ +ResearchFormalCoreR1.egm_HM (r : ℝ) (j : Fin 6 → ℝ) : Matrix (Fin 2) (Fin 2) ℝ +ResearchFormalCoreR1.egm_HS (r : ℝ) (j : Fin 6 → ℝ) : Matrix (Fin 2) (Fin 2) ℝ +ResearchFormalCoreR1.egm_weight (r : ℝ) (j : Fin 6 → ℝ) : ℝ +ResearchFormalCoreR1.egm_weight_measurable (r : ℝ) : Measurable (egm_weight r) +ResearchFormalCoreR1.egm_pointwise (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) (hr1 : r ≤ 1) : + egm_weight r j ^ 2 ≤ 16 * r ^ 4 * gm8Radius j ^ 8 +ResearchFormalCoreR1.egm_gaussian_moment.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] (μ : Measure Ω) + [IsProbabilityMeasure μ] (J : Ω → Fin 6 → ℝ) (hJ : Measurable J) (m : Fin 6 → ℝ) (v : Fin 6 → ℝ≥0) (B V r : ℝ) + (hB : 0 ≤ B) (hV : 0 ≤ V) (hr : 0 < r) (hr1 : r ≤ 1) (hmean : ∀ (i : Fin 6), |m i| ≤ B) + (hvar : ∀ (i : Fin 6), ↑(v i) ≤ V) (hlaw : ∀ (i : Fin 6), HasLaw (fun x => J x i) (gaussianReal (m i) (v i)) μ) : + Integrable (fun x => egm_weight r (J x) ^ 2) μ ∧ ∫ (x : Ω), egm_weight r (J x) ^ 2 ∂μ ≤ 16 * gm8FiniteM8 6 B V * r ^ 4 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/export_contract.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/export_contract.lean.txt new file mode 100644 index 000000000..38daab0cd --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/export_contract.lean.txt @@ -0,0 +1,26 @@ +import ResearchFormalCoreR1.EndpointGaussianMoments + +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +open scoped NNReal + +#check egm_HM +#check egm_HS +#check egm_weight +#check egm_weight_measurable +#check egm_pointwise +#check egm_gaussian_moment + +example : egm_weight (1 / 2) ![-1,0,-1,1,0,-1] = 1 / 4 := by + norm_num [egm_weight, egm_HM, egm_HS, dm2_negative_iff, Matrix.det_fin_two] + +example : egm_weight (1 / 2) ![1,0,-1,1,0,-1] = 0 := by + norm_num [egm_weight, egm_HM, egm_HS, dm2_negative_iff, Matrix.det_fin_two] + +example {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] + (J : Ω → Fin 6 → ℝ) (hJ : Measurable J) (m : Fin 6 → ℝ) (v : Fin 6 → ℝ≥0) + (B V r : ℝ) (hB : 0 ≤ B) (hV : 0 ≤ V) (hr : 0 < r) (hr1 : r ≤ 1) + (hmean : ∀ i, |m i| ≤ B) (hvar : ∀ i, (v i : ℝ) ≤ V) + (hlaw : ∀ i, HasLaw (fun x => J x i) (gaussianReal (m i) (v i)) μ) : + Integrable (fun x => egm_weight r (J x)^2) μ ∧ + (∫ x, egm_weight r (J x)^2 ∂μ) ≤ 16 * gm8FiniteM8 6 B V * r^4 := + egm_gaussian_moment μ J hJ m v B V r hB hV hr hr1 hmean hvar hlaw diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/initial_api.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/initial_api.lean.txt new file mode 100644 index 000000000..a9d325405 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-endpoint/initial_api.lean.txt @@ -0,0 +1,8 @@ +import ResearchFormalCoreR1.DeterminantMoment + +#check ResearchFormalCoreR1.egm_HM +#check ResearchFormalCoreR1.egm_HS +#check ResearchFormalCoreR1.egm_weight +#check ResearchFormalCoreR1.egm_weight_measurable +#check ResearchFormalCoreR1.egm_pointwise +#check ResearchFormalCoreR1.egm_gaussian_moment diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/01-vector-compile.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/01-vector-compile.log new file mode 100644 index 000000000..9e2f74c02 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/01-vector-compile.log @@ -0,0 +1,37 @@ +ResearchFormalCoreR1/GaussianJetMoments.lean:40:53: error(lean.synthInstanceFailed): failed to synthesize instance of type class + LE Type + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianJetMoments.lean:40:55: error(lean.synthInstanceFailed): failed to synthesize instance of type class + OfNat Type 0 +numerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is + Type +due to the absence of the instance above + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianJetMoments.lean:50:4: error: `MeasureTheory.integrable_finset_sum` has been deprecated: Use `MeasureTheory.integrable_finsetSum` instead +ResearchFormalCoreR1/GaussianJetMoments.lean:76:6: error: Type mismatch + add_le_add_left (Finset.sum_le_sum fun i x => hcoord i) 1 +has type + ∑ i ∈ ?m.926, ∫ (x : Ω), |J x i| ^ 8 ∂μ + 1 ≤ ∑ i ∈ ?m.926, 128 * (B ^ 8 + V ^ 4 * gm8C8) + 1 +but is expected to have type + 1 + ∑ i, ∫ (x : Ω), |J x i| ^ 8 ∂μ ≤ 1 + ∑ _i, 128 * (B ^ 8 + V ^ 4 * gm8C8) +ResearchFormalCoreR1/GaussianJetMoments.lean:83:41: error(lean.synthInstanceFailed): failed to synthesize instance of type class + LE Type + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianJetMoments.lean:83:43: error(lean.synthInstanceFailed): failed to synthesize instance of type class + OfNat Type 0 +numerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is + Type +due to the absence of the instance above + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianJetMoments.lean:96:2: error: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/02-vector-compile.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/02-vector-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/03-correlated-control.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/03-correlated-control.lean.txt new file mode 100644 index 000000000..03e43f962 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/03-correlated-control.lean.txt @@ -0,0 +1,12 @@ +import ResearchFormalCoreR1.GaussianJetMoments +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +open scoped NNReal +example : Integrable (fun z : ℝ => (1 + 9*|z|)^8) (gaussianReal 0 1) := by + have h := gm8_jet_integrable_bound (gaussianReal 0 1) + (fun z : ℝ => (fun _i : Fin 9 => z)) (by fun_prop) + (fun _ => 0) (fun _ => 1) 0 1 (by norm_num) (by norm_num) + (by intro i; norm_num) (by intro i; norm_num) + (by intro i; exact HasLaw.id) + simpa [dm2Radius, Finset.sum_const] using h.1 +example : (∫ x : ℝ, |x|^8 ∂gaussianReal 1 0) = 1 := by + norm_num [gaussianReal_zero_var] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/03-correlated-control.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/03-correlated-control.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/04-generic-vector-compile.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/04-generic-vector-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/TargetBefore.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/TargetBefore.lean.txt new file mode 100644 index 000000000..49de88960 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/TargetBefore.lean.txt @@ -0,0 +1,4 @@ +import ResearchFormalCoreR1 +#check ResearchFormalCoreR1.gm8_radius_power_bound +#check ResearchFormalCoreR1.gm8_jet_integrable_bound +#check ResearchFormalCoreR1.gm8_family_weight_moment diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/target-before.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/target-before.log new file mode 100644 index 000000000..539a40dd3 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-root/target-before.log @@ -0,0 +1,3 @@ +/workspace/scratch/gm8-root/TargetBefore.lean:2:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.gm8_radius_power_bound` +/workspace/scratch/gm8-root/TargetBefore.lean:3:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.gm8_jet_integrable_bound` +/workspace/scratch/gm8-root/TargetBefore.lean:4:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.gm8_family_weight_moment` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/00-initial-api.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/00-initial-api.log new file mode 100644 index 000000000..292b686f9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/00-initial-api.log @@ -0,0 +1,4 @@ +/workspace/scratch/gm8-scalar/initial_api.lean:3:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.gm8C8` +/workspace/scratch/gm8-scalar/initial_api.lean:4:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.gm8_standard_integrable` +/workspace/scratch/gm8-scalar/initial_api.lean:5:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.gm8_standard_nonneg` +/workspace/scratch/gm8-scalar/initial_api.lean:6:7: error(lean.unknownIdentifier): Unknown identifier `ResearchFormalCoreR1.gm8_scalar_integrable_bound` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/01-first-compile.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/01-first-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/01-first-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/01-first-compile.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/01-first-compile.log new file mode 100644 index 000000000..6b47b6aa0 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/01-first-compile.log @@ -0,0 +1,50 @@ +ResearchFormalCoreR1/GaussianScalarMoments.lean:27:44: error(lean.synthInstanceFailed): failed to synthesize instance of type class + LE Type + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianScalarMoments.lean:27:46: error(lean.synthInstanceFailed): failed to synthesize instance of type class + OfNat Type 0 +numerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is + Type +due to the absence of the instance above + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianScalarMoments.lean:39:44: error(lean.synthInstanceFailed): failed to synthesize instance of type class + LE Type + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianScalarMoments.lean:39:46: error(lean.synthInstanceFailed): failed to synthesize instance of type class + OfNat Type 0 +numerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is + Type +due to the absence of the instance above + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianScalarMoments.lean:54:50: error(lean.synthInstanceFailed): failed to synthesize instance of type class + LE Type + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianScalarMoments.lean:54:52: error(lean.synthInstanceFailed): failed to synthesize instance of type class + OfNat Type 0 +numerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is + Type +due to the absence of the instance above + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianScalarMoments.lean:67:6: error: Type mismatch: After simplification, term + add_pow_le hB (mul_nonneg (Real.sqrt_nonneg V) (abs_nonneg z)) 8 + has type + (B + √V * |z|) ^ 8 ≤ 2 ^ 7 * (B ^ 8 + (√V * |z|) ^ 8) +but is expected to have type + (B + √V * |z|) ^ 8 ≤ 128 * (B ^ 8 + (√V * |z|) ^ 8) +ResearchFormalCoreR1/GaussianScalarMoments.lean:73:53: error(lean.synthInstanceFailed): failed to synthesize instance of type class + LE Type + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +ResearchFormalCoreR1/GaussianScalarMoments.lean:73:55: error(lean.synthInstanceFailed): failed to synthesize instance of type class + OfNat Type 0 +numerals are polymorphic in Lean, but the numeral `0` cannot be used in a context where the expected type is + Type +due to the absence of the instance above + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/02-scope-compile.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/02-scope-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/02-scope-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/02-scope-compile.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/02-scope-compile.log new file mode 100644 index 000000000..24295b098 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/02-scope-compile.log @@ -0,0 +1,3 @@ +ResearchFormalCoreR1/GaussianScalarMoments.lean:68:87: warning: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/03-final-compile.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/03-final-compile.command new file mode 100644 index 000000000..1cbcc5817 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/03-final-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -o .lake/build/lib/lean/ResearchFormalCoreR1/GaussianScalarMoments.olean ResearchFormalCoreR1/GaussianScalarMoments.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/03-final-compile.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/03-final-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/03-final-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/03-final-compile.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/03-final-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/04-export-contract.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/04-export-contract.command new file mode 100644 index 000000000..9906844d9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/04-export-contract.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean /workspace/scratch/gm8-scalar/export_contract.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/04-export-contract.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/04-export-contract.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/04-export-contract.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/04-export-contract.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/04-export-contract.log new file mode 100644 index 000000000..a30d3398c --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/04-export-contract.log @@ -0,0 +1,7 @@ +ResearchFormalCoreR1.gm8C8 : ℝ +ResearchFormalCoreR1.gm8_standard_integrable : Integrable (fun z => |z| ^ 8) (gaussianReal 0 1) +ResearchFormalCoreR1.gm8_standard_nonneg : 0 ≤ gm8C8 +ResearchFormalCoreR1.gm8_scalar_integrable_bound (m B V : ℝ) (v : ℝ≥0) (hB : 0 ≤ B) (hV : 0 ≤ V) (hm : |m| ≤ B) + (hv : ↑v ≤ V) : + Integrable (fun x => |x| ^ 8) (gaussianReal m v) ∧ + ∫ (x : ℝ), |x| ^ 8 ∂gaussianReal m v ≤ 128 * (B ^ 8 + V ^ 4 * gm8C8) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/export_contract.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/export_contract.lean.txt new file mode 100644 index 000000000..9805195c5 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/export_contract.lean.txt @@ -0,0 +1,21 @@ +import ResearchFormalCoreR1.GaussianScalarMoments + +open MeasureTheory ProbabilityTheory ResearchFormalCoreR1 +open scoped NNReal + +#check gm8C8 +#check gm8_standard_integrable +#check gm8_standard_nonneg +#check gm8_scalar_integrable_bound + +example (m B V : ℝ) (v : ℝ≥0) (hB : 0 ≤ B) (hV : 0 ≤ V) + (hm : |m| ≤ B) (hv : (v : ℝ) ≤ V) : + Integrable (fun x : ℝ => |x| ^ 8) (gaussianReal m v) ∧ + (∫ x : ℝ, |x| ^ 8 ∂gaussianReal m v) ≤ + 128 * (B ^ 8 + V ^ 4 * gm8C8) := + gm8_scalar_integrable_bound m B V v hB hV hm hv + +example : Integrable (fun x : ℝ => |x| ^ 8) (gaussianReal 1 0) ∧ + (∫ x : ℝ, |x| ^ 8 ∂gaussianReal 1 0) ≤ 128 := by + simpa using gm8_scalar_integrable_bound 1 1 0 0 + (by norm_num) (by norm_num) (by norm_num) (by norm_num) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/initial_api.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/initial_api.lean.txt new file mode 100644 index 000000000..a379f422c --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/gm8-scalar/initial_api.lean.txt @@ -0,0 +1,6 @@ +import ResearchFormalCoreR1.DeterminantMoment + +#check ResearchFormalCoreR1.gm8C8 +#check ResearchFormalCoreR1.gm8_standard_integrable +#check ResearchFormalCoreR1.gm8_standard_nonneg +#check ResearchFormalCoreR1.gm8_scalar_integrable_bound diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe-01.log new file mode 100644 index 000000000..a84fbb813 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe-01.log @@ -0,0 +1,6 @@ +/workspace/scratch/normalizer-floor/fatou-probe.lean:23:53: error: Type mismatch + hnonneg r ω +has type + 0 ≤ f r ω +but is expected to have type + ?m.106 ω hω r ≤ g ω diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe-02.log new file mode 100644 index 000000000..4e629644f --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe-02.log @@ -0,0 +1,7 @@ +/workspace/scratch/normalizer-floor/fatou-probe.lean:40:9: error: Function expected at + ENNReal.ofReal_lt_ofReal_iff' +but this term has type + ENNReal.ofReal ?m.300 < ENNReal.ofReal ?m.301 ↔ ?m.300 < ?m.301 ∧ 0 < ?m.301 + +Note: Expected a function because this term is being applied to the argument + .mp diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe.lean.txt new file mode 100644 index 000000000..88077c4e7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/fatou-probe.lean.txt @@ -0,0 +1,41 @@ +import Mathlib.MeasureTheory.Integral.Bochner.Basic +import Mathlib.MeasureTheory.Integral.Lebesgue.Add +import Mathlib.Topology.Instances.ENNReal.Lemmas +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.Positivity + +open MeasureTheory Filter +open scoped ENNReal Topology +set_option autoImplicit false + +namespace ResearchFormalCoreR1 + +theorem fatou_floor {Ω ι : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (u : Filter ι) [IsCountablyGenerated u] [NeBot u] + (f : ι → Ω → ℝ) (g : Ω → ℝ) + (hmeas : ∀ r, Measurable (f r)) + (hnonneg : ∀ r ω, 0 ≤ f r ω) + (hg : Integrable g P) (hgpos : 0 < ∫ ω, g ω ∂P) + (hconv : ∀ᵐ ω ∂P, Tendsto (fun r => f r ω) u (𝓝 (g ω))) + (hfint : ∀ᶠ r in u, Integrable (f r) P) : + ∀ᶠ r in u, (∫ ω, g ω ∂P)/2 ≤ ∫ ω, f r ω ∂P := by + have hg0 : 0 ≤ᵐ[P] g := hconv.mono (fun ω hω => + ge_of_tendsto hω (Eventually.of_forall (fun r => hnonneg r ω))) + have hlim : (fun ω => liminf (fun r => ENNReal.ofReal (f r ω)) u) =ᵐ[P] + (fun ω => ENNReal.ofReal (g ω)) := by + filter_upwards [hconv] with ω hω + exact ((ENNReal.continuous_ofReal.tendsto (g ω)).comp hω).liminf_eq + have hfatou : ENNReal.ofReal (∫ ω, g ω ∂P) ≤ + liminf (fun r => ∫⁻ ω, ENNReal.ofReal (f r ω) ∂P) u := by + rw [ofReal_integral_eq_lintegral_ofReal hg hg0] + rw [← lintegral_congr_ae hlim] + exact lintegral_liminf_le (fun r => (hmeas r).ennreal_ofReal) + have hhalf : ENNReal.ofReal ((∫ ω, g ω ∂P)/2) < + ENNReal.ofReal (∫ ω, g ω ∂P) := by + exact (ENNReal.ofReal_lt_ofReal_iff hgpos).mpr (by linarith) + have hevent := eventually_lt_of_lt_liminf (hhalf.trans_le hfatou) + filter_upwards [hevent, hfint] with r hr hri + rw [← ofReal_integral_eq_lintegral_ofReal hri + (Eventually.of_forall (fun ω => hnonneg r ω))] at hr + exact (ENNReal.ofReal_lt_ofReal_iff'.mp hr).1.le +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-compile-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-compile-01.log new file mode 100644 index 000000000..1fef09779 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-compile-01.log @@ -0,0 +1,6 @@ +ResearchFormalCoreR1/NormalizerFloor.lean:118:2: error: Type mismatch + mul_le_mul_of_nonneg_left hbound (sq_nonneg r) +has type + r ^ 2 * ((∫ (ω : Ω), g ω ∂P) / 2) ≤ r ^ 2 * ∫ (ω : Ω), gn_normalizedWeight r (X r ω) ∂P +but is expected to have type + (∫ (ω : Ω), g ω ∂P) / 2 * r ^ 2 ≤ r ^ 2 * ∫ (a : Ω), gn_normalizedWeight r (X r a) ∂P diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-compile-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-compile-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-controls-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-controls-01.log new file mode 100644 index 000000000..8e75bcd03 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-controls-01.log @@ -0,0 +1,16 @@ +/workspace/scratch/normalizer-floor/normalizer-floor-controls.lean:22:59: warning: try 'simp' instead of 'simpa' + +Note: This linter can be disabled with `set_option linter.unnecessarySimpa false` +/workspace/scratch/normalizer-floor/normalizer-floor-controls.lean:27:8: warning: Variable name `q` is not explicitly referenced. + +Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning: + [apply] _q + +Note: This linter can be disabled with `set_option linter.unusedVariables false` +ResearchFormalCoreR1.gn_normalizer_floor_of_common_coupling.{u_1} {Ω : Type u_1} [MeasurableSpace Ω] (P : Measure Ω) + [IsProbabilityMeasure P] (X : ℝ → Ω → Fin 6 → ℝ) (hX : ∀ (r : ℝ), Measurable (X r)) (A Q : Ω → ℝ) (m : ℝ) (v : ℝ≥0) + (hv : v ≠ 0) (hQ : HasLaw Q (gaussianReal m v) P) + (hconv : ∀ᵐ (ω : Ω) ∂P, Tendsto (fun r => X r ω) (𝓝[>] 0) (𝓝 ![-1, 1, -A ω / 2, A ω / 2, Q ω, Q ω])) + (hint : ∀ᶠ (r : ℝ) in 𝓝[>] 0, Integrable (fun ω => gn_normalizedWeight r (X r ω)) P) : + ∃ c, 0 < c ∧ ∃ δ, 0 < δ ∧ ∀ (r : ℝ), 0 < r → r < δ → c * r ^ 2 ≤ ∫ (ω : Ω), r ^ 2 * gn_normalizedWeight r (X r ω) ∂P +'ResearchFormalCoreR1.gn_normalizer_floor_of_common_coupling' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-controls.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-controls.lean.txt new file mode 100644 index 000000000..1c9ecaa02 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/normalizer-floor-controls.lean.txt @@ -0,0 +1,36 @@ +import ResearchFormalCoreR1.NormalizerFloor +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter +open scoped Topology ENNReal NNReal + +-- A genuine noncentered Gaussian common coupling inhabits every premise. +example : ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → c*r^2 ≤ + ∫ q : ℝ, r^2 * gn_normalizedWeight r ![-1,1,0,0,q,q] ∂gaussianReal 3 2 := by + have hcontact (r q : ℝ) : + gn_normalizedWeight r ![-1,1,0,0,q,q] = (max (-q) 0)^2 := by + simp [gn_normalizedWeight, pow_two] + have hmeas (r : ℝ) : Measurable (fun q : ℝ => (![-1,1,0,0,q,q] : Fin 6 → ℝ)) := by + fun_prop + have hi : ∀ᶠ r in 𝓝[>] (0:ℝ), Integrable + (fun q : ℝ => gn_normalizedWeight r ![-1,1,0,0,q,q]) (gaussianReal 3 2) := by + filter_upwards [] with r + simp_rw [hcontact] + exact (gn_gaussian_reference_integrable_pos 3 2 (by norm_num)).1 + simpa using gn_normalizer_floor_of_common_coupling (gaussianReal 3 2) + (fun _r q => ![-1,1,0,0,q,q]) hmeas (fun _q => 0) id 3 2 (by norm_num) + (HasLaw.id) (Filter.Eventually.of_forall (fun _q => by simpa using tendsto_const_nhds)) hi + +-- Without nondegeneracy a positive constant Q can make every weight zero. +example : ¬ ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → c*r^2 ≤ + ∫ q : ℝ, r^2 * gn_normalizedWeight r ![-1,1,0,0,1,1] ∂gaussianReal 1 0 := by + rintro ⟨c, hc, δ, hδ, hfloor⟩ + have h := hfloor (δ/2) (by linarith) (by linarith) + have hp : 0 < c*(δ/2)^2 := mul_pos hc (sq_pos_of_pos (by linarith)) + norm_num [gn_normalizedWeight] at h + linarith + +#check gn_normalizer_floor_of_common_coupling +#print axioms gn_normalizer_floor_of_common_coupling +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe-01.log new file mode 100644 index 000000000..32e4adc18 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe-01.log @@ -0,0 +1,8 @@ +/workspace/scratch/normalizer-floor/pointwise-probe.lean:23:63: error: Application type mismatch: The argument + Tendsto.prodMk hr hX +has type + Tendsto (fun x => (x, X x)) (𝓝[>] 0) (𝓝 0 ×ˢ 𝓝 ![-1, 1, -a / 2, a / 2, q, q]) +but is expected to have type + Tendsto ?m.144 ?m.146 (𝓝 (0, ![-1, 1, -a / 2, a / 2, q, q])) +in the application + Tendsto.comp (Continuous.tendsto weight_continuous (0, ![-1, 1, -a / 2, a / 2, q, q])) (Tendsto.prodMk hr hX) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe-02.log new file mode 100644 index 000000000..e959a632e --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe-02.log @@ -0,0 +1,6 @@ +/workspace/scratch/normalizer-floor/pointwise-probe.lean:22:2: error: Type mismatch: After simplification, term + Tendsto.comp (Continuous.tendsto weight_continuous (0, ![-1, 1, -a / 2, a / 2, q, q])) (Tendsto.prodMk_nhds hr hX) + has type + @Tendsto ℝ ℝ ((fun p => gn_normalizedWeight p.1 p.2) ∘ fun c => (c, X c)) (𝓝[>] 0) (𝓝 (max (-q) 0 ^ 2)) +but is expected to have type + @Tendsto ℝ ℝ (fun r => gn_normalizedWeight r (X r)) (𝓝[>] 0) (𝓝 (max (-q) 0 ^ 2)) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe.lean.txt new file mode 100644 index 000000000..8aaf5fdc8 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/normalizer-floor/pointwise-probe.lean.txt @@ -0,0 +1,34 @@ +import ResearchFormalCoreR1.GaussianNormalizer +import Mathlib.Probability.HasLaw +import Mathlib.MeasureTheory.Integral.Lebesgue.Add +import Mathlib.Topology.Order.LeftRightNhds +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter +open scoped Topology ENNReal NNReal +set_option autoImplicit false +private theorem weight_continuous : + Continuous (fun p : ℝ × (Fin 6 → ℝ) => gn_normalizedWeight p.1 p.2) := by + unfold gn_normalizedWeight + fun_prop +private theorem contact_value (a q : ℝ) : + gn_normalizedWeight 0 ![-1,1,-a/2,a/2,q,q] = (max (-q) 0)^2 := by + simp [gn_normalizedWeight, pow_two] +private theorem weight_limit (X : ℝ → Fin 6 → ℝ) (a q : ℝ) + (hX : Tendsto X (𝓝[>] (0:ℝ)) (𝓝 ![-1,1,-a/2,a/2,q,q])) : + Tendsto (fun r => gn_normalizedWeight r (X r)) (𝓝[>] (0:ℝ)) + (𝓝 ((max (-q) 0)^2)) := by + have hr : Tendsto (fun r : ℝ => r) (𝓝[>] (0:ℝ)) (𝓝 0) := + continuousWithinAt_id + simpa only [contact_value, Function.comp_def] using + (weight_continuous.tendsto (0, ![-1,1,-a/2,a/2,q,q])).comp (hr.prodMk_nhds hX) +private theorem reference {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (Q : Ω → ℝ) (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) + (hlaw : HasLaw Q (gaussianReal m v) P) : + Integrable (fun ω => (max (-Q ω) 0)^2) P ∧ + 0 < ∫ ω, (max (-Q ω) 0)^2 ∂P := by + obtain ⟨hInt,hPos⟩ := gn_gaussian_reference_integrable_pos m v hv + refine ⟨hlaw.integrable_comp hInt, ?_⟩ + have heq := hlaw.integral_comp + (show AEStronglyMeasurable (fun q : ℝ => (max (-q) 0)^2) (gaussianReal m v) by fun_prop) + exact heq.trans_gt hPos +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/Api.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/Api.lean.txt new file mode 100644 index 000000000..39ce17e39 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/Api.lean.txt @@ -0,0 +1,26 @@ +import Mathlib.MeasureTheory.Function.L2Space +import Mathlib.MeasureTheory.Measure.WithDensity +import Mathlib.Tactic +open MeasureTheory +#check Real.holderConjugate_iff +#check Real.HolderConjugate.two_two +#check Real.holderConjugate_two_two +#check Real.rpow_two +#check Real.sqrt_eq_rpow +#check Real.sqrt_mul +#check Real.sqrt_sq +#check Real.sqrt_sq_eq_abs +#check integral_indicator +#check integral_indicator_one +#check integral_const +#check memLp_const +#check MemLp.indicator +#check Set.mul_indicator +#check Set.indicator_mul +#check integral_const_mul +#check integral_div +#check integral_mul_le_Lp_mul_Lq_of_nonneg +#check memLp_two_iff_integrable_sq +#check div_le_div_of_nonneg_left +#check div_le_iff₀ +#check div_le_div₀ diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/Controls.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/Controls.lean.txt new file mode 100644 index 000000000..818109051 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/Controls.lean.txt @@ -0,0 +1,43 @@ +import ResearchFormalCoreR1.PalmReweighting + +open MeasureTheory +open scoped ENNReal + +namespace PalmControls + +example {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] : + IsProbabilityMeasure (ResearchFormalCoreR1.prwMeasure μ (fun _ => (2 : ℝ))) := by + apply ResearchFormalCoreR1.prw_isProbabilityMeasure + · exact integrable_const _ + · intro ω; norm_num + · simp + +example {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] : + (ResearchFormalCoreR1.prwMeasure μ (fun _ => (2 : ℝ))).real Set.univ = 1 := by + rw [ResearchFormalCoreR1.prw_measure_real μ (fun _ => (2 : ℝ)) (integrable_const _) + (fun _ => by norm_num) (by simp) Set.univ MeasurableSet.univ] + simp + +example {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] + (E : Set Ω) (hE : MeasurableSet E) : + IsProbabilityMeasure (ResearchFormalCoreR1.prwMeasure μ (fun _ => (2 : ℝ))) ∧ + (ResearchFormalCoreR1.prwMeasure μ (fun _ => (2 : ℝ))).real E ≤ + (Real.sqrt 64 / 8) * Real.sqrt (μ.real E) := by + apply ResearchFormalCoreR1.prw_square_root_event_bound μ (fun _ => (2 : ℝ)) + measurable_const (fun _ => by norm_num) (by simp) 64 8 (1/2) + (by norm_num) (by norm_num) (by norm_num) (by norm_num) (by norm_num) E hE + +example {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) : + ¬ IsProbabilityMeasure (ResearchFormalCoreR1.prwMeasure μ (fun _ => (0 : ℝ))) := by + intro h + have hmass := h.measure_univ + simp [ResearchFormalCoreR1.prwMeasure] at hmass + +#print axioms ResearchFormalCoreR1.prw_integrable_of_integrable_sq +#print axioms ResearchFormalCoreR1.prw_measure_apply +#print axioms ResearchFormalCoreR1.prw_isProbabilityMeasure +#print axioms ResearchFormalCoreR1.prw_measure_real +#print axioms ResearchFormalCoreR1.prw_numerator_bound +#print axioms ResearchFormalCoreR1.prw_square_root_event_bound + +end PalmControls diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/FalseOmitNormalizer.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/FalseOmitNormalizer.lean.txt new file mode 100644 index 000000000..40ec8c432 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/FalseOmitNormalizer.lean.txt @@ -0,0 +1,11 @@ +import ResearchFormalCoreR1.PalmReweighting + +open MeasureTheory +open scoped ENNReal + +-- Intentionally false: a density of constant two without normalization +-- has total mass two, rather than one. +example : IsProbabilityMeasure + ((Measure.dirac (0 : ℝ)).withDensity (fun _ => ENNReal.ofReal (2 : ℝ))) := by + constructor + norm_num [withDensity_apply] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/FalseZeroNormalizer.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/FalseZeroNormalizer.lean.txt new file mode 100644 index 000000000..367755dc2 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/FalseZeroNormalizer.lean.txt @@ -0,0 +1,10 @@ +import ResearchFormalCoreR1.PalmReweighting + +open MeasureTheory + +-- Intentionally false: deleting positivity of the actual normalizer +-- cannot turn the zero density into a probability measure. +example : IsProbabilityMeasure + (ResearchFormalCoreR1.prwMeasure (Measure.dirac (0 : ℝ)) (fun _ => 0)) := by + constructor + norm_num [ResearchFormalCoreR1.prwMeasure] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/HANDOFF.json b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/HANDOFF.json new file mode 100644 index 000000000..eb1282630 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/HANDOFF.json @@ -0,0 +1,37 @@ +{ + "source": "ResearchFormalCoreR1/PalmReweighting.lean", + "sha256": "d34e422693f1d449bfcaf9c844c99c8fef4f19503312d3677ca525bd2d67c4d6", + "production_compile": "production-02", + "production_exit": 0, + "controls": "controls-04", + "controls_exit": 0, + "controls_scope": [ + "nonunit constant weight2 defines probability", + "actualeventformula has mass1", + "r=1/2,C=64,c=8 inhabits square-root event bound", + "zero actualnormalizer cannot define probability" + ], + "negative_controls": { + "false-omit-normalizer-01": { + "exit": 1, + "expected_diagnostic": "unsolved goals: False" + }, + "false-zero-normalizer-01": { + "exit": 1, + "expected_diagnostic": "unsolved goals: False" + } + }, + "allowed_axioms": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "author_lineage": [ + "OpenAI/Codex /root/palm_measure_transfer", + "OpenAI/Codex /root/palm_measure_transfer/density_api (scratch-only density proof/API helper)" + ], + "scientific_review": "author-side only; different-family review not supplied", + "old_verification": "Not repeated; all runs target new module or new controls", + "kernel_replay": "Not run separately; parent changed-closure gate owns replay", + "command_files_note": "Executable argv rendered from the actual commands above; exitcodes and raw logs retained from executions. Earlier control source changed in response to logged tactic failures; final Controls.lean is controls-04 source." +} diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/api-01.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/api-01.command new file mode 100644 index 000000000..d32e531b8 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/api-01.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean /workspace/scratch/palm-transfer/Api.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/api-01.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/api-01.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/api-01.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/api-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/api-01.log new file mode 100644 index 000000000..3c330c85d --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/api-01.log @@ -0,0 +1,39 @@ +Real.holderConjugate_iff {p q : ℝ} : p.HolderConjugate q ↔ 1 < p ∧ p⁻¹ + q⁻¹ = 1 +Real.HolderConjugate.two_two : Real.HolderConjugate 2 2 +/workspace/scratch/palm-transfer/Api.lean:7:7: error(lean.unknownIdentifier): Unknown constant `Real.holderConjugate_two_two` +Real.rpow_two (x : ℝ) : x ^ 2 = x ^ 2 +Real.sqrt_eq_rpow (x : ℝ) : √x = x ^ (1 / 2) +Real.sqrt_mul {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : √(x * y) = √x * √y +Real.sqrt_sq {x : ℝ} (h : 0 ≤ x) : √(x ^ 2) = x +Real.sqrt_sq_eq_abs (x : ℝ) : √(x ^ 2) = |x| +MeasureTheory.integral_indicator.{u_1, u_3} {X : Type u_1} {E : Type u_3} {mX : MeasurableSpace X} + [NormedAddCommGroup E] [NormedSpace ℝ E] {f : X → E} {s : Set X} {μ : Measure X} (hs : MeasurableSet s) : + ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ +MeasureTheory.integral_indicator_one.{u_1} {X : Type u_1} {mX : MeasurableSpace X} {μ : Measure X} ⦃s : Set X⦄ + (hs : MeasurableSet s) : ∫ (x : X), s.indicator 1 x ∂μ = μ.real s +MeasureTheory.integral_const.{u_1, u_2} {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] + {m : MeasurableSpace α} {μ : Measure α} [hE : CompleteSpace E] (c : E) : ∫ (x : α), c ∂μ = μ.real Set.univ • c +MeasureTheory.memLp_const.{u_1, u_5} {α : Type u_1} {E : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} + {μ : Measure α} [NormedAddCommGroup E] (c : E) [IsFiniteMeasure μ] : MemLp (fun x => c) p μ +MeasureTheory.MemLp.indicator.{u_1, u_6} {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} + {ε : Type u_6} [TopologicalSpace ε] [ESeminormedAddMonoid ε] {s : Set α} {f : α → ε} (hs : MeasurableSet s) + (hf : MemLp f p μ) : MemLp (s.indicator f) p μ +/workspace/scratch/palm-transfer/Api.lean:18:7: error(lean.unknownIdentifier): Unknown constant `Set.mul_indicator` +Set.indicator_mul.{u_1, u_4} {ι : Type u_1} {M₀ : Type u_4} [MulZeroClass M₀] (s : Set ι) (f g : ι → M₀) : + (s.indicator fun i => f i * g i) = fun i => s.indicator f i * s.indicator g i +MeasureTheory.integral_const_mul.{u_1, u_6} {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {L : Type u_6} + [RCLike L] (r : L) (f : α → L) : ∫ (a : α), r * f a ∂μ = r * ∫ (a : α), f a ∂μ +MeasureTheory.integral_div.{u_1, u_6} {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {L : Type u_6} [RCLike L] + (r : L) (f : α → L) : ∫ (a : α), f a / r ∂μ = (∫ (a : α), f a ∂μ) / r +MeasureTheory.integral_mul_le_Lp_mul_Lq_of_nonneg.{u_1} {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {p q : ℝ} + (hpq : p.HolderConjugate q) {f g : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f) (hg_nonneg : 0 ≤ᵐ[μ] g) + (hf : MemLp f (ENNReal.ofReal p) μ) (hg : MemLp g (ENNReal.ofReal q) μ) : + ∫ (a : α), f a * g a ∂μ ≤ (∫ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫ (a : α), g a ^ q ∂μ) ^ (1 / q) +MeasureTheory.memLp_two_iff_integrable_sq.{u_1} {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {f : α → ℝ} + (hf : AEStronglyMeasurable f μ) : MemLp f 2 μ ↔ Integrable (fun x => f x ^ 2) μ +div_le_div_of_nonneg_left.{u_3} {G₀ : Type u_3} [GroupWithZero G₀] [PartialOrder G₀] [PosMulReflectLT G₀] + [MulPosReflectLT G₀] {a b c : G₀} (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) : a / b ≤ a / c +div_le_iff₀.{u_3} {G₀ : Type u_3} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] {a b c : G₀} (hc : 0 < c) : + b / c ≤ a ↔ b ≤ a * c +div_le_div₀.{u_3} {G₀ : Type u_3} [GroupWithZero G₀] [PartialOrder G₀] [PosMulReflectLT G₀] [MulPosReflectLT G₀] + {a b c d : G₀} (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hdb : d ≤ b) : a / b ≤ c / d diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-01.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-01.command new file mode 100644 index 000000000..b7ddd3c6b --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-01.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/palm-transfer/Controls.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-01.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-01.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-01.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-01.log new file mode 100644 index 000000000..f185c54b9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-01.log @@ -0,0 +1,14 @@ +/workspace/scratch/palm-transfer/Controls.lean:28:47: error: unsolved goals +Ω : Type u_1 +inst✝¹ : MeasurableSpace Ω +μ : Measure Ω +inst✝ : IsProbabilityMeasure μ +E : Set Ω +hE : MeasurableSet E +⊢ 2 ^ 2 ≤ 4 +'ResearchFormalCoreR1.prw_integrable_of_integrable_sq' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_isProbabilityMeasure' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_real' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_numerator_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_square_root_event_bound' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-02.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-02.command new file mode 100644 index 000000000..b7ddd3c6b --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-02.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/palm-transfer/Controls.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-02.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-02.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-02.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-02.log new file mode 100644 index 000000000..ff7676ff7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-02.log @@ -0,0 +1,6 @@ +'ResearchFormalCoreR1.prw_integrable_of_integrable_sq' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_isProbabilityMeasure' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_real' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_numerator_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_square_root_event_bound' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-03.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-03.command new file mode 100644 index 000000000..b7ddd3c6b --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-03.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/palm-transfer/Controls.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-03.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-03.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-03.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-03.log new file mode 100644 index 000000000..0b88be4ff --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-03.log @@ -0,0 +1,14 @@ +/workspace/scratch/palm-transfer/Controls.lean:28:61: error: unsolved goals +Ω : Type u_1 +inst✝¹ : MeasurableSpace Ω +μ : Measure Ω +inst✝ : IsProbabilityMeasure μ +E : Set Ω +hE : MeasurableSet E +⊢ 8 * (2 ^ 2)⁻¹ ≤ 2 +'ResearchFormalCoreR1.prw_integrable_of_integrable_sq' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_isProbabilityMeasure' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_real' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_numerator_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_square_root_event_bound' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-04.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-04.command new file mode 100644 index 000000000..b7ddd3c6b --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-04.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/palm-transfer/Controls.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-04.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-04.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-04.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-04.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-04.log new file mode 100644 index 000000000..ff7676ff7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/controls-04.log @@ -0,0 +1,6 @@ +'ResearchFormalCoreR1.prw_integrable_of_integrable_sq' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_isProbabilityMeasure' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_measure_real' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_numerator_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'ResearchFormalCoreR1.prw_square_root_event_bound' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/DensityProbe-initial.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/DensityProbe-initial.lean.txt new file mode 100644 index 000000000..c72f112d1 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/DensityProbe-initial.lean.txt @@ -0,0 +1,42 @@ +import Mathlib.MeasureTheory.Measure.WithDensity +import Mathlib.MeasureTheory.Integral.Bochner.Set +import Mathlib.MeasureTheory.Measure.Real + +open MeasureTheory +open scoped ENNReal + +namespace DensityProbe + +noncomputable def tilt {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) : Measure Ω := + μ.withDensity (fun ω => ENNReal.ofReal (W ω / ∫ x, W x ∂μ)) + +theorem tilt_apply {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + tilt μ W E = ENNReal.ofReal ((∫ x in E, W x ∂μ) / ∫ x, W x ∂μ) := by + unfold tilt + rw [withDensity_apply _ hE] + rw [← ofReal_integral_eq_lintegral_ofReal + (hW.integrableOn.div_const _) (Filter.Eventually.of_forall (fun ω => + div_nonneg (hW0 ω) hZ.le))] + rw [integral_div_const] + +theorem tilt_probability {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) : + IsProbabilityMeasure (tilt μ W) := by + constructor + rw [tilt_apply μ W hW hW0 hZ Set.univ MeasurableSet.univ] + simp [hZ.ne'] + +theorem tilt_real {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + (tilt μ W).real E = (∫ x in E, W x ∂μ) / ∫ x, W x ∂μ := by + rw [measureReal_def, tilt_apply μ W hW hW0 hZ E hE, + ENNReal.toReal_ofReal (div_nonneg (integral_nonneg hW0) hZ.le)] + +end DensityProbe diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/DensityProbe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/DensityProbe.lean.txt new file mode 100644 index 000000000..d873b0688 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/DensityProbe.lean.txt @@ -0,0 +1,42 @@ +import Mathlib.MeasureTheory.Measure.WithDensity +import Mathlib.MeasureTheory.Integral.Bochner.Set +import Mathlib.MeasureTheory.Measure.Real + +open MeasureTheory +open scoped ENNReal + +namespace DensityProbe + +noncomputable def tilt {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) : Measure Ω := + μ.withDensity (fun ω => ENNReal.ofReal (W ω / ∫ x, W x ∂μ)) + +theorem tilt_apply {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + tilt μ W E = ENNReal.ofReal ((∫ x in E, W x ∂μ) / ∫ x, W x ∂μ) := by + unfold tilt + rw [withDensity_apply _ hE] + rw [← ofReal_integral_eq_lintegral_ofReal + (hW.integrableOn.div_const _) (Filter.Eventually.of_forall (fun ω => + div_nonneg (hW0 ω) hZ.le))] + rw [integral_div] + +theorem tilt_probability {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) : + IsProbabilityMeasure (tilt μ W) := by + constructor + rw [tilt_apply μ W hW hW0 hZ Set.univ MeasurableSet.univ] + simp [hZ.ne'] + +theorem tilt_real {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + (tilt μ W).real E = (∫ x in E, W x ∂μ) / ∫ x, W x ∂μ := by + rw [measureReal_def, tilt_apply μ W hW hW0 hZ E hE, + ENNReal.toReal_ofReal (div_nonneg (integral_nonneg hW0) hZ.le)] + +end DensityProbe diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-01.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-01.command new file mode 100644 index 000000000..420c37427 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-01.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean /workspace/scratch/palm-transfer/density-api/DensityProbe.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-01.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-01.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-01.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-01.log new file mode 100644 index 000000000..96ff9fa9d --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-01.log @@ -0,0 +1,13 @@ +/workspace/scratch/palm-transfer/density-api/DensityProbe.lean:24:6: error(lean.unknownIdentifier): Unknown identifier `integral_div_const` +/workspace/scratch/palm-transfer/density-api/DensityProbe.lean:18:70: error: unsolved goals +Ω : Type u_1 +inst✝ : MeasurableSpace Ω +μ : Measure Ω +W : Ω → ℝ +hW : Integrable W μ +hW0 : ∀ (ω : Ω), 0 ≤ W ω +hZ : 0 < ∫ (x : Ω), W x ∂μ +E : Set Ω +hE : MeasurableSet E +⊢ ENNReal.ofReal (∫ (x : Ω) in E, W x / ∫ (x : Ω), W x ∂μ ∂μ) = + ENNReal.ofReal ((∫ (x : Ω) in E, W x ∂μ) / ∫ (x : Ω), W x ∂μ) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-02.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-02.command new file mode 100644 index 000000000..420c37427 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-02.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean /workspace/scratch/palm-transfer/density-api/DensityProbe.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-02.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-02.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-02.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/density-api/density-probe-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-omit-normalizer-01.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-omit-normalizer-01.command new file mode 100644 index 000000000..0a6295170 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-omit-normalizer-01.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/palm-transfer/FalseOmitNormalizer.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-omit-normalizer-01.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-omit-normalizer-01.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-omit-normalizer-01.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-omit-normalizer-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-omit-normalizer-01.log new file mode 100644 index 000000000..58e780668 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-omit-normalizer-01.log @@ -0,0 +1,2 @@ +/workspace/scratch/palm-transfer/FalseOmitNormalizer.lean:9:79: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-zero-normalizer-01.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-zero-normalizer-01.command new file mode 100644 index 000000000..8cf8bbe28 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-zero-normalizer-01.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/palm-transfer/FalseZeroNormalizer.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-zero-normalizer-01.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-zero-normalizer-01.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-zero-normalizer-01.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-zero-normalizer-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-zero-normalizer-01.log new file mode 100644 index 000000000..17ec5899f --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/false-zero-normalizer-01.log @@ -0,0 +1,2 @@ +/workspace/scratch/palm-transfer/FalseZeroNormalizer.lean:8:78: error: unsolved goals +⊢ False diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-01.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-01.command new file mode 100644 index 000000000..6c1679fdc --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-01.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -o .lake/build/lib/lean/ResearchFormalCoreR1/PalmReweighting.olean ResearchFormalCoreR1/PalmReweighting.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-01.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-01.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-01.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-01.log new file mode 100644 index 000000000..f4608271b --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-01.log @@ -0,0 +1,26 @@ +ResearchFormalCoreR1/PalmReweighting.lean:41:2: warning: Unused tactic linter: `change |W ω| ≤ 1 + W ω ^ 2` does nothing + +Note: This linter can be disabled with `set_option linter.unusedTactic false` +ResearchFormalCoreR1/PalmReweighting.lean:100:44: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ∫ (x : Ω), E.indicator 1 x ∂?m.288 +in the target expression + ∫ (x : Ω) in E, W x ∂μ ≤ (∫ (a : Ω), W a ^ 2 ∂μ) ^ (1 / 2) * integral μ (E.indicator fun x => 1) ^ (1 / 2) + +Ω : Type u_1 +inst✝¹ : MeasurableSpace Ω +μ : Measure Ω +inst✝ : IsFiniteMeasure μ +W : Ω → ℝ +hW : Measurable W +hW0 : ∀ (ω : Ω), 0 ≤ W ω +hsq : Integrable (fun ω => W ω ^ 2) μ +E : Set Ω +hE : MeasurableSet E +I : Ω → ℝ := E.indicator fun x => 1 +hI0 : ∀ (ω : Ω), 0 ≤ I ω +hWLp : MemLp W (ENNReal.ofReal 2) μ +hILp : MemLp I (ENNReal.ofReal 2) μ +hCS : ∫ (x : Ω) in E, W x ∂μ ≤ (∫ (a : Ω), W a ^ 2 ∂μ) ^ (1 / 2) * integral μ (E.indicator fun x => 1) ^ (1 / 2) +hleft : (fun ω => W ω * I ω) = E.indicator W +hright : (fun ω => I ω ^ 2) = E.indicator fun x => 1 +⊢ ∫ (x : Ω) in E, W x ∂μ ≤ √(∫ (x : Ω), W x ^ 2 ∂μ) * √(μ.real E) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-02.command b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-02.command new file mode 100644 index 000000000..ec0cc3fee --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-02.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/ResearchFormalCoreR1/PalmReweighting.olean ResearchFormalCoreR1/PalmReweighting.lean diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-02.exitcode b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-02.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-02.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/palm-transfer/production-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe-01.log new file mode 100644 index 000000000..861cc44da --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe-01.log @@ -0,0 +1,3 @@ +/workspace/scratch/regression-kernel/DirectRegressionProbe.lean:84:35: warning: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe-02.log new file mode 100644 index 000000000..e9afdf81c --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe-02.log @@ -0,0 +1,22 @@ +/workspace/scratch/regression-kernel/DirectRegressionProbe.lean:86:4: error: Tactic `rfl` failed: The left-hand side + (L (X ω, Y ω) + c).1 +is not definitionally equal to the right-hand side + (X ω, Y ω - mY - B (X ω - mX)).1 + +case fst +Ω : Type u_1 +inst✝ : MeasurableSpace Ω +P : Measure Ω +n k : ℕ +X : Ω → Fin n → ℝ +Y : Ω → Fin k → ℝ +hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P +B : (Fin n → ℝ) →L[ℝ] Fin k → ℝ +mX : Fin n → ℝ +mY : Fin k → ℝ +L : (Fin n → ℝ) × (Fin k → ℝ) →L[ℝ] (Fin n → ℝ) × (Fin k → ℝ) := ⋯ +c : (Fin n → ℝ) × (Fin k → ℝ) := ⋯ +hL : HasGaussianLaw (fun ω => L (X ω, Y ω)) P +hshift : HasGaussianLaw (fun ω => L (X ω, Y ω) + c) P +ω : Ω +⊢ (L (X ω, Y ω) + c).1 = (X ω, Y ω - mY - B (X ω - mX)).1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe.lean.txt new file mode 100644 index 000000000..5efbc2d24 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/DirectRegressionProbe.lean.txt @@ -0,0 +1,225 @@ +import Mathlib.Probability.HasCondDistrib +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory BigOperators +set_option autoImplicit false + +namespace ResearchFormalCoreR1 + +lemma regression_map_coordinate {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (v : Fin n → ℝ) (j : Fin k) : + B v j = ∑ l, B (Pi.single l 1) j * v l := by + calc + B v j = B (∑ l, v l • (Pi.single l 1 : Fin n → ℝ)) j := + congrArg (fun w => B w j) (pi_eq_sum_univ' v) + _ = ∑ l, B (Pi.single l 1) j * v l := by + simp [map_sum, mul_comm] + +lemma regression_covariance_map {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {n k : ℕ} + {X : Ω → Fin n → ℝ} (hX : HasGaussianLaw X P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) + (i : Fin n) (j : Fin k) : + covariance (fun ω => X ω i) (fun ω => B (X ω - mX) j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P := by + have hcoord : (fun ω => B (X ω - mX) j) = + (fun ω => ∑ l, B (Pi.single l 1) j * (X ω l - mX l)) := by + funext ω + exact regression_map_coordinate B (X ω - mX) j + rw [hcoord, covariance_fun_sum_right] + · apply Finset.sum_congr rfl + intro l _ + rw [covariance_const_mul_right, covariance_sub_const_right (hX.eval l).integrable] + · intro l + exact ((hX.eval l).memLp_two.sub (memLp_const _)).const_mul _ + · exact (hX.eval i).memLp_two + +lemma regression_residual_zero_covariance {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (hnormal : ∀ i j, covariance (fun ω => X ω i) (fun ω => Y ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P) : + ∀ i j, covariance (fun ω => X ω i) + (fun ω => (Y ω - mY - B (X ω - mX)) j) P = 0 := by + have : IsProbabilityMeasure P := hXY.isProbabilityMeasure + intro i j + have hXi := (hXY.fst.eval i).memLp_two + have hYj := (hXY.snd.eval j).memLp_two + have hBj : MemLp (fun ω => B (X ω - mX) j) 2 P := by + have hcoord : (fun ω => B (X ω - mX) j) = + (fun ω => ∑ l, B (Pi.single l 1) j * (X ω l - mX l)) := by + funext ω + exact regression_map_coordinate B (X ω - mX) j + rw [hcoord] + exact memLp_finsetSum _ (fun l _ => + ((hXY.fst.eval l).memLp_two.sub (memLp_const _)).const_mul _) + simp only [Pi.sub_apply] + rw [covariance_fun_sub_right (Y := fun ω => Y ω j - mY j) hXi (hYj.sub (memLp_const _)) hBj, + covariance_sub_const_right (hXY.snd.eval j).integrable, + regression_covariance_map hXY.fst B mX i j, hnormal i j, sub_self] + +lemma regression_residual_jointGaussian {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {n k : ℕ} + {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasGaussianLaw (fun ω => (X ω, Y ω - mY - B (X ω - mX))) P := by + let L : ((Fin n → ℝ) × (Fin k → ℝ)) →L[ℝ] + ((Fin n → ℝ) × (Fin k → ℝ)) := + (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ)).prod + (ContinuousLinearMap.snd ℝ (Fin n → ℝ) (Fin k → ℝ) - + B.comp (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ))) + let c : (Fin n → ℝ) × (Fin k → ℝ) := (0, B mX - mY) + have hL : HasGaussianLaw (fun ω => L (X ω, Y ω)) P := hXY.map_fun L + have hshift : HasGaussianLaw (fun ω => L (X ω, Y ω) + c) P := { + aemeasurable := hL.aemeasurable.add aemeasurable_const + isGaussian_map := by + have : IsGaussian (P.map (fun ω => L (X ω, Y ω))) := hL.isGaussian_map + convert (inferInstance : IsGaussian + ((P.map (fun ω => L (X ω, Y ω))).map (fun p => p + c))) using 1 + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hL.aemeasurable] + rfl } + apply hshift.congr + filter_upwards [] with ω + apply Prod.ext + · simp [L, c] + · change Y ω - B (X ω) + (B mX - mY) = Y ω - mY - B (X ω - mX) + rw [map_sub] + abel + + + +/-- Copying the first coordinate and retaining an independent second coordinate. -/ +lemma regression_compProd_id_const {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] + (μ : Measure α) (ν : Measure β) [SFinite μ] [SFinite ν] : + μ ⊗ₘ (Kernel.id ×ₖ Kernel.const α ν) = + (μ.prod ν).map (fun p : α × β => (p.1, p)) := by + ext s hs + rw [Measure.compProd_apply hs, Measure.map_apply (by fun_prop) hs, + Measure.prod_apply (s := (fun p : α × β => (p.1, p)) ⁻¹' s) (hs.preimage (by fun_prop))] + simp_rw [Kernel.prod_apply, Kernel.id_apply, Kernel.const_apply, Measure.dirac_prod] + congr with x + rw [Measure.map_apply (by fun_prop) (measurable_prodMk_left hs)] + rfl + +/-- An independent residual paired with the observation has the explicit product kernel. -/ +lemma regression_pair_hasCondDistrib {Ω α β : Type*} [MeasurableSpace Ω] + [MeasurableSpace α] [MeasurableSpace β] {P : Measure Ω} [IsProbabilityMeasure P] + {X : Ω → α} {R : Ω → β} (hX : AEMeasurable X P) (hR : AEMeasurable R P) + (hXR : IndepFun X R P) : + HasCondDistrib (fun ω => (X ω, R ω)) X + (Kernel.id ×ₖ Kernel.const α (P.map R)) P := by + refine ⟨hX.prodMk (hX.prodMk hR), ?_⟩ + calc + P.map (fun ω => (X ω, (X ω, R ω))) = + (P.map (fun ω => (X ω, R ω))).map (fun p : α × β => (p.1, p)) := by + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) (hX.prodMk hR)] + rfl + _ = ((P.map X).prod (P.map R)).map (fun p : α × β => (p.1, p)) := by + rw [hXR.map_prod_eq_prod_map_map hX hR] + _ = (P.map X) ⊗ₘ (Kernel.id ×ₖ Kernel.const α (P.map R)) := + (regression_compProd_id_const (P.map X) (P.map R)).symm + + +/-- Affine reconstruction of the target from the observation and the regression residual. -/ +def regressionReconstruction {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (p : (Fin n → ℝ) × (Fin k → ℝ)) : Fin k → ℝ := + p.2 + mY + B (p.1 - mX) + +lemma regressionReconstruction_measurable {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Measurable (regressionReconstruction B mX mY) := by + unfold regressionReconstruction + fun_prop + +/-- The canonical regression law is specified at every observation, including null pins. -/ +noncomputable def regressionKernel {n k : ℕ} (ν : Measure (Fin k → ℝ)) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Kernel (Fin n → ℝ) (Fin k → ℝ) := + ((Kernel.id : Kernel (Fin n → ℝ) (Fin n → ℝ)) ×ₖ + Kernel.const (Fin n → ℝ) ν).map (regressionReconstruction B mX mY) + +instance regressionKernel_isMarkov {n k : ℕ} (ν : Measure (Fin k → ℝ)) + [IsProbabilityMeasure ν] (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) + (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + IsMarkovKernel (regressionKernel ν B mX mY) := by + unfold regressionKernel + exact Kernel.IsMarkovKernel.map _ (regressionReconstruction_measurable B mX mY) + +lemma regressionKernel_apply {n k : ℕ} (ν : Measure (Fin k → ℝ)) [SFinite ν] + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (x : Fin n → ℝ) : + regressionKernel ν B mX mY x = ν.map (fun r => r + mY + B (x - mX)) := by + rw [regressionKernel, Kernel.map_apply _ (regressionReconstruction_measurable B mX mY), + Kernel.prod_apply, Kernel.id_apply, Kernel.const_apply, Measure.dirac_prod, + Measure.map_map (regressionReconstruction_measurable B mX mY) (by fun_prop)] + rfl + +/-- Affine reconstruction from an independent residual has the canonical conditional kernel. -/ +lemma independentResidual_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {n k : ℕ} + {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hX : AEMeasurable X P) (hR : AEMeasurable R P) (hXR : IndepFun X R P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + exact (regression_pair_hasCondDistrib hX hR hXR).comp_left + (regressionReconstruction_measurable B mX mY) + +/-- Joint Gaussianity and vanishing observation/residual cross-covariance give the canonical +conditional law, without choosing an almost-everywhere version at a prescribed observation. -/ +theorem gaussianRegression_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hXR : HasGaussianLaw (fun ω => (X ω, R ω)) P) + (hzero : ∀ i j, covariance (fun ω => X ω i) (fun ω => R ω j) P = 0) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + have : IsProbabilityMeasure P := hXR.isProbabilityMeasure + exact independentResidual_hasCondDistrib hXR.fst.aemeasurable hXR.snd.aemeasurable + (hXR.indepFun_of_covariance_eval hzero) B mX mY + +/-- Kernel measurability is pointwise in the observation and does not require a Gaussian law. -/ +lemma regressionKernel_measurable {n k : ℕ} (ν : Measure (Fin k → ℝ)) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Measurable (regressionKernel ν B mX mY) := + Kernel.measurable _ + +/-- All parts of the canonical Gaussian conditional-law construction, including the literal +pushforward at every prescribed observation. Zero-dimensional and degenerate laws are allowed. -/ +theorem gaussianRegressionKernel_spec {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hXR : HasGaussianLaw (fun ω => (X ω, R ω)) P) + (hzero : ∀ i j, covariance (fun ω => X ω i) (fun ω => R ω j) P = 0) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + IsMarkovKernel (regressionKernel (P.map R) B mX mY) ∧ + Measurable (regressionKernel (P.map R) B mX mY) ∧ + (∀ x, regressionKernel (P.map R) B mX mY x = + (P.map R).map (fun r => r + mY + B (x - mX))) ∧ + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + have : IsProbabilityMeasure P := hXR.isProbabilityMeasure + exact ⟨inferInstance, regressionKernel_measurable _ B mX mY, + regressionKernel_apply _ B mX mY, gaussianRegression_hasCondDistrib hXR hzero B mX mY⟩ + + +/-- The actual target has the explicit regression conditional law when the coefficient map +satisfies the coordinate normal equations. The residual hypotheses are derived. -/ +theorem gaussianRegression_observed_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (hnormal : ∀ i j, covariance (fun ω => X ω i) (fun ω => Y ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P) : + HasCondDistrib Y X + (regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY) P := by + have hXR := regression_residual_jointGaussian hXY B mX mY + have hzero := regression_residual_zero_covariance hXY B mX mY hnormal + have hcond := gaussianRegression_hasCondDistrib hXR hzero B mX mY + convert hcond using 1 + funext ω + abel + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/GaussianRegressionKernel-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/GaussianRegressionKernel-01.log new file mode 100644 index 000000000..554f85843 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/GaussianRegressionKernel-01.log @@ -0,0 +1,16 @@ +ResearchFormalCoreR1/GaussianRegressionKernel.lean:115:2: warning: Try this: + haveI̵ + +The goal is a proposition, so `have` is preferred over `haveI`. +The difference between `have` and `haveI` is that `haveI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` +ResearchFormalCoreR1/GaussianRegressionKernel.lean:138:2: warning: Try this: + haveI̵ + +The goal is a proposition, so `have` is preferred over `haveI`. +The difference between `have` and `haveI` is that `haveI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/GaussianRegressionKernel-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/GaussianRegressionKernel-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/GaussianRegressionKernel-strengthened-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/GaussianRegressionKernel-strengthened-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-01.log new file mode 100644 index 000000000..bdd016ebd --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-01.log @@ -0,0 +1,12 @@ +/workspace/scratch/regression-kernel/KernelProbe.lean:17:27: error: `MeasurableSet + ((fun p => (p.1, p)) ⁻¹' s)` is not a `fun_prop` goal! Consider marking `MeasurableSet` with `@[fun_prop]`. +/workspace/scratch/regression-kernel/KernelProbe.lean:20:42: error: `MeasurableSet (Prod.mk x ⁻¹' s)` is not a `fun_prop` goal! Consider marking `MeasurableSet` with `@[fun_prop]`. +/workspace/scratch/regression-kernel/KernelProbe.lean:29:43: error(lean.unknownIdentifier): Unknown constant `MeasureTheory.Measure.isProbabilityMeasure_map` +/workspace/scratch/regression-kernel/KernelProbe.lean:29:2: warning: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-02.log new file mode 100644 index 000000000..bc52d0979 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-02.log @@ -0,0 +1,25 @@ +/workspace/scratch/regression-kernel/KernelProbe.lean:17:4: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + (Measure.prod ?m.90 ?m.91) ((fun a => (a.1, id a)) ⁻¹' s) +in the target expression + ∫⁻ (a : α), ((Kernel.id ×ₖ Kernel.const α ν) a) (Prod.mk a ⁻¹' s) ∂μ = (μ.prod ν) ((fun p => (p.1, p)) ⁻¹' s) + +α : Type u_1 +β : Type u_2 +inst✝³ : MeasurableSpace α +inst✝² : MeasurableSpace β +μ : Measure α +ν : Measure β +inst✝¹ : SFinite μ +inst✝ : SFinite ν +s : Set (α × α × β) +hs : MeasurableSet s +⊢ ∫⁻ (a : α), ((Kernel.id ×ₖ Kernel.const α ν) a) (Prod.mk a ⁻¹' s) ∂μ = (μ.prod ν) ((fun p => (p.1, p)) ⁻¹' s) +/workspace/scratch/regression-kernel/KernelProbe.lean:29:43: error(lean.unknownIdentifier): Unknown identifier `isProbabilityMeasure_map` +/workspace/scratch/regression-kernel/KernelProbe.lean:29:2: warning: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-03.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-04.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-04.log new file mode 100644 index 000000000..efe156b11 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe-04.log @@ -0,0 +1,8 @@ +/workspace/scratch/regression-kernel/KernelProbe.lean:98:2: warning: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe.lean.txt new file mode 100644 index 000000000..ecb6cf99c --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/KernelProbe.lean.txt @@ -0,0 +1,102 @@ +import Mathlib.Probability.HasCondDistrib +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory + +set_option autoImplicit false + +namespace ResearchFormalCoreR1 + +lemma regression_compProd_id_const {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] + (μ : Measure α) (ν : Measure β) [SFinite μ] [SFinite ν] : + μ ⊗ₘ (Kernel.id ×ₖ Kernel.const α ν) = + (μ.prod ν).map (fun p : α × β => (p.1, p)) := by + ext s hs + rw [Measure.compProd_apply hs, Measure.map_apply (by fun_prop) hs, + Measure.prod_apply (s := (fun p : α × β => (p.1, p)) ⁻¹' s) (hs.preimage (by fun_prop))] + simp_rw [Kernel.prod_apply, Kernel.id_apply, Kernel.const_apply, Measure.dirac_prod] + congr with x + rw [Measure.map_apply (by fun_prop) (measurable_prodMk_left hs)] + rfl + +lemma regression_pair_hasCondDistrib {Ω α β : Type*} [MeasurableSpace Ω] + [MeasurableSpace α] [MeasurableSpace β] {P : Measure Ω} [IsProbabilityMeasure P] + {X : Ω → α} {R : Ω → β} (hX : AEMeasurable X P) (hR : AEMeasurable R P) + (hXR : IndepFun X R P) : + HasCondDistrib (fun ω => (X ω, R ω)) X + (Kernel.id ×ₖ Kernel.const α (P.map R)) P := by + refine ⟨hX.prodMk (hX.prodMk hR), ?_⟩ + calc + P.map (fun ω => (X ω, (X ω, R ω))) = + (P.map (fun ω => (X ω, R ω))).map (fun p : α × β => (p.1, p)) := by + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) (hX.prodMk hR)] + rfl + _ = ((P.map X).prod (P.map R)).map (fun p : α × β => (p.1, p)) := by + rw [hXR.map_prod_eq_prod_map_map hX hR] + _ = (P.map X) ⊗ₘ (Kernel.id ×ₖ Kernel.const α (P.map R)) := + (regression_compProd_id_const (P.map X) (P.map R)).symm + +end ResearchFormalCoreR1 + +namespace ResearchFormalCoreR1 + +/-- Affine reconstruction of the target from the observation and the regression residual. -/ +def regressionReconstruction {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (p : (Fin n → ℝ) × (Fin k → ℝ)) : Fin k → ℝ := + p.2 + mY + B (p.1 - mX) + +lemma regressionReconstruction_measurable {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Measurable (regressionReconstruction B mX mY) := by + unfold regressionReconstruction + fun_prop + +/-- The canonical regression law is specified at every observation, including null pins. -/ +noncomputable def regressionKernel {n k : ℕ} (ν : Measure (Fin k → ℝ)) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Kernel (Fin n → ℝ) (Fin k → ℝ) := + ((Kernel.id : Kernel (Fin n → ℝ) (Fin n → ℝ)) ×ₖ + Kernel.const (Fin n → ℝ) ν).map (regressionReconstruction B mX mY) + +instance regressionKernel_isMarkov {n k : ℕ} (ν : Measure (Fin k → ℝ)) + [IsProbabilityMeasure ν] (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) + (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + IsMarkovKernel (regressionKernel ν B mX mY) := by + unfold regressionKernel + exact Kernel.IsMarkovKernel.map _ (regressionReconstruction_measurable B mX mY) + +lemma regressionKernel_apply {n k : ℕ} (ν : Measure (Fin k → ℝ)) [SFinite ν] + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (x : Fin n → ℝ) : + regressionKernel ν B mX mY x = ν.map (fun r => r + mY + B (x - mX)) := by + rw [regressionKernel, Kernel.map_apply _ (regressionReconstruction_measurable B mX mY), + Kernel.prod_apply, Kernel.id_apply, Kernel.const_apply, Measure.dirac_prod, + Measure.map_map (regressionReconstruction_measurable B mX mY) (by fun_prop)] + rfl + +lemma independentResidual_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {n k : ℕ} + {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hX : AEMeasurable X P) (hR : AEMeasurable R P) (hXR : IndepFun X R P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + exact (regression_pair_hasCondDistrib hX hR hXR).comp_left + (regressionReconstruction_measurable B mX mY) + +/-- Joint Gaussianity and vanishing observation/residual cross-covariance give the canonical +conditional law, without choosing an almost-everywhere version at a prescribed observation. -/ +theorem gaussianRegression_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hXR : HasGaussianLaw (fun ω => (X ω, R ω)) P) + (hzero : ∀ i j, covariance (fun ω => X ω i) (fun ω => R ω j) P = 0) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + letI : IsProbabilityMeasure P := hXR.isProbabilityMeasure + exact independentResidual_hasCondDistrib hXR.fst.aemeasurable hXR.snd.aemeasurable + (hXR.indepFun_of_covariance_eval hzero) B mX mY + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/ResidualCovarianceProbe-01.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/ResidualCovarianceProbe-01.log new file mode 100644 index 000000000..553a80b7f --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/ResidualCovarianceProbe-01.log @@ -0,0 +1,34 @@ +/workspace/scratch/regression-kernel/ResidualCovarianceProbe.lean:18:4: error: `simp` made no progress +/workspace/scratch/regression-kernel/ResidualCovarianceProbe.lean:29:11: error: Tactic `simp` failed with a nested error: +maximum recursion depth has been reached +use `set_option maxRecDepth ` to increase limit +use `set_option diagnostics true` to get diagnostic information +/workspace/scratch/regression-kernel/ResidualCovarianceProbe.lean:51:13: error: Tactic `simp` failed with a nested error: +maximum recursion depth has been reached +use `set_option maxRecDepth ` to increase limit +use `set_option diagnostics true` to get diagnostic information +/workspace/scratch/regression-kernel/ResidualCovarianceProbe.lean:55:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + cov[fun ω => X ω i, fun ω => ((fun ω => Y ω j) - fun x => ?m.207) ω - B (X ω - mX) j; P] +in the target expression + cov[fun ω => X ω i, fun ω => Y ω j - mY j - B (X ω - mX) j; P] = 0 + +Ω : Type u_1 +inst✝ : MeasurableSpace Ω +P : Measure Ω +n k : ℕ +X : Ω → Fin n → ℝ +Y : Ω → Fin k → ℝ +hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P +B : (Fin n → ℝ) →L[ℝ] Fin k → ℝ +mX : Fin n → ℝ +mY : Fin k → ℝ +hnormal : + ∀ (i : Fin n) (j : Fin k), + cov[fun ω => X ω i, fun ω => Y ω j; P] = ∑ l, B (Pi.single l 1) j * cov[fun ω => X ω i, fun ω => X ω l; P] +this : IsProbabilityMeasure P +i : Fin n +j : Fin k +hXi : MemLp (fun ω => X ω i) 2 P +hYj : MemLp (fun ω => Y ω j) 2 P +hBj : MemLp (fun ω => B (X ω - mX) j) 2 P +⊢ cov[fun ω => X ω i, fun ω => Y ω j - mY j - B (X ω - mX) j; P] = 0 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/ResidualCovarianceProbe-02.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/ResidualCovarianceProbe-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/ResidualCovarianceProbe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/ResidualCovarianceProbe.lean.txt new file mode 100644 index 000000000..23f08db1d --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/ResidualCovarianceProbe.lean.txt @@ -0,0 +1,63 @@ +import Mathlib.Probability.HasCondDistrib +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory BigOperators +set_option autoImplicit false + +namespace ResearchFormalCoreR1 + +lemma regression_map_coordinate {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (v : Fin n → ℝ) (j : Fin k) : + B v j = ∑ l, B (Pi.single l 1) j * v l := by + calc + B v j = B (∑ l, v l • (Pi.single l 1 : Fin n → ℝ)) j := + congrArg (fun w => B w j) (pi_eq_sum_univ' v) + _ = ∑ l, B (Pi.single l 1) j * v l := by + simp [map_sum, mul_comm] + +lemma regression_covariance_map {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {n k : ℕ} + {X : Ω → Fin n → ℝ} (hX : HasGaussianLaw X P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) + (i : Fin n) (j : Fin k) : + covariance (fun ω => X ω i) (fun ω => B (X ω - mX) j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P := by + have hcoord : (fun ω => B (X ω - mX) j) = + (fun ω => ∑ l, B (Pi.single l 1) j * (X ω l - mX l)) := by + funext ω + exact regression_map_coordinate B (X ω - mX) j + rw [hcoord, covariance_fun_sum_right] + · apply Finset.sum_congr rfl + intro l _ + rw [covariance_const_mul_right, covariance_sub_const_right (hX.eval l).integrable] + · intro l + exact ((hX.eval l).memLp_two.sub (memLp_const _)).const_mul _ + · exact (hX.eval i).memLp_two + +lemma regression_residual_zero_covariance {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (hnormal : ∀ i j, covariance (fun ω => X ω i) (fun ω => Y ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P) : + ∀ i j, covariance (fun ω => X ω i) + (fun ω => (Y ω - mY - B (X ω - mX)) j) P = 0 := by + have : IsProbabilityMeasure P := hXY.isProbabilityMeasure + intro i j + have hXi := (hXY.fst.eval i).memLp_two + have hYj := (hXY.snd.eval j).memLp_two + have hBj : MemLp (fun ω => B (X ω - mX) j) 2 P := by + have hcoord : (fun ω => B (X ω - mX) j) = + (fun ω => ∑ l, B (Pi.single l 1) j * (X ω l - mX l)) := by + funext ω + exact regression_map_coordinate B (X ω - mX) j + rw [hcoord] + exact memLp_finsetSum _ (fun l _ => + ((hXY.fst.eval l).memLp_two.sub (memLp_const _)).const_mul _) + simp only [Pi.sub_apply] + rw [covariance_fun_sub_right (Y := fun ω => Y ω j - mY j) hXi (hYj.sub (memLp_const _)) hBj, + covariance_sub_const_right (hXY.snd.eval j).integrable, + regression_covariance_map hXY.fst B mX i j, hnormal i j, sub_self] + +end ResearchFormalCoreR1 diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.failed1.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.failed1.lean.txt new file mode 100644 index 000000000..e4b3ea5d9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.failed1.lean.txt @@ -0,0 +1,30 @@ +import Mathlib.LinearAlgebra.Matrix.PosDef + +open Matrix +open scoped Matrix +set_option autoImplicit false + +namespace ContactSchurProbe + +theorem block_posDef_schur_pos (A : Matrix (Fin 6) (Fin 6) ℝ) + (B : Matrix (Fin 6) Unit ℝ) (D : Matrix Unit Unit ℝ) + (hG : (Matrix.fromBlocks A B Bᴴ D).PosDef) : + A.PosDef ∧ 0 < (D - Bᴴ * A⁻¹ * B) () () := by + have hA : A.PosDef := by + simpa [Matrix.submatrix] using hG.submatrix (e := Sum.inl) Sum.inl_injective + letI : Invertible A := hA.isUnit.invertible + let y : Unit → ℝ := fun _ => 1 + let x : Fin 6 → ℝ := -((A⁻¹ * B) *ᵥ y) + have hxy : Sum.elim x y ≠ 0 := by + intro h + have hh := congrFun h (Sum.inr ()) + simpa [y] using hh + have hp := hG.dotProduct_mulVec_pos hxy + rw [Matrix.dotProduct_mulVec, + Matrix.schur_complement_eq₁₁ B D x y hA.isHermitian] at hp + dsimp [x] at hp + rw [neg_add_cancel, dotProduct_zero, zero_add] at hp + refine ⟨hA, ?_⟩ + simpa [y, Matrix.vecMul, dotProduct] using hp + +end ContactSchurProbe diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.failed1.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.failed1.log new file mode 100644 index 000000000..a04fe3f32 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.failed1.log @@ -0,0 +1,17 @@ +/workspace/scratch/regression-kernel/contact-schur/SchurProbe.lean:14:4: error: Type mismatch: After simplification, term + PosDef.submatrix hG Sum.inl_injective + has type + @PosDef (Fin 6) ℝ Real.instRing Real.partialOrder instStarRingReal (of fun i j => A i j) +but is expected to have type + @PosDef (Fin 6) ℝ Real.instRing Real.partialOrder instStarRingReal A +/workspace/scratch/regression-kernel/contact-schur/SchurProbe.lean:15:2: warning: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` +/workspace/scratch/regression-kernel/contact-schur/SchurProbe.lean:21:4: warning: Try `simp at hh` instead of `simpa using hh` + +Note: This linter can be disabled with `set_option linter.unnecessarySimpa false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.lean.txt new file mode 100644 index 000000000..b14d8e445 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.lean.txt @@ -0,0 +1,32 @@ +import Mathlib.LinearAlgebra.Matrix.PosDef + +open Matrix +open scoped Matrix +set_option autoImplicit false + +namespace ContactSchurProbe + +theorem block_posDef_schur_pos (A : Matrix (Fin 6) (Fin 6) ℝ) + (B : Matrix (Fin 6) Unit ℝ) (D : Matrix Unit Unit ℝ) + (hG : (Matrix.fromBlocks A B Bᴴ D).PosDef) : + A.PosDef ∧ 0 < (D - Bᴴ * A⁻¹ * B) () () := by + have hA : A.PosDef := by + convert hG.submatrix (e := Sum.inl) Sum.inl_injective using 1 + ext i j + rfl + let _ : Invertible A := hA.isUnit.invertible + let y : Unit → ℝ := fun _ => 1 + let x : Fin 6 → ℝ := -((A⁻¹ * B) *ᵥ y) + have hxy : Sum.elim x y ≠ 0 := by + intro h + have hh := congrFun h (Sum.inr ()) + simp [y] at hh + have hp := hG.dotProduct_mulVec_pos hxy + rw [Matrix.dotProduct_mulVec, + Matrix.schur_complement_eq₁₁ B D x y hA.isHermitian] at hp + dsimp [x] at hp + rw [neg_add_cancel, dotProduct_zero, zero_add] at hp + refine ⟨hA, ?_⟩ + simpa [y, Matrix.vecMul, dotProduct] using hp + +end ContactSchurProbe diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/contact-schur/SchurProbe.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed1.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed1.lean.txt new file mode 100644 index 000000000..5233c40b7 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed1.lean.txt @@ -0,0 +1,28 @@ +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory +set_option autoImplicit false + +example {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {n k : ℕ} + {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasGaussianLaw (fun ω => (X ω, Y ω - mY - B (X ω - mX))) P := by + let L : ((Fin n → ℝ) × (Fin k → ℝ)) →L[ℝ] + ((Fin n → ℝ) × (Fin k → ℝ)) := + (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ)).prod + (ContinuousLinearMap.snd ℝ (Fin n → ℝ) (Fin k → ℝ) - + B.comp (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ))) + let c : (Fin n → ℝ) × (Fin k → ℝ) := (0, B mX - mY) + have hL : HasGaussianLaw (fun ω => L (X ω, Y ω)) P := hXY.map_fun L + have hshift : HasGaussianLaw (fun ω => L (X ω, Y ω) + c) P where + aemeasurable := hL.aemeasurable.add aemeasurable_const + isGaussian_map := by + have : IsGaussian (P.map (fun ω => L (X ω, Y ω))) := hL.isGaussian_map + convert (inferInstance : IsGaussian + ((P.map (fun ω => L (X ω, Y ω))).map (fun p => p + c))) using 1 + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hL.aemeasurable] + rfl + convert hshift using 1 + ext ω i <;> simp [L, c, map_sub] <;> abel diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed1.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed1.log new file mode 100644 index 000000000..df47663a2 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed1.log @@ -0,0 +1,20 @@ +/workspace/scratch/regression-kernel/gaussian-api/AffineProbe.lean:19:60: error: unexpected token 'where'; expected ':=' or '|' +/workspace/scratch/regression-kernel/gaussian-api/AffineProbe.lean:19:7: error: Missing cases: +/workspace/scratch/regression-kernel/gaussian-api/AffineProbe.lean:11:66: error: unsolved goals +Ω : Type u_1 +inst✝ : MeasurableSpace Ω +P : Measure Ω +n k : ℕ +X : Ω → Fin n → ℝ +Y : Ω → Fin k → ℝ +hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P +B : (Fin n → ℝ) →L[ℝ] Fin k → ℝ +mX : Fin n → ℝ +mY : Fin k → ℝ +L : (Fin n → ℝ) × (Fin k → ℝ) →L[ℝ] (Fin n → ℝ) × (Fin k → ℝ) := + (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ)).prod + (ContinuousLinearMap.snd ℝ (Fin n → ℝ) (Fin k → ℝ) - B ∘SL ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ)) +c : (Fin n → ℝ) × (Fin k → ℝ) := (0, B mX - mY) +hL : HasGaussianLaw (fun ω => L (X ω, Y ω)) P +hshift : HasGaussianLaw (fun ω => L (X ω, Y ω) + c) P +⊢ HasGaussianLaw (fun ω => (X ω, Y ω - mY - B (X ω - mX))) P diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed2.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed2.lean.txt new file mode 100644 index 000000000..205b4baa9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed2.lean.txt @@ -0,0 +1,28 @@ +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory +set_option autoImplicit false + +example {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {n k : ℕ} + {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasGaussianLaw (fun ω => (X ω, Y ω - mY - B (X ω - mX))) P := by + let L : ((Fin n → ℝ) × (Fin k → ℝ)) →L[ℝ] + ((Fin n → ℝ) × (Fin k → ℝ)) := + (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ)).prod + (ContinuousLinearMap.snd ℝ (Fin n → ℝ) (Fin k → ℝ) - + B.comp (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ))) + let c : (Fin n → ℝ) × (Fin k → ℝ) := (0, B mX - mY) + have hL : HasGaussianLaw (fun ω => L (X ω, Y ω)) P := hXY.map_fun L + have hshift : HasGaussianLaw (fun ω => L (X ω, Y ω) + c) P := { + aemeasurable := hL.aemeasurable.add aemeasurable_const + isGaussian_map := by + have : IsGaussian (P.map (fun ω => L (X ω, Y ω))) := hL.isGaussian_map + convert (inferInstance : IsGaussian + ((P.map (fun ω => L (X ω, Y ω))).map (fun p => p + c))) using 1 + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hL.aemeasurable] + rfl } + convert hshift using 1 + ext ω i <;> (simp [L, c, map_sub]; abel) diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed2.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed2.log new file mode 100644 index 000000000..50ae19cf9 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.failed2.log @@ -0,0 +1 @@ +/workspace/scratch/regression-kernel/gaussian-api/AffineProbe.lean:28:37: error: No goals to be solved diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.lean.txt new file mode 100644 index 000000000..9f5fef154 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.lean.txt @@ -0,0 +1,28 @@ +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory +set_option autoImplicit false + +example {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {n k : ℕ} + {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasGaussianLaw (fun ω => (X ω, Y ω - mY - B (X ω - mX))) P := by + let L : ((Fin n → ℝ) × (Fin k → ℝ)) →L[ℝ] + ((Fin n → ℝ) × (Fin k → ℝ)) := + (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ)).prod + (ContinuousLinearMap.snd ℝ (Fin n → ℝ) (Fin k → ℝ) - + B.comp (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ))) + let c : (Fin n → ℝ) × (Fin k → ℝ) := (0, B mX - mY) + have hL : HasGaussianLaw (fun ω => L (X ω, Y ω)) P := hXY.map_fun L + have hshift : HasGaussianLaw (fun ω => L (X ω, Y ω) + c) P := { + aemeasurable := hL.aemeasurable.add aemeasurable_const + isGaussian_map := by + have : IsGaussian (P.map (fun ω => L (X ω, Y ω))) := hL.isGaussian_map + convert (inferInstance : IsGaussian + ((P.map (fun ω => L (X ω, Y ω))).map (fun p => p + c))) using 1 + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hL.aemeasurable] + rfl } + convert hshift using 1 + ext ω i <;> simp [L, c, map_sub] <;> abel diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.log new file mode 100644 index 000000000..4edbaa120 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.log @@ -0,0 +1,3 @@ +/workspace/scratch/regression-kernel/gaussian-api/AffineProbe.lean:28:35: warning: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.warning1.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.warning1.lean.txt new file mode 100644 index 000000000..9f5fef154 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.warning1.lean.txt @@ -0,0 +1,28 @@ +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory +set_option autoImplicit false + +example {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {n k : ℕ} + {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasGaussianLaw (fun ω => (X ω, Y ω - mY - B (X ω - mX))) P := by + let L : ((Fin n → ℝ) × (Fin k → ℝ)) →L[ℝ] + ((Fin n → ℝ) × (Fin k → ℝ)) := + (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ)).prod + (ContinuousLinearMap.snd ℝ (Fin n → ℝ) (Fin k → ℝ) - + B.comp (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ))) + let c : (Fin n → ℝ) × (Fin k → ℝ) := (0, B mX - mY) + have hL : HasGaussianLaw (fun ω => L (X ω, Y ω)) P := hXY.map_fun L + have hshift : HasGaussianLaw (fun ω => L (X ω, Y ω) + c) P := { + aemeasurable := hL.aemeasurable.add aemeasurable_const + isGaussian_map := by + have : IsGaussian (P.map (fun ω => L (X ω, Y ω))) := hL.isGaussian_map + convert (inferInstance : IsGaussian + ((P.map (fun ω => L (X ω, Y ω))).map (fun p => p + c))) using 1 + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hL.aemeasurable] + rfl } + convert hshift using 1 + ext ω i <;> simp [L, c, map_sub] <;> abel diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.warning1.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.warning1.log new file mode 100644 index 000000000..4edbaa120 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/AffineProbe.warning1.log @@ -0,0 +1,3 @@ +/workspace/scratch/regression-kernel/gaussian-api/AffineProbe.lean:28:35: warning: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/Probe.lean.txt b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/Probe.lean.txt new file mode 100644 index 000000000..645277b46 --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/Probe.lean.txt @@ -0,0 +1,19 @@ +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory + +set_option autoImplicit false + +example {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) {n k : ℕ} + (X : Ω → Fin n → ℝ) (R : Ω → Fin k → ℝ) + (hXR : HasGaussianLaw (fun ω => (X ω, R ω)) P) + (hzero : ∀ i j, covariance (fun ω => X ω i) (fun ω => R ω j) P = 0) : + IndepFun X R P := by + exact hXR.indepFun_of_covariance_eval hzero + +example {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) {n k : ℕ} + (X : Ω → Fin n → ℝ) (R : Ω → Fin k → ℝ) + (hXR : HasGaussianLaw (fun ω => (X ω, R ω)) P) : + AEMeasurable X P ∧ AEMeasurable R P ∧ IsProbabilityMeasure P := by + exact ⟨hXR.fst.aemeasurable, hXR.snd.aemeasurable, hXR.isProbabilityMeasure⟩ diff --git a/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/Probe.log b/frontiers/determinant_moment_20261025/evidence/gaussian-history/regression-kernel/gaussian-api/Probe.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/determinant_moment_20261025/evidence/scientific-review/claude-01cz-scoped.json b/frontiers/determinant_moment_20261025/evidence/scientific-review/claude-01cz-scoped.json new file mode 100644 index 000000000..c27d909cb --- /dev/null +++ b/frontiers/determinant_moment_20261025/evidence/scientific-review/claude-01cz-scoped.json @@ -0,0 +1,53 @@ +{ + "schema_version": 1, + "scientific_effect": "NONE", + "native_request": "https://github.com/d6g8k5htny-coder/Math-/pull/512#issuecomment-6403871817", + "native_review": "https://github.com/d6g8k5htny-coder/Math-/pull/512#issuecomment-6403917675", + "reviewer_declared_identity": { + "provider": "Anthropic", + "family": "Claude", + "session": "session_01Cz7WZybv8znP64SpPj6sWY", + "nonauthor": true, + "organizational_independence": 0, + "blind_independence": 0 + }, + "source_commit": "16f81b973a02783f8be83081aa7f198724cf5767", + "manifest_sha256": "0008bee4d620206e78ad72ca3d4f39006a8c495f185f556d3b2ffa60e248ccbd", + "scope_sha256": "1105cc13a0ccfe382e9582dccd4cc1d07c215733f565dcf2f75801ab7b090ee4", + "disposition": "PASS_SCOPED", + "coverage": "Ten new modules: statement/proof reading, internal scope alignment; recorded build and axiom logs consumed", + "not_reviewed": [ + "13 scalar baseline statements", + "5 original DM2 statements", + "external GP-DER-047/LM006 contact normalization source join", + "new actual-source companion modules", + "independent Lean/kernel re-execution" + ], + "machine_ACCEPTED_alignment": "NOT_SUPPLIED", + "comments": [ + { + "id": 6403892136, + "url": "https://github.com/d6g8k5htny-coder/Math-/pull/512#issuecomment-6403892136", + "user": { + "login": "d6g8k5htny-coder", + "name": "d6g8k5htny-coder", + "email": null, + "avatar_url": "https://avatars.githubusercontent.com/u/249851642?v=4", + "id": 249851642 + }, + "body": "## PICKUP — REVIEW-MATH512-GAUSSIAN-66-SCOPED — Claude 01Cz\n\n**Request.** Math-#512/6403871817: a different-family scientific review of the fixed 66-target packet. I am Anthropic reviewing OpenAI/Codex authors, so the family rule is satisfied.\n\n**Contender check (21:58Z).** No other review pickup on Math-#512 or main#539. I am not touching Claude S's finite-bar review.\n\n**Exact cut.** Proof/source commit `16f81b973a02783f8be83081aa7f198724cf5767`, with these pins:\n- `manifest.json` sha256 `0008bee4…`\n- SCOPE sha256 `1105cc13…`\n- evidence at `af61a9c`, receipt `95e50247…`\n\n**Scope.** A statement/source/proof alignment review of the ten new modules, focused on the eight points named in the request:\n- GaussianScalarMoments, GaussianJetMoments, EndpointGaussianMoments;\n- GaussianRegressionKernel, GaussianCoupling;\n- GaussianNormalizer, GaussianNormalizerLimit, NormalizerFloor;\n- EndpointNormalizerBridge, PalmReweighting.\n\nThe unchanged 13-statement baseline and the 5 DM2 statements are not re-reviewed.\n\n**Not reproduced.** No Lean toolchain is available in this session, so I will not rerun the kernel build. I will say exactly what I read and what I relied on from the recorded logs.\n\n**Exposure.**\n- I reviewed the predecessor DM2-LEAN alignment (539/6402538781).\n- I authored band notes that consume similar Gaussian-conditioning facts (JOINT-HEIGHT, GENERIC-ASSEMBLY).\n- I am a nonauthor of this packet. Organisational and blind independence: 0.\n\n**Deadline.** About 23:00Z.\n\ncredit 0 · sci NONE · eng ≠ discharge · no flags · OBL OPEN\n\n\n---\n_Generated by [Claude Code](https://claude.ai/code)_" + }, + { + "id": 6403917675, + "url": "https://github.com/d6g8k5htny-coder/Math-/pull/512#issuecomment-6403917675", + "user": { + "login": "d6g8k5htny-coder", + "name": "d6g8k5htny-coder", + "email": null, + "avatar_url": "https://avatars.githubusercontent.com/u/249851642?v=4", + "id": 249851642 + }, + "body": "## RESULT + RELEASE — REVIEW-MATH512-GAUSSIAN-66-SCOPED: **PASS_SCOPED** — Claude 01Cz\n\n**Closes** PICKUP Math-#512/6403892136.\n\n**Reviewer identity.** Anthropic Claude, session `session_01Cz7WZybv8znP64SpPj6sWY`. The authors are OpenAI/Codex, so this is a different-family scientific review. I am a nonauthor; organisational and blind independence are 0. My exposure is as stated in the PICKUP.\n\n### Exact cut\n- Source commit `16f81b973a02783f8be83081aa7f198724cf5767`. Recomputed sha256: `manifest.json` = `0008bee4d620206e…` ✓; `SCOPE.md` = `1105cc13a0ccfe38…` ✓.\n- Evidence commit `af61a9c6fb4147b057cff9f1648e3be3298fd889`. Its `ResearchFormalCoreR1/` sources are byte-identical to `16f81b97` (empty diff).\n- `evidence/gaussian-final/axioms.log` has 66 entries, and all 66 depend on exactly `[propext, Classical.choice, Quot.sound]`.\n\n### What I did and did not do\n- **Did:** read in full the ten new modules (1,467 lines) and checked each public statement against the review points and the SCOPE table. `grep` finds no `sorry` and no `axiom` declarations in them.\n- **Did not:** rerun Lean, leanchecker or the gate, since no toolchain is available in this session. I relied on the recorded build and axiom logs for kernel acceptance. I did not open the external sources (GP-DER-047 §13, LM006 §§1–2) to recheck the contact normalisation; see N3. I did not re-review the 13 baseline statements or the 5 DM2 statements.\n\n### The eight review points\n1. **Nonzero means and zero variances — PASS.** `gm8_scalar_integrable_bound` takes `v : ℝ≥0` and an arbitrary `m`, via the affine pushforward of N(0,1). Zero variance is included.\n2. **Correlated finite moments — PASS.** `gm8_finite_integrable_bound` and `gm8_jet_integrable_bound` use only the per-coordinate marginal laws (`HasLaw … gaussianReal`). So correlation is allowed, and neither independence nor joint Gaussianity is needed. This is a stronger statement than the claim.\n3. **Actual six versus quartic nine — PASS.**\n - `egm_HM`/`egm_HS` rebuild H_M and H_S exactly from (fxx/r, fxy/r, fyy).\n - `egm_weight` is the strict maximum (`dm2Negative`) at M times the strict saddle (det < 0) at S, which is F₂(H_M)F₁(H_S).\n - weight² ≤ 16r⁴R⁸ uses only r ≤ 1. No midpoint-jet substitution occurs.\n - `enb_permutation` and `enb_weight_identity` give W = r²·(continuous normalised weight) exactly, for r > 0, including the type boundaries. I checked the equality case by case: on each boundary both sides are 0, because F_j is continuous there.\n4. **Every-pin canonical regression — PASS, note N1.** `gaussianRegression_observed_spec` needs only joint Gaussianity and the normal equations B·Σ_XX = Σ_XY coordinatewise. No inverse is used, so singular observation covariance is allowed. From these it *derives* that the residual is jointly Gaussian, the zero cross-covariance, independence (Mathlib `indepFun_of_covariance_eval`), and `HasCondDistrib`. The kernel is the literal affine pushforward at every x.\n5. **Singular PSD coupling — PASS.** `gc_coupledVector_tendsto` uses continuity of `CFC.sqrt` on the PSD cone. It assumes no fixed rank and no Lipschitz bound, and the means vary freely.\n6. **Gaussian-parameter Fatou floor — PASS.**\n - `nfloor_eventually_lower_of_fatou` is the correct nonnegative Fatou argument (lintegral liminf, then eventually above half).\n - Eventual integrability is a visible premise in `gn_normalizer_floor_of_common_coupling`. In `gnl_normalizer_floor_of_parameter_limits` it is *derived* from Gaussian L⁴ bounds.\n - The contact relations follow correctly from the zero variances.\n - The contact weight is (q⁻)² with q ~ N(m₀₄, S₀₄₄) and S₀₄₄ > 0, so it is positive.\n7. **Actual density and Cauchy–Schwarz transfer — PASS.** `prwMeasure` is the literal density W/∫W. Its event formula is proved, and `prw_square_root_event_bound` gives P_W(E) ≤ (√C/c)·√μ(E), with r⁴/r² cancelling exactly. Null events are allowed.\n8. **Concrete-model premises visible — PASS.** The parameter limits, the contact means (aM → −1, aS → +1, bM + bS → 0, cM = cS), the zero variances, the PSD conditions and S₀₄₄ > 0 are all explicit hypotheses. The module docstrings and SCOPE do not claim the field identification, the Palm identification, GT5 or parent closure.\n\n### Notes (non-blocking)\n- **N1.** `HasCondDistrib` determines the kernel only P∘X⁻¹-almost everywhere. The value at a prescribed null observation, such as the actual pin target, is the canonical continuous (affine) version. That is the standard convention, and the docstring's wording (\"specified value even where the observation has probability zero\") is accurate. Downstream Kac–Rice uses should cite it as a choice of version justified by continuity of x ↦ kernel(x), not as uniqueness.\n- **N2.** `gn_open_box_floor` is pointwise and claims no mass, which is correct. Its box constants are illustrative.\n- **N3.** The contact normalisation (aM → −1, aS → +1) is a premise. Its agreement with GP-DER-047/LM006 scaling is a source join that I did not recheck. Internally it is consistent: aM < 0 gives a maximum at M, and aS > 0 gives a saddle at S.\n- **N4.** I did not reproduce the kernel acceptance, as stated above.\n\n### Verdict\n**PASS_SCOPED** for statement, source and proof alignment of the ten new modules at `16f81b97`, given the recorded build and axiom logs. This is not a whole-field, Palm, GT5 or parent closure, and it is not machine ACCEPTED alignment, which remains a separate exact-coverage requirement. Scientific effect NONE. No status change. **RELEASE.**\n\ncredit 0 · sci NONE · eng ≠ discharge · no flags · OBL OPEN\n\n\n---\n_Generated by [Claude Code](https://claude.ai/code)_" + } + ] +} diff --git a/frontiers/determinant_moment_20261025/manifest.json b/frontiers/determinant_moment_20261025/manifest.json index 71a692415..9e178a94a 100644 --- a/frontiers/determinant_moment_20261025/manifest.json +++ b/frontiers/determinant_moment_20261025/manifest.json @@ -17,7 +17,17 @@ "source_modules": [ "ResearchFormalCoreR1/AlgebraV2.lean", "ResearchFormalCoreR1/ProbabilityCompanionsV2.lean", - "ResearchFormalCoreR1/DeterminantMoment.lean" + "ResearchFormalCoreR1/DeterminantMoment.lean", + "ResearchFormalCoreR1/GaussianScalarMoments.lean", + "ResearchFormalCoreR1/GaussianJetMoments.lean", + "ResearchFormalCoreR1/GaussianNormalizer.lean", + "ResearchFormalCoreR1/GaussianRegressionKernel.lean", + "ResearchFormalCoreR1/EndpointGaussianMoments.lean", + "ResearchFormalCoreR1/NormalizerFloor.lean", + 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"ResearchFormalCoreR1.gn_gaussian_reference_integrable_pos", + "ResearchFormalCoreR1.gn_open_box_floor", + "ResearchFormalCoreR1.regression_compProd_id_const", + "ResearchFormalCoreR1.regression_pair_hasCondDistrib", + "ResearchFormalCoreR1.regressionReconstruction_measurable", + "ResearchFormalCoreR1.regressionKernel_apply", + "ResearchFormalCoreR1.independentResidual_hasCondDistrib", + "ResearchFormalCoreR1.gaussianRegression_hasCondDistrib", + "ResearchFormalCoreR1.regressionKernel_measurable", + "ResearchFormalCoreR1.gaussianRegressionKernel_spec", + "ResearchFormalCoreR1.regression_map_coordinate", + "ResearchFormalCoreR1.regression_covariance_map", + "ResearchFormalCoreR1.regression_residual_zero_covariance", + "ResearchFormalCoreR1.regression_residual_jointGaussian", + "ResearchFormalCoreR1.gaussianRegression_observed_hasCondDistrib", + "ResearchFormalCoreR1.gaussianRegression_observed_spec", + "ResearchFormalCoreR1.egm_weight_measurable", + "ResearchFormalCoreR1.egm_pointwise", 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b/frontiers/source_derivative_gaussian_20261025/.gitignore @@ -0,0 +1 @@ +.lake/ diff --git a/frontiers/source_derivative_gaussian_20261025/AUTHOR_HISTORY_SUPPLEMENT.json b/frontiers/source_derivative_gaussian_20261025/AUTHOR_HISTORY_SUPPLEMENT.json new file mode 100644 index 000000000..90befdcb9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/AUTHOR_HISTORY_SUPPLEMENT.json @@ -0,0 +1,110 @@ +{ + "record_type": "author tool-history reports; not a machine-created execution receipt", + "scientific_effect": "NONE", + "reports": [ + { + "author": "OpenAI/Codex /root/uniform_jet_moment_sources", + "scope": [ + "SourceDerivativeGaussian", + "SourceContactCovariance", + "SourceDerivativeContinuity" + ], + "verbatim_report": "Author-history confirmation: all three exact SHA values you list are my final source bytes, unchanged by me since their recorded final compile/controls/axiom/kernel executions. No existing JSON receipt/sourcehash record file was created by me; SHA values were read by sha256sum after source freeze and reported in my source-bound handoffs to root. Treat that as author-history confirmation, not a machine-created execution receipt. Existing command/exit/log triples are the only machine artifacts I produced; please preserve those honest limits and do not rerun.", + "confirmed_source_hashes": { + "SourceDerivativeGaussian.lean": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "SourceContactCovariance.lean": "4189cbaf4c18c55f21e3af2c96b0b2842d83161f44596eb691c507dcc1a439c8", + "SourceDerivativeContinuity.lean": "552511e0aee227133b7d067faeceaab9c43c0094b07d68b8eddab0ab2419775e" + } + }, + { + "author": "OpenAI/Codex /root/regression_kernel", + "scope": [ + "ContactJetCovariance", + "SourceContactSchur" + ], + "control_cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "capture_shell_prelude": "set -o pipefail", + "controls": [ + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log", + "observed_exit": 0 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency-01.log", + "observed_exit": 1 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls-01.log", + "observed_exit": 0 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex-01.log", + "observed_exit": 1 + } + ], + "report_transcription": "Actual control commands, no rerun: cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025; each ran bash set -o pipefail then /home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log observed exit0; corresponding contact-gram RejectedZeroFrequency.lean / RejectedZeroAmplitude.lean with matching -01.log each observed exit1. Schur controls same command pattern under scratch/regression-kernel/source-contact-schur/: PositiveControls.lean to PositiveControls-01.log observedexit0; RejectedZeroPeriod.lean and RejectedWrongQIndex.lean to matching-01.log each observedexit1. These are actual tool-observed command/exits from my current transcript. Helper evidence: /workspace/scratch/regression-kernel/contact-schur/SchurProbe.{lean,log} final exit0 and failed1 retained; /workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.{lean,log} final exit0 and ResidualVariance.failed1.* retained. Helpers coauthored OpenAI/Codex gaussian_api. Earlier failedproduction ContactJetCovariance-01 was assembly-script ValueError/no sourcefile, exit1; ContactJetCovariance-02 success. SourceContactSchur-01/-02/-03 failed author attempts exit1; -04 success.", + "transcription_note": "Faithful plain-text transcription of author report; punctuation/arrow notation normalized. Not a raw machine log." + }, + { + "author": "OpenAI/Codex /root/normalizer_palm_research", + "scope": [ + "GaussianRegressionLimits" + ], + "record_type": "actual author tool-history report; not a machine-created execution receipt", + "verbatim_report": "Author tool-history report (not invented machine receipt): cwd /workspace/Math-/frontiers/source_derivative_gaussian_20261025. Final compile `lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean` exit 0, captured /workspace/scratch/regression-limits/regression-limits-03.log empty. Kernel `lake env leanchecker GaussianRegressionLimits` exit 0, regression-limits-leanchecker.log empty. Final controls `lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean` exit 0, regression-limits-controls-02.log empty. Imported-module #print axioms outputs for regression_parameter_limits and regression_uniform_bounds are in controls-01.log (that command exit 1 for unrelated untyped control index); both only propext, Classical.choice, Quot.sound. No production changes occurred after final successful compile/kernel. Current source SHA256 read-only measured 26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be. All three success logs empty SHA e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855; controls01 SHA c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be. Exact source compiled was this source, author-history association; no rerun.", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "confirmed_source_sha256": "26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be", + "source_unchanged_after_final_compile_kernel": true, + "runs": [ + { + "role": "final_compile", + "command": "lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-03.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "named_kernel", + "command": "lake env leanchecker GaussianRegressionLimits", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-leanchecker.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "corrected_controls", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-02.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "initial_controls_container_with_valid_printed_axioms", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 1, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-01.log", + "raw_log_sha256": "c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be" + } + ] + } + ], + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "reuse": "No checks repeated. These metadata limits do not invalidate completed applicable mathematical evidence or authorize a repeated proof campaign." +} diff --git a/frontiers/source_derivative_gaussian_20261025/AXIOM_COVERAGE_ADDENDUM.json b/frontiers/source_derivative_gaussian_20261025/AXIOM_COVERAGE_ADDENDUM.json new file mode 100644 index 000000000..7146909f5 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/AXIOM_COVERAGE_ADDENDUM.json @@ -0,0 +1,50 @@ +{ + "role": "read-only reuse of completed transitive axiom evidence, not a new execution receipt", + "coverage": [ + { + "target": "I4SourceContact.sourceContactEndpointSchur_posSemidef", + "kind": "TRANSITIVE_FROM_PRINTED_PARENT", + "source_path": "SourceContactEndpointParameters.lean", + "source_sha256": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "printed_parent": "I4SourceEndpointEstimates.sourceEndpoint_normalizer_floor", + "printed_parent_source_path": "SourceEndpointEstimates.lean", + "printed_parent_source_sha256": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_code_edge": { + "line": 259, + "text": "hm hS hPSD (sourceContactEndpointSchur_posSemidef hL)" + }, + "raw_log_path": "/workspace/scratch/dm2_landed_composition/endpoint_estimates/axioms.log", + "printed_parent_log_line": 38, + "permitted_axioms_by_parent_bound": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "qualification": "Applicable completed named-kernel and printed-parent transitive axiom evidence reused; no direct target #print output claimed.", + "raw_log_sha256": "ded91623f2200300ff7b8ab366f5aa64564a157dd98166cb3eedaa10c884caf1" + }, + { + "target": "I4SourceContact.sourceContactEndpointSchur_four_four_eq_schur", + "kind": "TRANSITIVE_FROM_PRINTED_PARENT", + "source_path": "SourceContactEndpointParameters.lean", + "source_sha256": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "printed_parent": "I4SourceContact.sourceContactEndpointSchur_referenceVariance_pos", + "printed_parent_source_path": "SourceContactEndpointParameters.lean", + "printed_parent_source_sha256": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "source_code_edge": { + "line": 322, + "text": "rw [sourceContactEndpointSchur_four_four_eq_schur]" + }, + "raw_log_path": "/workspace/scratch/contact-endpoint-parameters/contact-parameters-controls-01.log", + "printed_parent_log_line": 25, + "permitted_axioms_by_parent_bound": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "qualification": "Applicable completed named-kernel and printed-parent transitive axiom evidence reused; no direct target #print output claimed.", + "raw_log_sha256": "3eb74647a2d1698732992034708b9c24d542250b15358cac3db4130130c3183a" + } + ], + "owner_reuse_basis": "Two public helpers are consumed by already printed parents whose transitive reports allow only the three standard axioms. A direct-output process gap does not erase completed applicable mathematical evidence. No rebuild/re-audit solely to create clean output." +} diff --git a/frontiers/source_derivative_gaussian_20261025/ConfluentIntegralIdentities.lean b/frontiers/source_derivative_gaussian_20261025/ConfluentIntegralIdentities.lean new file mode 100644 index 000000000..eafb41ab0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/ConfluentIntegralIdentities.lean @@ -0,0 +1,204 @@ +import Mathlib.Analysis.Calculus.IteratedDeriv.Defs +import Mathlib.Analysis.Calculus.Deriv.Pow +import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts +import Mathlib.Tactic.FieldSimp +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.Ring + +/-! +# Exact integral formulas for confluent pins and adjusted endpoints + +All integrals use the ordinary oriented interval `[-1/2,1/2]`. The physical +radius can have either sign. These deterministic scalar formulas use only C3 +regularity of F and C2 regularity of H; no integral identity is a premise. +See SOURCE_CONFLUENT_INTEGRALS.md for the frozen source contract. +-/ + +noncomputable section + +open MeasureTheory Set +open scoped Interval + +namespace I4ConfluentIntegralIdentities + +def pin3 (F : ℝ → ℝ) (r : ℝ) : ℝ := + (deriv F (r / 2) - deriv F (-r / 2)) / r + +def pin4 (H : ℝ → ℝ) (r : ℝ) : ℝ := + (H (r / 2) - H (-r / 2)) / r + +def pin5 (F : ℝ → ℝ) (r : ℝ) : ℝ := + (12 / r ^ 2) * ((deriv F (r / 2) + deriv F (-r / 2)) / 2 - + (F (r / 2) - F (-r / 2)) / r) + +private theorem scaled_hasDerivAt (g g' : ℝ → ℝ) + (hg : ∀ x, HasDerivAt g (g' x) x) (r u : ℝ) : + HasDerivAt (fun u => g (r * u)) (r * g' (r * u)) u := by + have hru : HasDerivAt (fun u : ℝ => r*u) r u := by + simpa [id] using (hasDerivAt_id u).const_mul r + convert (hg (r*u)).comp u hru using 1 <;> simp [Function.comp_def, mul_comm] + +private theorem average_identity (g g' : ℝ → ℝ) + (hg : ∀ x, HasDerivAt g (g' x) x) (hc : Continuous g') (r : ℝ) (hr : r ≠ 0) : + (g (r / 2) - g (-r / 2)) / r = + ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), g' (r * u) := by + have hc' : Continuous (fun u => r * g' (r * u)) := + continuous_const.mul (hc.comp (continuous_const.mul continuous_id)) + have h := intervalIntegral.integral_eq_sub_of_hasDerivAt + (a := (-1/2 : ℝ)) (b := (1/2 : ℝ)) + (fun u _ => scaled_hasDerivAt g g' hg r u) (hc'.intervalIntegrable _ _) + rw [intervalIntegral.integral_const_mul] at h + have hleft : r * (-1/2) = -r / 2 := by ring + have hright : r * (1/2) = r / 2 := by ring + rw [hleft, hright] at h + apply (div_eq_iff hr).2 + nlinarith [h] + +private theorem left_endpoint_identity (g g' : ℝ → ℝ) + (hg : ∀ x, HasDerivAt g (g' x) x) (hc : Continuous g') (r : ℝ) (hr : r ≠ 0) : + (g (-r/2) - ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), g (r*u)) / r = + -(∫ u in (-1/2 : ℝ)..(1/2 : ℝ), (1/2-u) * g' (r*u)) := by + have hgc : Continuous g := + (show Differentiable ℝ g from fun x => (hg x).differentiableAt).continuous + have hc' : Continuous (fun u => r * g' (r*u)) := + continuous_const.mul (hc.comp (continuous_const.mul continuous_id)) + have h := intervalIntegral.integral_mul_deriv_eq_deriv_mul_of_hasDerivAt + (a := (-1/2 : ℝ)) (b := (1/2 : ℝ)) + (u := fun u : ℝ => 1/2-u) (u' := fun _ : ℝ => -1) + (v := fun u => g (r*u)) (v' := fun u => r * g' (r*u)) + (by fun_prop) (hgc.comp (continuous_const.mul continuous_id)).continuousOn + (fun u _ => by + convert (hasDerivAt_id u).const_sub (1/2 : ℝ) using 1 <;> simp [id]) + (fun u _ => scaled_hasDerivAt g g' hg r u) + (continuous_const.intervalIntegrable _ _) (hc'.intervalIntegrable _ _) + have hfun : (fun u : ℝ => (1/2-u) * (r * g' (r*u))) = + (fun u => r * ((1/2-u) * g' (r*u))) := by funext u; ring + rw [hfun, intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul] at h + norm_num [mul_div_assoc, mul_neg] at h + apply (div_eq_iff hr).2 + norm_num at h ⊢ + rw [show -(r * (1/2 : ℝ)) = -r/2 by ring] at h + nlinarith [h] + +private theorem right_endpoint_identity (g g' : ℝ → ℝ) + (hg : ∀ x, HasDerivAt g (g' x) x) (hc : Continuous g') (r : ℝ) (hr : r ≠ 0) : + (g (r/2) - ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), g (r*u)) / r = + ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), (u+1/2) * g' (r*u) := by + have hgc : Continuous g := + (show Differentiable ℝ g from fun x => (hg x).differentiableAt).continuous + have hc' : Continuous (fun u => r * g' (r*u)) := + continuous_const.mul (hc.comp (continuous_const.mul continuous_id)) + have h := intervalIntegral.integral_mul_deriv_eq_deriv_mul_of_hasDerivAt + (a := (-1/2 : ℝ)) (b := (1/2 : ℝ)) + (u := fun u : ℝ => u+1/2) (u' := fun _ : ℝ => 1) + (v := fun u => g (r*u)) (v' := fun u => r * g' (r*u)) + (by fun_prop) (hgc.comp (continuous_const.mul continuous_id)).continuousOn + (fun u _ => by simpa [id] using (hasDerivAt_id u).add_const (1/2 : ℝ)) + (fun u _ => scaled_hasDerivAt g g' hg r u) + (continuous_const.intervalIntegrable _ _) (hc'.intervalIntegrable _ _) + have hfun : (fun u : ℝ => (u+1/2) * (r * g' (r*u))) = + (fun u => r * ((u+1/2) * g' (r*u))) := by funext u; ring + rw [hfun, intervalIntegral.integral_const_mul] at h + norm_num [mul_div_assoc, mul_neg] at h + apply (div_eq_iff hr).2 + norm_num at h ⊢ + rw [show r * (1/2 : ℝ) = r/2 by ring] at h + nlinarith [h] + +private theorem peano_identity (f f1 f2 f3 : ℝ → ℝ) + (h0 : ∀ x, HasDerivAt f (f1 x) x) + (h1 : ∀ x, HasDerivAt f1 (f2 x) x) + (h2 : ∀ x, HasDerivAt f2 (f3 x) x) (hc3 : Continuous f3) + (r : ℝ) (hr : r ≠ 0) : + (12 / r ^ 2) * ((f1 (r/2) + f1 (-r/2))/2 - (f (r/2) - f (-r/2))/r) = + 6 * ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), (1/4-u^2) * f3 (r*u) := by + let A : ℝ → ℝ := fun u => + (1/4-u^2)*f2 (r*u) + (2/r)*u*f1 (r*u) - (2/r^2)*f (r*u) + have hA (u : ℝ) : HasDerivAt A (r*((1/4-u^2)*f3 (r*u))) u := by + have hru : HasDerivAt (fun u : ℝ => r*u) r u := by + simpa [id] using (hasDerivAt_id u).const_mul r + have hK : HasDerivAt (fun u : ℝ => 1/4-u^2) (-2*u) u := by + convert ((hasDerivAt_id u).pow 2).const_sub (1/4 : ℝ) using 1 <;> simp [id] + have hU : HasDerivAt (fun u : ℝ => (2/r)*u) (2/r) u := by + simpa [id] using (hasDerivAt_id u).const_mul (2/r) + have h := ((hK.mul ((h2 (r*u)).comp u hru)).add + (hU.mul ((h1 (r*u)).comp u hru))).sub (((h0 (r*u)).comp u hru).const_mul (2/r^2)) + convert h using 1 + · rfl + · dsimp [Function.comp_def] + field_simp [hr] + ring + have hcont : Continuous (fun u : ℝ => r*((1/4-u^2)*f3 (r*u))) := + (continuous_const.sub (continuous_id.pow 2)).mul + (hc3.comp (continuous_const.mul continuous_id)) |>.const_mul r + have hft := intervalIntegral.integral_eq_sub_of_hasDerivAt + (fun u _ => hA u) (hcont.intervalIntegrable (-1/2 : ℝ) (1/2 : ℝ)) + rw [intervalIntegral.integral_const_mul] at hft + norm_num [A, mul_div_assoc, mul_neg] at hft + field_simp [hr] at hft ⊢ + nlinarith [hft] + +private theorem iterated_hasDerivAt {F : ℝ → ℝ} {n : ℕ} + (hF : ContDiff ℝ n F) (k : ℕ) (hk : k < n) (x : ℝ) : + HasDerivAt (iteratedDeriv k F) (iteratedDeriv (k+1) F x) x := by + rw [iteratedDeriv_succ] + exact (hF.differentiable_iteratedDeriv k (by exact_mod_cast hk) x).hasDerivAt + +/-- The x-derivative divided difference is the exact average second derivative. -/ +theorem pin3_integral (F : ℝ → ℝ) (hF : ContDiff ℝ 3 F) (r : ℝ) (hr : r ≠ 0) : + pin3 F r = ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), iteratedDeriv 2 F (r*u) := by + simpa [pin3, iteratedDeriv_one] using average_identity (iteratedDeriv 1 F) + (iteratedDeriv 2 F) (iterated_hasDerivAt hF 1 (by norm_num)) + (hF.continuous_iteratedDeriv 2 (by norm_num)) r hr + +/-- H is the y-derivative curve: the pin uses H itself, not its derivative. -/ +theorem pin4_integral (H : ℝ → ℝ) (hH : ContDiff ℝ 2 H) (r : ℝ) (hr : r ≠ 0) : + pin4 H r = ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), deriv H (r*u) := by + simpa [pin4, iteratedDeriv_one] using average_identity (iteratedDeriv 0 H) + (iteratedDeriv 1 H) (iterated_hasDerivAt hH 0 (by norm_num)) + (hH.continuous_iteratedDeriv 1 (by norm_num)) r hr + +/-- The exact cubic confluent pin is its normalized Peano-kernel integral. -/ +theorem pin5_integral (F : ℝ → ℝ) (hF : ContDiff ℝ 3 F) (r : ℝ) (hr : r ≠ 0) : + pin5 F r = 6 * ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), + (1/4-u^2) * iteratedDeriv 3 F (r*u) := by + simpa [pin5, iteratedDeriv_one] using peano_identity (iteratedDeriv 0 F) + (iteratedDeriv 1 F) (iteratedDeriv 2 F) (iteratedDeriv 3 F) + (iterated_hasDerivAt hF 0 (by norm_num)) + (iterated_hasDerivAt hF 1 (by norm_num)) + (iterated_hasDerivAt hF 2 (by norm_num)) + (hF.continuous_iteratedDeriv 3 (by norm_num)) r hr + +theorem adjusted_second_left (F : ℝ → ℝ) (hF : ContDiff ℝ 3 F) (r : ℝ) (hr : r ≠ 0) : + (iteratedDeriv 2 F (-r/2) - pin3 F r) / r = + -(∫ u in (-1/2 : ℝ)..(1/2 : ℝ), (1/2-u) * iteratedDeriv 3 F (r*u)) := by + rw [pin3_integral F hF r hr] + exact left_endpoint_identity (iteratedDeriv 2 F) (iteratedDeriv 3 F) + (iterated_hasDerivAt hF 2 (by norm_num)) + (hF.continuous_iteratedDeriv 3 (by norm_num)) r hr + +theorem adjusted_second_right (F : ℝ → ℝ) (hF : ContDiff ℝ 3 F) (r : ℝ) (hr : r ≠ 0) : + (iteratedDeriv 2 F (r/2) - pin3 F r) / r = + ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), (u+1/2) * iteratedDeriv 3 F (r*u) := by + rw [pin3_integral F hF r hr] + exact right_endpoint_identity (iteratedDeriv 2 F) (iteratedDeriv 3 F) + (iterated_hasDerivAt hF 2 (by norm_num)) + (hF.continuous_iteratedDeriv 3 (by norm_num)) r hr + +theorem adjusted_transverse_left (H : ℝ → ℝ) (hH : ContDiff ℝ 2 H) (r : ℝ) (hr : r ≠ 0) : + (deriv H (-r/2) - pin4 H r) / r = + -(∫ u in (-1/2 : ℝ)..(1/2 : ℝ), (1/2-u) * iteratedDeriv 2 H (r*u)) := by + rw [pin4_integral H hH r hr] + simpa [iteratedDeriv_one] using left_endpoint_identity (iteratedDeriv 1 H) + (iteratedDeriv 2 H) (iterated_hasDerivAt hH 1 (by norm_num)) + (hH.continuous_iteratedDeriv 2 (by norm_num)) r hr + +theorem adjusted_transverse_right (H : ℝ → ℝ) (hH : ContDiff ℝ 2 H) (r : ℝ) (hr : r ≠ 0) : + (deriv H (r/2) - pin4 H r) / r = + ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), (u+1/2) * iteratedDeriv 2 H (r*u) := by + rw [pin4_integral H hH r hr] + simpa [iteratedDeriv_one] using right_endpoint_identity (iteratedDeriv 1 H) + (iteratedDeriv 2 H) (iterated_hasDerivAt hH 1 (by norm_num)) + (hH.continuous_iteratedDeriv 2 (by norm_num)) r hr + +end I4ConfluentIntegralIdentities diff --git a/frontiers/source_derivative_gaussian_20261025/ContactJetCovariance.lean b/frontiers/source_derivative_gaussian_20261025/ContactJetCovariance.lean new file mode 100644 index 000000000..a6614b817 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/ContactJetCovariance.lean @@ -0,0 +1,169 @@ +import Mathlib.LinearAlgebra.Matrix.PosDef +import Mathlib.LinearAlgebra.Matrix.Notation +import Mathlib.Analysis.Normed.Ring.InfiniteSum +import Mathlib.Analysis.Normed.Module.FiniteDimension +import Mathlib.Tactic.FinCases +import Mathlib.Tactic.Linarith + +/-! +# Seven-mode certificate for contact-jet covariance + +The row order is `(f, fx, fy, fxx, fxy, fxxx, fyy)`, with selected columns +`(DC, cos(1,0), cos(0,1), cos(1,1), sin(1,0), sin(2,0), sin(0,1))`. +Nonzero mode amplitudes are retained. The witness forces strict positivity of an +infinite coefficient Gram matrix by its sum-of-squares quadratic identity. + +This is a coefficient certificate. Applying it to the literal smooth source requires +the exact derivative-coefficient identities and physical covariance identification; +these are not premises that may be silently omitted. See `SOURCE_CONTACT_GRAM.md`. +-/ + +open Matrix +open scoped BigOperators Matrix +set_option autoImplicit false + +namespace I4ContactGram + +def contactWitness (c : ℝ) : Matrix (Fin 7) (Fin 7) ℝ := + !![1,1,1,1,0,0,0; + 0,0,0,0,c,2*c,0; + 0,0,0,0,0,0,c; + 0,-c^2,0,-c^2,0,0,0; + 0,0,0,-c^2,0,0,0; + 0,0,0,0,-c^3,-8*c^3,0; + 0,0,-c^2,-c^2,0,0,0] + +lemma contactWitness_transpose_mulVec_eq_zero {c : ℝ} (hc : c ≠ 0) + (z : Fin 7 → ℝ) (h : (contactWitness c)ᵀ *ᵥ z = 0) : z = 0 := by + have hs (s : Fin 7) := congrFun h s + have h0 : z 0 = 0 := by + simpa [contactWitness, Matrix.mulVec, dotProduct, Fin.sum_univ_succ] using hs 0 + have h1 := hs 1 + have h2 := hs 2 + have h3 := hs 3 + have h4 := hs 4 + have h5 := hs 5 + have h6 := hs 6 + simp [contactWitness, Matrix.mulVec, dotProduct, Fin.sum_univ_succ, h0] at h1 h2 h3 h4 h5 h6 + have hc2 : c^2 ≠ 0 := pow_ne_zero _ hc + have hc3 : c^3 ≠ 0 := pow_ne_zero _ hc + have hz3 : z 3 = 0 := h1.resolve_left hc + have hz6 : z 6 = 0 := h2.resolve_left hc + have hz4 : z 4 = 0 := (mul_eq_zero.mp (show c^2 * z 4 = 0 by nlinarith [h3])).resolve_left hc2 + have hz5 : z 5 = 0 := (mul_eq_zero.mp (show c^3 * z 5 = 0 by nlinarith [h4, h5])).resolve_left hc3 + have hz1 : z 1 = 0 := (mul_eq_zero.mp (show c * z 1 = 0 by nlinarith [h4])).resolve_left hc + have hz2 : z 2 = 0 := h6.resolve_left hc + funext i + fin_cases i <;> assumption + +lemma contactWitness_transpose_injective {c : ℝ} (hc : c ≠ 0) : + Function.Injective ((contactWitness c)ᵀ).mulVec := by + intro x y h + apply sub_eq_zero.mp + apply contactWitness_transpose_mulVec_eq_zero hc + simp [Matrix.mulVec_sub, h] + +lemma contactWitness_amplitude_transpose_injective {c : ℝ} (hc : c ≠ 0) + (a : Fin 7 → ℝ) (ha : ∀ s, a s ≠ 0) : + Function.Injective ((contactWitness c * diagonal a)ᵀ).mulVec := by + intro x y h + apply contactWitness_transpose_injective hc + funext s + have hs := congrFun h s + simp only [Matrix.transpose_mul, Matrix.diagonal_transpose, ← Matrix.mulVec_mulVec, + Matrix.mulVec_diagonal] at hs + exact mul_left_cancel₀ (ha s) hs + +noncomputable def coefficientGram {ι : Type*} {n : ℕ} (coeff : Fin n → ι → ℝ) : + Matrix (Fin n) (Fin n) ℝ := fun i j => ∑' m, coeff i m * coeff j m + +lemma coefficient_product_summable {ι : Type*} {a b : ι → ℝ} + (ha : Summable a) (hb : Summable b) : Summable (fun m => a m * b m) := by + have hp := summable_mul_of_summable_norm ha.norm hb.norm + simpa only [Function.comp_def] using hp.comp_injective + (show Function.Injective (fun m : ι => (m,m)) from + fun i j h => congrArg Prod.fst h) + +lemma coefficientGram_quadratic_hasSum {ι : Type*} {n : ℕ} (coeff : Fin n → ι → ℝ) + (hc : ∀ i, Summable (coeff i)) (z : Fin n → ℝ) : + HasSum (fun m => (∑ i, z i * coeff i m)^2) + (star z ⬝ᵥ (coefficientGram coeff *ᵥ z)) := by + have hs : HasSum (fun m => ∑ i, ∑ j, z i * (coeff i m * coeff j m) * z j) + (∑ i, ∑ j, z i * coefficientGram coeff i j * z j) := + hasSum_sum (fun i _ => hasSum_sum (fun j _ => + (((coefficient_product_summable (hc i) (hc j)).hasSum).mul_left (z i)).mul_right (z j))) + convert hs using 1 + · funext m + rw [pow_two, Finset.sum_mul_sum] + apply Finset.sum_congr rfl + intro i _ + apply Finset.sum_congr rfl + intro j _ + ring + · simp [Matrix.mulVec, dotProduct, Finset.mul_sum, mul_assoc] + +lemma coefficientGram_posDef_of_separating {ι : Type*} {n : ℕ} + (coeff : Fin n → ι → ℝ) (hc : ∀ i, Summable (coeff i)) + (hsep : ∀ z : Fin n → ℝ, z ≠ 0 → ∃ m, (∑ i, z i * coeff i m) ≠ 0) : + (coefficientGram coeff).PosDef := by + apply Matrix.PosDef.of_dotProduct_mulVec_pos + · ext i j + simp only [coefficientGram, Matrix.conjTranspose_apply, star_trivial] + apply tsum_congr + intro m + ring + · intro z hz + have hs := coefficientGram_quadratic_hasSum coeff hc z + rw [← hs.tsum_eq] + obtain ⟨m, hm⟩ := hsep z hz + exact hs.summable.tsum_pos (fun m => sq_nonneg _) m (sq_pos_of_ne_zero hm) + +/-- Seven explicit independent mode columns force positivity of the entire infinite Gram sum. +No positive-definiteness or invertibility conclusion is an input premise. -/ +theorem contact_coefficientGram_posDef {ι : Type*} (coeff : Fin 7 → ι → ℝ) + (hcoeff : ∀ i, Summable (coeff i)) {c : ℝ} (hc : c ≠ 0) + (mode : Fin 7 → ι) (a : Fin 7 → ℝ) (ha : ∀ s, a s ≠ 0) + (hcolumns : ∀ i s, coeff i (mode s) = a s * contactWitness c i s) : + (coefficientGram coeff).PosDef := by + apply coefficientGram_posDef_of_separating coeff hcoeff + intro z hz + classical + by_cases hex : ∃ m, (∑ i, z i * coeff i m) ≠ 0 + · exact hex + · exfalso + apply hz + apply contactWitness_amplitude_transpose_injective hc a ha + rw [Matrix.mulVec_zero] + funext s + have hm : (∑ i, z i * coeff i (mode s)) = 0 := + not_ne_iff.mp (not_exists.mp hex (mode s)) + simp_rw [hcolumns] at hm + simpa [Matrix.mulVec, dotProduct, Matrix.mul_diagonal, mul_comm, mul_left_comm, + mul_assoc] using hm + +/-- A positive-definite six-plus-one block covariance has strictly positive scalar Schur complement. -/ +theorem contact_schur_pos (A : Matrix (Fin 6) (Fin 6) ℝ) + (B : Matrix (Fin 6) Unit ℝ) (D : Matrix Unit Unit ℝ) + (hG : (Matrix.fromBlocks A B Bᴴ D).PosDef) : + A.PosDef ∧ 0 < (D - Bᴴ * A⁻¹ * B) () () := by + have hA : A.PosDef := by + convert hG.submatrix (e := Sum.inl) Sum.inl_injective using 1 + ext i j + rfl + let _ : Invertible A := hA.isUnit.invertible + let y : Unit → ℝ := fun _ => 1 + let x : Fin 6 → ℝ := -((A⁻¹ * B) *ᵥ y) + have hxy : Sum.elim x y ≠ 0 := by + intro h + have hh := congrFun h (Sum.inr ()) + simp [y] at hh + have hp := hG.dotProduct_mulVec_pos hxy + rw [Matrix.dotProduct_mulVec, + Matrix.schur_complement_eq₁₁ B D x y hA.isHermitian] at hp + dsimp [x] at hp + rw [neg_add_cancel, dotProduct_zero, zero_add] at hp + refine ⟨hA, ?_⟩ + simpa [y, Matrix.vecMul, dotProduct] using hp + + +end I4ContactGram diff --git a/frontiers/source_derivative_gaussian_20261025/EVIDENCE_CATALOG.json b/frontiers/source_derivative_gaussian_20261025/EVIDENCE_CATALOG.json new file mode 100644 index 000000000..163460e1b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/EVIDENCE_CATALOG.json @@ -0,0 +1,9352 @@ +{ + "schema_version": 1, + "role": "read-only projection of existing mathematical evidence; no new execution", + "scientific_effect": "NONE", + "goal_status": "FULL_ORIGINAL_RESEARCH_GOAL_ACTIVE", + "source_root": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "scope": [ + "SourceDerivativeGaussian", + "ContactJetCovariance", + "SourceContactCovariance", + "SourceConfluentPins", + "SourceContactSchur", + "SourceDerivativeContinuity", + "ConfluentIntegralIdentities", + "SourceEndpointRegression", + "GaussianRegressionLimits", + "SourceEndpointParameters", + "SourceContactEndpointParameters", + "SourceConfluentLimits", + "SourcePhysicalEndpointKernel", + "SourceEndpointEstimates" + ], + "not_in_this_completed_inventory": [], + "source_inputs_binding": { + "path": "INPUT_BINDING.json", + "sha256": 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optional Schur and generic L2 residual-variance proof support", + "OpenAI/Codex /root/dm2_landed_composition: confluent pin equivalence and integral identities", + "OpenAI/Codex /root/dm2_landed_composition/peano_strategy: actual generic Peano scratch proof helper", + "OpenAI/Codex /root/palm_measure_transfer: read-only evidence projection", + "OpenAI/Codex /root/palm_measure_transfer/density_api: read-only README/SCOPE draft assistance", + "OpenAI/Codex /root/normalizer_palm_research: Gaussian regression parameter limits and uniform bounds", + "OpenAI/Codex /root/regression_kernel/gaussian_api: actual SourceEndpointRegression matrix normal-equations API support", + "OpenAI/Codex /root/regression_kernel/endpoint_means: actual endpoint unconditional zero-mean proofs, retained means/Means.lean and Means-03.log", + "OpenAI/Codex /root/regression_kernel/gaussian_api: actual SourceEndpointParameters residual covariance proof support, retained covariance/ResidualCovariance.lean/.log", + "OpenAI/Codex /root/normalizer_palm_research/gaussian_reference_probe: incorporated private generic contact residual covariance scratch helper; no reviewer credit", + "OpenAI/Codex /root/dm2_landed_composition: actual mode-curve, weighted-integral and regularized-continuity proofs incorporated into SourceConfluentLimits", + "OpenAI/Codex /root/normalizer_palm_research: actual contact coefficient/regularized-zero scratch proofs incorporated into SourceConfluentLimits", + "OpenAI/Codex /root/dm2_landed_composition/peano_strategy: actual normalizer-floor assembly draft incorporated into SourceEndpointEstimates" + ], + "review": "All above work is OpenAI author-side. Different-family scientific alignment review remains pending.", + "verification_reuse": "All saved compilation, named kernel, axiom and controls evidence reused. No Lean, kernel replay, tests, or old verification executed during this projection.", + "packaging_note": "Original raw logs, commands, metadata, failures and control/helper source bytes preserved. Lean source artifacts have .lean.txt projection suffix; actual original paths remain in records. This index is not a hosted receipt or accepted scientific alignment record.", + "author_history_supplement": { + "path": "AUTHOR_HISTORY_SUPPLEMENT.json", + "sha256": "e3c4a4fbf56f5938f65138d8c61c632f921ee84cc6e9f1b6ebd7af70f8da8052", + "record": { + "record_type": "author tool-history reports; not a machine-created execution receipt", + "scientific_effect": "NONE", + "reports": [ + { + "author": "OpenAI/Codex /root/uniform_jet_moment_sources", + "scope": [ + "SourceDerivativeGaussian", + "SourceContactCovariance", + "SourceDerivativeContinuity" + ], + "verbatim_report": "Author-history confirmation: all three exact SHA values you list are my final source bytes, unchanged by me since their recorded final compile/controls/axiom/kernel executions. No existing JSON receipt/sourcehash record file was created by me; SHA values were read by sha256sum after source freeze and reported in my source-bound handoffs to root. Treat that as author-history confirmation, not a machine-created execution receipt. Existing command/exit/log triples are the only machine artifacts I produced; please preserve those honest limits and do not rerun.", + "confirmed_source_hashes": { + "SourceDerivativeGaussian.lean": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "SourceContactCovariance.lean": "4189cbaf4c18c55f21e3af2c96b0b2842d83161f44596eb691c507dcc1a439c8", + "SourceDerivativeContinuity.lean": "552511e0aee227133b7d067faeceaab9c43c0094b07d68b8eddab0ab2419775e" + } + }, + { + "author": "OpenAI/Codex /root/regression_kernel", + "scope": [ + "ContactJetCovariance", + "SourceContactSchur" + ], + "control_cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "capture_shell_prelude": "set -o pipefail", + "controls": [ + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log", + "observed_exit": 0 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency-01.log", + "observed_exit": 1 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls-01.log", + "observed_exit": 0 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex-01.log", + "observed_exit": 1 + } + ], + "report_transcription": "Actual control commands, no rerun: cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025; each ran bash set -o pipefail then /home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log observed exit0; corresponding contact-gram RejectedZeroFrequency.lean / RejectedZeroAmplitude.lean with matching -01.log each observed exit1. Schur controls same command pattern under scratch/regression-kernel/source-contact-schur/: PositiveControls.lean to PositiveControls-01.log observedexit0; RejectedZeroPeriod.lean and RejectedWrongQIndex.lean to matching-01.log each observedexit1. These are actual tool-observed command/exits from my current transcript. Helper evidence: /workspace/scratch/regression-kernel/contact-schur/SchurProbe.{lean,log} final exit0 and failed1 retained; /workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.{lean,log} final exit0 and ResidualVariance.failed1.* retained. Helpers coauthored OpenAI/Codex gaussian_api. Earlier failedproduction ContactJetCovariance-01 was assembly-script ValueError/no sourcefile, exit1; ContactJetCovariance-02 success. SourceContactSchur-01/-02/-03 failed author attempts exit1; -04 success.", + "transcription_note": "Faithful plain-text transcription of author report; punctuation/arrow notation normalized. Not a raw machine log." + }, + { + "author": "OpenAI/Codex /root/normalizer_palm_research", + "scope": [ + "GaussianRegressionLimits" + ], + "record_type": "actual author tool-history report; not a machine-created execution receipt", + "verbatim_report": "Author tool-history report (not invented machine receipt): cwd /workspace/Math-/frontiers/source_derivative_gaussian_20261025. Final compile `lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean` exit 0, captured /workspace/scratch/regression-limits/regression-limits-03.log empty. Kernel `lake env leanchecker GaussianRegressionLimits` exit 0, regression-limits-leanchecker.log empty. Final controls `lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean` exit 0, regression-limits-controls-02.log empty. Imported-module #print axioms outputs for regression_parameter_limits and regression_uniform_bounds are in controls-01.log (that command exit 1 for unrelated untyped control index); both only propext, Classical.choice, Quot.sound. No production changes occurred after final successful compile/kernel. Current source SHA256 read-only measured 26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be. All three success logs empty SHA e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855; controls01 SHA c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be. Exact source compiled was this source, author-history association; no rerun.", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "confirmed_source_sha256": "26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be", + "source_unchanged_after_final_compile_kernel": true, + "runs": [ + { + "role": "final_compile", + "command": "lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-03.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "named_kernel", + "command": "lake env leanchecker GaussianRegressionLimits", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-leanchecker.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "corrected_controls", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-02.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "initial_controls_container_with_valid_printed_axioms", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 1, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-01.log", + "raw_log_sha256": "c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be" + } + ] + } + ], + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "reuse": "No checks repeated. These metadata limits do not invalidate completed applicable mathematical evidence or authorize a repeated proof campaign." + } + }, + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "append_scope": "The exact ten-module85-target snapshot/catalog8be and published manifestb3f are preserved in historical-ten-module/. Only the four newly qualified modules and actual helper evidence are appended. No proof or check rerun.", + "target_inventory_identity": { + "package": "frontiers/source_derivative_gaussian_20261025", + "public_theorem_lemma_count": 182, + "distinct_from": "Sibling determinant-moment packet has a separate 66-target inventory and scoped reviews; no review transfer." + }, + "historical_catalogs": [ + { + "path": "historical-seven-module/EVIDENCE_CATALOG.json", + "sha256": "f4977b7a28a714112df72a9249dbdb0c81f80539e5f19f43a848c59e9393f963" + }, + { + "path": "historical-nine-module/EVIDENCE_CATALOG.json", + "sha256": "74509aaf3eb1e17bf8fd3b10880aa2eedc9e14b257dc0cb0b693cd76a4770cbb" + }, + { + "path": "historical-ten-module/EVIDENCE_CATALOG.json", + "sha256": "8be0a858dddf374b081542b4c180dd069d3f3ff5a36b576b1c915aa196fd55aa" + } + ], + "published_ten_module_manifest_snapshot": { + "path": "historical-ten-module/published-package-metadata/SOURCE_MANIFEST.json", + "sha256": "b3f276d1ac5dab86b9dbaf1dfae4eda122089ea7a1fc0fbd81b37532b1da679c", + "scope": "published ten-module85-target review scope retained unchanged" + }, + "actual_endpoint_result": { + "theorem": "I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events", + "only_research_inputs": [ + "L != 0", + "fixed arbitrary real height b" + ], + "derived": [ + "actual current pin invertibility for all sufficiently small positive radii", + "actual physical canonical every-pin Gaussian law with derived parameters", + "actual physical determinant second moment <= C*r^4 and integrability", + "positive physical normalizer >= c*r^2", + "literal weighted physical endpoint probability and square-root transfer for every measurable endpoint event" + ], + "still_in_original_goal": [ + "whole-field geometric Kac-Rice/Palm identification", + "arbitrary field-dependent geometric events", + "GT5 and further parents" + ], + "not_asserted": [ + "uniformity over unbounded b", + "independent review or scientific acceptance" + ] + }, + "transitive_axiom_coverage": [ + { + "target": "I4SourceContact.sourceContactEndpointSchur_posSemidef", + "kind": "TRANSITIVE_FROM_PRINTED_PARENT", + "source_path": "SourceContactEndpointParameters.lean", + "source_sha256": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "printed_parent": "I4SourceEndpointEstimates.sourceEndpoint_normalizer_floor", + "printed_parent_source_path": "SourceEndpointEstimates.lean", + "printed_parent_source_sha256": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_code_edge": { + "line": 259, + "text": "hm hS hPSD (sourceContactEndpointSchur_posSemidef hL)" + }, + "raw_log_path": "/workspace/scratch/dm2_landed_composition/endpoint_estimates/axioms.log", + "printed_parent_log_line": 38, + "permitted_axioms_by_parent_bound": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "qualification": "Applicable completed named-kernel and printed-parent transitive axiom evidence reused; no direct target #print output claimed.", + "raw_log_sha256": "ded91623f2200300ff7b8ab366f5aa64564a157dd98166cb3eedaa10c884caf1" + }, + { + "target": "I4SourceContact.sourceContactEndpointSchur_four_four_eq_schur", + "kind": "TRANSITIVE_FROM_PRINTED_PARENT", + "source_path": "SourceContactEndpointParameters.lean", + "source_sha256": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "printed_parent": "I4SourceContact.sourceContactEndpointSchur_referenceVariance_pos", + "printed_parent_source_path": "SourceContactEndpointParameters.lean", + "printed_parent_source_sha256": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "source_code_edge": { + "line": 322, + "text": "rw [sourceContactEndpointSchur_four_four_eq_schur]" + }, + "raw_log_path": "/workspace/scratch/contact-endpoint-parameters/contact-parameters-controls-01.log", + "printed_parent_log_line": 25, + "permitted_axioms_by_parent_bound": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "qualification": "Applicable completed named-kernel and printed-parent transitive axiom evidence reused; no direct target #print output claimed.", + "raw_log_sha256": "3eb74647a2d1698732992034708b9c24d542250b15358cac3db4130130c3183a" + } + ], + "direct_report_omissions": [ + { + "module": "SourceContactEndpointParameters", + "targets": [ + "I4SourceContact.sourceContactEndpointSchur_posSemidef", + "I4SourceContact.sourceContactEndpointSchur_four_four_eq_schur" + ], + "resolution": "TRANSITIVE_FROM_PRINTED_PARENT; not a missing mathematical obligation" + } + ], + "axiom_coverage_summary": { + "public_theorem_lemma_targets": 182, + "extra_audited_inverse_definition": 1, + "directly_printed_declarations": 181, + "transitively_covered_public_helpers": 2, + "total_declaration_coverage": 183, + "all_direct": false, + "permitted_axioms": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "addendum_path": "AXIOM_COVERAGE_ADDENDUM.json", + "addendum_sha256": "238f838a7ddd2caf09631122b079b9a81ff4eb6399f30d8ad86e9906634a92f9" + } +} diff --git a/frontiers/source_derivative_gaussian_20261025/EVIDENCE_SUMMARY.md b/frontiers/source_derivative_gaussian_20261025/EVIDENCE_SUMMARY.md new file mode 100644 index 000000000..0d672f1c6 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/EVIDENCE_SUMMARY.md @@ -0,0 +1,96 @@ +# Fourteen-module actual physical endpoint evidence + +The final theorem +`I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events` +takes only `L != 0` and fixed arbitrary real height `b`. It derives one interval +`0 < r < delta <= 1` with literal physical weight square integrability, +`integral W^2 <= C*r^4`, `integral W >= c*r^2` for `C >= 0,c > 0`, and the +explicit weighted physical endpoint probability/event square-root bound. +Actual covariance limits, current invertibility, Gaussian law/parameters and +normalizer positivity are conclusions, not final model premises. Eventual +actual physical `HasCondDistrib` is also exported. + +The catalog covers **182 public theorem/lemma targets across fourteen modules**. +Axiom coverage comprises **181 directly printed declarations**, including the +actual inverse definition, plus **two public helpers covered transitively by +already printed parents**, for **183 declarations of coverage**. All permitted +axioms are `propext`, `Classical.choice`, `Quot.sound`. This does not claim that +all targets have direct print output. All fourteen named module kernel replay +and final compilation records report exit 0. No proof/check was rerun during +packaging. + +| Module | Public theorem/lemma targets | Principal saved evidence | +|---|---:|---| +| SourceDerivativeGaussian | 5 | Final compile05, controls06, axioms07, named kernel08 | +| ContactJetCovariance | 8 | EVIDENCE.json; compile02, kernel, positive/false controls | +| SourceContactCovariance | 7 | Final compile05, controls06, axioms07, named kernel08 | +| SourceConfluentPins | 8 | Qualification receipt; exact target/signed-radius controls | +| SourceContactSchur | 10 | EVIDENCE.json; compile04, kernel, actual residual variance controls | +| SourceDerivativeContinuity | 6 | Final compile02, corrected controls05, axioms04, kernel06 | +| ConfluentIntegralIdentities | 7 | Qualification receipt; seven literal formulas and false controls | +| SourceEndpointRegression | 13 | EVIDENCE.json; compile03, kernel, thirteen printed reports and controls | +| GaussianRegressionLimits | 2 | Compile03, kernel, corrected controls02; valid axiom reports in failed control container01 | +| SourceEndpointParameters | 19 | EVIDENCE.json; compile02, kernel, nineteen printed reports and controls | +| SourceContactEndpointParameters | 6 | Actual compile author-history, kernel/control subprocess receipts, four direct reports plus two transitive coverage records | +| SourceConfluentLimits | 57 | Namespace-aware targets.json; compile18, genuine subprocess records19/20/21 with final source SHA before/after | +| SourcePhysicalEndpointKernel | 9 | EVIDENCE.json; compile04, kernel, nine printed reports, physical/null-pin correction controls | +| SourceEndpointEstimates | 25 | Final physical compile, qualification receipt, kernel, types, twenty-five printed reports, positive/false controls | + +`EVIDENCE_CATALOG.json` retains exact source hashes, public declarations, +recorded commands/exits and an index of **411 original raw/helper evidence +files**, with **103 execution records**. The 57 ConfluentLimits names retain +their four actual namespaces. Initial failures and corrected control attempts +remain included. Lean control/helper projections end in `.lean.txt`; original +paths and bytes remain bound. The integral module’s harmless linter warning +and the physical-kernel controls’ harmless unused-simp warnings remain recorded. + +`AXIOM_COVERAGE_ADDENDUM.json` resolves the two direct-output omissions by +`TRANSITIVE_FROM_PRINTED_PARENT`, with exact source edges and printed logs: + +- `sourceContactEndpointSchur_posSemidef` is consumed by the printed + `sourceEndpoint_normalizer_floor` at SourceEndpointEstimates.lean:259. +- `sourceContactEndpointSchur_four_four_eq_schur` is consumed by the printed + `sourceContactEndpointSchur_referenceVariance_pos` at + SourceContactEndpointParameters.lean:322. + +This reuses applicable completed mathematical evidence. It is not a new direct +print, fresh audit process or historical machine receipt. No missing mathematical +coverage is inferred from a separate direct-output process distinction. + +`AUTHOR_HISTORY_SUPPLEMENT.json` preserves the honest sparse-metadata limits. +Three original derivative/contact/continuity groups have command/exit/raw-log +triples with author-confirmed unchanged final source association. The generic +regression-limits command/source association is also author tool history. Its +initial controls container exited 1 for an unrelated control elaboration error +while printing two valid imported-module axiom reports; corrected controls02 +separately exited 0. No clean axiom-only process is invented. Newer genuine +subprocess receipts retain their actual before/after source hashes separately. + +The actual `gaussian_api`, `endpoint_means`, `peano_strategy` and +`gaussian_reference_probe` helper contributions are OpenAI author-side work. +The incorporated actual mode-curve, weighted-integral, regularized-continuity, +contact-coefficient and regularized-zero scratch proofs and their original +metadata are preserved. None receives independent-review credit. + +The current input binding remains **23 reused files**: twelve landed smooth +source files and eleven exact sibling determinant-packet modules at commit +`16f81b973a02783f8be83081aa7f198724cf5767`, execution receipt +`95e502472e3d748c8b5dca3328a8644a9cb7cfc8fe40d5f9cce95a32814e5ec6`. +INPUT_BINDING SHA256 remains +`42292a58cf84e118b34f4b8b952fa3b650126dc3d167cee85d8048b9bc6d4b13`. +Their applicable completed verification is reused unchanged. + +The ten-module 85-target catalog snapshot retains SHA256 +`8be0a858dddf374b081542b4c180dd069d3f3ff5a36b576b1c915aa196fd55aa`. +Its published manifest retains SHA256 +`b3f276d1ac5dab86b9dbaf1dfae4eda122089ea7a1fc0fbd81b37532b1da679c`. +Both are preserved under `historical-ten-module/`; their original review scope +is not silently broadened. Seven- and nine-module snapshots also remain intact. + +The fourteen-module source inventory is distinct from the sibling 66-target +packet and its scoped scientific reviews. Different-family scientific review +of this actual-source result remains separate. Scientific effect NONE; no parent +acceptance, hosted execution or status-register change follows from this catalog. +The original goal remains active for whole-field geometric Kac–Rice/Palm +identification, arbitrary field-dependent geometric events, GT5 and further +parents. No uniformity over unbounded height is asserted. diff --git a/frontiers/source_derivative_gaussian_20261025/GaussianRegressionLimits.lean b/frontiers/source_derivative_gaussian_20261025/GaussianRegressionLimits.lean new file mode 100644 index 000000000..4e5485667 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/GaussianRegressionLimits.lean @@ -0,0 +1,159 @@ +import Mathlib.Topology.Instances.Matrix +import Mathlib.Topology.Algebra.GroupWithZero +import Mathlib.Topology.Algebra.Ring.Real +import Mathlib.Topology.Order.LeftRightNhds +import Mathlib.Analysis.Normed.Group.Real +import Mathlib.Analysis.Normed.Group.Constructions +import Mathlib.Analysis.Normed.Group.Continuity +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.Positivity +import Mathlib.Tactic.FunProp + +/-! +# Regression parameter limits and uniform small-radius bounds + +Only the limiting pin covariance is assumed invertible. Current inverses, +regression means/covariances and their uniform bounds are consequences of the +input block limits. No target covariance inverse or zero-mean premise occurs. +See SOURCE_REGRESSION_LIMITS.md for the frozen source contract. +-/ + +noncomputable section +namespace I4GaussianRegressionLimits +open Matrix Filter +open scoped Topology +set_option autoImplicit false + +/-- The actual regression mean for a centered prior and target d. -/ +def regressionMean {p t : Type*} [Fintype p] [DecidableEq p] + (G : Matrix p p ℝ) (C : Matrix t p ℝ) (d : p → ℝ) : t → ℝ := + (C * G⁻¹) *ᵥ d + +/-- The Schur covariance; no inverse of this target block is taken. -/ +def regressionCovariance {p t : Type*} [Fintype p] [DecidableEq p] + (G : Matrix p p ℝ) (C : Matrix t p ℝ) (B : Matrix t t ℝ) : Matrix t t ℝ := + B - C * G⁻¹ * C.transpose + +private theorem tendsto_matrix_mul_explicit {α p t q : Type*} [Fintype t] + {u : Filter α} {A : α → Matrix p t ℝ} {A0 : Matrix p t ℝ} + {B : α → Matrix t q ℝ} {B0 : Matrix t q ℝ} + (hA : Tendsto A u (𝓝 A0)) (hB : Tendsto B u (𝓝 B0)) : + Tendsto (fun r => A r * B r) u (𝓝 (A0 * B0)) := by + have hAij (i : p) (j : t) : Tendsto (fun r => A r i j) u (𝓝 (A0 i j)) := + ((continuous_apply j).tendsto (A0 i)).comp + (((continuous_apply i).tendsto A0).comp hA) + have hBij (i : t) (j : q) : Tendsto (fun r => B r i j) u (𝓝 (B0 i j)) := + ((continuous_apply j).tendsto (B0 i)).comp + (((continuous_apply i).tendsto B0).comp hB) + apply tendsto_pi_nhds.mpr + intro i + apply tendsto_pi_nhds.mpr + intro j + exact tendsto_finsetSum Finset.univ (fun k _ => (hAij i k).mul (hBij k j)) + +private theorem tendsto_matrix_mulVec_explicit {α p t : Type*} [Fintype t] + {u : Filter α} {A : α → Matrix p t ℝ} {A0 : Matrix p t ℝ} + {b : α → (t → ℝ)} {b0 : t → ℝ} + (hA : Tendsto A u (𝓝 A0)) (hb : Tendsto b u (𝓝 b0)) : + Tendsto (fun r => A r *ᵥ b r) u (𝓝 (A0 *ᵥ b0)) := by + have hAij (i : p) (j : t) : Tendsto (fun r => A r i j) u (𝓝 (A0 i j)) := + ((continuous_apply j).tendsto (A0 i)).comp + (((continuous_apply i).tendsto A0).comp hA) + have hbi (i : t) : Tendsto (fun r => b r i) u (𝓝 (b0 i)) := + ((continuous_apply i).tendsto b0).comp hb + apply tendsto_pi_nhds.mpr + intro i + exact tendsto_finsetSum Finset.univ (fun j _ => (hAij i j).mul (hbi j)) + +/-- Continuous pin/target blocks give current invertibility and convergence +of the inverse, noncentered regression mean and possibly singular covariance. -/ +theorem regression_parameter_limits + {p t a : Type*} [Fintype p] [DecidableEq p] [Fintype t] + (u : Filter a) + (G : a → Matrix p p ℝ) (C : a → Matrix t p ℝ) + (B : a → Matrix t t ℝ) (d : a → p → ℝ) + (G0 : Matrix p p ℝ) (C0 : Matrix t p ℝ) (B0 : Matrix t t ℝ) (d0 : p → ℝ) + (hG : Tendsto G u (𝓝 G0)) (hC : Tendsto C u (𝓝 C0)) + (hB : Tendsto B u (𝓝 B0)) (hd : Tendsto d u (𝓝 d0)) (hG0 : IsUnit G0) : + (∀ᶠ r in u, IsUnit (G r)) ∧ + Tendsto (fun r => (G r)⁻¹) u (𝓝 G0⁻¹) ∧ + Tendsto (fun r => regressionMean (G r) (C r) (d r)) u + (𝓝 (regressionMean G0 C0 d0)) ∧ + Tendsto (fun r => regressionCovariance (G r) (C r) (B r)) u + (𝓝 (regressionCovariance G0 C0 B0)) := by + have hdet0 : G0.det ≠ 0 := ((Matrix.isUnit_iff_isUnit_det G0).mp hG0).ne_zero + have hdet : Tendsto (fun r => (G r).det) u (𝓝 G0.det) := + (continuous_id.matrix_det.tendsto G0).comp hG + have hunit : ∀ᶠ r in u, IsUnit (G r) := by + filter_upwards [hdet.eventually_ne hdet0] with r hr + exact (Matrix.isUnit_iff_isUnit_det (G r)).mpr (isUnit_iff_ne_zero.mpr hr) + have hInv : Tendsto (fun r => (G r)⁻¹) u (𝓝 G0⁻¹) := by + have hc : ContinuousAt (Ring.inverse : ℝ → ℝ) G0.det := by + simpa only [Ring.inverse_eq_inv'] using (continuousAt_inv₀ hdet0) + exact (continuousAt_matrix_inv G0 hc).tendsto.comp hG + have hK : Tendsto (fun r => C r * (G r)⁻¹) u (𝓝 (C0 * G0⁻¹)) := + tendsto_matrix_mul_explicit hC hInv + have hm : Tendsto (fun r => regressionMean (G r) (C r) (d r)) u + (𝓝 (regressionMean G0 C0 d0)) := + tendsto_matrix_mulVec_explicit hK hd + have htrans : Tendsto (fun r => (C r).transpose) u (𝓝 C0.transpose) := by + have hc : Continuous (fun D : Matrix t p ℝ => D.transpose) := by fun_prop + exact (hc.tendsto C0).comp hC + have hcov := hB.sub (tendsto_matrix_mul_explicit hK htrans) + exact ⟨hunit, hInv, hm, hcov⟩ + +private theorem bounds_from_limits {ι : Type*} [Fintype ι] + (f : ℝ → (ι → ℝ)) (g : ℝ → Matrix ι ι ℝ) + (f0 : ι → ℝ) (g0 : Matrix ι ι ℝ) + (hf : Tendsto f (𝓝[>] (0 : ℝ)) (𝓝 f0)) + (hg : Tendsto g (𝓝[>] (0 : ℝ)) (𝓝 g0)) : + ∃ B V δ : ℝ, 0 ≤ B ∧ 0 ≤ V ∧ 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → + (∀ i, |f r i| ≤ B) ∧ (∀ i, |g r i i| ≤ V) := by + have hfb : ∀ᶠ r in 𝓝[>] (0 : ℝ), ‖f r‖ < ‖f0‖ + 1 := + hf.norm.eventually (gt_mem_nhds (by linarith)) + have hgg : Tendsto (fun r => fun i j => g r i j) (𝓝[>] (0 : ℝ)) + (𝓝 (fun i j => g0 i j)) := hg + have hgb : ∀ᶠ r in 𝓝[>] (0 : ℝ), ‖fun i j => g r i j‖ < ‖fun i j => g0 i j‖ + 1 := + hgg.norm.eventually (gt_mem_nhds (by linarith)) + obtain ⟨δ, hδ, hbounds⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp (hfb.and hgb) + refine ⟨‖f0‖+1, ‖fun i j => g0 i j‖+1, δ, by positivity, by positivity, hδ, ?_⟩ + intro r hr hrδ + have hb := hbounds ⟨hr, hrδ⟩ + constructor + · intro i + exact (show |f r i| ≤ ‖f r‖ by simpa only [Real.norm_eq_abs] using norm_le_pi_norm (f r) i).trans hb.1.le + · intro i + exact (show |g r i i| ≤ ‖fun i j => g r i j‖ by + exact (show |g r i i| ≤ ‖g r i‖ by simpa only [Real.norm_eq_abs] using norm_le_pi_norm (g r i) i).trans + (norm_le_pi_norm (fun i j => g r i j) i)).trans hb.2.le + + +/-- One small-radius interval supplies current invertibility and simultaneous +mean/diagonal-covariance bounds, derived solely from the block limits. -/ +theorem regression_uniform_bounds + {p t : Type*} [Fintype p] [DecidableEq p] [Fintype t] + (G : ℝ → Matrix p p ℝ) (C : ℝ → Matrix t p ℝ) + (B : ℝ → Matrix t t ℝ) (d : ℝ → p → ℝ) + (G0 : Matrix p p ℝ) (C0 : Matrix t p ℝ) (B0 : Matrix t t ℝ) (d0 : p → ℝ) + (hG : Tendsto G (𝓝[>] (0 : ℝ)) (𝓝 G0)) + (hC : Tendsto C (𝓝[>] (0 : ℝ)) (𝓝 C0)) + (hB : Tendsto B (𝓝[>] (0 : ℝ)) (𝓝 B0)) + (hd : Tendsto d (𝓝[>] (0 : ℝ)) (𝓝 d0)) (hG0 : IsUnit G0) : + ∃ M V δ : ℝ, 0 ≤ M ∧ 0 ≤ V ∧ 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → IsUnit (G r) ∧ + (∀ i, |regressionMean (G r) (C r) (d r) i| ≤ M) ∧ + (∀ i, |regressionCovariance (G r) (C r) (B r) i i| ≤ V) := by + obtain ⟨hunit, _hInv, hm, hcov⟩ := regression_parameter_limits + (𝓝[>] (0 : ℝ)) G C B d G0 C0 B0 d0 hG hC hB hd hG0 + obtain ⟨M, V, δB, hM, hV, hδB, hb⟩ := bounds_from_limits + (fun r => regressionMean (G r) (C r) (d r)) + (fun r => regressionCovariance (G r) (C r) (B r)) + (regressionMean G0 C0 d0) (regressionCovariance G0 C0 B0) hm hcov + obtain ⟨δU, hδU, hu⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hunit + refine ⟨M, V, min δB δU, hM, hV, lt_min hδB hδU, ?_⟩ + intro r hr hrδ + exact ⟨hu ⟨hr, lt_of_lt_of_le hrδ (min_le_right _ _)⟩, + hb r hr (lt_of_lt_of_le hrδ (min_le_left _ _))⟩ + +end I4GaussianRegressionLimits diff --git a/frontiers/source_derivative_gaussian_20261025/INPUT_BINDING.json b/frontiers/source_derivative_gaussian_20261025/INPUT_BINDING.json new file mode 100644 index 000000000..199dd5e71 --- 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This manifest alone grants no runtime admission." +} diff --git a/frontiers/source_derivative_gaussian_20261025/ORIGIN_SOURCE_SHA256SUMS b/frontiers/source_derivative_gaussian_20261025/ORIGIN_SOURCE_SHA256SUMS new file mode 100644 index 000000000..b991528fe --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/ORIGIN_SOURCE_SHA256SUMS @@ -0,0 +1,20 @@ +fde850d24186b6e99c8c0debd641fb3ab85198b109eb66f71a1fea02bd6ea1b3 .gitignore +18f486b9a1e94d449ab078fccdb5bfaec7bad0075c3a8e8d26f96775b68e10f7 DESIGN.md +feea566d5aeba58459bd4a1f20f070bdcd80ee30712817324cdd594e3e232c95 GaussianSeriesFoundation.lean +d3fe096b830c4c5bf2fff47a2a2aa528ae2a79d4cb9816afb6be536685d8d9ca SOURCE_MANIFEST.json +b15e53b825c2e2d6af2af18182da227d8794dddcc3e1b29611a913f89a889397 SourceC4.lean +30560009582659a100637b5510c657cf1fa762f68a37bae8fab42c1eeee6aee5 SourceCovariance.lean +5f145e8e8693bfe4d1554092c8258dbda4f17f8e078c503029e4e8ff2c018014 SourceFourierSeries.lean +a60ab6099d3cef0eeecaefa9e35d95ec14ec54ffd6c89e42dfcf6170907166e8 SourceGaussianLaw.lean +5fa3abc8fb0db0c14c67b4ee840e659cdf80f7d22c9fd4cb5e6c1c7b204fbd21 SourceKernel.lean +c644305a426bc876f609c055ce999b8008f028c67a1091639d919f4122981b7b SourceLattice.lean +12ad7263267a970bd5d5c95d7c7aa62d9c971f7beebf28f1aa7ed16d3a112f1b SourceSmooth.lean +4dfa901029fab71d127c9f4c8cf30c937607f4444e6af64f429890d9d141d591 SourceSmoothAxiomAudit.lean +5defdf223e85e7ccd23b5a903a1a306fe7f723727eb6aabf62eef523ef8639ed SourceSmoothCarrier.lean +8eca5ecdb5e2a0c1cb786468b340e4516df653468f958e535f156baafef088b4 SourceSmoothControls.lean +02857b44552ab0e51a756b3b095c4ba43f8168ac332b069acb5bc8146d337eef SourceTorus.lean +b67b13030389dea4521e8e2afcac30ea44f84abd65c374bf208067b288fc64a1 SourceTorusC4.lean +92af14a2d83304de3f8576ee10b3d376587e93c4464f8c715b8d2e32360f44b0 SpectralWeights.lean +4cc60e1f98fc3a46bccafe6b8315445d950ca1db55ceba26fdebe3a45eafe59a lake-manifest.json +7782dd2fef85bf956a0ccc76e4e6420b8818eabb3bc133cade7252ea7d1d92e6 lakefile.toml +d5edba4e4b8faad9c1baeadb265716d20d03be4d1a2647dc5e35b0c0325bea7b lean-toolchain diff --git a/frontiers/source_derivative_gaussian_20261025/README.md b/frontiers/source_derivative_gaussian_20261025/README.md new file mode 100644 index 000000000..cf8cd96d5 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/README.md @@ -0,0 +1,34 @@ +# Actual Fourier-source derivatives and contact conditioning prerequisites + +This additive Lean package derives the actual canonical physical endpoint estimates from the existing smooth Fourier source. For every nonzero period parameter `L` and fixed real height `b`, it proves one small-radius interval with physical determinant second moment `<= C*r^4`, positive normalizer `>= c*r^2`, and the square-root bound for every measurable endpoint event under the explicitly weighted endpoint probability. The final theorem assumes no covariance limit, conditional Gaussian law, invertibility, moment bound or normalizer floor. The full original research goal remains active: whole-field geometric Palm identification, field-dependent events, GT5 and parent conclusions still require their actual remaining proofs. + +The source is `I4SmoothCarrier.smoothSourceField` under its unchanged `smoothLaw`. `INPUT_BINDING.json` pins twelve byte-identical smooth-source inputs to Math- commit `e75cb81e8961d178f43aa9967dddd11cd579dc8a`. It separately pins eleven exact input modules from the sibling determinant-moment packet at `16f81b973a02783f8be83081aa7f198724cf5767`, under its existing execution receipt `95e502472e3d748c8b5dca3328a8644a9cb7cfc8fe40d5f9cce95a32814e5ec6`. The current binding has 23 reused inputs in total; their completed applicable verification is reused. The historical sources and reviews remain unchanged. Lean is pinned to v4.34.1 and Mathlib to `d13f23b723b8a846827a245b89c10fc7d3f11612`. + +The fourteen completed modules provide: + +| Module | Result and principal hypotheses | +|---|---| +| `SourceDerivativeGaussian` | For every natural dimension, nonzero period, and arbitrary finite collection of derivative orders, points and direction lists, the actual observations are jointly Gaussian. Every smooth sample satisfies the exact differentiated Fourier-series identity. Covariances equal convergent products of the actual differentiated coefficients. Empty families, order zero and repeated observations are retained. | +| `ContactJetCovariance` | A seven-mode polynomial witness forces positivity of a row-summable infinite coefficient Gram matrix. The coefficient identities and nonzero mode amplitudes are explicit hypotheses of this generic certificate. | +| `SourceContactCovariance` | Instantiates that certificate on the literal origin observations `(f,fx,fy,fxx,fxy,fxxx,fyy)` in dimension two. Canonical half-lattice membership, exact selected derivative coefficients and positive actual Fourier amplitudes are proved. For `L != 0`, the full covariance is positive definite. | +| `SourceConfluentPins` | The actual six endpoint pins admit an explicit continuous linear equivalence for `r != 0`. The literal target `(b,b-r^3/6,0,0,0,0)` maps to `(b-r^3/12,0,0,0,0,2)`. The raw pins, confluent pins and joint twelve-vector of raw pins plus endpoint Hessians have their Gaussianity derived from actual derivatives. | +| `SourceContactSchur` | For `L != 0`, the contact pin covariance of `J=(f,fx,fy,fxx,fxy,fxxx)` is positive definite and invertible. Its exact `6+1` decomposition with `Q=fyy` yields a positive Schur complement, identified with the actual least-squares curvature residual variance. | +| `SourceDerivativeContinuity` | Derivative covariance is continuous in both evaluation points, using point-independent summable coefficient majorants. Every actual directional derivative has zero unconditional mean. This does not substitute zero for conditional regression means. | +| `ConfluentIntegralIdentities` | For real scalar curves `F` of class C3 and `H` of class C2, exact average and Peano-kernel formulas remove the pin and adjusted-endpoint divided differences. The identities preserve signed nonzero `r`. | +| `GaussianRegressionLimits` | From finite block limits for `G,C,B,d` and invertibility of `G0`, derives eventual current `G` invertibility, convergence of `C G^-1 d` and `B-C G^-1 C^T`, and uniform small-radius mean/diagonal bounds. Empty dimensions and singular target covariance are retained; concrete source block limits remain an application obligation. | +| `SourceEndpointRegression` | For the actual source, derives joint Gaussianity of confluent pins and adjusted endpoints, proves the actual matrix normal equations, and constructs the explicit measurable Markov residual kernel at every pin value. `HasCondDistrib` uses the explicit current actual-pin covariance invertibility premise. On the prescribed fiber, adjusted endpoints equal the physical normalized six-vector. | +| `SourceEndpointParameters` | Derives actual zero unconditional means, Gaussian residual/posterior laws, the exact regression mean and Schur covariance, and PSD of that covariance. Under current actual pin invertibility, the canonical all-pin kernel becomes the literal `multivariateGaussian` after the specified Euclidean coordinate transport, including singular laws and noncentered prescribed-pin means. | +| `SourceContactEndpointParameters` | Derives the actual limiting endpoint mean relations, singular PSD covariance, contact zero-variance relations and strictly positive `Q` reference variance from literal origin derivatives. Arbitrary height and the actual torus regression mean are retained. | +| `SourceConfluentLimits` | Proves exact differentiated Fourier and Peano representations, uniform summable mode majorants, contact coefficient identities, and the three actual pin/cross/endpoint covariance limits as `r -> 0+`. Its 57 exported theorems span four namespaces and are inventoried by their full names. | +| `SourcePhysicalEndpointKernel` | Constructs the physical normalized-endpoint kernel using the actual common residual and the pin-dependent correction. Proves its actual `HasCondDistrib` under current pin invertibility and literal equality with the adjusted kernel at the prescribed null pin. Away from that pin, the correction is retained. | +| `SourceEndpointEstimates` | Composes the actual limits, contact parameters, canonical physical law and reused moment/normalizer/reweighting proofs. From only `L != 0` and fixed `b`, derives eventual actual current invertibility/physical `HasCondDistrib`, uniform parameters, `E W^2 <= C*r^4`, `E W >= c*r^2`, and the explicit weighted physical endpoint probability/event bound. | + +The preimplementation contracts are the `SOURCE_*.md` files. `I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events` is the final composition theorem. Its constants may depend on `L` and the fixed height `b`; no uniformity over unbounded height is asserted. Strict maximum/saddle typing and zero/type boundaries are retained, with an explicit coordinate permutation between the two six-vector conventions. A midpoint quartic nine-vector is not substituted for the physical endpoint vector. + +The remaining original-goal work is whole-field geometric Kac–Rice/Palm identification, transfer for arbitrary field-dependent geometric events, GT5 and further parents. An endpoint-marginal theorem does not identify those events. Scientific-status and parent acceptance do not follow from this package. + +All implementation and technical assistance are OpenAI author-side work, including `/root`, `/root/uniform_jet_moment_sources`, `/root/regression_kernel`, `/root/dm2_landed_composition`, `/root/normalizer_palm_research`, and the `gaussian_api`, `endpoint_means`, `peano_strategy` and `gaussian_reference_probe` helpers. The mode-chain, weighted-integral, regularized-continuity and contact-coefficient helpers incorporated into the new modules are credited in the evidence catalog. Different-family scientific alignment review remains pending. Existing module compilation, named kernel replay, axiom audits and mathematical controls are recorded in the evidence catalog and reused. This packaging adds no combined hosted run or independent scientific acceptance. Scientific effect: NONE. No parent or scientific-status acceptance is claimed. + +This actual-source package has 182 public theorem/lemma targets across fourteen modules. Its inventory is distinct from the sibling determinant-moment packet’s 66 targets. Earlier sibling scientific reviews retain their original scope and do not cover this actual-source inventory. + +The evidence catalog distinguishes direct axiom reports from two helper exports covered by their exact already printed parents; `AXIOM_COVERAGE_ADDENDUM.json` records that reuse. No verification was repeated for packaging. diff --git a/frontiers/source_derivative_gaussian_20261025/SCOPE.md b/frontiers/source_derivative_gaussian_20261025/SCOPE.md new file mode 100644 index 000000000..2cc18b168 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SCOPE.md @@ -0,0 +1,41 @@ +# Exact fourteen-module research scope + +This document describes the fourteen completed modules named below. Later modules need their own exact statements and execution evidence before being added. The full original research goal remains active. + +`SourceDerivativeGaussian` works with the literal `SmoothSample d L`, `smoothLaw d L` and `smoothSourceField d L`. Its model premise is `L != 0`; dimension, family size, orders, points and direction lists are arbitrary. Differentiated deterministic coefficient summability and the every-sample derivative-series identity are conclusions. Joint Gaussianity is derived from the preserved infinite coefficient law. The covariance conclusion is the infinite differentiated-coefficient pairing; it is not yet an identification with derivatives of a theta-kernel expression. + +`ContactJetCovariance` is a generic coefficient certificate. Row summability, seven specified witness columns, nonzero column amplitudes and nonzero frequency scale imply positive definiteness of the full infinite Gram matrix. The proof uses the sum-of-squares quadratic form and the explicit witness's separating columns. It does not replace the infinite source by those seven modes. + +`SourceContactCovariance` supplies those hypotheses for the actual two-dimensional source. Its origin coordinate order is `(f,fx,fy,fxx,fxy,fxxx,fyy)`. The selected modes are DC, cos(1,0), cos(0,1), cos(1,1), sin(1,0), sin(2,0), sin(0,1), with their actual source amplitudes and canonical half-lattice membership. From `L != 0` it derives positive definiteness of the literal contact covariance; no covariance positivity is assumed. + +`SourceContactSchur` defines `J=(f,fx,fy,fxx,fxy,fxxx)`, `Q=fyy`, their covariance blocks `G,b,d`, and `d-b dotProduct (G inverse mulVec b)`. An exact coordinate equivalence proves the `6+1` block identity. From `L != 0` it derives contact `G` positivity/invertibility, Schur positivity and equality with the variance of the actual curvature residual `Q-(G inverse mulVec b) dotProduct J`. These are contact claims. They do not alone prove current invertibility at all sufficiently small nonzero separations or select a conditional distribution at a prescribed null pin. + +`SourceConfluentPins` fixes endpoints `(-r/2,0),(r/2,0)` and raw order `(fM,fS,fxM,fxS,fyM,fyS)`. Its explicit transform is invertible for `r != 0`; the Gaussian laws are consequences of actual derivative observations. Arbitrary real height `b` is retained in the exact target `(b-r^3/12,0,0,0,0,2)`. The twelve-observation Gaussian statement includes endpoint Hessian order `(fxxM,fxyM,fyyM,fxxS,fxyS,fyyS)` jointly with the raw pins. It does not assume independence or a physical-endpoint identity with a midpoint quartic jet. + +`SourceDerivativeContinuity` proves continuity of every fixed pair of directional-derivative covariances in its evaluation points using global summable bounds. Zero means refer to the unchanged unconditional source law. Conditional torus means are not replaced by zero or a planar reference value. + +`ConfluentIntegralIdentities` is a deterministic scalar result for `F : Real -> Real` of class C3, `H : Real -> Real` of class C2 and signed `r != 0`. It proves the exact integral formulas for the second and transverse divided differences, cubic Peano pin, and left/right adjusted endpoints. The transverse curve `H` denotes the y-derivative curve, so its pin uses `H` itself. Neither an integral representation nor a quotient limit is assumed. Actual-source instantiation and justified infinite-sum limiting arguments are supplied by `SourceConfluentLimits` below. + +`GaussianRegressionLimits` is a deterministic matrix theorem. Given convergence of finite pin, cross, target and prescribed-value blocks `G,C,B,d` to `G0,C0,B0,d0`, and only `IsUnit G0`, it derives eventual current invertibility, inverse convergence, regression mean/covariance convergence and one interval of simultaneous coordinate bounds. The formulas describe a centered prior with arbitrary prescribed target `d`; conditional means need not vanish. It assumes neither target covariance invertibility nor current pin invertibility. Actual source block convergence is not supplied by this generic theorem. + +`SourceEndpointRegression` uses the literal actual source observations. In LM006 order, its adjusted endpoint vector is `((xxM-X3)/r,(xxS-X3)/r,(xyM-X4)/r,(xyS-X4)/r,yyM,yyS)`. Joint Gaussianity comes from the actual twelve-derivative vector and a proved continuous linear image. For current actual pin covariance `G`, the explicit premise `IsUnit G` supplies matrix inverse normal equations; residual Gaussianity and independence are derived through the reused canonical regression theorem. The resulting Markov measurable kernel has its literal residual pushforward at every pin value, including the prescribed null pin, and satisfies `HasCondDistrib`. The physical endpoint identity is proved deterministically on the `X3=X4=0` fiber. This does not assert eventual current invertibility or a canonical whole-field conditioning theorem. + +`SourceEndpointParameters` derives actual unconditional zero means rather than assuming them, then proves residual and all-pin posterior Gaussianity. Under the explicit current actual pin covariance invertibility premise, covariance bilinearity and the normal equations identify the residual covariance with `B-C G^-1 C^T`. PSD follows from that actual covariance; no residual nondegeneracy is assumed. The literal canonical kernel has mean `C G^-1 x` at every pin value and, after the specified `toLp 2` Euclidean coordinate transport, equals the pinned `multivariateGaussian` of that mean and Schur covariance. Singular covariance, arbitrary prescribed pin and nonzero posterior mean are preserved. The included generic finite-Gaussian parameter lemma also retains dimension zero. These local current-invertibility premises are discharged eventually by the actual covariance-limit and estimate modules below. + +`SourceContactEndpointParameters` defines the literal limiting adjusted endpoints `(-J5/2,J5/2,-A/2,A/2,Q,Q)`, with `A=fxxy`, `Q=fyy` and prescribed contact pins `(b,0,0,0,0,2)`. It derives mean coordinates `-1,+1`, the opposite mixed-coordinate relation, duplicate `Q` relation, PSD covariance and the necessary zero variances. The reference covariance entry equals the already proved actual curvature Schur complement and is strictly positive. Singular covariance is preserved; neither a planar law nor a centered posterior is substituted. + +`SourceConfluentLimits` derives the actual three block limits. Differentiated coefficient series represent the actual pin/adjusted-endpoint observations. Exact mode-curve and Peano identities remove the singular quotients; point-independent summable majorants and justified infinite-sum limits give continuity of regularized coefficient Grams at zero. Contact coefficient series identify those zero values with the literal contact blocks. Values of the raw divided-difference definitions at `r=0` are not silently equated to their continuous extensions. The 57 public theorem names span `I4SourceConfluentLimits`, `I4ModeCurveChains`, `I4WeightedModeIntegrals` and `I4ContactCoefficientLimits`; their namespace-aware inventory is retained. + +`SourcePhysicalEndpointKernel` proves the deterministic relation between actual physical and adjusted endpoint vectors with correction `D_r x=(x3/r,x3/r,x4/r,x4/r,0,0)`. The physical canonical kernel uses the same actual residual law and regression map plus `D_r`. Its every-pin formula and actual physical `HasCondDistrib` are derived; at the prescribed pin, `D_r` vanishes and the physical and adjusted kernels are literally equal measures. This fixes the probability-zero pin join without choosing an arbitrary conditional-law version. The kernels can differ at other pins, and the correction is not discarded globally. + +`SourceEndpointEstimates` completes the actual endpoint composition. Its final theorem `sourcePhysicalEndpoint_estimates_and_weighted_events` takes only `hL : L != 0` and a fixed arbitrary real height `b`. It derives `C >= 0`, `c > 0` and `0 < delta <= 1` such that every `0 < r < delta` has the literal physical weight square integrable with integral at most `C*r^4`, normalizer at least `c*r^2`, and the explicitly normalized weight-density probability. Every measurable endpoint event satisfies the bound `sqrt(C)/c * sqrt(K_r.real E)` under that weighted law. Actual current invertibility, covariance/mean limits, Gaussian parameters, PSD and the positive normalizer are conclusions of the composition, not final model premises. Eventual actual physical `HasCondDistrib` is also exported. + +Still required within the original goal: whole-field geometric Kac–Rice/Palm identification, arbitrary field-dependent geometric-event transfer, GT5 and later parent conclusions. The final endpoint theorem does not assert uniformity over unbounded `b`. These remaining obligations are not removed from the original research scope. The 14-module inventory now includes the completed actual finite-endpoint joins that earlier checkpoints left pending. + +Applicable completed verification on unchanged sources is reusable. Completed module compilations, named kernel replays, transitive axiom audits and meaningful controls are recorded separately from hosted execution and different-family alignment. This packaging does not rerun or reclassify those checks. All present authors/helpers are OpenAI; `gaussian_api`, `endpoint_means`, `peano_strategy` and `gaussian_reference_probe` assistance is author-side. Different-family scientific review remains pending. Scientific effect NONE; no status-register changes or parent acceptance follow from this packet. + +The 182 public theorem/lemma names inventoried here belong to this fourteen-module actual-source package. The sibling determinant-moment packet has a separate 66-target inventory and its reviews retain their own exact scope. Neither execution receipts nor scientific review transfer between those inventories. + +The current input binding covers 23 reused sources: twelve landed smooth-source files plus eleven exact sibling-packet modules at commit `16f81b973a02783f8be83081aa7f198724cf5767`, under execution receipt `95e502472e3d748c8b5dca3328a8644a9cb7cfc8fe40d5f9cce95a32814e5ec6`. This metadata expansion does not repeat or broaden the completed verification or its scientific review. + +Axiom coverage distinguishes 181 directly printed declarations from two additional public helpers covered transitively through their exact printed parents and source-code dependency edges. `AXIOM_COVERAGE_ADDENDUM.json` records this reuse; no direct output or new audit process is invented for those two helpers. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_INTEGRALS.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_INTEGRALS.md new file mode 100644 index 000000000..8f938e023 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_INTEGRALS.md @@ -0,0 +1,37 @@ +# Exact integral formulas for confluent pins and adjusted endpoints + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/dm2_landed_composition. Source LM006 endpoint-extension NOTE section 3, +immutable identity already given in SOURCE_CONTRACT's sibling Gaussian coupling +source contract. Scientific effect NONE. + +Sole new ConfluentIntegralIdentities.lean. For scalar curves F C3 and H C2 on +the real line, r!=0, I=[-1/2,1/2], define +P3=(F'(r/2)-F'(-r/2))/r, +P4=(H(r/2)-H(-r/2))/r, +P5=(12/r^2)*((F'(r/2)+F'(-r/2))/2-(F(r/2)-F(-r/2))/r). +Prove the literal Peano-kernel identities + + P3 = integral_I F''(r*u), + P4 = integral_I H'(r*u), + P5 = 6*integral_I (1/4-u^2)*F'''(r*u). + +Also prove the adjusted endpoint identities + + (F''(-r/2)-P3)/r = -integral_I (1/2-u)*F'''(r*u), + (F''( r/2)-P3)/r = integral_I (u+1/2)*F'''(r*u), + (H'(-r/2)-P4)/r = -integral_I (1/2-u)*H''(r*u), + (H'( r/2)-P4)/r = integral_I (u+1/2)*H''(r*u). + +Here H is the y-derivative curve, so its divided difference uses H itself. +Use ordinary oriented interval integrals; for I increasing these equal the +restricted-measure Bochner integrals in the source. Do not assume any integral +identity, divided-difference limit or Taylor remainder as a premise. FTOC, +integration by parts and the stated smoothness may be used. Preserve signed +nonzero r. The kernel integrals are 1, 1/2 and the normalized cubic mass1; +derive associated coefficient bounds or limits only if short. + +These deterministic identities will be instantiated on actual differentiated +source modes. No stochastic/concrete covariance conclusion follows without +that instantiation and a justified infinite-sum limit. Reuse completed +verification and leave historical sources unchanged. Root integrates evidence. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_LIMITS.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_LIMITS.md new file mode 100644 index 000000000..f86ad0af4 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_LIMITS.md @@ -0,0 +1,38 @@ +# Actual confluent pin and adjusted endpoint covariance limits + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/uniform_jet_moment_sources. Source LM006 endpoint-extension NOTE section 3, +immutable identity in sibling Gaussian coupling source contract. Scientific effect NONE. + +Sole new SourceConfluentLimits.lean, depending on SourceEndpointRegression's +actual observation definitions and completed ConfluentIntegralIdentities. +For the actual smoothSourceField 2 L, L!=0, let X_r be its LM006 confluent +six pins and V_r its pin-adjusted LM006 endpoint six-vector. Prove as r->0+ +the actual covariance blocks G_r=Cov(X_r,X_r), C_r=Cov(V_r,X_r), +B_r=Cov(V_r,V_r) tend to the covariance blocks of (J,T), where + + J=(f,fx,fy,fxx,fxy,fxxx) at the origin, + T=(-fxxx/2,fxxx/2,-fxxy/2,fxxy/2,fyy,fyy) at the origin. + +No assumed coefficient/integral representation, covariance convergence, +mean-square convergence or uniform moment premise is permitted. Derive the +actual observation coefficient arrays from the literal derivative-series +identities and finite pin transforms. Apply the scalar Peano identities to +the actual deterministic mode curves, prove summable mode majorants independent +of sufficiently small positive r, and justify the infinite-sum limit. Existing +actual derivative covariance continuity may be reused for the integrands. +Do not pass unadjusted Hessians/r to a limit without subtracting the pin rows. + +Public intermediate actual coefficient representations, bounds and convergence +are useful mathematical results and may be exported; do not label the full +block limit proved until it is derived. A staged module with precise completed +theorems is preferable to a theorem adding the desired limit as a hypothesis. +The all-small-r current invertibility and regression-parameter bounds then +follow by the separately proved actual contact positivity and regression +continuity, not from an assumed matrix inverse at every r. + +Coordinate exact definition names with /root/regression_kernel, who alone owns +SourceEndpointRegression. Definitions can be extended arbitrarily at r=0 while +preserving the literal nonzero-radius formulas; right limits do not identify +the singular raw quotient with its extension at0. Reuse all completed proof +evidence and check only this new module. Root integrates Lake/evidence/review. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_PINS.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_PINS.md new file mode 100644 index 000000000..92ca5781b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_PINS.md @@ -0,0 +1,41 @@ +# Actual six pins and an invertible confluent frame + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/dm2_landed_composition. Scientific effect NONE. +Source: LM006 endpoint-extension NOTE section 2 (immutable identity in sibling +SOURCE_GAUSSIAN_COUPLING.md). This is that particular equivalent frame, not the +different GP-DER-044 historical frame. + +Sole new SourceConfluentPins.lean. In raw pin order +(fM,fS,fxM,fxS,fyM,fyS), define for r!=0 + + P0=(fM+fS)/2, P1=(fxM+fxS)/2, P2=(fyM+fyS)/2, + P3=(fxS-fxM)/r, P4=(fyS-fyM)/r, + P5=(12/r^2)*(P1-(fS-fM)/r). + +Prove this is an actual continuous linear equivalence with inverse + + fM=P0-r*P1/2+r^3*P5/24, + fS=P0+r*P1/2-r^3*P5/24, + fxM=P1-r*P3/2, fxS=P1+r*P3/2, + fyM=P2-r*P4/2, fyS=P2+r*P4/2. + +For literal prescribed raw pins (b,b-r^3/6,0,0,0,0), prove the transformed +value is exactly (b-r^3/12,0,0,0,0,2). Preserve arbitrary real b and r!=0; +do not substitute an incorrect nonzero P1 or the planar conditional mean. + +Define these raw observations as the actual values and x/y first derivatives +of smoothSourceField 2 L at (-r/2,0),(r/2,0). Derive their joint Gaussianity +from the completed heterogeneous derivative theorem. Define actual confluent +pins by the proved linear equivalence and derive Gaussianity as a CLM image. +Optional joint Gaussianity with actual endpoint second derivatives may be +included by one finite heterogeneous derivative vector, not by assuming +Gaussianity of the endpoint variables or independence of observations. + +This proves the literal pin transform and prescribed target. Covariance +limits/invertibility, integral formulas for adjusted endpoints, regression +block identification and actual Palm events remain original research work. + +Reuse completed verification. Narrow pinned imports and only this new module; +root integrates Lakefile/source inventory/evidence. No old-source edits, +invented scientific acceptance, custom axioms or assumed conclusions. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_ENDPOINT_PARAMETERS.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_ENDPOINT_PARAMETERS.md new file mode 100644 index 000000000..3066daeb4 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_ENDPOINT_PARAMETERS.md @@ -0,0 +1,33 @@ +# Actual limiting endpoint mean and covariance contact relations + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/normalizer_palm_research. Scientific effect NONE. +Source LM006 endpoint-extension NOTE sections 3–4 and stability NOTE sections +2,5, with immutable identities in the sibling Gaussian source contracts. + +Sole new SourceContactEndpointParameters.lean. On the actual smooth source2L, +let J=(f,fx,fy,fxx,fxy,fxxx), A=fxxy and Q=fyy at0, and define +T=(-J5/2,J5/2,-A/2,A/2,Q,Q). Set G0=Cov(J,J), C0=Cov(T,J), +B0=Cov(T,T), d0=(b,0,0,0,0,2), +m0=C0 G0^(-1)d0, S0=B0-C0 G0^(-1)C0^T. + +For L!=0 and arbitrary real b, prove G0 unit from completed actual contact +positivity, S0 PSD from an actual Gaussian/residual covariance, and exactly +the GaussianNormalizerLimit contact relations: + + m0_0=-1, m0_1=1, m0_2+m0_3=0, m0_4-m0_5=0, + S0_00=0, S0_11=0, + S0_22+2*S0_23+S0_33=0, + S0_44-2*S0_45+S0_55=0, and S0_44>0. + +The last strict inequality must be joined to the actual curvature residual +Schur variance already proved in SourceContactSchur, not assumed. Mean and +zero-variance identities follow from literal linear relations in T and the +conditioned cubic pin value2, not an assumed contact vector law. Preserve +the actual noncentered Q mean; no planar N(-b,2) substitution. An actual +contact Gaussian kernel/law identification is useful but not needed to +replace the concrete matrix/contact conclusions. + +Coordinate the literal T,C0,B0 definition names with the SourceConfluentLimits +owner. Reuse the completed regression/Schur/covariance proofs and verify only +this new module. Root owns integration/evidence and different-family review. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_GRAM.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_GRAM.md new file mode 100644 index 000000000..e25908585 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_GRAM.md @@ -0,0 +1,48 @@ +# Contact pin covariance certificate: frozen before code + +25 October 2026. OpenAI/Codex root and /root/regression_kernel. +This is a prerequisite of the actual conditioning research, scientific effect NONE. + +The primary GP-DER-044 source identity is recorded in the sibling determinant +packet's SOURCE_GAUSSIAN_MOMENTS.md. The concrete Fourier definitions and their +immutable hashes are in this directory's INPUT_BINDING.json. SourceFourierSeries +uses false=cos and true=sin on halfLattice modes with positive amplitudes. +SourceCovariance.scalarSeries_covariance gives the exact coefficient Gram sum. +SourceDerivativeGaussian's separately frozen contract connects differentiated +mode coefficients to the literal smooth source. + +Frozen sole new ContactJetCovariance.lean: in row order +(f,fx,fy,fxx,fxy,fxxx,fyy), use columns +(DC,cos(1,0),cos(0,1),cos(1,1),sin(1,0),sin(2,0),sin(0,1)). +Before multiplying each column by its nonzero amplitude, the explicit rows are + + [1,1,1,1,0,0,0] + [0,0,0,0,c,2*c,0] + [0,0,0,0,0,0,c] + [0,-c^2,0,-c^2,0,0,0] + [0,0,0,-c^2,0,0,0] + [0,0,0,0,-c^3,-8*c^3,0] + [0,0,-c^2,-c^2,0,0,0]. + +Prove injectivity of its transpose (including arbitrary nonzero column +amplitudes) for c!=0. A determinant formula is optional; it is not a premise. +Prove the infinite Gram consequence: any coefficient array coeff : Fin 7 -> +iota -> Real with each row summable, and seven selected columns exactly equal +to the displayed columns times nonzero amplitudes, has positive-definite +Gram matrix G_ij = sum' m, coeff_i(m)*coeff_j(m). No independence, positivity +or invertibility of G may be assumed. Derive the needed summability and the +sum-of-squares quadratic-form identity. A 6+1 block Schur-complement positivity +consequence may be added, proved from PosDef rather than assumed. + +In the actual source c=2*pi/L!=0, DC amplitude=sqrt(1/sourceZ), and other +amplitudes=sqrt(2*sourceWeight/sourceZ)>0. The listed lattice modes satisfy the +canonical half-lattice convention. These coefficient-row identities and exact +physical covariance identification must still be proved for an actual-source +PosDef claim; a generic certificate alone does not discharge that application. +Do not claim the physical pin Gram or reference variance from an uninstantiated +certificate. Contact A=fxxy is allowed correlated or degenerate; only Q's +conditional variance must be positive for the normalizer reference theorem. + +Narrow pinned imports, no old-source changes, no custom axioms or assumed +conclusion. Preserve initial failed attempts and reuse existing verification. +Root alone integrates source inventory/evidence; scientific review is separate. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_SCHUR.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_SCHUR.md new file mode 100644 index 000000000..3ef6fd93a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_SCHUR.md @@ -0,0 +1,26 @@ +# Literal six-pin covariance and reference Schur variance + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/regression_kernel. Scientific effect NONE. + +Sole new SourceContactSchur.lean, depending on the completed actual-source +SourceContactCovariance and generic ContactJetCovariance certificate. Define +the actual principal six-pin covariance G, cross vector b and Q variance d +from sourceContactCovariance in order (f,fx,fy,fxx,fxy,fxxx; Q=fyy). +From L!=0 alone, prove G positive definite and invertible, and +d - b dotProduct (G inverse mulVec b) > 0. Supply the exact reindexing between +the actual seven-observation covariance and its 6+1 block representation; +no assumed block identity, pin invertibility, Schur positivity or variance +lower bound is allowed. The existing contact_schur_pos may be applied after +the literal block identity and positivity transport are proved. + +If short, express this as the actual variance of Q minus its linear least +squares prediction from J using the proved covariance formulas and Gaussian +L2 integrability. This may be a consequence, not a premise. Preserve arbitrary +period L!=0, and do not identify a torus conditional mean with planar -b or +a variance with 2. Finite-r invertibility and parameter limits remain separate +original research obligations. + +Reuse all completed module/kernel/axiom/control evidence on unchanged files. +Check only this new module and its own mathematical controls. Root owns +Lake/source/evidence integration and different-family scientific review. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_SOURCE.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_SOURCE.md new file mode 100644 index 000000000..049eafecc --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTACT_SOURCE.md @@ -0,0 +1,34 @@ +# Instantiate the contact covariance for the literal Fourier field + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/uniform_jet_moment_sources; scientific effect NONE. + +Sole new SourceContactCovariance.lean module, using the immutable inputs and +SourceDerivativeGaussian. Define the actual seven derivative observations at +the origin of smoothSourceField 2 L in order +(f,fx,fy,fxx,fxy,fxxx,fyy), by orders (0,1,1,2,2,3,2) and coordinate direction +lists. This is the same concrete source and common smooth carrier as before. +For L!=0, prove their joint Gaussianity and identify each covariance with the +Gram sum of the actual differentiated Fourier coefficients using the already +proved derivative covariance formula. + +Prove the seven selected coefficient columns are EXACTLY the polynomial +columns in SOURCE_CONTACT_GRAM.md, with c=sourceC L=2*pi/L and the actual +positive source amplitudes. Use canonical source modes, preserving the +half-lattice convention and false=cos/true=sin tags. Prove mode membership, +nonzero frequency and amplitude positivity; do not assume these selected +columns or a finite-mode source replacement. Use the generic seven-mode Gram +certificate from ContactJetCovariance to conclude positive definiteness of +the literal seven-observation covariance. No PosDef/invertible pin covariance +or conditional reference variance may be assumed in this instantiation. + +The six-pin contact covariance is its principal block. A positive reference +Schur variance follows once the 6+1 algebra is supplied. This is a contact +claim only: do not infer finite-r confluent invertibility or uniform parameter +bounds without proving the needed covariance convergence. + +All input and SourceDerivativeGaussian verification already completed is +reusable. Compile/check only this new affected module/closure as necessary. +Retain commands/exits, initial failures and meaningful controls. Root owns +Lake/source inventory/evidence and different-family review routing. No edits +to the landed source or completed derivative theorem are needed. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTRACT.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTRACT.md new file mode 100644 index 000000000..dd80d5fee --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_CONTRACT.md @@ -0,0 +1,47 @@ +# Actual source derivative joint Gaussianity: frozen before code + +25 October 2026. OpenAI/Codex root and /root/uniform_jet_moment_sources. +Scientific effect NONE. This is an actual-field prerequisite in the original +research scope; it does not close the full parent theorem. + +Use byte-identical inputs from the landed smooth Fourier source at Math-main +e75cb81e8961d178f43aa9967dddd11cd579dc8a, directory +frontiers/source_smooth_realization_20261011. Original immutable file hashes +are retained in INPUT_BINDING.json, with the source's own manifest/hash list +as provenance. Do not modify the historical package or its recorded reviews. +SourceSmooth proves all-order mode derivative bounds, summability and a common +good sample event. SourceSmoothCarrier supplies the literal every-sample +smoothSourceField and smoothLaw, preserving the entire product coefficient law. +SourceGaussianLaw proves scalarSeries_map_gaussian for summable deterministic +coefficients. These existing proofs and applicable verification are reused. + +Frozen new SourceDerivativeGaussian.lean target: for d,n natural, L != 0, +orders k : Fin n -> Nat, points x : Fin n -> (Fin d -> Real), and direction +lists u : forall j, Fin(k j) -> (Fin d -> Real), prove HasGaussianLaw of +omega |-> (j |-> iteratedFDeriv Real (k j) +(smoothSourceField d L omega) (x j) (u j)) under smoothLaw d L. +No derivative Gaussianity, derivative series identity, conditional-law result, +covariance nonsingularity or desired moment bound may be assumed. + +Prove deterministic derivative coefficient summability from the existing +all-order operator-norm bounds. Prove the literal derivative-series identity +for EVERY SmoothSample using pinned Mathlib SmoothSeries.iteratedFDeriv_tsum_apply +and ContinuousMultilinearMap.tsum_eval, not a finite-mode replacement. Prove +joint Gaussianity of arbitrary finite arrays of summable scalar series, then +transfer through sampleInclusion_map and substitute the literal identity. +Preserve orders zero, repeated observations, zero direction lists, empty +families, arbitrary finite dimension and period; no unproved independence of +observations is introduced. Optional coordinate laws/finite CLM images must be +proved consequences. Actual six-pin/endpoint pairs follow for each nonzero r +by deterministic finite linear maps, not by assuming an endpoint law. + +This target does not itself compute the derivative covariances, prove pin-Gram +positivity, bound the confluent regression uniformly in r, or identify a +whole-field Palm law. Those joins remain in the original research scope. + +Production changes are confined to this new additive directory. Only necessary +dependency compilation is permitted when existing artifacts are unavailable; +it does not justify rerunning old tests or review campaigns. Check the new +module with narrow pinned imports, kernel replay and a transitive axiom audit. +Retain initial failures, exact commands/exits and mathematical controls. The +root integrates source inventory/evidence and seeks a different-family review. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_COVARIANCE_CONTINUITY.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_COVARIANCE_CONTINUITY.md new file mode 100644 index 000000000..8278ac9a1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_COVARIANCE_CONTINUITY.md @@ -0,0 +1,25 @@ +# Continuous actual derivative covariance before singular pin quotients + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/uniform_jet_moment_sources. Scientific effect NONE. + +Sole new SourceDerivativeContinuity.lean. Using the actual differentiated +Fourier coefficient identity and all-order summable global bounds, prove +continuity of the covariance of any two fixed-order directional derivatives +of smoothSourceField d L as their evaluation points vary, for L!=0. Directions +may be fixed finite lists; preserve dimension/order zero and repeated points. +No assumed derivative covariance continuity or mean-square continuity premise. +Use the convergent actual coefficient-product sum, prove a summable bound +independent of the points, and justify passing continuity through the infinite +sum. The source all-order bounds hold globally, so a compact or finite-mode +replacement is unnecessary. Derive zero means or L2 continuity if short and +useful, without changing the completed SourceDerivativeGaussian module. + +This supplies regularity of the literal covariance blocks used by the LM006 +endpoint-extension note sections 3–4. It alone does not remove the confluent +1/r quotients. Their integral representations and dominated limiting arguments +remain explicit subsequent mathematical joins; do not infer their convergence +from continuity of unadjusted endpoint entries. + +Preserve prior source/kernel/control receipts and check only this new module. +Root owns Lake/source/evidence integration and provider-distinct review routing. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_ENDPOINT_ESTIMATES.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_ENDPOINT_ESTIMATES.md new file mode 100644 index 000000000..08dd5b80c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_ENDPOINT_ESTIMATES.md @@ -0,0 +1,42 @@ +# Actual source endpoint moment, normalizer and weighted-event estimates + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/dm2_landed_composition. Scientific effect NONE. + +Sole new SourceEndpointEstimates.lean. Combine the actual source confluent +covariance limits, contact pin/Schur positivity and contact parameter relations, +canonical every-pin kernel Gaussian law, and regression limits with the already +verified moment/normalizer/Palm-reweighting components. Those components are +byte-identical inputs from the 66-target packet at16f81b973a02783f8be83081aa7f198724cf5767, +whose source hashes and execution receipt are recorded in INPUT_BINDING.json. + +For every L!=0 and fixed real height b, derive constants C>=0,c>0,delta>0, +delta<=1 such that for all0 v+Bx). Prove measurable Markov status, its literal +formula at EVERY x and HasCondDistrib V X. Do not assume residual Gaussianity, +zero covariance, independence or a conditional-law conclusion. Reuse the +verified GaussianRegressionKernel from the 66-target packet as an immutable +input; its source commit/hash is recorded in INPUT_BINDING.json. + +The actual prescribed value is d_r=(b-r^3/12,0,0,0,0,2). Prove the literal +deterministic reconstruction: when X3=X4=0, adjusted V is exactly the six +physical normalized endpoint entries in LM006 order. This is an algebraic +pin-fiber statement; no arbitrary-version RCD may be substituted at d_r. +If short, derive the kernel's Gaussianity/mean/Schur covariance, retaining +arbitrary means or deriving the actual unconditional zero means. The core +conclusion must be about actual source observations, not an assumed model. + +Eventual invertibility is supplied only after the actual confluent covariance +limit is proved and joined to contact positivity. The present current-unit +premise does not claim that missing source result. Whole-field/geometric Palm +identification and event-tail parents remain original research obligations. + +Reuse unchanged source/66-target verification and artifacts. Check only this +new module, preserve failures and controls. Root owns package/evidence/review. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_MANIFEST.json b/frontiers/source_derivative_gaussian_20261025/SOURCE_MANIFEST.json new file mode 100644 index 000000000..f878ec356 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_MANIFEST.json @@ -0,0 +1,710 @@ +{ + "schema_version": 1, + "scientific_effect": "NONE", + "goal_status": "FULL_ORIGINAL_RESEARCH_GOAL_ACTIVE", + "alignment_status": "PENDING_DIFFERENT_FAMILY_REVIEW", + "qualification_role": "fourteen named modules individually locally kernel checked; no combined hosted result", + "source_modules": [ + "SourceDerivativeGaussian.lean", + "ContactJetCovariance.lean", + "SourceContactCovariance.lean", + "SourceConfluentPins.lean", + "SourceContactSchur.lean", + "SourceDerivativeContinuity.lean", + "ConfluentIntegralIdentities.lean", + "SourceEndpointRegression.lean", + "GaussianRegressionLimits.lean", + "SourceEndpointParameters.lean", + "SourceContactEndpointParameters.lean", + "SourceConfluentLimits.lean", + "SourcePhysicalEndpointKernel.lean", + "SourceEndpointEstimates.lean" + ], + "targets": [ + "I4SourceDerivativeGaussian.derivativeCoefficient_summable", + "I4SourceDerivativeGaussian.smoothSourceField_derivative_series", + "I4SourceDerivativeGaussian.scalarSeries_finiteVector_hasGaussianLaw", + "I4SourceDerivativeGaussian.smoothSourceField_derivativeVector_hasGaussianLaw", + "I4SourceDerivativeGaussian.smoothSourceField_derivative_covariance", + "I4ContactGram.contactWitness_transpose_mulVec_eq_zero", + "I4ContactGram.contactWitness_transpose_injective", + "I4ContactGram.contactWitness_amplitude_transpose_injective", + "I4ContactGram.coefficient_product_summable", + "I4ContactGram.coefficientGram_quadratic_hasSum", + "I4ContactGram.coefficientGram_posDef_of_separating", + "I4ContactGram.contact_coefficientGram_posDef", + "I4ContactGram.contact_schur_pos", + "I4SourceContact.sourceContactVector_hasGaussianLaw", + "I4SourceContact.sourceContactCoefficient_summable", + "I4SourceContact.sourceContactCovariance_eq_coefficientGram", + "I4SourceContact.sourceContactMode_nonzero_frequency", + "I4SourceContact.sourceContactAmplitude_pos", + "I4SourceContact.sourceContact_selected_columns", + "I4SourceContact.sourceContactCovariance_posDef", + "I4SourceConfluentPins.rawTransform_confluentTransform", + "I4SourceConfluentPins.confluentTransform_rawTransform", + "I4SourceConfluentPins.confluentEquiv_apply", + "I4SourceConfluentPins.confluentEquiv_symm_apply", + "I4SourceConfluentPins.prescribed_target", + "I4SourceConfluentPins.rawPins_hasGaussianLaw", + "I4SourceConfluentPins.confluentPins_hasGaussianLaw", + "I4SourceConfluentPins.rawPinsWithHessians_hasGaussianLaw", + "I4SourceContact.sourceContactBlockEquiv_inl", + "I4SourceContact.sourceContactBlockEquiv_inr", + "I4SourceContact.sourceContactCovariance_block_identity", + "I4SourceContact.sourcePinCovariance_posDef", + "I4SourceContact.sourcePinCovariance_isUnit", + "I4SourceContact.sourceContactQVariance_eq_variance", + "I4SourceContact.sourceContactSchur_pos", + "I4SourceContact.contact_residual_variance_of_isUnit_covariance", + "I4SourceContact.sourceContactResidual_variance", + "I4SourceContact.sourceContactResidual_variance_pos", + "I4SourceDerivativeContinuity.derivativeMajorant_nonneg", + "I4SourceDerivativeContinuity.derivativeMajorant_summable", + "I4SourceDerivativeContinuity.derivativeCoefficient_bound", + "I4SourceDerivativeContinuity.derivativeCoefficient_continuous", + "I4SourceDerivativeContinuity.smoothSourceField_derivative_covariance_continuous", + "I4SourceDerivativeContinuity.smoothSourceField_derivative_integral_zero", + "I4ConfluentIntegralIdentities.pin3_integral", + "I4ConfluentIntegralIdentities.pin4_integral", + "I4ConfluentIntegralIdentities.pin5_integral", + "I4ConfluentIntegralIdentities.adjusted_second_left", + "I4ConfluentIntegralIdentities.adjusted_second_right", + "I4ConfluentIntegralIdentities.adjusted_transverse_left", + "I4ConfluentIntegralIdentities.adjusted_transverse_right", + "I4SourceEndpointRegression.sourceConfluentPinsAt_eq", + "I4SourceEndpointRegression.endpointObservationMap_actual", + "I4SourceEndpointRegression.sourcePins_adjusted_hasGaussianLaw", + "I4SourceEndpointRegression.sourceRegressionMapAt_apply", + "I4SourceEndpointRegression.sourceRegressionMapAt_single", + "I4SourceEndpointRegression.sourceRegression_normal_equations", + "I4SourceEndpointRegression.sourceEndpointKernel_markov", + "I4SourceEndpointRegression.sourceEndpointKernel_measurable", + "I4SourceEndpointRegression.sourceEndpointKernel_apply", + "I4SourceEndpointRegression.sourceEndpointKernel_hasCondDistrib", + "I4SourceEndpointRegression.sourceEndpointKernel_spec", + "I4SourceEndpointRegression.sourceAdjustedEndpoints_on_pin_fiber", + "I4SourceEndpointRegression.sourceAdjustedEndpoints_on_prescribed", + "I4GaussianRegressionLimits.regression_parameter_limits", + "I4GaussianRegressionLimits.regression_uniform_bounds", + "I4SourceEndpointParameters.smoothSourceField_derivative_integral_zero", + "I4SourceEndpointParameters.rawPinsWithHessians_integral_zero", + "I4SourceEndpointParameters.rawPinsWithHessians_integral_vector_zero", + "I4SourceEndpointParameters.sourcePins_adjusted_integral_zero", + "I4SourceEndpointParameters.sourceConfluentPinsAt_integral_zero", + "I4SourceEndpointParameters.sourceAdjustedEndpoints_integral_zero", + "I4SourceEndpointParameters.sourceEndpointResidual_integral_zero", + "I4SourceEndpointParameters.sourceEndpointResidual_covariance", + "I4SourceEndpointParameters.finiteGaussian_law_parameters", + "I4SourceEndpointParameters.sourceEndpointResidual_hasGaussianLaw", + "I4SourceEndpointParameters.sourceEndpointPosterior_hasGaussianLaw", + "I4SourceEndpointParameters.sourceEndpointPosterior_integral", + "I4SourceEndpointParameters.sourceEndpointPosterior_covariance", + "I4SourceEndpointParameters.sourceEndpointKernel_eq_map_posterior", + "I4SourceEndpointParameters.sourceEndpointKernel_isGaussian", + "I4SourceEndpointParameters.sourceEndpointSchurAt_posSemidef", + "I4SourceEndpointParameters.sourceEndpointKernel_eq_multivariateGaussian", + "I4SourceEndpointParameters.sourceEndpointKernel_integral", + "I4SourceEndpointParameters.sourceEndpointKernel_covariance", + "I4SourceContact.sourceContactEndpointMean_contact", + "I4SourceContact.sourceContactEndpointSchur_covariance", + "I4SourceContact.sourceContactEndpointSchur_posSemidef", + "I4SourceContact.sourceContactEndpointSchur_contact", + "I4SourceContact.sourceContactEndpointSchur_four_four_eq_schur", + "I4SourceContact.sourceContactEndpointSchur_referenceVariance_pos", + "I4SourceConfluentLimits.rawPinsWithHessians_series", + "I4SourceConfluentLimits.pinCoefficient_summable", + "I4SourceConfluentLimits.adjustedCoefficient_summable", + "I4SourceConfluentLimits.sourceConfluentPinsAt_series", + "I4SourceConfluentLimits.sourceAdjustedEndpoints_series", + "I4SourceConfluentLimits.sourcePinCovarianceAt_coefficient", + "I4SourceConfluentLimits.sourceCrossCovarianceAt_coefficient", + "I4SourceConfluentLimits.sourceEndpointCovarianceAt_coefficient", + "I4ModeCurveChains.mode_smooth", + "I4ModeCurveChains.axis_iteratedDeriv", + "I4ModeCurveChains.axis_transverse_smooth", + "I4ModeCurveChains.axis_transverse_iteratedDeriv", + "I4ModeCurveChains.modeH_eq", + "I4ModeCurveChains.modeF_C3", + "I4ModeCurveChains.modeH_C2", + "I4ModeCurveChains.modeF_deriv", + "I4ModeCurveChains.modeF_second", + "I4ModeCurveChains.modeF_third", + "I4ModeCurveChains.modeH_deriv", + "I4ModeCurveChains.modeH_second", + "I4SourceConfluentLimits.rawDerivativeCoefficient_explicit", + "I4SourceConfluentLimits.pinCoefficient_explicit", + "I4SourceConfluentLimits.adjustedCoefficient_explicit", + "I4WeightedModeIntegrals.weighted_integrand_bound", + "I4WeightedModeIntegrals.weightedDerivativeIntegral_continuous", + 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Two helper axiom coverages are explicitly transitive from printed parents, not invented direct reports.", + "not_qualified_by_this_manifest": [] +} diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_PHYSICAL_ENDPOINT_KERNEL.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_PHYSICAL_ENDPOINT_KERNEL.md new file mode 100644 index 000000000..b517abade --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_PHYSICAL_ENDPOINT_KERNEL.md @@ -0,0 +1,29 @@ +# Literal physical endpoint canonical law at the prescribed null pin + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/regression_kernel. Scientific effect NONE. + +Sole new SourcePhysicalEndpointKernel.lean. A deterministic identity on the +zero-probability pin fiber alone must not be used to identify arbitrary RCD +versions at that pin. Supply the full canonical affine-kernel argument. + +Let X=sourceConfluentPinsAt, V=sourceAdjustedEndpoints, Y=sourceNormalizedEndpoints +in the same LM006 order, and define the literal CLM +D_r(x)=(x3/r,x3/r,x4/r,x4/r,0,0). Prove Y=V+D_r X pointwise. +Using the actual residual R=V-BX, define the physical kernel with the SAME +residual law and affine map B+D_r. Under L!=0 and current G invertible, derive +HasCondDistrib Y X, Markov/measurability and the literal pushforward at every x. +Residual joint Gaussianity/zero covariance/independence must be derived/reused +from actual joint(X,V) and normal equations, not assumed. + +For every prescribed d_r=(b-r^3/12,0,0,0,0,2), prove D_r(d_r)=0 and the +physical kernel at d_r equals the already constructed adjusted kernel there, +as literal measures, including null pins. Preserve arbitrary b and allow the +target covariance singular. This proves the actual physical endpoint law +needed by SourceEndpointEstimates; do not infer it merely from an a.e. +conditional-distribution equivalence or a zero-measure fiber identity. + +Reuse completed source/regression verification and artifacts. Check only this +new module, with a control at a nonprescribed pin where D_r(x)!=0 to ensure +the two kernels are not silently equated everywhere. Root integrates evidence +and different-family source review. Whole-field Palm/event joins remain open. diff --git a/frontiers/source_derivative_gaussian_20261025/SOURCE_REGRESSION_LIMITS.md b/frontiers/source_derivative_gaussian_20261025/SOURCE_REGRESSION_LIMITS.md new file mode 100644 index 000000000..044882304 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SOURCE_REGRESSION_LIMITS.md @@ -0,0 +1,27 @@ +# Continuous regression blocks and uniform small-radius parameters + +Frozen before code, 25 October 2026. OpenAI/Codex root and +/root/normalizer_palm_research. Source LM006 endpoint-extension NOTE section 4, +immutable identity in sibling SOURCE_GAUSSIAN_COUPLING.md. Scientific effect NONE. + +Sole new GaussianRegressionLimits.lean. For arbitrary finite pin/target sizes, +let G_r,C_r,B_r,d_r converge as r->0+ to G0,C0,B0,d0, with G0 invertible. +Define the actual regression formulas +m_r=C_r*G_r^(-1)*d_r and S_r=B_r-C_r*G_r^(-1)*C_r^T. +Prove eventual invertibility of G_r and convergence of m_r,S_r to those same +formulas at0. Do not assume current invertibility, regression parameter limits +or boundedness as premises. Matrix determinant/inverse continuity on its unit +domain may be used. No inverse or nondegeneracy is required of target S_r/S0. + +Derive existence of B,V>=0 and delta>0 such that every0 if i.val < 4 then directionX else if i.val < 6 then directionY + else if i.val % 3 = 0 then directionX + else if i.val % 3 = 1 ∧ a.val = 0 then directionX else directionY + +def rawDerivativeCoefficient (L r : ℝ) (i : Fin 12) (m : SourceMode 2) : ℝ := + derivativeCoefficient 2 L (rawDerivativeOrder i) (rawDerivativePoint r i) + (rawDerivativeDirections i) m + +def pinObservationMap (r : ℝ) : (Fin 12 → ℝ) →L[ℝ] (Fin 6 → ℝ) := + (ContinuousLinearMap.fst ℝ (Fin 6 → ℝ) (Fin 6 → ℝ)).comp (endpointObservationMap r) + +def adjustedObservationMap (r : ℝ) : (Fin 12 → ℝ) →L[ℝ] (Fin 6 → ℝ) := + (ContinuousLinearMap.snd ℝ (Fin 6 → ℝ) (Fin 6 → ℝ)).comp (endpointObservationMap r) + +def pinCoefficient (L r : ℝ) (i : Fin 6) (m : SourceMode 2) : ℝ := + pinObservationMap r (fun j => rawDerivativeCoefficient L r j m) i + +def adjustedCoefficient (L r : ℝ) (i : Fin 6) (m : SourceMode 2) : ℝ := + adjustedObservationMap r (fun j => rawDerivativeCoefficient L r j m) i + +private theorem derivativeCoefficient_weighted_summable {L : ℝ} (hL : L ≠ 0) + (ω : SmoothSample 2 L) (k : ℕ) (x : Fin 2 → ℝ) (u : Fin k → (Fin 2 → ℝ)) : + Summable (fun m => derivativeCoefficient 2 L k x u m * sampleInclusion 2 L ω m) := by + apply ((ω.property k).mul_right (∏ j, ‖u j‖)).of_norm_bounded + intro m + rw [norm_mul, Real.norm_eq_abs] + calc + _ ≤ derivativeMajorant 2 L k u m * |sampleInclusion 2 L ω m| := + mul_le_mul_of_nonneg_right (derivativeCoefficient_bound 2 hL k u m x) (abs_nonneg _) + _ = (orderModeBound k 2 L m * |sampleInclusion 2 L ω m|) * ∏ j, ‖u j‖ := by + unfold derivativeMajorant + ring + +private theorem finiteSeries_map {ι : Type*} {n q : ℕ} + (a : Fin n → ι → ℝ) (ω : ι → ℝ) + (ha : ∀ i, Summable (fun m => a i m * ω m)) + (T : (Fin n → ℝ) →L[ℝ] (Fin q → ℝ)) : + T (fun i => scalarSeries (a i) ω) = + (fun j => scalarSeries (fun m => T (fun i => a i m) j) ω) := by + have hs : HasSum (fun m i => a i m * ω m) (fun i => scalarSeries (a i) ω) := + Pi.hasSum.mpr (fun i => (ha i).hasSum) + have ht := hs.map T.toAddMonoidHom T.continuous + funext j + have hj : HasSum (fun m => T (fun i => a i m) j * ω m) + (T (fun i => scalarSeries (a i) ω) j) := by + convert Pi.hasSum.mp ht j using 1 + · funext m + have hm : (fun i => a i m * ω m) = ω m • (fun i => a i m) := by + funext i + simp only [Pi.smul_apply, smul_eq_mul, mul_comm] + change T (fun i => a i m) j * ω m = T (fun i => a i m * ω m) j + rw [hm, map_smul] + exact mul_comm _ _ + · rfl + exact hj.tsum_eq.symm + +private theorem finiteCoefficient_map_summable {ι : Type*} {n q : ℕ} + (a : Fin n → ι → ℝ) (ha : ∀ i, Summable (a i)) + (T : (Fin n → ℝ) →L[ℝ] (Fin q → ℝ)) (j : Fin q) : + Summable (fun m => T (fun i => a i m) j) := by + have hs : HasSum (fun m i => a i m) (fun i => ∑' m, a i m) := + Pi.hasSum.mpr (fun i => (ha i).hasSum) + exact (Pi.hasSum.mp (hs.map T.toAddMonoidHom T.continuous) j).summable + +theorem rawPinsWithHessians_series {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (ω : SmoothSample 2 L) : + rawPinsWithHessians L r ω = + (fun i => scalarSeries (rawDerivativeCoefficient L r i) (sampleInclusion 2 L ω)) := by + funext i + unfold rawDerivativeCoefficient + rw [← smoothSourceField_derivative_series 2 hL ω] + fin_cases i <;> + dsimp [rawPinsWithHessians, rawPins, rawDerivativeCoefficient, rawDerivativeOrder, + rawDerivativePoint, rawDerivativeDirections] + all_goals first + | rfl + | exact (iteratedFDeriv_one_apply (rawDerivativeDirections 2)).symm + | exact (iteratedFDeriv_one_apply (rawDerivativeDirections 3)).symm + | exact (iteratedFDeriv_one_apply (rawDerivativeDirections 4)).symm + | exact (iteratedFDeriv_one_apply (rawDerivativeDirections 5)).symm + | apply congrArg (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointM r)) + ext a + fin_cases a <;> rfl + | apply congrArg (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) + ext a + fin_cases a <;> rfl + +theorem pinCoefficient_summable {L : ℝ} (hL : L ≠ 0) (r : ℝ) (i : Fin 6) : + Summable (pinCoefficient L r i) := + finiteCoefficient_map_summable (rawDerivativeCoefficient L r) + (fun j => derivativeCoefficient_summable 2 hL (rawDerivativeOrder j) + (rawDerivativePoint r j) (rawDerivativeDirections j)) (pinObservationMap r) i + +theorem adjustedCoefficient_summable {L : ℝ} (hL : L ≠ 0) (r : ℝ) (i : Fin 6) : + Summable (adjustedCoefficient L r i) := + finiteCoefficient_map_summable (rawDerivativeCoefficient L r) + (fun j => derivativeCoefficient_summable 2 hL (rawDerivativeOrder j) + (rawDerivativePoint r j) (rawDerivativeDirections j)) (adjustedObservationMap r) i + +theorem sourceConfluentPinsAt_series {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (ω : SmoothSample 2 L) : + sourceConfluentPinsAt L r ω = + (fun i => scalarSeries (pinCoefficient L r i) (sampleInclusion 2 L ω)) := by + have hactual := congrArg Prod.fst (endpointObservationMap_actual L r ω) + change pinObservationMap r (rawPinsWithHessians L r ω) = sourceConfluentPinsAt L r ω + at hactual + rw [← hactual, rawPinsWithHessians_series hL] + exact finiteSeries_map (rawDerivativeCoefficient L r) (sampleInclusion 2 L ω) + (fun j => derivativeCoefficient_weighted_summable hL ω (rawDerivativeOrder j) + (rawDerivativePoint r j) (rawDerivativeDirections j)) (pinObservationMap r) + +theorem sourceAdjustedEndpoints_series {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (ω : SmoothSample 2 L) : + sourceAdjustedEndpoints L r ω = + (fun i => scalarSeries (adjustedCoefficient L r i) (sampleInclusion 2 L ω)) := by + have hactual := congrArg Prod.snd (endpointObservationMap_actual L r ω) + change adjustedObservationMap r (rawPinsWithHessians L r ω) = sourceAdjustedEndpoints L r ω + at hactual + rw [← hactual, rawPinsWithHessians_series hL] + exact finiteSeries_map (rawDerivativeCoefficient L r) (sampleInclusion 2 L ω) + (fun j => derivativeCoefficient_weighted_summable hL ω (rawDerivativeOrder j) + (rawDerivativePoint r j) (rawDerivativeDirections j)) (adjustedObservationMap r) + +theorem sourcePinCovarianceAt_coefficient {L : ℝ} (hL : L ≠ 0) (r : ℝ) (i j : Fin 6) : + sourcePinCovarianceAt L r i j = ∑' m, pinCoefficient L r i m * pinCoefficient L r j m := by + unfold sourcePinCovarianceAt + simp_rw [sourceConfluentPinsAt_series hL] + rw [covariance_comp_inclusion 2 hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (pinCoefficient_summable hL r i) (pinCoefficient_summable hL r j) + +theorem sourceCrossCovarianceAt_coefficient {L : ℝ} (hL : L ≠ 0) (r : ℝ) (i j : Fin 6) : + sourceCrossCovarianceAt L r i j = + ∑' m, adjustedCoefficient L r i m * pinCoefficient L r j m := by + unfold sourceCrossCovarianceAt + simp_rw [sourceConfluentPinsAt_series hL, sourceAdjustedEndpoints_series hL] + rw [covariance_comp_inclusion 2 hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (adjustedCoefficient_summable hL r i) (pinCoefficient_summable hL r j) + +theorem sourceEndpointCovarianceAt_coefficient {L : ℝ} (hL : L ≠ 0) (r : ℝ) (i j : Fin 6) : + sourceEndpointCovarianceAt L r i j = + ∑' m, adjustedCoefficient L r i m * adjustedCoefficient L r j m := by + unfold sourceEndpointCovarianceAt + simp_rw [sourceAdjustedEndpoints_series hL] + rw [covariance_comp_inclusion 2 hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (adjustedCoefficient_summable hL r i) (adjustedCoefficient_summable hL r j) + +end I4SourceConfluentLimits + + +/- The following source-mode chain qualifications were coauthored by +OpenAI/Codex /root/dm2_landed_composition and supplied as compiled scratch +ModeCurveChains.lean; this is proof authorship, not independent review. -/ + + +open I4Source I4Weights I4SourceDerivativeGaussian I4SourceConfluentPins + +namespace I4ModeCurveChains + +def axisMap : ℝ →L[ℝ] (Fin 2 → ℝ) where + toFun t := ![t,0] + map_add' s t := by ext i; fin_cases i <;> simp + map_smul' c t := by ext i; fin_cases i <;> simp + cont := by fun_prop + +theorem mode_smooth (L : ℝ) (m : SourceMode 2) : + ContDiff ℝ (⊤ : ℕ∞) (modeCoefficient 2 L m) := by + cases m with + | none => + change ContDiff ℝ (⊤ : ℕ∞) (fun _ : Fin 2 → ℝ => Real.sqrt (1 / sourceZ 2 L)) + fun_prop + | some p => + rcases p with ⟨k,b⟩ + change ContDiff ℝ (⊤ : ℕ∞) (fun x : Fin 2 → ℝ => + Real.sqrt (2 * sourceWeight 2 L k / sourceZ 2 L) * + if b then Real.sin (phase 2 L k x) else Real.cos (phase 2 L k x)) + cases b <;> simp only [Bool.false_eq_true, ite_false, ite_true] <;> + unfold phase <;> fun_prop + +theorem axis_iteratedDeriv (f : (Fin 2 → ℝ) → ℝ) + (hf : ContDiff ℝ (⊤ : ℕ∞) f) (k : ℕ) (t : ℝ) : + iteratedDeriv k (fun s => f ![s,0]) t = + iteratedFDeriv ℝ k f ![t,0] (fun _ => directionX) := by + rw [iteratedDeriv_eq_iteratedFDeriv] + have h := axisMap.iteratedFDeriv_comp_right hf t (i := k) (by simp) + have hh := congrArg (fun A => A (fun _ => (1 : ℝ))) h + simpa [Function.comp_def, axisMap, directionX, + ContinuousMultilinearMap.compContinuousLinearMap_apply] using hh + +theorem axis_transverse_smooth (f : (Fin 2 → ℝ) → ℝ) + (hf : ContDiff ℝ (⊤ : ℕ∞) f) : + ContDiff ℝ (⊤ : ℕ∞) (fun t => fderiv ℝ f ![t,0] directionY) := by + have hdf : ContDiff ℝ (⊤ : ℕ∞) (fderiv ℝ f) := hf.fderiv_right (by simp) + exact (hdf.clm_apply contDiff_const).comp axisMap.contDiff + +theorem axis_transverse_iteratedDeriv (f : (Fin 2 → ℝ) → ℝ) + (hf : ContDiff ℝ (⊤ : ℕ∞) f) (k : ℕ) (t : ℝ) : + iteratedDeriv k (fun s => fderiv ℝ f ![s,0] directionY) t = + iteratedFDeriv ℝ (k+1) f ![t,0] (Fin.snoc (fun _ => directionX) directionY) := by + have hdf : ContDiff ℝ (⊤ : ℕ∞) (fderiv ℝ f) := hf.fderiv_right (by simp) + have hfun : ContDiff ℝ (⊤ : ℕ∞) (fun x => fderiv ℝ f x directionY) := + hdf.clm_apply contDiff_const + calc + _ = iteratedFDeriv ℝ k (fun x => fderiv ℝ f x directionY) ![t,0] + (fun _ => directionX) := axis_iteratedDeriv _ hfun k t + _ = iteratedFDeriv ℝ k (fderiv ℝ f) ![t,0] (fun _ => directionX) directionY := + iteratedFDeriv_clm_apply_const_apply hdf (by simp) + _ = _ := by + rw [iteratedFDeriv_succ_apply_right, Fin.init_snoc, Fin.snoc_last] + +def modeF (L : ℝ) (m : SourceMode 2) (t : ℝ) : ℝ := modeCoefficient 2 L m ![t,0] +def modeH (L : ℝ) (m : SourceMode 2) (t : ℝ) : ℝ := + derivativeCoefficient 2 L 1 ![t,0] ![directionY] m + +theorem modeH_eq (L : ℝ) (m : SourceMode 2) : + modeH L m = fun t => fderiv ℝ (modeCoefficient 2 L m) ![t,0] directionY := by + funext t + exact iteratedFDeriv_one_apply _ + +theorem modeF_C3 (L : ℝ) (m : SourceMode 2) : ContDiff ℝ 3 (modeF L m) := + ((mode_smooth L m).comp axisMap.contDiff).of_le (by simp) + +theorem modeH_C2 (L : ℝ) (m : SourceMode 2) : ContDiff ℝ 2 (modeH L m) := by + rw [modeH_eq] + exact (axis_transverse_smooth _ (mode_smooth L m)).of_le (by simp) + +theorem modeF_deriv (L : ℝ) (m : SourceMode 2) (t : ℝ) : + deriv (modeF L m) t = derivativeCoefficient 2 L 1 ![t,0] ![directionX] m := by + change deriv (fun s => modeCoefficient 2 L m ![s,0]) t = _ + have h := axis_iteratedDeriv _ (mode_smooth L m) 1 t + simpa [derivativeCoefficient, iteratedDeriv_one] using h + +theorem modeF_second (L : ℝ) (m : SourceMode 2) (t : ℝ) : + iteratedDeriv 2 (modeF L m) t = + derivativeCoefficient 2 L 2 ![t,0] ![directionX,directionX] m := by + change iteratedDeriv 2 (fun s => modeCoefficient 2 L m ![s,0]) t = _ + have h := axis_iteratedDeriv _ (mode_smooth L m) 2 t + have hd : (fun _ : Fin 2 => directionX) = ![directionX,directionX] := by + ext i; fin_cases i <;> rfl + simpa only [derivativeCoefficient, hd] using h + +theorem modeF_third (L : ℝ) (m : SourceMode 2) (t : ℝ) : + iteratedDeriv 3 (modeF L m) t = + derivativeCoefficient 2 L 3 ![t,0] ![directionX,directionX,directionX] m := by + change iteratedDeriv 3 (fun s => modeCoefficient 2 L m ![s,0]) t = _ + have h := axis_iteratedDeriv _ (mode_smooth L m) 3 t + have hd : (fun _ : Fin 3 => directionX) = ![directionX,directionX,directionX] := by + ext i; fin_cases i <;> rfl + simpa only [derivativeCoefficient, hd] using h + +theorem modeH_deriv (L : ℝ) (m : SourceMode 2) (t : ℝ) : + deriv (modeH L m) t = derivativeCoefficient 2 L 2 ![t,0] ![directionX,directionY] m := by + rw [modeH_eq] + have h := axis_transverse_iteratedDeriv _ (mode_smooth L m) 1 t + have hd : Fin.snoc (fun _ : Fin 1 => directionX) directionY = + ![directionX,directionY] := by ext i; fin_cases i <;> rfl + simpa only [iteratedDeriv_one, derivativeCoefficient, hd] using h + +theorem modeH_second (L : ℝ) (m : SourceMode 2) (t : ℝ) : + iteratedDeriv 2 (modeH L m) t = + derivativeCoefficient 2 L 3 ![t,0] ![directionX,directionX,directionY] m := by + rw [modeH_eq] + have h := axis_transverse_iteratedDeriv _ (mode_smooth L m) 2 t + have hd : Fin.snoc (fun _ : Fin 2 => directionX) directionY = + ![directionX,directionX,directionY] := by ext i; fin_cases i <;> rfl + simpa only [derivativeCoefficient, hd] using h + +end I4ModeCurveChains + +namespace I4SourceConfluentLimits + +open I4ModeCurveChains + +def modeYY (L : ℝ) (m : SourceMode 2) (t : ℝ) : ℝ := + derivativeCoefficient 2 L 2 ![t,0] ![directionY,directionY] m + +def modeRawValues (L r : ℝ) (m : SourceMode 2) : Fin 12 → ℝ := + ![modeF L m (-r/2), modeF L m (r/2), + deriv (modeF L m) (-r/2), deriv (modeF L m) (r/2), + modeH L m (-r/2), modeH L m (r/2), + iteratedDeriv 2 (modeF L m) (-r/2), deriv (modeH L m) (-r/2), modeYY L m (-r/2), + iteratedDeriv 2 (modeF L m) (r/2), deriv (modeH L m) (r/2), modeYY L m (r/2)] + +private theorem rawDirections2 : rawDerivativeDirections 2 = ![directionX] := by + funext a + fin_cases a + rfl + +private theorem rawDirections3 : rawDerivativeDirections 3 = ![directionX] := by + funext a + fin_cases a + rfl + +private theorem rawDirections4 : rawDerivativeDirections 4 = ![directionY] := by + funext a + fin_cases a + rfl + +private theorem rawDirections5 : rawDerivativeDirections 5 = ![directionY] := by + funext a + fin_cases a + rfl + +private theorem rawDirections6 : rawDerivativeDirections 6 = ![directionX,directionX] := by + funext a + fin_cases a <;> rfl + +private theorem rawDirections7 : rawDerivativeDirections 7 = ![directionX,directionY] := by + funext a + fin_cases a <;> rfl + +private theorem rawDirections8 : rawDerivativeDirections 8 = ![directionY,directionY] := by + funext a + fin_cases a <;> rfl + +private theorem rawDirections9 : rawDerivativeDirections 9 = ![directionX,directionX] := by + funext a + fin_cases a <;> rfl + +private theorem rawDirections10 : rawDerivativeDirections 10 = ![directionX,directionY] := by + funext a + fin_cases a <;> rfl + +private theorem rawDirections11 : rawDerivativeDirections 11 = ![directionY,directionY] := by + funext a + fin_cases a <;> rfl + +theorem rawDerivativeCoefficient_explicit (L r : ℝ) (m : SourceMode 2) : + (fun i => rawDerivativeCoefficient L r i m) = modeRawValues L r m := by + funext i + fin_cases i + all_goals try rfl + all_goals first + | change rawDerivativeCoefficient L r 2 m = modeRawValues L r m 2 + | change rawDerivativeCoefficient L r 3 m = modeRawValues L r m 3 + | change rawDerivativeCoefficient L r 4 m = modeRawValues L r m 4 + | change rawDerivativeCoefficient L r 5 m = modeRawValues L r m 5 + | change rawDerivativeCoefficient L r 6 m = modeRawValues L r m 6 + | change rawDerivativeCoefficient L r 7 m = modeRawValues L r m 7 + | change rawDerivativeCoefficient L r 8 m = modeRawValues L r m 8 + | change rawDerivativeCoefficient L r 9 m = modeRawValues L r m 9 + | change rawDerivativeCoefficient L r 10 m = modeRawValues L r m 10 + | change rawDerivativeCoefficient L r 11 m = modeRawValues L r m 11 + all_goals unfold rawDerivativeCoefficient + all_goals first + | rw [rawDirections2] + | rw [rawDirections3] + | rw [rawDirections4] + | rw [rawDirections5] + | rw [rawDirections6] + | rw [rawDirections7] + | rw [rawDirections8] + | rw [rawDirections9] + | rw [rawDirections10] + | rw [rawDirections11] + all_goals dsimp only [rawDerivativeOrder, rawDerivativePoint, modeRawValues] + all_goals try simp only [modeF_deriv, modeF_second, modeH_deriv] + all_goals rfl + +def modePinValues (L r : ℝ) (m : SourceMode 2) : Fin 6 → ℝ := + ![(modeF L m (-r/2) + modeF L m (r/2))/2, + (deriv (modeF L m) (-r/2) + deriv (modeF L m) (r/2))/2, + (modeH L m (-r/2) + modeH L m (r/2))/2, + pin3 (modeF L m) r, pin4 (modeH L m) r, pin5 (modeF L m) r] + +theorem pinCoefficient_explicit (L r : ℝ) (m : SourceMode 2) : + (fun i => pinCoefficient L r i m) = modePinValues L r m := by + unfold pinCoefficient + rw [rawDerivativeCoefficient_explicit] + funext i + fin_cases i <;> + simp [pinObservationMap, endpointObservationMap, confluentTransform, modeRawValues, + modePinValues, pin3, pin4, pin5, add_comm] + +def modeAdjustedValues (L r : ℝ) (m : SourceMode 2) : Fin 6 → ℝ := + ![(iteratedDeriv 2 (modeF L m) (-r/2) - pin3 (modeF L m) r)/r, + (iteratedDeriv 2 (modeF L m) (r/2) - pin3 (modeF L m) r)/r, + (deriv (modeH L m) (-r/2) - pin4 (modeH L m) r)/r, + (deriv (modeH L m) (r/2) - pin4 (modeH L m) r)/r, + modeYY L m (-r/2), modeYY L m (r/2)] + +theorem adjustedCoefficient_explicit (L r : ℝ) (m : SourceMode 2) : + (fun i => adjustedCoefficient L r i m) = modeAdjustedValues L r m := by + unfold adjustedCoefficient + rw [rawDerivativeCoefficient_explicit] + funext i + fin_cases i <;> + simp [adjustedObservationMap, endpointObservationMap, confluentTransform, modeRawValues, + modeAdjustedValues, pin3, pin4] + +end I4SourceConfluentLimits + +/- Weighted-integral support coauthored by OpenAI/Codex +/root/dm2_landed_composition, supplied as compiled scratch +WeightedModeIntegrals.lean; no scientific-review credit. -/ + + +open I4Source I4SourceDerivativeGaussian I4SourceDerivativeContinuity +open MeasureTheory Set +open scoped Interval + +namespace I4WeightedModeIntegrals + +def weightedDerivativeIntegral (L : ℝ) (k : ℕ) (u : Fin k → (Fin 2 → ℝ)) + (m : SourceMode 2) (w : ℝ → ℝ) (r : ℝ) : ℝ := + ∫ t in (-1/2 : ℝ)..(1/2 : ℝ), w t * derivativeCoefficient 2 L k ![r*t,0] u m + +theorem weighted_integrand_bound {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) (m : SourceMode 2) (w : ℝ → ℝ) + (K : ℝ) (hK : 0 ≤ K) + (hwbound : ∀ t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ), |w t| ≤ K) + (r t : ℝ) (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : + ‖w t * derivativeCoefficient 2 L k ![r*t,0] u m‖ ≤ + K * derivativeMajorant 2 L k u m := by + rw [norm_mul, Real.norm_eq_abs] + exact mul_le_mul (hwbound t ht) (derivativeCoefficient_bound 2 hL k u m ![r*t,0]) + (norm_nonneg _) hK + +theorem weightedDerivativeIntegral_continuous {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) (m : SourceMode 2) (w : ℝ → ℝ) (hw : Continuous w) + (K : ℝ) (hK : 0 ≤ K) + (hwbound : ∀ t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ), |w t| ≤ K) : + Continuous (weightedDerivativeIntegral L k u m w) := by + unfold weightedDerivativeIntegral + apply intervalIntegral.continuous_of_dominated_interval + (bound := fun _ => K * derivativeMajorant 2 L k u m) + · intro r + have hc : Continuous (fun t => w t * derivativeCoefficient 2 L k ![r*t,0] u m) := + hw.mul ((derivativeCoefficient_continuous 2 L k u m).comp (by fun_prop)) + exact hc.aestronglyMeasurable + · intro r + exact Filter.Eventually.of_forall (fun t ht => + weighted_integrand_bound hL k u m w K hK hwbound r t ht) + · exact continuous_const.intervalIntegrable _ _ + · exact Filter.Eventually.of_forall (fun t _ => + continuous_const.mul ((derivativeCoefficient_continuous 2 L k u m).comp (by fun_prop))) + +theorem weightedDerivativeIntegral_norm_le {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) (m : SourceMode 2) (w : ℝ → ℝ) + (K : ℝ) (hK : 0 ≤ K) + (hwbound : ∀ t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ), |w t| ≤ K) (r : ℝ) : + ‖weightedDerivativeIntegral L k u m w r‖ ≤ K * derivativeMajorant 2 L k u m := by + have h := intervalIntegral.norm_integral_le_of_norm_le_const + (fun t ht => weighted_integrand_bound hL k u m w K hK hwbound r t ht) + norm_num at h + simpa only [weightedDerivativeIntegral, Real.norm_eq_abs, neg_div] using h + +theorem weightedDerivativeIntegral_zero (L : ℝ) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) (m : SourceMode 2) (w : ℝ → ℝ) : + weightedDerivativeIntegral L k u m w 0 = + (∫ t in (-1/2 : ℝ)..(1/2 : ℝ), w t) * derivativeCoefficient 2 L k 0 u m := by + unfold weightedDerivativeIntegral + have hzero : (![0,0] : Fin 2 → ℝ) = 0 := by ext i; fin_cases i <;> rfl + simp_rw [zero_mul, hzero] + exact intervalIntegral.integral_mul_const _ _ + +end I4WeightedModeIntegrals + +namespace I4SourceConfluentLimits + +open I4ModeCurveChains I4WeightedModeIntegrals + +def peanoLeft (t : ℝ) : ℝ := 1/2-t +def peanoRight (t : ℝ) : ℝ := t+1/2 +def peanoCubic (t : ℝ) : ℝ := 1/4-t^2 + +private theorem peano_one_bound (t : ℝ) + (_ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |(1 : ℝ)| ≤ 1 := by norm_num + +private theorem peano_left_bound (t : ℝ) + (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |peanoLeft t| ≤ 1 := by + rw [Set.uIoc_of_le (by norm_num)] at ht + unfold peanoLeft + exact abs_le.mpr ⟨by linarith [ht.2], by linarith [ht.1]⟩ + +private theorem peano_right_bound (t : ℝ) + (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |peanoRight t| ≤ 1 := by + rw [Set.uIoc_of_le (by norm_num)] at ht + unfold peanoRight + exact abs_le.mpr ⟨by linarith [ht.1], by linarith [ht.2]⟩ + +private theorem peano_cubic_bound (t : ℝ) + (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |peanoCubic t| ≤ 1 := by + rw [Set.uIoc_of_le (by norm_num)] at ht + have habs : |t| ≤ (1/2 : ℝ) := abs_le.mpr ⟨by linarith [ht.1], ht.2⟩ + have hsq := (sq_le_sq₀ (abs_nonneg t) (by norm_num : (0 : ℝ) ≤ 1/2)).2 habs + rw [sq_abs] at hsq + unfold peanoCubic + exact abs_le.mpr ⟨by nlinarith, by nlinarith [sq_nonneg t]⟩ + +theorem peano_kernel_masses : + (∫ _t in (-1/2 : ℝ)..(1/2 : ℝ), (1 : ℝ)) = 1 ∧ + (∫ t in (-1/2 : ℝ)..(1/2 : ℝ), peanoLeft t) = 1/2 ∧ + (∫ t in (-1/2 : ℝ)..(1/2 : ℝ), peanoRight t) = 1/2 ∧ + 6 * (∫ t in (-1/2 : ℝ)..(1/2 : ℝ), peanoCubic t) = 1 := by + have hi : IntervalIntegrable (fun t : ℝ => t) volume (-1/2 : ℝ) (1/2 : ℝ) := + continuous_id.intervalIntegrable _ _ + have hp : IntervalIntegrable (fun t : ℝ => t^2) volume (-1/2 : ℝ) (1/2 : ℝ) := + (continuous_id.pow 2).intervalIntegrable _ _ + refine ⟨by norm_num, ?_, ?_, ?_⟩ + · unfold peanoLeft + rw [intervalIntegral.integral_sub (continuous_const.intervalIntegrable _ _) hi] + norm_num [integral_id] + · unfold peanoRight + rw [intervalIntegral.integral_add hi (continuous_const.intervalIntegrable _ _)] + norm_num [integral_id] + · unfold peanoCubic + rw [intervalIntegral.integral_sub (continuous_const.intervalIntegrable _ _) hp] + norm_num [integral_pow] + +/-- The Peano formulas define a genuine continuous extension of each pin mode, +while preserving the singular raw formulas at every nonzero radius. -/ +def regularizedPinCoefficient (L r : ℝ) (i : Fin 6) (m : SourceMode 2) : ℝ := + ![(modeF L m (-r/2) + modeF L m (r/2))/2, + (deriv (modeF L m) (-r/2) + deriv (modeF L m) (r/2))/2, + (modeH L m (-r/2) + modeH L m (r/2))/2, + weightedDerivativeIntegral L 2 ![directionX,directionX] m (fun _ => 1) r, + weightedDerivativeIntegral L 2 ![directionX,directionY] m (fun _ => 1) r, + 6 * weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] m peanoCubic r] i + +def regularizedAdjustedCoefficient (L r : ℝ) (i : Fin 6) (m : SourceMode 2) : ℝ := + ![-weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] m peanoLeft r, + weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] m peanoRight r, + -weightedDerivativeIntegral L 3 ![directionX,directionX,directionY] m peanoLeft r, + weightedDerivativeIntegral L 3 ![directionX,directionX,directionY] m peanoRight r, + modeYY L m (-r/2), modeYY L m (r/2)] i + +theorem pinCoefficient_eq_regularized (L r : ℝ) (hr : r ≠ 0) (i : Fin 6) + (m : SourceMode 2) : pinCoefficient L r i m = regularizedPinCoefficient L r i m := by + rw [show pinCoefficient L r i m = modePinValues L r m i from + congrFun (pinCoefficient_explicit L r m) i] + fin_cases i + · rfl + · rfl + · rfl + · change pin3 (modeF L m) r = _ + rw [pin3_integral _ (modeF_C3 L m) r hr] + simp_rw [modeF_second] + simp [regularizedPinCoefficient, weightedDerivativeIntegral] + · change pin4 (modeH L m) r = _ + rw [pin4_integral _ (modeH_C2 L m) r hr] + simp_rw [modeH_deriv] + simp [regularizedPinCoefficient, weightedDerivativeIntegral] + · change pin5 (modeF L m) r = _ + rw [pin5_integral _ (modeF_C3 L m) r hr] + simp_rw [modeF_third] + rfl + +theorem adjustedCoefficient_eq_regularized (L r : ℝ) (hr : r ≠ 0) (i : Fin 6) + (m : SourceMode 2) : adjustedCoefficient L r i m = regularizedAdjustedCoefficient L r i m := by + rw [show adjustedCoefficient L r i m = modeAdjustedValues L r m i from + congrFun (adjustedCoefficient_explicit L r m) i] + fin_cases i + · change (iteratedDeriv 2 (modeF L m) (-r/2) - pin3 (modeF L m) r)/r = _ + rw [adjusted_second_left _ (modeF_C3 L m) r hr] + simp_rw [modeF_third] + rfl + · change (iteratedDeriv 2 (modeF L m) (r/2) - pin3 (modeF L m) r)/r = _ + rw [adjusted_second_right _ (modeF_C3 L m) r hr] + simp_rw [modeF_third] + rfl + · change (deriv (modeH L m) (-r/2) - pin4 (modeH L m) r)/r = _ + rw [adjusted_transverse_left _ (modeH_C2 L m) r hr] + simp_rw [modeH_second] + rfl + · change (deriv (modeH L m) (r/2) - pin4 (modeH L m) r)/r = _ + rw [adjusted_transverse_right _ (modeH_C2 L m) r hr] + simp_rw [modeH_second] + rfl + · rfl + · rfl + +end I4SourceConfluentLimits + +/- Literal contact coefficient identification coauthored by OpenAI/Codex +/root/normalizer_palm_research, supplied as compiled scratch +ContactCoefficientLimits.lean; no independent-review credit. -/ + +open I4Foundation I4Source I4SourceGaussian I4SmoothCarrier I4SourceDerivativeGaussian +open I4SourceContact I4SourceConfluentPins MeasureTheory ProbabilityTheory +open scoped BigOperators +set_option autoImplicit false + +namespace I4ContactCoefficientLimits + +def contactPinCoefficient (L : ℝ) (i : Fin 6) (m : SourceMode 2) : ℝ := + sourceContactCoefficient L i.castSucc m + +def contactAdjustedCoefficient (L : ℝ) (i : Fin 6) (m : SourceMode 2) : ℝ := + ![-derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m / 2, + derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m / 2, + -derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionY] m / 2, + derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionY] m / 2, + derivativeCoefficient 2 L 2 0 ![directionY,directionY] m, + derivativeCoefficient 2 L 2 0 ![directionY,directionY] m] i + +theorem contactPinCoefficient_summable {L : ℝ} (hL : L ≠ 0) (i : Fin 6) : + Summable (contactPinCoefficient L i) := + sourceContactCoefficient_summable hL i.castSucc + +theorem contactAdjustedCoefficient_summable {L : ℝ} (hL : L ≠ 0) (i : Fin 6) : + Summable (contactAdjustedCoefficient L i) := by + change Summable (fun m => contactAdjustedCoefficient L i m) + have hx := derivativeCoefficient_summable 2 hL 3 0 ![directionX,directionX,directionX] + have ha := derivativeCoefficient_summable 2 hL 3 0 ![directionX,directionX,directionY] + have hq := derivativeCoefficient_summable 2 hL 2 0 ![directionY,directionY] + fin_cases i + · simpa [contactAdjustedCoefficient, div_eq_mul_inv, mul_comm] using + hx.neg.mul_left ((2 : ℝ)⁻¹) + · simpa [contactAdjustedCoefficient, div_eq_mul_inv, mul_comm] using + hx.mul_left ((2 : ℝ)⁻¹) + · simpa [contactAdjustedCoefficient, div_eq_mul_inv, mul_comm] using + ha.neg.mul_left ((2 : ℝ)⁻¹) + · simpa [contactAdjustedCoefficient, div_eq_mul_inv, mul_comm] using + ha.mul_left ((2 : ℝ)⁻¹) + · exact hq + · exact hq + +theorem sourcePinVector_eq_contactSeries {L : ℝ} (hL : L ≠ 0) + (ω : SmoothSample 2 L) (i : Fin 6) : + sourcePinVector L ω i = + scalarSeries (contactPinCoefficient L i) (sampleInclusion 2 L ω) := + smoothSourceField_derivative_series 2 hL ω (contactOrder i.castSucc) 0 + (contactDirections i.castSucc) + +private theorem scalarSeries_div {ι : Type*} (a : ι → ℝ) (ω : ι → ℝ) (c : ℝ) : + scalarSeries (fun m => a m / c) ω = scalarSeries a ω / c := by + simp [scalarSeries, div_mul_eq_mul_div, tsum_div_const] + +private theorem scalarSeries_neg {ι : Type*} (a : ι → ℝ) (ω : ι → ℝ) : + scalarSeries (fun m => -a m) ω = -scalarSeries a ω := by + simp [scalarSeries, neg_mul, tsum_neg] + +theorem sourceContactEndpointVector_eq_contactSeries {L : ℝ} (hL : L ≠ 0) + (ω : SmoothSample 2 L) (i : Fin 6) : + sourceContactEndpointVector L ω i = + scalarSeries (contactAdjustedCoefficient L i) (sampleInclusion 2 L ω) := by + have hx := smoothSourceField_derivative_series 2 hL ω 3 0 + ![directionX,directionX,directionX] + have ha := smoothSourceField_derivative_series 2 hL ω 3 0 + ![directionX,directionX,directionY] + have hq := smoothSourceField_derivative_series 2 hL ω 2 0 ![directionY,directionY] + fin_cases i + · change -iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0 + ![directionX,directionX,directionX] / 2 = + scalarSeries (fun m => -derivativeCoefficient 2 L 3 0 + ![directionX,directionX,directionX] m / 2) (sampleInclusion 2 L ω) + rw [scalarSeries_div, scalarSeries_neg, hx] + · change iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0 + ![directionX,directionX,directionX] / 2 = + scalarSeries (fun m => derivativeCoefficient 2 L 3 0 + ![directionX,directionX,directionX] m / 2) (sampleInclusion 2 L ω) + rw [scalarSeries_div, hx] + · change -iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0 + ![directionX,directionX,directionY] / 2 = + scalarSeries (fun m => -derivativeCoefficient 2 L 3 0 + ![directionX,directionX,directionY] m / 2) (sampleInclusion 2 L ω) + rw [scalarSeries_div, scalarSeries_neg, ha] + · change iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0 + ![directionX,directionX,directionY] / 2 = + scalarSeries (fun m => derivativeCoefficient 2 L 3 0 + ![directionX,directionX,directionY] m / 2) (sampleInclusion 2 L ω) + rw [scalarSeries_div, ha] + · exact hq + · exact hq + +theorem sourcePinCovariance_eq_contactCoefficientGram {L : ℝ} (hL : L ≠ 0) + (i j : Fin 6) : sourcePinCovariance L i j = + ∑' m, contactPinCoefficient L i m * contactPinCoefficient L j m := by + change covariance (fun ω => sourcePinVector L ω i) (fun ω => sourcePinVector L ω j) + (smoothLaw 2 L) = _ + simp_rw [sourcePinVector_eq_contactSeries hL] + rw [covariance_comp_inclusion 2 hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (contactPinCoefficient_summable hL i) (contactPinCoefficient_summable hL j) + +theorem sourceContactEndpointCrossCovariance_eq_contactCoefficientGram {L : ℝ} + (hL : L ≠ 0) (i j : Fin 6) : sourceContactEndpointCrossCovariance L i j = + ∑' m, contactAdjustedCoefficient L i m * contactPinCoefficient L j m := by + unfold sourceContactEndpointCrossCovariance + simp_rw [sourceContactEndpointVector_eq_contactSeries hL, sourcePinVector_eq_contactSeries hL] + rw [covariance_comp_inclusion 2 hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (contactAdjustedCoefficient_summable hL i) (contactPinCoefficient_summable hL j) + +theorem sourceContactEndpointCovariance_eq_contactCoefficientGram {L : ℝ} + (hL : L ≠ 0) (i j : Fin 6) : sourceContactEndpointCovariance L i j = + ∑' m, contactAdjustedCoefficient L i m * contactAdjustedCoefficient L j m := by + unfold sourceContactEndpointCovariance + simp_rw [sourceContactEndpointVector_eq_contactSeries hL] + rw [covariance_comp_inclusion 2 hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (contactAdjustedCoefficient_summable hL i) (contactAdjustedCoefficient_summable hL j) + +end I4ContactCoefficientLimits + +namespace I4SourceConfluentLimits + +open I4ModeCurveChains I4WeightedModeIntegrals I4ContactCoefficientLimits + +def modeComponentMajorant (L : ℝ) (m : SourceMode 2) : Fin 8 → ℝ := + ![derivativeMajorant 2 L 0 ![] m, + derivativeMajorant 2 L 1 ![directionX] m, + derivativeMajorant 2 L 1 ![directionY] m, + derivativeMajorant 2 L 2 ![directionX,directionX] m, + derivativeMajorant 2 L 2 ![directionX,directionY] m, + derivativeMajorant 2 L 3 ![directionX,directionX,directionX] m, + derivativeMajorant 2 L 3 ![directionX,directionX,directionY] m, + derivativeMajorant 2 L 2 ![directionY,directionY] m] + +def confluentModeMajorant (L : ℝ) (m : SourceMode 2) : ℝ := + 6 * ∑ s, modeComponentMajorant L m s + +private theorem modeComponentMajorant_nonneg (L : ℝ) (m : SourceMode 2) (s : Fin 8) : + 0 ≤ modeComponentMajorant L m s := by + fin_cases s <;> dsimp [modeComponentMajorant] + all_goals exact derivativeMajorant_nonneg 2 L _ _ m + +theorem confluentModeMajorant_nonneg (L : ℝ) (m : SourceMode 2) : + 0 ≤ confluentModeMajorant L m := + mul_nonneg (by norm_num) (Finset.sum_nonneg (fun s _ => modeComponentMajorant_nonneg L m s)) + +theorem confluentModeMajorant_summable {L : ℝ} (hL : L ≠ 0) : + Summable (confluentModeMajorant L) := by + apply Summable.mul_left 6 + apply summable_sum + intro s _ + fin_cases s <;> dsimp [modeComponentMajorant] + all_goals exact derivativeMajorant_summable 2 hL _ _ + +private theorem six_component_le_majorant (L : ℝ) (m : SourceMode 2) (s : Fin 8) : + 6 * modeComponentMajorant L m s ≤ confluentModeMajorant L m := by + apply mul_le_mul_of_nonneg_left _ (by norm_num) + exact Finset.single_le_sum (fun t _ => modeComponentMajorant_nonneg L m t) (Finset.mem_univ s) + +private theorem component_le_majorant (L : ℝ) (m : SourceMode 2) (s : Fin 8) : + modeComponentMajorant L m s ≤ confluentModeMajorant L m := by + have h := six_component_le_majorant L m s + have hn := modeComponentMajorant_nonneg L m s + linarith + +private theorem average_norm_le (a b M : ℝ) (ha : ‖a‖ ≤ M) (hb : ‖b‖ ≤ M) : + ‖(a+b)/2‖ ≤ M := by + have h := (norm_add_le a b).trans (add_le_add ha hb) + calc + _ = ‖a+b‖ / 2 := by rw [norm_div]; norm_num + _ ≤ (M+M)/2 := div_le_div_of_nonneg_right h (by norm_num) + _ = M := by ring + +private theorem integral_bound_one {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) (m : SourceMode 2) (w : ℝ → ℝ) + (hw : ∀ t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ), |w t| ≤ 1) (r : ℝ) : + ‖weightedDerivativeIntegral L k u m w r‖ ≤ derivativeMajorant 2 L k u m := by + have h := weightedDerivativeIntegral_norm_le hL k u m w 1 (by norm_num) hw r + simpa only [one_mul] using h + +private theorem modeF_norm_bound {L : ℝ} (hL : L ≠ 0) (m : SourceMode 2) (t : ℝ) : + ‖modeF L m t‖ ≤ derivativeMajorant 2 L 0 ![] m := by + simpa only [derivativeCoefficient, iteratedFDeriv_zero_apply, modeF] using + derivativeCoefficient_bound 2 hL 0 ![] m ![t,0] + +theorem regularizedPinCoefficient_bound {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (i : Fin 6) (m : SourceMode 2) : + ‖regularizedPinCoefficient L r i m‖ ≤ confluentModeMajorant L m := by + fin_cases i + · exact (average_norm_le _ _ _ (modeF_norm_bound hL m (-r/2)) + (modeF_norm_bound hL m (r/2))).trans (component_le_majorant L m 0) + · change ‖(deriv (modeF L m) (-r/2) + deriv (modeF L m) (r/2))/2‖ ≤ _ + simp_rw [modeF_deriv] + exact (average_norm_le _ _ _ (derivativeCoefficient_bound 2 hL 1 ![directionX] m _) + (derivativeCoefficient_bound 2 hL 1 ![directionX] m _)).trans (component_le_majorant L m 1) + · exact (average_norm_le _ _ _ (derivativeCoefficient_bound 2 hL 1 ![directionY] m _) + (derivativeCoefficient_bound 2 hL 1 ![directionY] m _)).trans (component_le_majorant L m 2) + · exact (integral_bound_one hL 2 + ![directionX,directionX] m (fun _ => 1) peano_one_bound r).trans + (component_le_majorant L m 3) + · exact (integral_bound_one hL 2 + ![directionX,directionY] m (fun _ => 1) peano_one_bound r).trans + (component_le_majorant L m 4) + · change ‖6 * weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] m peanoCubic r‖ ≤ _ + rw [norm_mul, show ‖(6 : ℝ)‖ = 6 from by norm_num] + have h := integral_bound_one hL 3 ![directionX,directionX,directionX] m + peanoCubic peano_cubic_bound r + calc + _ ≤ 6 * derivativeMajorant 2 L 3 ![directionX,directionX,directionX] m := + mul_le_mul_of_nonneg_left h (by norm_num) + _ ≤ confluentModeMajorant L m := six_component_le_majorant L m 5 + +theorem regularizedAdjustedCoefficient_bound {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (i : Fin 6) (m : SourceMode 2) : + ‖regularizedAdjustedCoefficient L r i m‖ ≤ confluentModeMajorant L m := by + fin_cases i + · change ‖-weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] m peanoLeft r‖ ≤ _ + rw [norm_neg] + exact (integral_bound_one hL 3 + ![directionX,directionX,directionX] m peanoLeft peano_left_bound r).trans + (component_le_majorant L m 5) + · exact (integral_bound_one hL 3 + ![directionX,directionX,directionX] m peanoRight peano_right_bound r).trans + (component_le_majorant L m 5) + · change ‖-weightedDerivativeIntegral L 3 ![directionX,directionX,directionY] m peanoLeft r‖ ≤ _ + rw [norm_neg] + exact (integral_bound_one hL 3 + ![directionX,directionX,directionY] m peanoLeft peano_left_bound r).trans + (component_le_majorant L m 6) + · exact (integral_bound_one hL 3 + ![directionX,directionX,directionY] m peanoRight peano_right_bound r).trans + (component_le_majorant L m 6) + · exact (derivativeCoefficient_bound 2 hL 2 ![directionY,directionY] m _).trans + (component_le_majorant L m 7) + · exact (derivativeCoefficient_bound 2 hL 2 ![directionY,directionY] m _).trans + (component_le_majorant L m 7) + +end I4SourceConfluentLimits + +/- Coauthor support: OpenAI/Codex /root/dm2_landed_composition: regularized coefficient continuity. +Copied from the supplied compiled scratch proof; no review credit. -/ + + +open I4Source I4SourceDerivativeGaussian I4SourceDerivativeContinuity +open I4SourceConfluentPins I4ModeCurveChains I4WeightedModeIntegrals +open MeasureTheory Set +open scoped Interval + +namespace I4SourceConfluentLimits + +private theorem helper_left_bound (t : ℝ) + (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |peanoLeft t| ≤ 1 := by + rw [Set.uIoc_of_le (by norm_num)] at ht + unfold peanoLeft + exact abs_le.mpr ⟨by linarith [ht.2], by linarith [ht.1]⟩ + +private theorem helper_right_bound (t : ℝ) + (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |peanoRight t| ≤ 1 := by + rw [Set.uIoc_of_le (by norm_num)] at ht + unfold peanoRight + exact abs_le.mpr ⟨by linarith [ht.1], by linarith [ht.2]⟩ + +private theorem helper_cubic_bound (t : ℝ) + (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |peanoCubic t| ≤ 1 := by + rw [Set.uIoc_of_le (by norm_num)] at ht + have habs : |t| ≤ (1/2 : ℝ) := abs_le.mpr ⟨by linarith [ht.1], ht.2⟩ + have hsq := (sq_le_sq₀ (abs_nonneg t) (by norm_num : (0 : ℝ) ≤ 1/2)).2 habs + rw [sq_abs] at hsq + unfold peanoCubic + exact abs_le.mpr ⟨by nlinarith, by nlinarith [sq_nonneg t]⟩ + +theorem regularizedPinCoefficient_continuous {L : ℝ} (hL : L ≠ 0) + (i : Fin 6) (m : SourceMode 2) : + Continuous (fun r => regularizedPinCoefficient L r i m) := by + have hF := (modeF_C3 L m).continuous + have hH := (modeH_C2 L m).continuous + have hF' : Continuous (deriv (modeF L m)) := by + simpa only [iteratedDeriv_one] using + (modeF_C3 L m).continuous_iteratedDeriv 1 (by norm_num) + have hpoly : Continuous peanoCubic := by unfold peanoCubic; fun_prop + fin_cases i + · change Continuous (fun r : ℝ => (modeF L m (-r/2) + modeF L m (r/2))/2) + exact ((hF.comp (by fun_prop)).add (hF.comp (by fun_prop))).div_const 2 + · change Continuous (fun r : ℝ => (deriv (modeF L m) (-r/2) + deriv (modeF L m) (r/2))/2) + exact ((hF'.comp (by fun_prop)).add (hF'.comp (by fun_prop))).div_const 2 + · change Continuous (fun r : ℝ => (modeH L m (-r/2) + modeH L m (r/2))/2) + exact ((hH.comp (by fun_prop)).add (hH.comp (by fun_prop))).div_const 2 + · exact weightedDerivativeIntegral_continuous hL 2 ![directionX,directionX] m + (fun _ => 1) continuous_const 1 (by norm_num) (by intro t ht; norm_num) + · exact weightedDerivativeIntegral_continuous hL 2 ![directionX,directionY] m + (fun _ => 1) continuous_const 1 (by norm_num) (by intro t ht; norm_num) + · exact (weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionX] m + peanoCubic hpoly 1 (by norm_num) helper_cubic_bound).const_mul 6 + +theorem regularizedAdjustedCoefficient_continuous {L : ℝ} (hL : L ≠ 0) + (i : Fin 6) (m : SourceMode 2) : + Continuous (fun r => regularizedAdjustedCoefficient L r i m) := by + have hleft : Continuous peanoLeft := by unfold peanoLeft; fun_prop + have hright : Continuous peanoRight := by unfold peanoRight; fun_prop + have hYY : Continuous (modeYY L m) := + (derivativeCoefficient_continuous 2 L 2 ![directionY,directionY] m).comp (by fun_prop) + fin_cases i + · exact (weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionX] m + peanoLeft hleft 1 (by norm_num) helper_left_bound).neg + · exact weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionX] m + peanoRight hright 1 (by norm_num) helper_right_bound + · exact (weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionY] m + peanoLeft hleft 1 (by norm_num) helper_left_bound).neg + · exact weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionY] m + peanoRight hright 1 (by norm_num) helper_right_bound + · exact hYY.comp (by fun_prop) + · exact hYY.comp (by fun_prop) + +end I4SourceConfluentLimits + +/- Coauthor support: OpenAI/Codex /root/normalizer_palm_research: exact contact coefficient values. +Copied from the supplied compiled scratch proof; no review credit. -/ + +open I4Source I4SourceDerivativeGaussian I4SourceConfluentPins I4SourceContact +open I4SourceConfluentLimits I4ModeCurveChains I4WeightedModeIntegrals I4ContactCoefficientLimits +open MeasureTheory +open scoped BigOperators Interval +set_option autoImplicit false + +namespace I4SourceConfluentLimits + +theorem regularizedPinCoefficient_zero (L : ℝ) (i : Fin 6) (m : SourceMode 2) : + regularizedPinCoefficient L 0 i m = contactPinCoefficient L i m := by + have hpoint : (![0,0] : Fin 2 → ℝ) = 0 := by ext j; fin_cases j <;> rfl + have hm := peano_kernel_masses + fin_cases i + · change (modeF L m (-0/2) + modeF L m (0/2))/2 = + derivativeCoefficient 2 L 0 0 ![] m + simp [modeF, derivativeCoefficient, iteratedFDeriv_zero_apply, hpoint] + · change (deriv (modeF L m) (-0/2) + deriv (modeF L m) (0/2))/2 = + derivativeCoefficient 2 L 1 0 ![directionX] m + simp only [neg_zero, zero_div, modeF_deriv, hpoint] + ring + · change (modeH L m (-0/2) + modeH L m (0/2))/2 = + derivativeCoefficient 2 L 1 0 ![directionY] m + simp only [neg_zero, zero_div, modeH, hpoint] + ring + · change weightedDerivativeIntegral L 2 ![directionX,directionX] m (fun _ => 1) 0 = + derivativeCoefficient 2 L 2 0 ![directionX,directionX] m + rw [weightedDerivativeIntegral_zero, hm.1, one_mul] + · change weightedDerivativeIntegral L 2 ![directionX,directionY] m (fun _ => 1) 0 = + derivativeCoefficient 2 L 2 0 ![directionX,directionY] m + rw [weightedDerivativeIntegral_zero, hm.1, one_mul] + · change 6 * weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] + m peanoCubic 0 = derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m + rw [weightedDerivativeIntegral_zero, ← mul_assoc, hm.2.2.2, one_mul] + +theorem regularizedAdjustedCoefficient_zero (L : ℝ) (i : Fin 6) (m : SourceMode 2) : + regularizedAdjustedCoefficient L 0 i m = contactAdjustedCoefficient L i m := by + have hpoint : (![0,0] : Fin 2 → ℝ) = 0 := by ext j; fin_cases j <;> rfl + have hm := peano_kernel_masses + fin_cases i + · change -weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] + m peanoLeft 0 = -derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m / 2 + rw [weightedDerivativeIntegral_zero, hm.2.1] + ring + · change weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] + m peanoRight 0 = derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m / 2 + rw [weightedDerivativeIntegral_zero, hm.2.2.1] + ring + · change -weightedDerivativeIntegral L 3 ![directionX,directionX,directionY] + m peanoLeft 0 = -derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionY] m / 2 + rw [weightedDerivativeIntegral_zero, hm.2.1] + ring + · change weightedDerivativeIntegral L 3 ![directionX,directionX,directionY] + m peanoRight 0 = derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionY] m / 2 + rw [weightedDerivativeIntegral_zero, hm.2.2.1] + ring + · change modeYY L m (-0/2) = derivativeCoefficient 2 L 2 0 ![directionY,directionY] m + simp only [neg_zero, zero_div, modeYY, hpoint] + · change modeYY L m (0/2) = derivativeCoefficient 2 L 2 0 ![directionY,directionY] m + simp only [zero_div, modeYY, hpoint] + +end I4SourceConfluentLimits + +namespace I4SourceConfluentLimits + +open I4ContactCoefficientLimits + +def regularizedPinCovariance (L r : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + fun i j => ∑' m, regularizedPinCoefficient L r i m * regularizedPinCoefficient L r j m + +def regularizedCrossCovariance (L r : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + fun i j => ∑' m, regularizedAdjustedCoefficient L r i m * regularizedPinCoefficient L r j m + +def regularizedEndpointCovariance (L r : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + fun i j => ∑' m, regularizedAdjustedCoefficient L r i m * regularizedAdjustedCoefficient L r j m + +private theorem regularizedGram_continuous {L : ℝ} (hL : L ≠ 0) + (a b : ℝ → Fin 6 → SourceMode 2 → ℝ) + (ha : ∀ i m, Continuous (fun r => a r i m)) + (hb : ∀ i m, Continuous (fun r => b r i m)) + (hba : ∀ r i m, ‖a r i m‖ ≤ confluentModeMajorant L m) + (hbb : ∀ r i m, ‖b r i m‖ ≤ confluentModeMajorant L m) : + Continuous (fun r => (fun i j => ∑' m, a r i m * b r j m : Matrix (Fin 6) (Fin 6) ℝ)) := by + apply continuous_pi + intro i + apply continuous_pi + intro j + apply continuous_tsum + · intro m + exact (ha i m).mul (hb j m) + · exact I4ContactGram.coefficient_product_summable + (confluentModeMajorant_summable hL) (confluentModeMajorant_summable hL) + · intro m r + rw [norm_mul] + exact mul_le_mul (hba r i m) (hbb r j m) (norm_nonneg _) + (confluentModeMajorant_nonneg L m) + +theorem regularizedPinCovariance_continuous {L : ℝ} (hL : L ≠ 0) : + Continuous (regularizedPinCovariance L) := + regularizedGram_continuous hL (regularizedPinCoefficient L) (regularizedPinCoefficient L) + (regularizedPinCoefficient_continuous hL) (regularizedPinCoefficient_continuous hL) + (regularizedPinCoefficient_bound hL) (regularizedPinCoefficient_bound hL) + +theorem regularizedCrossCovariance_continuous {L : ℝ} (hL : L ≠ 0) : + Continuous (regularizedCrossCovariance L) := + regularizedGram_continuous hL (regularizedAdjustedCoefficient L) (regularizedPinCoefficient L) + (regularizedAdjustedCoefficient_continuous hL) (regularizedPinCoefficient_continuous hL) + (regularizedAdjustedCoefficient_bound hL) (regularizedPinCoefficient_bound hL) + +theorem regularizedEndpointCovariance_continuous {L : ℝ} (hL : L ≠ 0) : + Continuous (regularizedEndpointCovariance L) := + regularizedGram_continuous hL (regularizedAdjustedCoefficient L) (regularizedAdjustedCoefficient L) + (regularizedAdjustedCoefficient_continuous hL) (regularizedAdjustedCoefficient_continuous hL) + (regularizedAdjustedCoefficient_bound hL) (regularizedAdjustedCoefficient_bound hL) + +theorem sourcePinCovarianceAt_eq_regularized {L : ℝ} (hL : L ≠ 0) (r : ℝ) (hr : r ≠ 0) : + sourcePinCovarianceAt L r = regularizedPinCovariance L r := by + ext i j + rw [sourcePinCovarianceAt_coefficient hL] + unfold regularizedPinCovariance + apply tsum_congr + intro m + rw [pinCoefficient_eq_regularized L r hr i m, pinCoefficient_eq_regularized L r hr j m] + +theorem sourceCrossCovarianceAt_eq_regularized {L : ℝ} (hL : L ≠ 0) (r : ℝ) (hr : r ≠ 0) : + sourceCrossCovarianceAt L r = regularizedCrossCovariance L r := by + ext i j + rw [sourceCrossCovarianceAt_coefficient hL] + unfold regularizedCrossCovariance + apply tsum_congr + intro m + rw [adjustedCoefficient_eq_regularized L r hr i m, pinCoefficient_eq_regularized L r hr j m] + +theorem sourceEndpointCovarianceAt_eq_regularized {L : ℝ} (hL : L ≠ 0) (r : ℝ) (hr : r ≠ 0) : + sourceEndpointCovarianceAt L r = regularizedEndpointCovariance L r := by + ext i j + rw [sourceEndpointCovarianceAt_coefficient hL] + unfold regularizedEndpointCovariance + apply tsum_congr + intro m + rw [adjustedCoefficient_eq_regularized L r hr i m, adjustedCoefficient_eq_regularized L r hr j m] + +theorem regularizedPinCovariance_zero {L : ℝ} (hL : L ≠ 0) : + regularizedPinCovariance L 0 = sourcePinCovariance L := by + ext i j + rw [sourcePinCovariance_eq_contactCoefficientGram hL] + unfold regularizedPinCovariance + simp_rw [regularizedPinCoefficient_zero] + +theorem regularizedCrossCovariance_zero {L : ℝ} (hL : L ≠ 0) : + regularizedCrossCovariance L 0 = sourceContactEndpointCrossCovariance L := by + ext i j + rw [sourceContactEndpointCrossCovariance_eq_contactCoefficientGram hL] + unfold regularizedCrossCovariance + simp_rw [regularizedPinCoefficient_zero, regularizedAdjustedCoefficient_zero] + +theorem regularizedEndpointCovariance_zero {L : ℝ} (hL : L ≠ 0) : + regularizedEndpointCovariance L 0 = sourceContactEndpointCovariance L := by + ext i j + rw [sourceContactEndpointCovariance_eq_contactCoefficientGram hL] + unfold regularizedEndpointCovariance + simp_rw [regularizedAdjustedCoefficient_zero] + +/-- The actual confluent six-pin covariance converges to the literal contact +pin covariance. Only the source period is assumed nonzero. -/ +theorem sourcePinCovarianceAt_tendsto {L : ℝ} (hL : L ≠ 0) : + Tendsto (sourcePinCovarianceAt L) (𝓝[>] (0 : ℝ)) (𝓝 (sourcePinCovariance L)) := by + have h := (regularizedPinCovariance_continuous hL).tendsto 0 + rw [regularizedPinCovariance_zero hL] at h + apply (h.mono_left nhdsWithin_le_nhds).congr' + filter_upwards [self_mem_nhdsWithin] with r hr + exact (sourcePinCovarianceAt_eq_regularized hL r (ne_of_gt hr)).symm + +/-- The actual adjusted-endpoint/pin cross covariance has the stated literal +contact limit, with the pin corrections retained before division. -/ +theorem sourceCrossCovarianceAt_tendsto {L : ℝ} (hL : L ≠ 0) : + Tendsto (sourceCrossCovarianceAt L) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactEndpointCrossCovariance L)) := by + have h := (regularizedCrossCovariance_continuous hL).tendsto 0 + rw [regularizedCrossCovariance_zero hL] at h + apply (h.mono_left nhdsWithin_le_nhds).congr' + filter_upwards [self_mem_nhdsWithin] with r hr + exact (sourceCrossCovarianceAt_eq_regularized hL r (ne_of_gt hr)).symm + +/-- The actual pin-adjusted endpoint covariance converges to Cov(T,T), where +T=(-fxxx/2,fxxx/2,-fxxy/2,fxxy/2,fyy,fyy) at the origin. -/ +theorem sourceEndpointCovarianceAt_tendsto {L : ℝ} (hL : L ≠ 0) : + Tendsto (sourceEndpointCovarianceAt L) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactEndpointCovariance L)) := by + have h := (regularizedEndpointCovariance_continuous hL).tendsto 0 + rw [regularizedEndpointCovariance_zero hL] at h + apply (h.mono_left nhdsWithin_le_nhds).congr' + filter_upwards [self_mem_nhdsWithin] with r hr + exact (sourceEndpointCovarianceAt_eq_regularized hL r (ne_of_gt hr)).symm + +end I4SourceConfluentLimits diff --git a/frontiers/source_derivative_gaussian_20261025/SourceConfluentPins.lean b/frontiers/source_derivative_gaussian_20261025/SourceConfluentPins.lean new file mode 100644 index 000000000..6ee0be6db --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourceConfluentPins.lean @@ -0,0 +1,152 @@ +import SourceDerivativeGaussian +import Mathlib.Tactic.FieldSimp +import Mathlib.Tactic.FinCases +import Mathlib.Tactic.Ring + +/-! +# Actual six endpoint pins and their confluent frame + +The raw order is `(fM,fS,fxM,fxS,fyM,fyS)`. This is the literal LM006 +endpoint-extension frame, including its arbitrary prescribed height. +See SOURCE_CONFLUENT_PINS.md for the frozen preimplementation contract. +-/ + +noncomputable section + +open I4SmoothCarrier I4SourceDerivativeGaussian +open MeasureTheory ProbabilityTheory +open scoped Topology + +namespace I4SourceConfluentPins + +/-- Literal raw-to-confluent transform, defined also at zero for notation. -/ +def confluentTransform (r : ℝ) (v : Fin 6 → ℝ) : Fin 6 → ℝ := + ![(v 0 + v 1) / 2, (v 2 + v 3) / 2, (v 4 + v 5) / 2, + (v 3 - v 2) / r, (v 5 - v 4) / r, + (12 / r ^ 2) * ((v 2 + v 3) / 2 - (v 1 - v 0) / r)] + +/-- The literal inverse, with the cubic correction to the two field values. -/ +def rawTransform (r : ℝ) (p : Fin 6 → ℝ) : Fin 6 → ℝ := + ![p 0 - r * p 1 / 2 + r ^ 3 * p 5 / 24, + p 0 + r * p 1 / 2 - r ^ 3 * p 5 / 24, + p 1 - r * p 3 / 2, p 1 + r * p 3 / 2, + p 2 - r * p 4 / 2, p 2 + r * p 4 / 2] + +theorem rawTransform_confluentTransform (r : ℝ) (hr : r ≠ 0) (v : Fin 6 → ℝ) : + rawTransform r (confluentTransform r v) = v := by + ext i + fin_cases i <;> simp [rawTransform, confluentTransform] + all_goals field_simp [hr] + all_goals ring + +theorem confluentTransform_rawTransform (r : ℝ) (hr : r ≠ 0) (p : Fin 6 → ℝ) : + confluentTransform r (rawTransform r p) = p := by + ext i + fin_cases i <;> simp [rawTransform, confluentTransform] + all_goals field_simp [hr] + all_goals ring + +/-- For every nonzero separation, the confluent frame is an actual continuous +linear equivalence. No covariance invertibility is assumed or asserted. -/ +def confluentEquiv (r : ℝ) (hr : r ≠ 0) : (Fin 6 → ℝ) ≃L[ℝ] (Fin 6 → ℝ) where + toFun := confluentTransform r + invFun := rawTransform r + left_inv := rawTransform_confluentTransform r hr + right_inv := confluentTransform_rawTransform r hr + map_add' v w := by + ext i + fin_cases i <;> simp [confluentTransform] <;> ring + map_smul' c v := by + ext i + fin_cases i <;> simp [confluentTransform] <;> ring + continuous_toFun := by unfold confluentTransform; fun_prop + continuous_invFun := by unfold rawTransform; fun_prop + +@[simp] theorem confluentEquiv_apply (r : ℝ) (hr : r ≠ 0) (v : Fin 6 → ℝ) : + confluentEquiv r hr v = confluentTransform r v := rfl + +@[simp] theorem confluentEquiv_symm_apply (r : ℝ) (hr : r ≠ 0) (p : Fin 6 → ℝ) : + (confluentEquiv r hr).symm p = rawTransform r p := rfl + +/-- The literal prescribed raw pins keep arbitrary height and have zero first +confluent derivative. The normalized third confluent pin is exactly two. -/ +theorem prescribed_target (r b : ℝ) (hr : r ≠ 0) : + confluentEquiv r hr ![b, b - r ^ 3 / 6, 0, 0, 0, 0] = + ![b - r ^ 3 / 12, 0, 0, 0, 0, 2] := by + ext i + fin_cases i <;> simp [confluentTransform] <;> field_simp [hr] <;> ring + +def endpointM (r : ℝ) : Fin 2 → ℝ := ![-r / 2, 0] +def endpointS (r : ℝ) : Fin 2 → ℝ := ![r / 2, 0] +def directionX : Fin 2 → ℝ := ![1, 0] +def directionY : Fin 2 → ℝ := ![0, 1] + +/-- Actual field values and actual directional x/y first derivatives of the +unchanged smooth Fourier source, in the literal raw pin order. -/ +def rawPins (L r : ℝ) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + ![smoothSourceField 2 L ω (endpointM r), + smoothSourceField 2 L ω (endpointS r), + fderiv ℝ (smoothSourceField 2 L ω) (endpointM r) directionX, + fderiv ℝ (smoothSourceField 2 L ω) (endpointS r) directionX, + fderiv ℝ (smoothSourceField 2 L ω) (endpointM r) directionY, + fderiv ℝ (smoothSourceField 2 L ω) (endpointS r) directionY] + +/-- Joint Gaussianity is derived from one heterogeneous vector of actual +order-zero and order-one observations; no observation independence is used. -/ +theorem rawPins_hasGaussianLaw {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + HasGaussianLaw (rawPins L r) (smoothLaw 2 L) := by + let k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] + let x : Fin 6 → (Fin 2 → ℝ) := + ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] + let u : ∀ j : Fin 6, Fin (k j) → (Fin 2 → ℝ) := + fun j _ => if j.val < 4 then directionX else directionY + have hG := smoothSourceField_derivativeVector_hasGaussianLaw 2 hL 6 k x u + apply hG.congr + filter_upwards with ω + ext i + fin_cases i <;> dsimp [rawPins, k, x, u] + all_goals first | rfl | exact iteratedFDeriv_one_apply _ + +/-- Actual confluent observations are the proved linear image of raw pins. -/ +def confluentPins (L r : ℝ) (hr : r ≠ 0) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + confluentEquiv r hr (rawPins L r ω) + +theorem confluentPins_hasGaussianLaw {L : ℝ} (hL : L ≠ 0) (r : ℝ) (hr : r ≠ 0) : + HasGaussianLaw (confluentPins L r hr) (smoothLaw 2 L) := + (rawPins_hasGaussianLaw hL r).map_equiv_fun (confluentEquiv r hr) + +/-- One literal vector containing the raw six pins followed by endpoint Hessian +entries `(fxxM,fxyM,fyyM,fxxS,fxyS,fyyS)`. No r-dependent midpoint jet is used. -/ +def rawPinsWithHessians (L r : ℝ) (ω : SmoothSample 2 L) : Fin 12 → ℝ := + ![rawPins L r ω 0, rawPins L r ω 1, rawPins L r ω 2, + rawPins L r ω 3, rawPins L r ω 4, rawPins L r ω 5, + iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointM r) ![directionX, directionX], + iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointM r) ![directionX, directionY], + iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointM r) ![directionY, directionY], + iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r) ![directionX, directionX], + iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r) ![directionX, directionY], + iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r) ![directionY, directionY]] + +/-- The pin and Hessian observations are jointly Gaussian, derived together +from their single actual derivative vector rather than separate marginal laws. -/ +theorem rawPinsWithHessians_hasGaussianLaw {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + HasGaussianLaw (rawPinsWithHessians L r) (smoothLaw 2 L) := by + let k : Fin 12 → ℕ := ![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] + let x : Fin 12 → (Fin 2 → ℝ) := + ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, + endpointM r, endpointM r, endpointM r, endpointS r, endpointS r, endpointS r] + let u : ∀ j : Fin 12, Fin (k j) → (Fin 2 → ℝ) := fun j a => + if j.val < 4 then directionX else if j.val < 6 then directionY + else if j.val % 3 = 0 then directionX + else if j.val % 3 = 1 ∧ a.val = 0 then directionX else directionY + have hG := smoothSourceField_derivativeVector_hasGaussianLaw 2 hL 12 k x u + apply hG.congr + filter_upwards with ω + ext i + fin_cases i <;> dsimp [rawPinsWithHessians, rawPins, k, x, u] + all_goals first + | rfl + | exact iteratedFDeriv_one_apply _ + | congr 1; ext a; fin_cases a <;> rfl + +end I4SourceConfluentPins diff --git a/frontiers/source_derivative_gaussian_20261025/SourceContactCovariance.lean b/frontiers/source_derivative_gaussian_20261025/SourceContactCovariance.lean new file mode 100644 index 000000000..b64390842 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourceContactCovariance.lean @@ -0,0 +1,180 @@ +import SourceDerivativeGaussian +import ContactJetCovariance +import Mathlib.Tactic.FinCases +import Mathlib.Tactic.NormNum +import Mathlib.Tactic.Ring + +/-! The actual origin contact covariance of the literal Fourier source. +Rows are (f,fx,fy,fxx,fxy,fxxx,fyy). Selected canonical Fourier modes are +proved to give the seven-mode witness with their actual positive amplitudes. -/ + +noncomputable section +open I4Source I4Weights I4C4 I4SmoothCarrier I4SourceDerivativeGaussian I4ContactGram +open MeasureTheory ProbabilityTheory Matrix +open scoped BigOperators Matrix NNReal + +namespace I4SourceContact + +def contactOrder : Fin 7 → ℕ := ![0,1,1,2,2,3,2] + +def contactE0 : Fin 2 → ℝ := ![1,0] +def contactE1 : Fin 2 → ℝ := ![0,1] + +def contactDirections : (i : Fin 7) → Fin (contactOrder i) → (Fin 2 → ℝ) := + Fin.cases ![] + (Fin.cases ![contactE0] + (Fin.cases ![contactE1] + (Fin.cases ![contactE0,contactE0] + (Fin.cases ![contactE0,contactE1] + (Fin.cases ![contactE0,contactE0,contactE0] + (Fin.cases ![contactE1,contactE1] (fun i => Fin.elim0 i))))))) + +def sourceContactVector (L : ℝ) (ω : SmoothSample 2 L) (i : Fin 7) : ℝ := + iteratedFDeriv ℝ (contactOrder i) (smoothSourceField 2 L ω) 0 (contactDirections i) + +def sourceContactCoefficient (L : ℝ) (i : Fin 7) (m : SourceMode 2) : ℝ := + derivativeCoefficient 2 L (contactOrder i) 0 (contactDirections i) m + +def sourceContactCovariance (L : ℝ) : Matrix (Fin 7) (Fin 7) ℝ := + fun i j => cov[fun ω => sourceContactVector L ω i, + fun ω => sourceContactVector L ω j; smoothLaw 2 L] + +theorem sourceContactVector_hasGaussianLaw {L : ℝ} (hL : L ≠ 0) : + HasGaussianLaw (sourceContactVector L) (smoothLaw 2 L) := + smoothSourceField_derivativeVector_hasGaussianLaw 2 hL 7 contactOrder (fun _ => 0) + contactDirections + +theorem sourceContactCoefficient_summable {L : ℝ} (hL : L ≠ 0) (i : Fin 7) : + Summable (sourceContactCoefficient L i) := + derivativeCoefficient_summable 2 hL (contactOrder i) 0 (contactDirections i) + +theorem sourceContactCovariance_eq_coefficientGram {L : ℝ} (hL : L ≠ 0) : + sourceContactCovariance L = coefficientGram (sourceContactCoefficient L) := by + ext i j + exact smoothSourceField_derivative_covariance 2 hL (contactOrder i) (contactOrder j) + 0 0 (contactDirections i) (contactDirections j) + +private def frequency10 : ↥(halfLattice 2) := ⟨![1,0], by + refine ⟨0, by norm_num, ?_⟩ + intro j hj + fin_cases j <;> norm_num at hj⟩ + +private def frequency01 : ↥(halfLattice 2) := ⟨![0,1], by + refine ⟨1, by norm_num, ?_⟩ + intro j hj + fin_cases j <;> norm_num at *⟩ + +private def frequency11 : ↥(halfLattice 2) := ⟨![1,1], by + refine ⟨0, by norm_num, ?_⟩ + intro j hj + fin_cases j <;> norm_num at hj⟩ + +private def frequency20 : ↥(halfLattice 2) := ⟨![2,0], by + refine ⟨0, by norm_num, ?_⟩ + intro j hj + fin_cases j <;> norm_num at hj⟩ + +/-- DC, cos(1,0), cos(0,1), cos(1,1), sin(1,0), sin(2,0), sin(0,1). +The frequency subtypes retain the canonical half-lattice membership proofs. -/ +def sourceContactMode : Fin 7 → SourceMode 2 := + ![none, some (frequency10,false), some (frequency01,false), some (frequency11,false), + some (frequency10,true), some (frequency20,true), some (frequency01,true)] + +theorem sourceContactMode_nonzero_frequency (s : Fin 7) : + match sourceContactMode s with + | none => True + | some (k, _) => (k : Fin 2 → ℤ) ≠ 0 := by + cases sourceContactMode s with + | none => trivial + | some p => exact I4Lattice.halfLattice_ne_zero 2 p.1.property + +def sourceModeAmplitude (L : ℝ) (m : SourceMode 2) : ℝ := match m with + | none => Real.sqrt (1 / sourceZ 2 L) + | some (k, _) => Real.sqrt (2 * sourceWeight 2 L k / sourceZ 2 L) + +def sourceContactAmplitude (L : ℝ) (s : Fin 7) : ℝ := + sourceModeAmplitude L (sourceContactMode s) + +theorem sourceContactAmplitude_pos {L : ℝ} (hL : L ≠ 0) (s : Fin 7) : + 0 < sourceContactAmplitude L s := by + unfold sourceContactAmplitude sourceModeAmplitude + cases sourceContactMode s with + | none => exact Real.sqrt_pos.mpr (one_div_pos.mpr (sourceZ_pos 2 hL)) + | some p => + apply Real.sqrt_pos.mpr + apply div_pos _ (sourceZ_pos 2 hL) + apply mul_pos (by norm_num) + rw [sourceWeight_eq_physical] + exact Real.exp_pos _ + +private theorem derivativeCoefficient_none (L : ℝ) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) : + derivativeCoefficient 2 L k 0 u none = if k = 0 then Real.sqrt (1 / sourceZ 2 L) else 0 := by + cases k with + | zero => simp [derivativeCoefficient, modeCoefficient, iteratedFDeriv_zero_apply] + | succ k => + simp only [Nat.succ_ne_zero, ite_false] + change (iteratedFDeriv ℝ (k + 1) + (fun _ : Fin 2 → ℝ => Real.sqrt (1 / sourceZ 2 L)) 0) u = 0 + rw [iteratedFDeriv_succ_const] + rfl + +private theorem derivativeCoefficient_some (L : ℝ) (k : ↥(halfLattice 2)) (b : Bool) + (n : ℕ) (u : Fin n → (Fin 2 → ℝ)) : + derivativeCoefficient 2 L n 0 u (some (k,b)) = + Real.sqrt (2 * sourceWeight 2 L k / sourceZ 2 L) * + ((∏ j, phaseCLM 2 L k (u j)) * + iteratedDeriv n (if b then Real.sin else Real.cos) 0) := by + let A := Real.sqrt (2 * sourceWeight 2 L k / sourceZ 2 L) + let g : ℝ → ℝ := if b then Real.sin else Real.cos + let ell := phaseCLM 2 L k + have hg : ContDiff ℝ n g := by + cases b <;> simp only [g, Bool.false_eq_true, ite_false, ite_true] <;> fun_prop + have hcomp : ContDiff ℝ n (g ∘ ell) := hg.comp_continuousLinearMap + have heq : modeCoefficient 2 L (some (k,b)) = A • (g ∘ ell) := by + funext x + cases b <;> simp [modeCoefficient, A, g, ell, phaseCLM_apply, Function.comp_apply] + unfold derivativeCoefficient + rw [heq, iteratedFDeriv_const_smul_apply hcomp.contDiffAt, + ell.iteratedFDeriv_comp_right hg 0 le_rfl] + simp only [_root_.smul_apply, ContinuousMultilinearMap.compContinuousLinearMap_apply, + map_zero, iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod, smul_eq_mul] + rfl + +set_option maxHeartbeats 2000000 in +/-- These are literal differentiated source columns, with the actual Fourier +amplitudes, rather than assumed witness columns or a finite-mode replacement. -/ +theorem sourceContact_selected_columns (L : ℝ) (i s : Fin 7) : + sourceContactCoefficient L i (sourceContactMode s) = + sourceContactAmplitude L s * contactWitness (sourceC L) i s := by + fin_cases i + all_goals first + | change derivativeCoefficient 2 L 0 0 ![] (sourceContactMode s) = _ + | change derivativeCoefficient 2 L 1 0 ![contactE0] (sourceContactMode s) = _ + | change derivativeCoefficient 2 L 1 0 ![contactE1] (sourceContactMode s) = _ + | change derivativeCoefficient 2 L 2 0 ![contactE0,contactE0] (sourceContactMode s) = _ + | change derivativeCoefficient 2 L 2 0 ![contactE0,contactE1] (sourceContactMode s) = _ + | change derivativeCoefficient 2 L 3 0 ![contactE0,contactE0,contactE0] (sourceContactMode s) = _ + | change derivativeCoefficient 2 L 2 0 ![contactE1,contactE1] (sourceContactMode s) = _ + all_goals fin_cases s <;> + norm_num [sourceContactCoefficient, sourceContactMode, sourceContactAmplitude, + sourceModeAmplitude, contactOrder, contactE0, contactE1, + derivativeCoefficient_none, derivativeCoefficient_some, contactWitness, + phaseCLM_apply, phase, frequency10, frequency01, frequency11, frequency20, + Fin.prod_univ_succ, Real.iteratedDeriv_add_one_sin, Real.iteratedDeriv_add_one_cos, + iteratedDeriv_zero] <;> ring_nf <;> simp + +/-- Positive definiteness of the actual contact covariance, obtained from its +infinite coefficient Gram sum and seven proved canonical source columns. -/ +theorem sourceContactCovariance_posDef {L : ℝ} (hL : L ≠ 0) : + (sourceContactCovariance L).PosDef := by + rw [sourceContactCovariance_eq_coefficientGram hL] + apply contact_coefficientGram_posDef (sourceContactCoefficient L) + (sourceContactCoefficient_summable hL) + (show sourceC L ≠ 0 from div_ne_zero + (mul_ne_zero (by norm_num) (ne_of_gt Real.pi_pos)) hL) + sourceContactMode (sourceContactAmplitude L) + (fun s => (sourceContactAmplitude_pos hL s).ne') + exact sourceContact_selected_columns L + +end I4SourceContact diff --git a/frontiers/source_derivative_gaussian_20261025/SourceContactEndpointParameters.lean b/frontiers/source_derivative_gaussian_20261025/SourceContactEndpointParameters.lean new file mode 100644 index 000000000..ba4c8e5a3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourceContactEndpointParameters.lean @@ -0,0 +1,325 @@ +import SourceContactSchur +import Mathlib.Probability.Moments.CovarianceBilin + +/-! +# Actual limiting adjusted-endpoint contact parameters + +The endpoint order is `(aM,aS,bM,bS,cM,cS)`. All covariance blocks below +come from literal origin derivatives of the unchanged smooth Fourier source. +The six pins are `(f,fx,fy,fxx,fxy,fxxx)` and their prescribed cubic value is +`(b,0,0,0,0,2)`. See `SOURCE_CONTACT_ENDPOINT_PARAMETERS.md`. +-/ + +noncomputable section + +open I4SmoothCarrier I4SourceDerivativeGaussian +open MeasureTheory ProbabilityTheory Matrix +open scoped BigOperators Matrix RealInnerProductSpace + +set_option autoImplicit false + +namespace I4SourceContact + +/-- The literal mixed third derivative A=fxxy at the origin. -/ +def sourceContactA (L : ℝ) (ω : SmoothSample 2 L) : ℝ := + iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0 ![contactE0,contactE0,contactE1] + +/-- The actual limiting adjusted endpoints, in the normalizer's six-coordinate order. -/ +def sourceContactEndpointVector (L : ℝ) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + ![-sourcePinVector L ω 5 / 2, sourcePinVector L ω 5 / 2, + -sourceContactA L ω / 2, sourceContactA L ω / 2, + sourceContactQ L ω, sourceContactQ L ω] + +/-- Literal limiting endpoint/pin cross covariance C0. -/ +def sourceContactEndpointCrossCovariance (L : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + fun i j => covariance (fun ω => sourceContactEndpointVector L ω i) + (fun ω => sourcePinVector L ω j) (smoothLaw 2 L) + +/-- Literal limiting endpoint covariance B0. -/ +def sourceContactEndpointCovariance (L : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + fun i j => covariance (fun ω => sourceContactEndpointVector L ω i) + (fun ω => sourceContactEndpointVector L ω j) (smoothLaw 2 L) + +def sourceContactEndpointRegression (L : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + sourceContactEndpointCrossCovariance L * (sourcePinCovariance L)⁻¹ + +/-- The prescribed limiting pin retains an arbitrary level b. -/ +def sourceContactPinValue (b : ℝ) : Fin 6 → ℝ := ![b,0,0,0,0,2] + +def sourceContactEndpointMean (L b : ℝ) : Fin 6 → ℝ := + sourceContactEndpointRegression L *ᵥ sourceContactPinValue b + +def sourceContactEndpointSchur (L : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + sourceContactEndpointCovariance L - + sourceContactEndpointCrossCovariance L * (sourcePinCovariance L)⁻¹ * + (sourceContactEndpointCrossCovariance L)ᵀ + +def sourceContactEndpointResidual (L : ℝ) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + sourceContactEndpointVector L ω - + sourceContactEndpointRegression L *ᵥ sourcePinVector L ω + +private theorem contact_generic_residual_covariance {Ω ι κ : Type*} [MeasurableSpace Ω] + [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] + {P : Measure Ω} [IsFiniteMeasure P] + (J : Ω → ι → ℝ) (T : Ω → κ → ℝ) + (hJ : ∀ i, MemLp (fun ω => J ω i) 2 P) + (hT : ∀ i, MemLp (fun ω => T ω i) 2 P) + (hG : IsUnit (Matrix.of (fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P) : Matrix ι ι ℝ)) : + let G : Matrix ι ι ℝ := fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P + let C : Matrix κ ι ℝ := fun i j => covariance (fun ω => T ω i) (fun ω => J ω j) P + let B : Matrix κ κ ℝ := fun i j => covariance (fun ω => T ω i) (fun ω => T ω j) P + let D := C * G⁻¹ + let R := fun ω => T ω - D *ᵥ J ω + (∀ i j, covariance (fun ω => R ω i) (fun ω => R ω j) P = (B - C * G⁻¹ * Cᵀ) i j) ∧ + (B - C * G⁻¹ * Cᵀ).PosSemidef := by + classical + intro G C B D R + have hG' : IsUnit G := hG + let _ : Invertible G := hG'.invertible + let U := fun ω => D *ᵥ J ω + have hU (i : κ) : MemLp (fun ω => U ω i) 2 P := + memLp_finsetSum Finset.univ (fun a _ => (hJ a).const_mul (D i a)) + have hR (i : κ) : MemLp (fun ω => R ω i) 2 P := (hT i).sub (hU i) + have hnormal : D * G = C := by + dsimp [D] + rw [Matrix.mul_assoc, Matrix.inv_mul_of_invertible, Matrix.mul_one] + have hUJ (i : κ) (j : ι) : covariance (fun ω => U ω i) (fun ω => J ω j) P = C i j := by + change covariance (fun ω => ∑ a, D i a * J ω a) (fun ω => J ω j) P = C i j + rw [covariance_fun_sum_left (fun a => (hJ a).const_mul _) (hJ j)] + simp_rw [covariance_const_mul_left] + exact congrFun (congrFun hnormal i) j + have hJR (i : ι) (j : κ) : covariance (fun ω => J ω i) (fun ω => R ω j) P = 0 := by + change covariance (fun ω => J ω i) (fun ω => T ω j - U ω j) P = 0 + rw [covariance_fun_sub_right (hJ i) (hT j) (hU j), + covariance_comm (fun ω => J ω i) (fun ω => U ω j), hUJ] + change covariance (fun ω => J ω i) (fun ω => T ω j) P - + covariance (fun ω => T ω j) (fun ω => J ω i) P = 0 + rw [covariance_comm, sub_self] + have hUR (i j : κ) : covariance (fun ω => U ω i) (fun ω => R ω j) P = 0 := by + change covariance (fun ω => ∑ a, D i a * J ω a) (fun ω => R ω j) P = 0 + rw [covariance_fun_sum_left (fun a => (hJ a).const_mul _) (hR j)] + simp_rw [covariance_const_mul_left, hJR, mul_zero] + exact Finset.sum_const_zero + have hUT (i j : κ) : covariance (fun ω => U ω i) (fun ω => T ω j) P = (D * Cᵀ) i j := by + change covariance (fun ω => ∑ a, D i a * J ω a) (fun ω => T ω j) P = _ + rw [covariance_fun_sum_left (fun a => (hJ a).const_mul _) (hT j)] + simp_rw [covariance_const_mul_left] + simp only [Matrix.mul_apply, C] + apply Finset.sum_congr rfl + intro a _ + rw [covariance_comm (fun ω => J ω a)] + rfl + have hcov (i j : κ) : covariance (fun ω => R ω i) (fun ω => R ω j) P = (B - C * G⁻¹ * Cᵀ) i j := by + change covariance (fun ω => R ω i) (fun ω => T ω j - U ω j) P = _ + rw [covariance_fun_sub_right (hR i) (hT j) (hU j), + covariance_comm (fun ω => R ω i) (fun ω => U ω j), hUR, sub_zero] + change covariance (fun ω => T ω i - U ω i) (fun ω => T ω j) P = _ + rw [covariance_fun_sub_left (hT i) (hU i) (hT j), hUT] + rfl + refine ⟨hcov, ?_⟩ + apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg + · ext i j + simp only [Matrix.conjTranspose_apply, star_trivial] + rw [← hcov i j, ← hcov j i] + exact covariance_comm _ _ + · intro x + have hp := covarianceBilin_self_nonneg (μ := P.map (fun ω => WithLp.toLp 2 (R ω))) (WithLp.toLp 2 x) + rw [covarianceBilin_apply_pi hR] at hp + simpa [hcov, dotProduct, Matrix.mulVec, Finset.mul_sum, mul_assoc, mul_left_comm, mul_comm] using hp + +private theorem sourceContactA_memLp {L : ℝ} (hL : L ≠ 0) : + MemLp (sourceContactA L) 2 (smoothLaw 2 L) := by + exact ((smoothSourceField_derivativeVector_hasGaussianLaw 2 hL 1 (fun _ => 3) + (fun _ => 0) (fun _ => ![contactE0,contactE0,contactE1])).eval 0).memLp_two + +private theorem sourcePin_memLp {L : ℝ} (hL : L ≠ 0) (i : Fin 6) : + MemLp (fun ω => sourcePinVector L ω i) 2 (smoothLaw 2 L) := + ((sourceContactVector_hasGaussianLaw hL).eval i.castSucc).memLp_two + +private theorem sourceContactEndpoint_memLp {L : ℝ} (hL : L ≠ 0) (i : Fin 6) : + MemLp (fun ω => sourceContactEndpointVector L ω i) 2 (smoothLaw 2 L) := by + have hQ := ((sourceContactVector_hasGaussianLaw hL).eval 6).memLp_two + fin_cases i + · simpa [sourceContactEndpointVector, div_eq_mul_inv, mul_comm] using + ((sourcePin_memLp hL 5).neg).const_mul ((2 : ℝ)⁻¹) + · simpa [sourceContactEndpointVector, div_eq_mul_inv, mul_comm] using + (sourcePin_memLp hL 5).const_mul ((2 : ℝ)⁻¹) + · simpa [sourceContactEndpointVector, div_eq_mul_inv, mul_comm] using + (sourceContactA_memLp hL).neg.const_mul ((2 : ℝ)⁻¹) + · simpa [sourceContactEndpointVector, div_eq_mul_inv, mul_comm] using + (sourceContactA_memLp hL).const_mul ((2 : ℝ)⁻¹) + · exact hQ + · exact hQ + +private theorem cross_row_zero (L : ℝ) (j : Fin 6) : + sourceContactEndpointCrossCovariance L 0 j = -sourcePinCovariance L 5 j / 2 := by + change covariance (fun ω => -sourcePinVector L ω 5 / 2) + (fun ω => sourcePinVector L ω j) (smoothLaw 2 L) = _ + rw [covariance_fun_div_left, covariance_fun_neg_left] + rfl + +private theorem cross_row_one (L : ℝ) (j : Fin 6) : + sourceContactEndpointCrossCovariance L 1 j = sourcePinCovariance L 5 j / 2 := by + change covariance (fun ω => sourcePinVector L ω 5 / 2) + (fun ω => sourcePinVector L ω j) (smoothLaw 2 L) = _ + rw [covariance_fun_div_left] + rfl + +private theorem cross_row_two (L : ℝ) (j : Fin 6) : + sourceContactEndpointCrossCovariance L 2 j = + -sourceContactEndpointCrossCovariance L 3 j := by + change covariance (fun ω => -sourceContactA L ω / 2) + (fun ω => sourcePinVector L ω j) (smoothLaw 2 L) = + -covariance (fun ω => sourceContactA L ω / 2) + (fun ω => sourcePinVector L ω j) (smoothLaw 2 L) + rw [covariance_fun_div_left, covariance_fun_div_left, covariance_fun_neg_left] + ring + +private theorem cross_row_four (L : ℝ) (j : Fin 6) : + sourceContactEndpointCrossCovariance L 4 j = + sourceContactEndpointCrossCovariance L 5 j := rfl + +private theorem regression_pin_coordinates {L : ℝ} (hL : L ≠ 0) (x : Fin 6 → ℝ) : + (sourceContactEndpointRegression L *ᵥ x) 0 = -x 5 / 2 ∧ + (sourceContactEndpointRegression L *ᵥ x) 1 = x 5 / 2 := by + let _ : Invertible (sourcePinCovariance L) := (sourcePinCovariance_isUnit hL).invertible + have hcancel : sourcePinCovariance L *ᵥ ((sourcePinCovariance L)⁻¹ *ᵥ x) = x := by + rw [Matrix.mulVec_mulVec, Matrix.mul_inv_of_invertible, Matrix.one_mulVec] + have hc := congrFun hcancel 5 + rw [sourceContactEndpointRegression, ← Matrix.mulVec_mulVec] + change (∑ j, sourceContactEndpointCrossCovariance L 0 j * + ((sourcePinCovariance L)⁻¹ *ᵥ x) j) = -x 5 / 2 ∧ + (∑ j, sourceContactEndpointCrossCovariance L 1 j * + ((sourcePinCovariance L)⁻¹ *ᵥ x) j) = x 5 / 2 + simp_rw [cross_row_zero, cross_row_one] + change (∑ j, sourcePinCovariance L 5 j * + ((sourcePinCovariance L)⁻¹ *ᵥ x) j) = x 5 at hc + constructor + · calc + _ = (-1 / 2 : ℝ) * (∑ j, sourcePinCovariance L 5 j * + ((sourcePinCovariance L)⁻¹ *ᵥ x) j) := by + rw [Finset.mul_sum] + apply Finset.sum_congr rfl + intro j _ + ring + _ = -x 5 / 2 := by rw [hc]; ring + · calc + _ = (1 / 2 : ℝ) * (∑ j, sourcePinCovariance L 5 j * + ((sourcePinCovariance L)⁻¹ *ᵥ x) j) := by + rw [Finset.mul_sum] + apply Finset.sum_congr rfl + intro j _ + ring + _ = x 5 / 2 := by rw [hc]; ring + +private theorem regression_other_coordinates (L : ℝ) (x : Fin 6 → ℝ) : + (sourceContactEndpointRegression L *ᵥ x) 2 = + -(sourceContactEndpointRegression L *ᵥ x) 3 ∧ + (sourceContactEndpointRegression L *ᵥ x) 4 = + (sourceContactEndpointRegression L *ᵥ x) 5 := by + rw [sourceContactEndpointRegression, ← Matrix.mulVec_mulVec] + constructor + · simp only [Matrix.mulVec, dotProduct, cross_row_two, neg_mul, Finset.sum_neg_distrib] + · simp only [Matrix.mulVec, dotProduct, cross_row_four] + +theorem sourceContactEndpointMean_contact {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + sourceContactEndpointMean L b 0 = -1 ∧ + sourceContactEndpointMean L b 1 = 1 ∧ + sourceContactEndpointMean L b 2 + sourceContactEndpointMean L b 3 = 0 ∧ + sourceContactEndpointMean L b 4 - sourceContactEndpointMean L b 5 = 0 := by + have hp := regression_pin_coordinates hL (sourceContactPinValue b) + have ho := regression_other_coordinates L (sourceContactPinValue b) + change _ ∧ _ ∧ _ ∧ _ + dsimp only [sourceContactEndpointMean] + refine ⟨?_, ?_, ?_, ?_⟩ + · simpa [sourceContactPinValue] using hp.1 + · simpa [sourceContactPinValue] using hp.2 + · rw [ho.1]; ring + · rw [ho.2]; ring + +private theorem sourceContactEndpointResidual_contact {L : ℝ} (hL : L ≠ 0) + (ω : SmoothSample 2 L) : + sourceContactEndpointResidual L ω 0 = 0 ∧ + sourceContactEndpointResidual L ω 1 = 0 ∧ + sourceContactEndpointResidual L ω 2 = -sourceContactEndpointResidual L ω 3 ∧ + sourceContactEndpointResidual L ω 4 = sourceContactEndpointResidual L ω 5 := by + have hp := regression_pin_coordinates hL (sourcePinVector L ω) + have ho := regression_other_coordinates L (sourcePinVector L ω) + dsimp only [sourceContactEndpointResidual, Pi.sub_apply, sourceContactEndpointVector, + Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val] + refine ⟨?_, ?_, ?_, ?_⟩ + · rw [hp.1]; ring + · rw [hp.2]; ring + · rw [ho.1]; ring + · rw [ho.2] + +/-- The limiting Schur matrix is the actual least-squares endpoint residual covariance. -/ +theorem sourceContactEndpointSchur_covariance {L : ℝ} (hL : L ≠ 0) : + (∀ i j, covariance (fun ω => sourceContactEndpointResidual L ω i) + (fun ω => sourceContactEndpointResidual L ω j) (smoothLaw 2 L) = + sourceContactEndpointSchur L i j) ∧ (sourceContactEndpointSchur L).PosSemidef := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + have hG : IsUnit (Matrix.of (fun i j => + covariance (fun ω => sourcePinVector L ω i) (fun ω => sourcePinVector L ω j) + (smoothLaw 2 L)) : Matrix (Fin 6) (Fin 6) ℝ) := by + convert sourcePinCovariance_isUnit hL using 1 + ext i j + rfl + exact contact_generic_residual_covariance (sourcePinVector L) (sourceContactEndpointVector L) + (sourcePin_memLp hL) (sourceContactEndpoint_memLp hL) hG + +theorem sourceContactEndpointSchur_posSemidef {L : ℝ} (hL : L ≠ 0) : + (sourceContactEndpointSchur L).PosSemidef := + (sourceContactEndpointSchur_covariance hL).2 + +/-- The deterministic endpoint relations become exactly the four required zero variances. -/ +theorem sourceContactEndpointSchur_contact {L : ℝ} (hL : L ≠ 0) : + sourceContactEndpointSchur L 0 0 = 0 ∧ + sourceContactEndpointSchur L 1 1 = 0 ∧ + sourceContactEndpointSchur L 2 2 + 2 * sourceContactEndpointSchur L 2 3 + + sourceContactEndpointSchur L 3 3 = 0 ∧ + sourceContactEndpointSchur L 4 4 - 2 * sourceContactEndpointSchur L 4 5 + + sourceContactEndpointSchur L 5 5 = 0 := by + have hc := (sourceContactEndpointSchur_covariance hL).1 + have hzero : (fun ω => sourceContactEndpointResidual L ω 0) = 0 := + funext fun ω => (sourceContactEndpointResidual_contact hL ω).1 + have hone : (fun ω => sourceContactEndpointResidual L ω 1) = 0 := + funext fun ω => (sourceContactEndpointResidual_contact hL ω).2.1 + have htwo : (fun ω => sourceContactEndpointResidual L ω 2) = + -(fun ω => sourceContactEndpointResidual L ω 3) := + funext fun ω => (sourceContactEndpointResidual_contact hL ω).2.2.1 + have hfour : (fun ω => sourceContactEndpointResidual L ω 4) = + (fun ω => sourceContactEndpointResidual L ω 5) := + funext fun ω => (sourceContactEndpointResidual_contact hL ω).2.2.2 + refine ⟨?_, ?_, ?_, ?_⟩ + · rw [← hc 0 0, hzero]; simp + · rw [← hc 1 1, hone]; simp + · rw [← hc 2 2, ← hc 2 3, ← hc 3 3, htwo] + simp only [covariance_neg_left, covariance_neg_right] + ring + · rw [← hc 4 4, ← hc 4 5, ← hc 5 5, hfour] + ring + +private theorem cross_row_Q (L : ℝ) (j : Fin 6) : + sourceContactEndpointCrossCovariance L 4 j = sourceContactCross L j := by + exact covariance_comm _ _ + +/-- The reference entry is the already established actual torus curvature Schur variance. -/ +theorem sourceContactEndpointSchur_four_four_eq_schur (L : ℝ) : + sourceContactEndpointSchur L 4 4 = sourceContactSchur L := by + have heq : (sourceContactEndpointCrossCovariance L * (sourcePinCovariance L)⁻¹ * + (sourceContactEndpointCrossCovariance L)ᵀ) 4 4 = + sourceContactCross L ⬝ᵥ ((sourcePinCovariance L)⁻¹ *ᵥ sourceContactCross L) := by + simp only [Matrix.mul_apply, Matrix.transpose_apply, cross_row_Q, dotProduct, Matrix.mulVec] + simp_rw [Finset.sum_mul] + rw [Finset.sum_comm] + simp only [Finset.mul_sum, mul_assoc] + change sourceContactQVariance L - _ = sourceContactQVariance L - _ + rw [heq] + +theorem sourceContactEndpointSchur_referenceVariance_pos {L : ℝ} (hL : L ≠ 0) : + 0 < sourceContactEndpointSchur L 4 4 := by + rw [sourceContactEndpointSchur_four_four_eq_schur] + exact sourceContactSchur_pos hL + +end I4SourceContact diff --git a/frontiers/source_derivative_gaussian_20261025/SourceContactSchur.lean b/frontiers/source_derivative_gaussian_20261025/SourceContactSchur.lean new file mode 100644 index 000000000..11019ab39 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourceContactSchur.lean @@ -0,0 +1,206 @@ +import SourceContactCovariance + +/-! +# Literal six-pin covariance and positive reference Schur variance + +The six pin coordinates are `(f, fx, fy, fxx, fxy, fxxx)` and the remaining +coordinate is `Q = fyy`, all taken from the literal origin contact vector. +The block identity below uses an explicit equivalence of the seven coordinates. +Positive definiteness, invertibility and strict Schur positivity follow from +`L ≠ 0`; no covariance or positivity conclusion is an input hypothesis. + +These statements concern the torus source with its actual period. They do not +identify its regression mean or variance with a planar reference constant. +See `SOURCE_CONTACT_SCHUR.md` for the frozen scope. +-/ + +noncomputable section + +open I4SmoothCarrier I4ContactGram MeasureTheory ProbabilityTheory Matrix +open scoped BigOperators Matrix + +set_option autoImplicit false + +namespace I4SourceContact + +/-- Literal six-pin observations in their source order. -/ +def sourcePinVector (L : ℝ) (ω : SmoothSample 2 L) (i : Fin 6) : ℝ := + sourceContactVector L ω i.castSucc + +/-- Literal remaining curvature coordinate Q=fyy. -/ +def sourceContactQ (L : ℝ) (ω : SmoothSample 2 L) : ℝ := + sourceContactVector L ω 6 + +/-- Principal covariance of the literal six origin pin coordinates. -/ +def sourcePinCovariance (L : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + (sourceContactCovariance L).submatrix Fin.castSucc Fin.castSucc + +/-- Covariance of each literal pin coordinate with the remaining curvature Q=fyy. -/ +def sourceContactCross (L : ℝ) (i : Fin 6) : ℝ := + sourceContactCovariance L i.castSucc 6 + +/-- Literal Q covariance with itself. -/ +def sourceContactQVariance (L : ℝ) : ℝ := sourceContactCovariance L 6 6 + +def sourceContactCrossMatrix (L : ℝ) : Matrix (Fin 6) Unit ℝ := + fun i _ => sourceContactCross L i + +def sourceContactQMatrix (L : ℝ) : Matrix Unit Unit ℝ := + fun _ _ => sourceContactQVariance L + +/-- The scalar Schur complement of the actual six-pin covariance. -/ +def sourceContactSchur (L : ℝ) : ℝ := + sourceContactQVariance L - + sourceContactCross L ⬝ᵥ ((sourcePinCovariance L)⁻¹ *ᵥ sourceContactCross L) + +/-- Exact 6+1 splitting of the source's seven origin coordinates. -/ +def sourceContactBlockEquiv : (Fin 6 ⊕ Unit) ≃ Fin 7 := + (Equiv.sumCongr (Equiv.refl (Fin 6)) finOneEquiv.symm).trans finSumFinEquiv + +@[simp] lemma sourceContactBlockEquiv_inl (i : Fin 6) : + sourceContactBlockEquiv (Sum.inl i) = i.castSucc := rfl + +@[simp] lemma sourceContactBlockEquiv_inr (u : Unit) : + sourceContactBlockEquiv (Sum.inr u) = 6 := by + cases u + rfl + +/-- The literal covariance equals the stated six-pin/Q blocks under the exact coordinate split. -/ +theorem sourceContactCovariance_block_identity (L : ℝ) : + (sourceContactCovariance L).submatrix sourceContactBlockEquiv sourceContactBlockEquiv = + Matrix.fromBlocks (sourcePinCovariance L) (sourceContactCrossMatrix L) + (sourceContactCrossMatrix L)ᴴ (sourceContactQMatrix L) := by + ext i j + cases i with + | inl i => + cases j with + | inl j => rfl + | inr u => cases u; rfl + | inr u => + cases u + cases j with + | inl j => + change covariance (fun ω => sourceContactVector L ω 6) + (fun ω => sourceContactVector L ω j.castSucc) (smoothLaw 2 L) = + covariance (fun ω => sourceContactVector L ω j.castSucc) + (fun ω => sourceContactVector L ω 6) (smoothLaw 2 L) + exact covariance_comm _ _ + | inr v => cases v; rfl + +theorem sourcePinCovariance_posDef {L : ℝ} (hL : L ≠ 0) : + (sourcePinCovariance L).PosDef := + (sourceContactCovariance_posDef hL).submatrix (Fin.castSucc_injective 6) + +theorem sourcePinCovariance_isUnit {L : ℝ} (hL : L ≠ 0) : + IsUnit (sourcePinCovariance L) := + (sourcePinCovariance_posDef hL).isUnit + +/-- An actual inverse of the six-pin covariance, with no invertibility premise. -/ +@[instance_reducible] def sourcePinCovarianceInvertible {L : ℝ} (hL : L ≠ 0) : + Invertible (sourcePinCovariance L) := + (sourcePinCovariance_isUnit hL).invertible + +theorem sourceContactQVariance_eq_variance {L : ℝ} (hL : L ≠ 0) : + sourceContactQVariance L = + Var[fun ω => sourceContactVector L ω 6; smoothLaw 2 L] := by + exact covariance_self ((sourceContactVector_hasGaussianLaw hL).eval 6).aemeasurable + +/-- Strict positivity of the actual curvature Schur variance follows from the literal contact law. -/ +theorem sourceContactSchur_pos {L : ℝ} (hL : L ≠ 0) : + 0 < sourceContactSchur L := by + have hblock : (Matrix.fromBlocks (sourcePinCovariance L) (sourceContactCrossMatrix L) + (sourceContactCrossMatrix L)ᴴ (sourceContactQMatrix L)).PosDef := by + rw [← sourceContactCovariance_block_identity L] + exact (sourceContactCovariance_posDef hL).submatrix sourceContactBlockEquiv.injective + have hs := (contact_schur_pos (sourcePinCovariance L) (sourceContactCrossMatrix L) + (sourceContactQMatrix L) hblock).2 + have heq : ((sourceContactCrossMatrix L)ᴴ * (sourcePinCovariance L)⁻¹ * + sourceContactCrossMatrix L) () () = + sourceContactCross L ⬝ᵥ ((sourcePinCovariance L)⁻¹ *ᵥ sourceContactCross L) := by + simp only [Matrix.mul_apply, Matrix.conjTranspose_apply, sourceContactCrossMatrix, + star_trivial, dotProduct, Matrix.mulVec] + simp_rw [Finset.sum_mul] + rw [Finset.sum_comm] + simp only [Finset.mul_sum, mul_assoc] + change 0 < sourceContactQVariance L - + ((sourceContactCrossMatrix L)ᴴ * (sourcePinCovariance L)⁻¹ * sourceContactCrossMatrix L) () () at hs + rw [heq] at hs + exact hs + +/-- Finite Gaussian applications use this L² covariance identity; its normal equations are derived. -/ +lemma contact_residual_variance_of_isUnit_covariance {Ω ι : Type*} [MeasurableSpace Ω] + [Fintype ι] [DecidableEq ι] {P : Measure Ω} [IsFiniteMeasure P] + (J : Ω → ι → ℝ) (Q : Ω → ℝ) + (hJ : ∀ i, MemLp (fun ω => J ω i) 2 P) (hQ : MemLp Q 2 P) + (hG : IsUnit (Matrix.of (fun i j => + covariance (fun ω => J ω i) (fun ω => J ω j) P) : Matrix ι ι ℝ)) : + let G : Matrix ι ι ℝ := fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P + let b : ι → ℝ := fun i => covariance (fun ω => J ω i) Q P + variance (fun ω => Q ω - (G⁻¹ *ᵥ b) ⬝ᵥ J ω) P = + variance Q P - b ⬝ᵥ (G⁻¹ *ᵥ b) := by + classical + let G : Matrix ι ι ℝ := fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P + let b : ι → ℝ := fun i => covariance (fun ω => J ω i) Q P + let a : ι → ℝ := G⁻¹ *ᵥ b + let U : Ω → ℝ := fun ω => a ⬝ᵥ J ω + change variance (fun ω => Q ω - U ω) P = variance Q P - b ⬝ᵥ a + have hG' : IsUnit G := hG + let _ : Invertible G := hG'.invertible + have hnormal : G *ᵥ a = b := by + dsimp [a] + rw [Matrix.mulVec_mulVec, Matrix.mul_inv_of_invertible, Matrix.one_mulVec] + have hterms (i : ι) : MemLp (fun ω => a i * J ω i) 2 P := + (hJ i).const_mul _ + have hU : MemLp U 2 P := memLp_finsetSum Finset.univ (fun i _ => hterms i) + have hJU (i : ι) : covariance (fun ω => J ω i) U P = b i := by + change covariance (fun ω => J ω i) (fun ω => ∑ j, a j * J ω j) P = b i + rw [covariance_fun_sum_right hterms (hJ i)] + simp_rw [covariance_const_mul_right] + simpa [G, Matrix.mulVec, dotProduct, mul_comm] using congrFun hnormal i + have hQU : covariance Q U P = a ⬝ᵥ b := by + change covariance Q (fun ω => ∑ i, a i * J ω i) P = a ⬝ᵥ b + rw [covariance_fun_sum_right hterms hQ] + simp_rw [covariance_const_mul_right] + simp only [b, dotProduct, covariance_comm Q] + have hvarU : variance U P = a ⬝ᵥ b := by + rw [← covariance_self hU.aemeasurable] + change covariance (fun ω => ∑ i, a i * J ω i) U P = a ⬝ᵥ b + rw [covariance_fun_sum_left hterms hU] + simp_rw [covariance_const_mul_left, hJU] + rfl + rw [variance_fun_sub hQ hU, hQU, hvarU, dotProduct_comm b a] + ring + +/-- The actual least-squares curvature residual, without replacing the torus covariance. -/ +def sourceContactResidual (L : ℝ) (ω : SmoothSample 2 L) : ℝ := + sourceContactQ L ω - + ((sourcePinCovariance L)⁻¹ *ᵥ sourceContactCross L) ⬝ᵥ sourcePinVector L ω + +/-- The strictly positive Schur complement is the actual curvature residual variance. -/ +theorem sourceContactResidual_variance {L : ℝ} (hL : L ≠ 0) : + Var[sourceContactResidual L; smoothLaw 2 L] = sourceContactSchur L := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + have hJ (i : Fin 6) : MemLp (fun ω => sourcePinVector L ω i) 2 (smoothLaw 2 L) := + ((sourceContactVector_hasGaussianLaw hL).eval i.castSucc).memLp_two + have hQ : MemLp (sourceContactQ L) 2 (smoothLaw 2 L) := + ((sourceContactVector_hasGaussianLaw hL).eval 6).memLp_two + have hG : IsUnit (Matrix.of (fun i j => + covariance (fun ω => sourcePinVector L ω i) (fun ω => sourcePinVector L ω j) + (smoothLaw 2 L)) : Matrix (Fin 6) (Fin 6) ℝ) := by + convert sourcePinCovariance_isUnit hL using 1 + ext i j + rfl + have hv := contact_residual_variance_of_isUnit_covariance + (sourcePinVector L) (sourceContactQ L) hJ hQ hG + change Var[sourceContactResidual L; smoothLaw 2 L] = + Var[sourceContactQ L; smoothLaw 2 L] - + sourceContactCross L ⬝ᵥ ((sourcePinCovariance L)⁻¹ *ᵥ sourceContactCross L) at hv + rw [sourceContactSchur, sourceContactQVariance_eq_variance hL] + exact hv + +theorem sourceContactResidual_variance_pos {L : ℝ} (hL : L ≠ 0) : + 0 < Var[sourceContactResidual L; smoothLaw 2 L] := by + rw [sourceContactResidual_variance hL] + exact sourceContactSchur_pos hL + +end I4SourceContact diff --git a/frontiers/source_derivative_gaussian_20261025/SourceDerivativeContinuity.lean b/frontiers/source_derivative_gaussian_20261025/SourceDerivativeContinuity.lean new file mode 100644 index 000000000..c618c20e0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourceDerivativeContinuity.lean @@ -0,0 +1,99 @@ +import SourceDerivativeGaussian +import Mathlib.Analysis.Normed.Group.FunctionSeries + +/-! Continuity of the literal derivative covariance via its actual infinite +Fourier pairing. The summable bounds hold globally in both evaluation points. +No confluent difference quotient or conditional covariance limit is inferred. -/ + +noncomputable section +open I4Foundation I4Source I4Smooth I4SourceGaussian I4SmoothCarrier I4SourceDerivativeGaussian +open MeasureTheory ProbabilityTheory +open scoped BigOperators NNReal + +namespace I4SourceDerivativeContinuity + +/-- A global, point-independent bound for a fixed directional derivative mode. -/ +def derivativeMajorant (d : ℕ) (L : ℝ) (k : ℕ) (u : Fin k → (Fin d → ℝ)) + (m : SourceMode d) : ℝ := + orderModeBound k d L m * ∏ j, ‖u j‖ + +theorem derivativeMajorant_nonneg (d : ℕ) (L : ℝ) (k : ℕ) + (u : Fin k → (Fin d → ℝ)) (m : SourceMode d) : + 0 ≤ derivativeMajorant d L k u m := + mul_nonneg (orderModeBound_nonneg k d L m) + (Finset.prod_nonneg (fun _ _ => norm_nonneg _)) + +theorem derivativeMajorant_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin d → ℝ)) : Summable (derivativeMajorant d L k u) := + (orderModeBound_summable k d hL).mul_right _ + +theorem derivativeCoefficient_bound (d : ℕ) {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin d → ℝ)) (m : SourceMode d) (x : Fin d → ℝ) : + ‖derivativeCoefficient d L k x u m‖ ≤ derivativeMajorant d L k u m := by + exact ((iteratedFDeriv ℝ k (modeCoefficient d L m) x).le_opNorm u).trans + (mul_le_mul_of_nonneg_right (modeCoefficient_iteratedFDeriv_order_bound d hL m k x) + (Finset.prod_nonneg (fun _ _ => norm_nonneg _))) + +theorem derivativeCoefficient_continuous (d : ℕ) (L : ℝ) (k : ℕ) + (u : Fin k → (Fin d → ℝ)) (m : SourceMode d) : + Continuous (fun x => derivativeCoefficient d L k x u m) := by + have hf : ContDiff ℝ k (modeCoefficient d L m) := by + cases m with + | none => + change ContDiff ℝ k (fun _ : Fin d → ℝ => Real.sqrt (1 / I4Weights.sourceZ d L)) + fun_prop + | some p => + rcases p with ⟨q, b⟩ + change ContDiff ℝ k (fun x : Fin d → ℝ => + Real.sqrt (2 * I4Weights.sourceWeight d L q / I4Weights.sourceZ d L) * + if b then Real.sin (phase d L q x) else Real.cos (phase d L q x)) + cases b <;> simp only [Bool.false_eq_true, ite_false, ite_true] <;> + unfold phase <;> fun_prop + exact (continuous_eval_const u).comp + (hf.continuous_iteratedFDeriv le_rfl) + +private theorem derivativeMajorant_product_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (k l : ℕ) (u : Fin k → (Fin d → ℝ)) (v : Fin l → (Fin d → ℝ)) : + Summable (fun m => derivativeMajorant d L k u m * derivativeMajorant d L l v m) := by + have h := summable_mul_of_summable_norm + (derivativeMajorant_summable d hL k u).norm + (derivativeMajorant_summable d hL l v).norm + simpa only [Function.comp_def] using h.comp_injective + (show Function.Injective (fun m : SourceMode d => (m,m)) from + fun _ _ h => congrArg Prod.fst h) + +/-- The covariance of two actual fixed-order directional derivatives is +continuous in their two evaluation points, including coincident points. -/ +theorem smoothSourceField_derivative_covariance_continuous (d : ℕ) {L : ℝ} + (hL : L ≠ 0) (k l : ℕ) (u : Fin k → (Fin d → ℝ)) + (v : Fin l → (Fin d → ℝ)) : + Continuous (fun xy : (Fin d → ℝ) × (Fin d → ℝ) => + cov[fun ω => iteratedFDeriv ℝ k (smoothSourceField d L ω) xy.1 u, + fun ω => iteratedFDeriv ℝ l (smoothSourceField d L ω) xy.2 v; smoothLaw d L]) := by + simp_rw [smoothSourceField_derivative_covariance d hL] + apply continuous_tsum + · intro m + exact ((derivativeCoefficient_continuous d L k u m).comp continuous_fst).mul + ((derivativeCoefficient_continuous d L l v m).comp continuous_snd) + · exact derivativeMajorant_product_summable d hL k l u v + · intro m xy + rw [norm_mul] + exact mul_le_mul (derivativeCoefficient_bound d hL k u m xy.1) + (derivativeCoefficient_bound d hL l v m xy.2) (norm_nonneg _) + (derivativeMajorant_nonneg d L k u m) + +/-- Every actual directional derivative is centered under the unchanged source +law. Noncentered conditional means are not identified with these means. -/ +theorem smoothSourceField_derivative_integral_zero (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (k : ℕ) (x : Fin d → ℝ) (u : Fin k → (Fin d → ℝ)) : + (∫ ω, iteratedFDeriv ℝ k (smoothSourceField d L ω) x u ∂smoothLaw d L) = 0 := by + simp_rw [smoothSourceField_derivative_series d hL] + have h := scalarSeries_integral_zero (derivativeCoefficient d L k x u) + (derivativeCoefficient_summable d hL k x u) + change (∫ ω, scalarSeries (derivativeCoefficient d L k x u) ω ∂sourceLaw d) = 0 at h + rw [← sampleInclusion_map d hL, + integral_map (sampleInclusion_measurable d L).aemeasurable + (scalarSeries_measurable _).aestronglyMeasurable] at h + exact h + +end I4SourceDerivativeContinuity diff --git a/frontiers/source_derivative_gaussian_20261025/SourceDerivativeGaussian.lean b/frontiers/source_derivative_gaussian_20261025/SourceDerivativeGaussian.lean new file mode 100644 index 000000000..c40511553 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourceDerivativeGaussian.lean @@ -0,0 +1,188 @@ +import SourceSmoothCarrier +import Mathlib.Analysis.Calculus.SmoothSeries +import Mathlib.Topology.Algebra.Module.Multilinear.Topology + +/-! Actual derivative observations of the literal smooth Fourier source. +All derivative orders are differentiated termwise on the existing SmoothSample +carrier. Joint Gaussianity follows from summable deterministic coefficient arrays, +without an assumed derivative law, conditional law or covariance lower bound. -/ + +noncomputable section + +open I4Foundation I4Source I4Smooth I4SourceGaussian I4SmoothCarrier +open MeasureTheory ProbabilityTheory +open scoped BigOperators ENNReal NNReal Topology + +namespace I4SourceDerivativeGaussian + +private theorem modeCoefficient_smooth (d : ℕ) (L : ℝ) (m : SourceMode d) : + ContDiff ℝ (⊤ : ℕ∞) (modeCoefficient d L m) := by + cases m with + | none => + change ContDiff ℝ (⊤ : ℕ∞) (fun _ : Fin d → ℝ => Real.sqrt (1 / I4Weights.sourceZ d L)) + fun_prop + | some p => + rcases p with ⟨k, b⟩ + change ContDiff ℝ (⊤ : ℕ∞) (fun x : Fin d → ℝ => + Real.sqrt (2 * I4Weights.sourceWeight d L k / I4Weights.sourceZ d L) * + if b then Real.sin (phase d L k x) else Real.cos (phase d L k x)) + cases b <;> simp only [Bool.false_eq_true, ite_false, ite_true] <;> + unfold phase <;> fun_prop + +private theorem weighted_iteratedFDeriv (d : ℕ) (L : ℝ) (m : SourceMode d) + (c : ℝ) (k : ℕ) (x : Fin d → ℝ) : + iteratedFDeriv ℝ k (fun y => modeCoefficient d L m y * c) x = + c • iteratedFDeriv ℝ k (modeCoefficient d L m) x := by + have hf : ContDiff ℝ k (modeCoefficient d L m) := + (modeCoefficient_smooth d L m).of_le + (by exact_mod_cast (show (k : ℕ∞) ≤ ⊤ from le_top)) + have heq : (fun y => modeCoefficient d L m y * c) = c • modeCoefficient d L m := by + funext y + simp only [Pi.smul_apply, smul_eq_mul, mul_comm] + rw [heq, iteratedFDeriv_const_smul_apply hf.contDiffAt] + +private theorem weighted_iteratedFDeriv_bound (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (m : SourceMode d) (c : ℝ) (k : ℕ) (x : Fin d → ℝ) : + ‖iteratedFDeriv ℝ k (fun y => modeCoefficient d L m y * c) x‖ ≤ + orderModeBound k d L m * |c| := by + rw [weighted_iteratedFDeriv, norm_smul, Real.norm_eq_abs] + simpa only [mul_comm] using mul_le_mul_of_nonneg_left + (modeCoefficient_iteratedFDeriv_order_bound d hL m k x) (abs_nonneg c) + +/-- Deterministic Fourier coefficient for an actual directional derivative. -/ +def derivativeCoefficient (d : ℕ) (L : ℝ) (k : ℕ) (x : Fin d → ℝ) + (u : Fin k → (Fin d → ℝ)) (m : SourceMode d) : ℝ := + iteratedFDeriv ℝ k (modeCoefficient d L m) x u + +theorem derivativeCoefficient_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (k : ℕ) (x : Fin d → ℝ) (u : Fin k → (Fin d → ℝ)) : + Summable (derivativeCoefficient d L k x u) := by + apply ((orderModeBound_summable k d hL).mul_right (∏ i, ‖u i‖)).of_norm_bounded + intro m + calc + ‖derivativeCoefficient d L k x u m‖ ≤ + ‖iteratedFDeriv ℝ k (modeCoefficient d L m) x‖ * ∏ i, ‖u i‖ := + (iteratedFDeriv ℝ k (modeCoefficient d L m) x).le_opNorm u + _ ≤ orderModeBound k d L m * ∏ i, ‖u i‖ := + mul_le_mul_of_nonneg_right (modeCoefficient_iteratedFDeriv_order_bound d hL m k x) + (Finset.prod_nonneg (fun _ _ => norm_nonneg _)) + +/-- This is a literal identity for every sample in the existing smooth carrier, +including order zero. It is not a finite-mode approximation or a new field. -/ +theorem smoothSourceField_derivative_series (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (ω : SmoothSample d L) (k : ℕ) (x : Fin d → ℝ) + (u : Fin k → (Fin d → ℝ)) : + iteratedFDeriv ℝ k (smoothSourceField d L ω) x u = + scalarSeries (derivativeCoefficient d L k x u) (sampleInclusion d L ω) := by + let f : SourceMode d → (Fin d → ℝ) → ℝ := + fun m y => modeCoefficient d L m y * sampleInclusion d L ω m + let major : ℕ → SourceMode d → ℝ := + fun j m => orderModeBound j d L m * |sampleInclusion d L ω m| + have hf (m : SourceMode d) : ContDiff ℝ (k : ℕ∞) (f m) := + ((modeCoefficient_smooth d L m).of_le + (by exact_mod_cast (show (k : ℕ∞) ≤ ⊤ from le_top))).mul contDiff_const + have hmajor (j : ℕ) (_hj : (j : ℕ∞) ≤ k) : Summable (major j) := + ω.property j + have hbound (j : ℕ) (m : SourceMode d) (y : Fin d → ℝ) + (_hj : (j : ℕ∞) ≤ k) : ‖iteratedFDeriv ℝ j (f m) y‖ ≤ major j m := + weighted_iteratedFDeriv_bound d hL m (sampleInclusion d L ω m) j y + have hsum : Summable (fun m => iteratedFDeriv ℝ k (f m) x) := + (hmajor k le_rfl).of_norm_bounded (fun m => hbound k m x le_rfl) + have hdiff := iteratedFDeriv_tsum_apply hf hmajor hbound (k := k) le_rfl x + change iteratedFDeriv ℝ k (fun y => ∑' m, f m y) x u = + ∑' m, derivativeCoefficient d L k x u m * sampleInclusion d L ω m + rw [hdiff, ContinuousMultilinearMap.tsum_eval hsum] + apply tsum_congr + intro m + dsimp only [f] + rw [weighted_iteratedFDeriv] + exact mul_comm _ _ + +private theorem scalarSeries_linearCombination_hasGaussianLaw + {ι : Type*} [Countable ι] (n : ℕ) (a : Fin n → ι → ℝ) + (ha : ∀ j, Summable (a j)) (c : Fin n → ℝ) : + HasGaussianLaw (fun ω => ∑ j, c j * scalarSeries (a j) ω) (coefficientLaw ι) := by + classical + let b : ι → ℝ := fun m => ∑ j, c j * a j m + have hb : Summable b := summable_sum (fun j _ => (ha j).mul_left (c j)) + have hG : HasGaussianLaw (scalarSeries b) (coefficientLaw ι) := by + constructor + · exact (scalarSeries_measurable b).aemeasurable + · rw [scalarSeries_map_gaussian b hb] + infer_instance + apply hG.congr + filter_upwards [ae_all_iff.2 (fun j : Fin n => scalarSeries_hasSum_ae (a j) (ha j)), + scalarSeries_hasSum_ae b hb] with ω hω hsum + have heq : HasSum (fun m => ∑ j, c j * a j m * ω m) + (∑ j, c j * scalarSeries (a j) ω) := by + simpa only [mul_assoc] using hasSum_sum (fun j _ => (hω j).mul_left (c j)) + exact hsum.unique (by simpa only [b, Finset.sum_mul] using heq) + +/-- Arbitrary finite arrays of summable coefficient series are jointly Gaussian. +No independence of the resulting observations is asserted or assumed. -/ +theorem scalarSeries_finiteVector_hasGaussianLaw + {ι : Type*} [Countable ι] (n : ℕ) (a : Fin n → ι → ℝ) + (ha : ∀ j, Summable (a j)) : + HasGaussianLaw (fun ω j => scalarSeries (a j) ω) (coefficientLaw ι) := by + classical + have hm : Measurable (fun ω j => scalarSeries (a j) ω) := + Measurable.of_eval (fun j => scalarSeries_measurable (a j)) + refine ⟨hm.aemeasurable, ?_⟩ + apply isGaussian_of_map_eq_gaussianReal + intro ell + let c : Fin n → ℝ := fun j => ell (Pi.single j 1) + have hrep (z : Fin n → ℝ) : ell z = ∑ j, c j * z j := by + rw [← LinearMap.sum_single_apply (fun _ : Fin n => ℝ) z, map_sum] + apply Finset.sum_congr rfl + intro j _ + have hs : Pi.single j (z j) = z j • Pi.single j (1 : ℝ) := by + rw [← Pi.single_smul, smul_eq_mul, mul_one] + rw [hs, map_smul] + simp [c, smul_eq_mul, mul_comm] + have hG : HasGaussianLaw (fun ω => ell (fun j => scalarSeries (a j) ω)) + (coefficientLaw ι) := + (scalarSeries_linearCombination_hasGaussianLaw n a ha c).congr + (Filter.Eventually.of_forall (fun ω => (hrep _).symm)) + refine ⟨(coefficientLaw ι)[fun ω => ell (fun j => scalarSeries (a j) ω)], + Var[fun ω => ell (fun j => scalarSeries (a j) ω); coefficientLaw ι].toNNReal, ?_⟩ + rw [Measure.map_map ell.continuous.measurable hm] + exact hG.map_eq_gaussianReal + +/-- Heterogeneous derivative orders, points and directions of the actual smooth +field are jointly Gaussian under its unchanged literal law. Empty families, +order zero and repeated observations are retained. -/ +theorem smoothSourceField_derivativeVector_hasGaussianLaw (d : ℕ) {L : ℝ} + (hL : L ≠ 0) (n : ℕ) (k : Fin n → ℕ) (x : Fin n → (Fin d → ℝ)) + (u : ∀ j, Fin (k j) → (Fin d → ℝ)) : + HasGaussianLaw (fun ω j => + iteratedFDeriv ℝ (k j) (smoothSourceField d L ω) (x j) (u j)) (smoothLaw d L) := by + let a : Fin n → SourceMode d → ℝ := fun j => derivativeCoefficient d L (k j) (x j) (u j) + have ha (j : Fin n) : Summable (a j) := + derivativeCoefficient_summable d hL (k j) (x j) (u j) + have hm : Measurable (fun η j => scalarSeries (a j) η) := + Measurable.of_eval (fun j => scalarSeries_measurable (a j)) + have hG : HasGaussianLaw + (fun ω j => scalarSeries (a j) (sampleInclusion d L ω)) (smoothLaw d L) := by + refine ⟨(hm.comp (sampleInclusion_measurable d L)).aemeasurable, ?_⟩ + rw [observation_map_eq d hL _ hm] + exact (scalarSeries_finiteVector_hasGaussianLaw n a ha).isGaussian_map + apply hG.congr + filter_upwards with ω + funext j + exact (smoothSourceField_derivative_series d hL ω (k j) (x j) (u j)).symm + +/-- Actual derivative covariance as a convergent Fourier coefficient pairing. +This does not yet identify the pairing with derivatives of the theta kernel. -/ +theorem smoothSourceField_derivative_covariance (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (k l : ℕ) (x y : Fin d → ℝ) (u : Fin k → (Fin d → ℝ)) + (v : Fin l → (Fin d → ℝ)) : + cov[fun ω => iteratedFDeriv ℝ k (smoothSourceField d L ω) x u, + fun ω => iteratedFDeriv ℝ l (smoothSourceField d L ω) y v; smoothLaw d L] = + ∑' m, derivativeCoefficient d L k x u m * derivativeCoefficient d L l y v m := by + simp_rw [smoothSourceField_derivative_series d hL] + rw [covariance_comp_inclusion d hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (derivativeCoefficient_summable d hL k x u) (derivativeCoefficient_summable d hL l y v) + +end I4SourceDerivativeGaussian diff --git a/frontiers/source_derivative_gaussian_20261025/SourceEndpointEstimates.lean b/frontiers/source_derivative_gaussian_20261025/SourceEndpointEstimates.lean new file mode 100644 index 000000000..5beba8edd --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourceEndpointEstimates.lean @@ -0,0 +1,366 @@ +import SourceEndpointParameters +import SourceConfluentLimits +import SourceContactEndpointParameters +import SourcePhysicalEndpointKernel +import GaussianRegressionLimits +import ResearchFormalCoreR1.GaussianNormalizerLimit +import ResearchFormalCoreR1.PalmReweighting + +/-! +# Actual canonical endpoint estimates and reweighting + +The LM006 endpoint vector is explicitly converted to the literal physical +maximum/saddle determinant weight. These composition lemmas retain the actual +canonical all-pin kernel and its derived Gaussian parameters. +See SOURCE_ENDPOINT_ESTIMATES.md for the frozen source contract. +-/ + +noncomputable section + +open I4SmoothCarrier I4SourceConfluentPins I4SourceEndpointRegression +open I4SourceEndpointParameters ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Matrix WithLp Filter +open scoped Topology NNReal + +set_option autoImplicit false + +namespace I4SourceEndpointEstimates + +/-- LM006 `(aM,aS,bM,bS,cM,cS)` to actual `(aM,bM,cM,aS,bS,cS)` order. -/ +def sourcePhysicalCoordinates (v : Fin 6 → ℝ) : Fin 6 → ℝ := + ![v 0,v 2,v 4,v 1,v 3,v 5] + +def sourcePhysicalIndex : Fin 6 → Fin 6 := ![0,2,4,1,3,5] + +theorem sourcePhysicalCoordinates_apply (v : Fin 6 → ℝ) (i : Fin 6) : + sourcePhysicalCoordinates v i = v (sourcePhysicalIndex i) := by + fin_cases i <;> rfl + +theorem sourcePhysicalCoordinates_continuous : Continuous sourcePhysicalCoordinates := by + unfold sourcePhysicalCoordinates + fun_prop + +theorem sourcePhysicalCoordinates_inverse (v : Fin 6 → ℝ) : + enb_permutation (sourcePhysicalCoordinates v) = v := by + ext i + fin_cases i <;> rfl + +/-- The literal strict typed physical endpoint determinant weight. -/ +def sourceEndpointWeight (r : ℝ) (v : Fin 6 → ℝ) : ℝ := + egm_weight r (sourcePhysicalCoordinates v) + +theorem sourceEndpointWeight_measurable (r : ℝ) : Measurable (sourceEndpointWeight r) := + (egm_weight_measurable r).comp sourcePhysicalCoordinates_continuous.measurable + +theorem sourceEndpointWeight_nonneg (r : ℝ) (v : Fin 6 → ℝ) : + 0 ≤ sourceEndpointWeight r v := by + unfold sourceEndpointWeight egm_weight + split_ifs + · exact abs_nonneg _ + · exact le_rfl + +theorem sourceEndpointWeight_eq_continuous (r : ℝ) (hr : 0 < r) (v : Fin 6 → ℝ) : + sourceEndpointWeight r v = r^2 * gn_normalizedWeight r v := by + simpa only [sourceEndpointWeight, sourcePhysicalCoordinates_inverse] using + enb_weight_identity r (sourcePhysicalCoordinates v) hr + +def sourceEndpointLaw (L r b : ℝ) : Measure (Fin 6 → ℝ) := + sourceEndpointKernel L r (sourcePrescribedPins r b) + +def sourceEndpointMean (L r b : ℝ) : Fin 6 → ℝ := + sourceRegressionMapAt L r (sourcePrescribedPins r b) + +theorem sourceEndpointLaw_isProbabilityMeasure {L : ℝ} (hL : L ≠ 0) (r b : ℝ) : + IsProbabilityMeasure (sourceEndpointLaw L r b) := by + unfold sourceEndpointLaw + have : IsMarkovKernel (sourceEndpointKernel L r) := sourceEndpointKernel_markov hL r + exact inferInstance + +/-- The actual canonical every-pin kernel has the derived finite Gaussian law. -/ +theorem sourceEndpointLaw_euclidean_hasLaw {L : ℝ} (hL : L ≠ 0) (r b : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) : + HasLaw (toLp 2) (multivariateGaussian (toLp 2 (sourceEndpointMean L r b)) + (sourceEndpointSchurAt L r)) (sourceEndpointLaw L r b) where + aemeasurable := by fun_prop + map_eq := sourceEndpointKernel_eq_multivariateGaussian hL r hG (sourcePrescribedPins r b) + +/-- Every physical coordinate marginal is derived from the actual kernel, +including singular residual covariances and noncentered prescribed heights. -/ +theorem sourceEndpointLaw_physical_marginal {L : ℝ} (hL : L ≠ 0) (r b : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (i : Fin 6) : + HasLaw (fun v => sourcePhysicalCoordinates v i) + (gaussianReal (sourceEndpointMean L r b (sourcePhysicalIndex i)) + ((sourceEndpointSchurAt L r) (sourcePhysicalIndex i) (sourcePhysicalIndex i)).toNNReal) + (sourceEndpointLaw L r b) := by + have hPSD := sourceEndpointSchurAt_posSemidef hL r hG + have h := (measurePreserving_eval_multivariateGaussian + (μ := toLp 2 (sourceEndpointMean L r b)) hPSD (i := sourcePhysicalIndex i)).hasLaw.fun_comp + (sourceEndpointLaw_euclidean_hasLaw hL r b hG) + exact h.congr (Eventually.of_forall (fun v => sourcePhysicalCoordinates_apply v i)) + +/-- Integrals of the literal physical weight transfer to the exact finite +Gaussian parameter law without an assumed integral representation. -/ +theorem sourceEndpointLaw_weight_integral {L : ℝ} (hL : L ≠ 0) (r b : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) : + (∫ v, sourceEndpointWeight r v ∂sourceEndpointLaw L r b) = + ∫ x : EuclideanSpace ℝ (Fin 6), egm_weight r ![x 0,x 2,x 4,x 1,x 3,x 5] + ∂multivariateGaussian (toLp 2 (sourceEndpointMean L r b)) (sourceEndpointSchurAt L r) := by + have h := (sourceEndpointLaw_euclidean_hasLaw hL r b hG).integral_comp + (f := fun x : EuclideanSpace ℝ (Fin 6) => egm_weight r ![x 0,x 2,x 4,x 1,x 3,x 5]) + (show AEStronglyMeasurable + (fun x : EuclideanSpace ℝ (Fin 6) => egm_weight r ![x 0,x 2,x 4,x 1,x 3,x 5]) + (multivariateGaussian (toLp 2 (sourceEndpointMean L r b)) (sourceEndpointSchurAt L r)) by + exact ((egm_weight_measurable r).comp (by fun_prop)).aestronglyMeasurable) + simpa only [sourceEndpointWeight, sourcePhysicalCoordinates, Function.comp_def, + ofLp_toLp] using h + +/-- Per-radius moment composition from explicit parameter bounds. The later +actual-source family theorem derives these bounds from the actual block limits. -/ +theorem sourceEndpointLaw_second_moment_of_parameter_bounds {L : ℝ} (hL : L ≠ 0) + (r b B V : ℝ) (hG : IsUnit (sourcePinCovarianceAt L r)) + (hB : 0 ≤ B) (hV : 0 ≤ V) (hr : 0 < r) (hr1 : r ≤ 1) + (hmean : ∀ i, |sourceEndpointMean L r b i| ≤ B) + (hvar : ∀ i, |sourceEndpointSchurAt L r i i| ≤ V) : + Integrable (fun v => sourceEndpointWeight r v ^ 2) (sourceEndpointLaw L r b) ∧ + (∫ v, sourceEndpointWeight r v ^ 2 ∂sourceEndpointLaw L r b) ≤ + 16 * gm8FiniteM8 6 B V * r ^ 4 := by + have := sourceEndpointLaw_isProbabilityMeasure hL r b + have hPSD := sourceEndpointSchurAt_posSemidef hL r hG + apply egm_gaussian_moment (sourceEndpointLaw L r b) sourcePhysicalCoordinates + sourcePhysicalCoordinates_continuous.measurable + (fun i => sourceEndpointMean L r b (sourcePhysicalIndex i)) + (fun i => ((sourceEndpointSchurAt L r) (sourcePhysicalIndex i) (sourcePhysicalIndex i)).toNNReal) + B V r hB hV hr hr1 + · exact fun i => hmean (sourcePhysicalIndex i) + · intro i + rw [Real.coe_toNNReal _ hPSD.diag_nonneg] + exact (le_abs_self _).trans (hvar (sourcePhysicalIndex i)) + · exact sourceEndpointLaw_physical_marginal hL r b hG + +/-- Actual physical square integrability needs no supplied parameter bound. +Finite current Gaussian parameters provide their own finite coordinate bounds. -/ +theorem sourceEndpointLaw_weight_square_integrable {L : ℝ} (hL : L ≠ 0) + (r b : ℝ) (hG : IsUnit (sourcePinCovarianceAt L r)) (hr : 0 < r) (hr1 : r ≤ 1) : + Integrable (fun v => sourceEndpointWeight r v ^ 2) (sourceEndpointLaw L r b) := by + apply (sourceEndpointLaw_second_moment_of_parameter_bounds hL r b + ‖sourceEndpointMean L r b‖ ‖fun i => sourceEndpointSchurAt L r i i‖ + hG (norm_nonneg _) (norm_nonneg _) hr hr1 ?_ ?_).1 + · intro i + simpa only [Real.norm_eq_abs] using norm_le_pi_norm (sourceEndpointMean L r b) i + · intro i + simpa only [Real.norm_eq_abs] using + norm_le_pi_norm (fun i => sourceEndpointSchurAt L r i i) i + +theorem sourceEndpointLaw_weight_integrable {L : ℝ} (hL : L ≠ 0) + (r b : ℝ) (hG : IsUnit (sourcePinCovarianceAt L r)) (hr : 0 < r) (hr1 : r ≤ 1) : + Integrable (sourceEndpointWeight r) (sourceEndpointLaw L r b) := by + have := sourceEndpointLaw_isProbabilityMeasure hL r b + exact prw_integrable_of_integrable_sq _ _ (sourceEndpointWeight_measurable r) + (sourceEndpointLaw_weight_square_integrable hL r b hG hr hr1) + +/-- The literal prescribed height has the actual contact pin as its right limit. -/ +theorem sourcePrescribedPins_tendsto (b : ℝ) : + Tendsto (fun r => sourcePrescribedPins r b) (𝓝[>] (0 : ℝ)) + (𝓝 ![b,0,0,0,0,2]) := by + have hc : Continuous (fun r => sourcePrescribedPins r b) := by + unfold sourcePrescribedPins + fun_prop + simpa [sourcePrescribedPins] using (hc.tendsto 0).mono_left nhdsWithin_le_nhds + +/-- Explicit endpoint probability with the literal normalized weight density. -/ +def sourceEndpointWeightedProbability (L r b : ℝ) : Measure (Fin 6 → ℝ) := + prwMeasure (sourceEndpointLaw L r b) (sourceEndpointWeight r) + +/-- Endpoint reweighting glue; its estimates are derived in the final actual +source theorem after the actual covariance-limit exports are available. -/ +theorem sourceEndpoint_weighted_event_bound_of_estimates {L : ℝ} (hL : L ≠ 0) + (r b C c : ℝ) (hC : 0 ≤ C) (hc : 0 < c) (hr : 0 < r) + (hsq : Integrable (fun v => sourceEndpointWeight r v ^ 2) (sourceEndpointLaw L r b)) + (hmoment : (∫ v, sourceEndpointWeight r v ^ 2 ∂sourceEndpointLaw L r b) ≤ C*r^4) + (hfloor : c*r^2 ≤ ∫ v, sourceEndpointWeight r v ∂sourceEndpointLaw L r b) + (E : Set (Fin 6 → ℝ)) (hE : MeasurableSet E) : + IsProbabilityMeasure (sourceEndpointWeightedProbability L r b) ∧ + (sourceEndpointWeightedProbability L r b).real E ≤ + (Real.sqrt C/c) * Real.sqrt ((sourceEndpointLaw L r b).real E) := by + have := sourceEndpointLaw_isProbabilityMeasure hL r b + exact prw_square_root_event_bound (sourceEndpointLaw L r b) (sourceEndpointWeight r) + (sourceEndpointWeight_measurable r) (sourceEndpointWeight_nonneg r) hsq + C c r hC hc hr hmoment hfloor E hE + +open I4SourceContact I4SourceConfluentLimits I4GaussianRegressionLimits + +theorem sourceEndpoint_parameter_limits {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + (∀ᶠ r in 𝓝[>] (0 : ℝ), IsUnit (sourcePinCovarianceAt L r)) ∧ + Tendsto (fun r => toLp 2 (sourceEndpointMean L r b)) (𝓝[>] (0 : ℝ)) + (𝓝 (toLp 2 (sourceContactEndpointMean L b))) ∧ + Tendsto (sourceEndpointSchurAt L) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactEndpointSchur L)) := by + have hd : Tendsto (fun r => sourcePrescribedPins r b) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactPinValue b)) := by + simpa only [sourceContactPinValue] using sourcePrescribedPins_tendsto b + have hp := regression_parameter_limits (𝓝[>] (0 : ℝ)) + (sourcePinCovarianceAt L) (sourceCrossCovarianceAt L) (sourceEndpointCovarianceAt L) + (fun r => sourcePrescribedPins r b) + (sourcePinCovariance L) (sourceContactEndpointCrossCovariance L) + (sourceContactEndpointCovariance L) (sourceContactPinValue b) + (sourcePinCovarianceAt_tendsto hL) (sourceCrossCovarianceAt_tendsto hL) + (sourceEndpointCovarianceAt_tendsto hL) hd (sourcePinCovariance_isUnit hL) + refine ⟨hp.1, ?_, ?_⟩ + · have hm : Tendsto (fun r => sourceEndpointMean L r b) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactEndpointMean L b)) := by + simpa only [regressionMean, sourceEndpointMean, sourceRegressionMapAt_apply, + sourceRegressionMatrixAt, sourceContactEndpointMean, sourceContactEndpointRegression] + using hp.2.2.1 + exact ((PiLp.continuous_toLp 2 (fun _ : Fin 6 => ℝ)).tendsto _).comp hm + · exact hp.2.2.2 + +theorem sourceEndpoint_uniform_parameter_bounds {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + ∃ M V δ : ℝ, 0 ≤ M ∧ 0 ≤ V ∧ 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → IsUnit (sourcePinCovarianceAt L r) ∧ + (∀ i, |sourceEndpointMean L r b i| ≤ M) ∧ + (∀ i, |sourceEndpointSchurAt L r i i| ≤ V) := by + have hd : Tendsto (fun r => sourcePrescribedPins r b) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactPinValue b)) := by + simpa only [sourceContactPinValue] using sourcePrescribedPins_tendsto b + obtain ⟨M,V,δ,hM,hV,hδ,hbounds⟩ := regression_uniform_bounds + (sourcePinCovarianceAt L) (sourceCrossCovarianceAt L) (sourceEndpointCovarianceAt L) + (fun r => sourcePrescribedPins r b) + (sourcePinCovariance L) (sourceContactEndpointCrossCovariance L) + (sourceContactEndpointCovariance L) (sourceContactPinValue b) + (sourcePinCovarianceAt_tendsto hL) (sourceCrossCovarianceAt_tendsto hL) + (sourceEndpointCovarianceAt_tendsto hL) hd (sourcePinCovariance_isUnit hL) + refine ⟨M,V,δ,hM,hV,hδ,?_⟩ + intro r hr hrδ + obtain ⟨hunit,hm,hcov⟩ := hbounds r hr hrδ + refine ⟨hunit,?_,?_⟩ + · simpa only [regressionMean, sourceEndpointMean, sourceRegressionMapAt_apply, + sourceRegressionMatrixAt] using hm + · simpa only [regressionCovariance, sourceEndpointSchurAt] using hcov + + +/- Normalizer assembly coauthored by actual OpenAI/Codex helper +/root/dm2_landed_composition/peano_strategy; author-side only. -/ +/-- The actual every-pin source law has a positive small-radius physical +normalizer coefficient, with arbitrary fixed real prescribed height. -/ +theorem sourceEndpoint_normalizer_floor {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → + c * r^2 ≤ ∫ v, sourceEndpointWeight r v ∂sourceEndpointLaw L r b := by + obtain ⟨hunit, hm, hS⟩ := sourceEndpoint_parameter_limits hL b + have hPSD : ∀ᶠ r in 𝓝[>] (0 : ℝ), (sourceEndpointSchurAt L r).PosSemidef := by + filter_upwards [hunit] with r hr + exact sourceEndpointSchurAt_posSemidef hL r hr + have hmC := sourceContactEndpointMean_contact hL b + have hSC := sourceContactEndpointSchur_contact hL + obtain ⟨c, hc, δN, hδN, hfloor⟩ := + gnl_physical_normalizer_floor_of_parameter_limits + (fun r => toLp 2 (sourceEndpointMean L r b)) (sourceEndpointSchurAt L) + (toLp 2 (sourceContactEndpointMean L b)) (sourceContactEndpointSchur L) + hm hS hPSD (sourceContactEndpointSchur_posSemidef hL) + hmC.1 hmC.2.1 hmC.2.2.1 hmC.2.2.2 + hSC.1 hSC.2.1 hSC.2.2.1 hSC.2.2.2 + (sourceContactEndpointSchur_referenceVariance_pos hL) + obtain ⟨δG, hδG, hGinterval⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hunit + refine ⟨c, hc, min δN δG, lt_min hδN hδG, ?_⟩ + intro r hr hrδ + have hG : IsUnit (sourcePinCovarianceAt L r) := + hGinterval ⟨hr, lt_of_lt_of_le hrδ (min_le_right _ _)⟩ + rw [sourceEndpointLaw_weight_integral hL r b hG] + exact hfloor r hr (lt_of_lt_of_le hrδ (min_le_left _ _)) + +theorem sourceEndpoint_estimates {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + ∃ C c δ : ℝ, 0 ≤ C ∧ 0 < c ∧ 0 < δ ∧ δ ≤ 1 ∧ + ∀ r : ℝ, 0 < r → r < δ → + Integrable (fun v => sourceEndpointWeight r v ^ 2) (sourceEndpointLaw L r b) ∧ + (∫ v, sourceEndpointWeight r v ^ 2 ∂sourceEndpointLaw L r b) ≤ C*r^4 ∧ + c*r^2 ≤ ∫ v, sourceEndpointWeight r v ∂sourceEndpointLaw L r b := by + obtain ⟨M,V,δM,hM,hV,hδM,hbounds⟩ := sourceEndpoint_uniform_parameter_bounds hL b + obtain ⟨c,hc,δF,hδF,hfloor⟩ := sourceEndpoint_normalizer_floor hL b + refine ⟨16 * gm8FiniteM8 6 M V,c,min δM (min δF 1), + mul_nonneg (by norm_num) (gm8FiniteM8_nonneg 6 M V),hc, + lt_min hδM (lt_min hδF (by norm_num)), + (min_le_right _ _).trans (min_le_right _ _),?_⟩ + intro r hr hrδ + have hrM : r < δM := lt_of_lt_of_le hrδ (min_le_left _ _) + have hrF : r < δF := lt_of_lt_of_le hrδ ((min_le_right _ _).trans (min_le_left _ _)) + have hr1 : r ≤ 1 := (le_of_lt hrδ).trans ((min_le_right _ _).trans (min_le_right _ _)) + obtain ⟨hunit,hm,hv⟩ := hbounds r hr hrM + obtain ⟨hsq,hbound⟩ := sourceEndpointLaw_second_moment_of_parameter_bounds hL r b M V + hunit hM hV hr hr1 hm hv + exact ⟨hsq,hbound,hfloor r hr hrF⟩ + +theorem sourceEndpoint_estimates_and_weighted_events {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + ∃ C c δ : ℝ, 0 ≤ C ∧ 0 < c ∧ 0 < δ ∧ δ ≤ 1 ∧ + ∀ r : ℝ, 0 < r → r < δ → + Integrable (fun v => sourceEndpointWeight r v ^ 2) (sourceEndpointLaw L r b) ∧ + (∫ v, sourceEndpointWeight r v ^ 2 ∂sourceEndpointLaw L r b) ≤ C*r^4 ∧ + c*r^2 ≤ (∫ v, sourceEndpointWeight r v ∂sourceEndpointLaw L r b) ∧ + IsProbabilityMeasure (sourceEndpointWeightedProbability L r b) ∧ + ∀ E : Set (Fin 6 → ℝ), MeasurableSet E → + (sourceEndpointWeightedProbability L r b).real E ≤ + (Real.sqrt C/c) * Real.sqrt ((sourceEndpointLaw L r b).real E) := by + obtain ⟨C,c,δ,hC,hc,hδ,hδ1,he⟩ := sourceEndpoint_estimates hL b + refine ⟨C,c,δ,hC,hc,hδ,hδ1,?_⟩ + intro r hr hrδ + obtain ⟨hsq,hm,hf⟩ := he r hr hrδ + have hp := sourceEndpoint_weighted_event_bound_of_estimates hL r b C c hC hc hr + hsq hm hf Set.univ MeasurableSet.univ + refine ⟨hsq,hm,hf,hp.1,?_⟩ + intro E hE + exact (sourceEndpoint_weighted_event_bound_of_estimates hL r b C c hC hc hr + hsq hm hf E hE).2 + + +open I4SourcePhysicalEndpointKernel + +/-- The canonical kernel of the actual physical normalized Hessian entries. -/ +def sourcePhysicalEndpointLaw (L r b : ℝ) : Measure (Fin 6 → ℝ) := + sourcePhysicalEndpointKernel L r (sourcePrescribedPins r b) + +/-- Literal equality of the specified kernels at the prescribed pins, derived +from their common residual law and the proved zero correction there. -/ +theorem sourcePhysicalEndpointLaw_eq {L : ℝ} (hL : L ≠ 0) (r b : ℝ) : + sourcePhysicalEndpointLaw L r b = sourceEndpointLaw L r b := + sourcePhysicalEndpointKernel_at_prescribed hL r b + +theorem sourcePhysicalEndpointLaw_isProbabilityMeasure {L : ℝ} (hL : L ≠ 0) (r b : ℝ) : + IsProbabilityMeasure (sourcePhysicalEndpointLaw L r b) := by + rw [sourcePhysicalEndpointLaw_eq hL r b] + exact sourceEndpointLaw_isProbabilityMeasure hL r b + +/-- Actual block limits supply current invertibility, hence the physical +kernel is a conditional law at every sufficiently small positive radius. -/ +theorem sourcePhysicalEndpointKernel_eventually_hasCondDistrib {L : ℝ} (hL : L ≠ 0) : + ∀ᶠ r in 𝓝[>] (0 : ℝ), + HasCondDistrib (sourceNormalizedEndpoints L r) (sourceConfluentPinsAt L r) + (sourcePhysicalEndpointKernel L r) (smoothLaw 2 L) := by + filter_upwards [(sourceEndpoint_parameter_limits hL 0).1] with r hr + exact sourcePhysicalEndpointKernel_hasCondDistrib hL r hr + +/-- The explicit probability is weighted from the actual physical endpoint law. -/ +def sourcePhysicalEndpointWeightedProbability (L r b : ℝ) : Measure (Fin 6 → ℝ) := + prwMeasure (sourcePhysicalEndpointLaw L r b) (sourceEndpointWeight r) + +theorem sourcePhysicalEndpointWeightedProbability_eq {L : ℝ} (hL : L ≠ 0) (r b : ℝ) : + sourcePhysicalEndpointWeightedProbability L r b = sourceEndpointWeightedProbability L r b := by + unfold sourcePhysicalEndpointWeightedProbability sourceEndpointWeightedProbability + rw [sourcePhysicalEndpointLaw_eq hL r b] + +/-- Actual physical canonical endpoint moment, normalizer and measurable-event +estimates, at arbitrary fixed height. No desired law, covariance limit, +parameter bound or positive normalizer is a premise. -/ +theorem sourcePhysicalEndpoint_estimates_and_weighted_events {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + ∃ C c δ : ℝ, 0 ≤ C ∧ 0 < c ∧ 0 < δ ∧ δ ≤ 1 ∧ + ∀ r : ℝ, 0 < r → r < δ → + Integrable (fun v => sourceEndpointWeight r v ^ 2) (sourcePhysicalEndpointLaw L r b) ∧ + (∫ v, sourceEndpointWeight r v ^ 2 ∂sourcePhysicalEndpointLaw L r b) ≤ C*r^4 ∧ + c*r^2 ≤ (∫ v, sourceEndpointWeight r v ∂sourcePhysicalEndpointLaw L r b) ∧ + IsProbabilityMeasure (sourcePhysicalEndpointWeightedProbability L r b) ∧ + ∀ E : Set (Fin 6 → ℝ), MeasurableSet E → + (sourcePhysicalEndpointWeightedProbability L r b).real E ≤ + (Real.sqrt C/c) * Real.sqrt ((sourcePhysicalEndpointLaw L r b).real E) := by + simpa only [sourcePhysicalEndpointLaw_eq hL, + sourcePhysicalEndpointWeightedProbability_eq hL] using + sourceEndpoint_estimates_and_weighted_events hL b + +end I4SourceEndpointEstimates diff --git a/frontiers/source_derivative_gaussian_20261025/SourceEndpointParameters.lean b/frontiers/source_derivative_gaussian_20261025/SourceEndpointParameters.lean new file mode 100644 index 000000000..dac364319 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourceEndpointParameters.lean @@ -0,0 +1,291 @@ +import SourceEndpointRegression +import Mathlib.Probability.Distributions.Gaussian.Multivariate + +/-! +# Actual canonical endpoint-kernel Gaussian parameters + +Unconditional means are derived from the literal differentiated source series. +The residual covariance is derived from bilinearity and the actual regression +normal equations. Its Schur matrix is positive semidefinite because it is an +actual Gaussian covariance; no positivity premise is added. + +At every pin the canonical kernel, transported from the Pi space to Euclidean +space, equals the literal `multivariateGaussian` with mean `C G⁻¹ x` and Schur +covariance. Singular residual covariance is retained. Current pin covariance +invertibility is explicit pending the separate covariance-limit join. +See `SOURCE_ENDPOINT_PARAMETERS.md` for the frozen source contract. +-/ + +noncomputable section + +open I4Foundation I4Source I4SourceGaussian I4SmoothCarrier I4SourceDerivativeGaussian +open I4SourceConfluentPins I4SourceEndpointRegression ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Matrix WithLp +open scoped BigOperators Matrix RealInnerProductSpace + +set_option autoImplicit false + +namespace I4SourceEndpointParameters + +theorem smoothSourceField_derivative_integral_zero (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (k : ℕ) (x : Fin d → ℝ) (u : Fin k → (Fin d → ℝ)) : + (∫ ω, iteratedFDeriv ℝ k (smoothSourceField d L ω) x u ∂smoothLaw d L) = 0 := by + simp_rw [smoothSourceField_derivative_series d hL] + have hm := integral_map (μ := smoothLaw d L) (sampleInclusion_measurable d L).aemeasurable + (scalarSeries_measurable (derivativeCoefficient d L k x u)).aestronglyMeasurable + rw [sampleInclusion_map d hL] at hm + rw [← hm] + exact scalarSeries_integral_zero _ (derivativeCoefficient_summable d hL k x u) + +theorem rawPinsWithHessians_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) (i : Fin 12) : + (∫ ω, rawPinsWithHessians L r ω i ∂smoothLaw 2 L) = 0 := by + fin_cases i + all_goals dsimp [rawPinsWithHessians, rawPins] + all_goals first + | exact smoothSourceField_derivative_integral_zero 2 hL 2 _ _ + | simpa only [iteratedFDeriv_zero_apply] using + smoothSourceField_derivative_integral_zero 2 hL 0 _ ![] + | simpa [iteratedFDeriv_one_apply] using + smoothSourceField_derivative_integral_zero 2 hL 1 _ ![directionX] + | simpa [iteratedFDeriv_one_apply] using + smoothSourceField_derivative_integral_zero 2 hL 1 _ ![directionY] + +theorem rawPinsWithHessians_integral_vector_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, rawPinsWithHessians L r ω ∂smoothLaw 2 L) = 0 := by + ext i + have hh := (ContinuousLinearMap.proj i : (Fin 12 → ℝ) →L[ℝ] ℝ).integral_comp_comm + (rawPinsWithHessians_hasGaussianLaw hL r).integrable + change (∫ ω, rawPinsWithHessians L r ω i ∂smoothLaw 2 L) = + (∫ ω, rawPinsWithHessians L r ω ∂smoothLaw 2 L) i at hh + change (∫ ω, rawPinsWithHessians L r ω ∂smoothLaw 2 L) i = 0 + rw [← hh] + exact rawPinsWithHessians_integral_zero hL r i + +theorem sourcePins_adjusted_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, (sourceConfluentPinsAt L r ω, sourceAdjustedEndpoints L r ω) ∂smoothLaw 2 L) = 0 := by + have hh := (endpointObservationMap r).integral_comp_comm + (rawPinsWithHessians_hasGaussianLaw hL r).integrable + simp_rw [endpointObservationMap_actual] at hh + rw [rawPinsWithHessians_integral_vector_zero hL r, map_zero] at hh + exact hh + +theorem sourceConfluentPinsAt_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, sourceConfluentPinsAt L r ω ∂smoothLaw 2 L) = 0 := by + have hh := congrArg Prod.fst (sourcePins_adjusted_integral_zero hL r) + rw [fst_integral (sourcePins_adjusted_hasGaussianLaw hL r).integrable] at hh + exact hh + +theorem sourceAdjustedEndpoints_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, sourceAdjustedEndpoints L r ω ∂smoothLaw 2 L) = 0 := by + have hh := congrArg Prod.snd (sourcePins_adjusted_integral_zero hL r) + rw [snd_integral (sourcePins_adjusted_hasGaussianLaw hL r).integrable] at hh + exact hh + +theorem sourceEndpointResidual_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, sourceEndpointResidual L r ω ∂smoothLaw 2 L) = 0 := by + have hG := sourcePins_adjusted_hasGaussianLaw hL r + unfold sourceEndpointResidual + rw [integral_sub hG.snd.integrable (hG.fst.map_fun (sourceRegressionMapAt L r)).integrable, + (sourceRegressionMapAt L r).integral_comp_comm hG.fst.integrable, + sourceAdjustedEndpoints_integral_zero hL r, sourceConfluentPinsAt_integral_zero hL r, + map_zero, sub_zero] + +theorem sourceEndpointResidual_covariance {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (i j : Fin 6) : + covariance (fun ω => sourceEndpointResidual L r ω i) + (fun ω => sourceEndpointResidual L r ω j) (smoothLaw 2 L) = + sourceEndpointCovarianceAt L r i j - + (sourceCrossCovarianceAt L r * (sourcePinCovarianceAt L r)⁻¹ * + (sourceCrossCovarianceAt L r)ᵀ) i j := by + let X := sourceConfluentPinsAt L r + let V := sourceAdjustedEndpoints L r + let B := sourceRegressionMapAt L r + let R := sourceEndpointResidual L r + let P := smoothLaw 2 L + have hXV := sourcePins_adjusted_hasGaussianLaw hL r + have : IsProbabilityMeasure P := hXV.isProbabilityMeasure + have hXi (l : Fin 6) : MemLp (fun ω => X ω l) 2 P := (hXV.fst.eval l).memLp_two + have hVi (l : Fin 6) : MemLp (fun ω => V ω l) 2 P := (hXV.snd.eval l).memLp_two + have hBi (l : Fin 6) : MemLp (fun ω => B (X ω) l) 2 P := + ((hXV.fst.map_fun B).eval l).memLp_two + have hRi (l : Fin 6) : MemLp (fun ω => R ω l) 2 P := by + exact (hVi l).sub (hBi l) + have hXR (l t : Fin 6) : + covariance (fun ω => X ω l) (fun ω => R ω t) P = 0 := by + simpa only [sub_zero, X, R, P, sourceEndpointResidual] using + regression_residual_zero_covariance hXV B 0 0 + (sourceRegression_normal_equations L r hG) l t + have hBR (l t : Fin 6) : + covariance (fun ω => B (X ω) l) (fun ω => R ω t) P = 0 := by + have heq : (fun ω => B (X ω) l) = + (fun ω => ∑ a, B (Pi.single a 1) l * X ω a) := by + funext ω + exact regression_map_coordinate B (X ω) l + rw [heq, covariance_fun_sum_left (fun a => (hXi a).const_mul _) (hRi t)] + simp_rw [covariance_const_mul_left, hXR, mul_zero] + exact Finset.sum_const_zero + have hBV (l t : Fin 6) : + covariance (fun ω => B (X ω) l) (fun ω => V ω t) P = + (sourceRegressionMatrixAt L r * (sourceCrossCovarianceAt L r)ᵀ) l t := by + have heq : (fun ω => B (X ω) l) = + (fun ω => ∑ a, B (Pi.single a 1) l * X ω a) := by + funext ω + exact regression_map_coordinate B (X ω) l + rw [heq, covariance_fun_sum_left (fun a => (hXi a).const_mul _) (hVi t)] + simp_rw [covariance_const_mul_left] + change (∑ a, sourceRegressionMapAt L r (Pi.single a 1) l * + covariance (fun ω => X ω a) (fun ω => V ω t) P) = _ + simp only [sourceRegressionMapAt_single, Matrix.mul_apply, Matrix.transpose_apply] + apply Finset.sum_congr rfl + intro a _ + congr 1 + exact covariance_comm _ _ + change covariance (fun ω => R ω i) (fun ω => V ω j - B (X ω) j) P = _ + rw [covariance_fun_sub_right (hRi i) (hVi j) (hBi j), + covariance_comm (fun ω => R ω i) (fun ω => B (X ω) j), hBR j i, sub_zero] + change covariance (fun ω => V ω i - B (X ω) i) (fun ω => V ω j) P = _ + rw [covariance_fun_sub_left (hVi i) (hBi i) (hVi j), hBV i j] + rfl + +lemma finiteGaussian_law_parameters {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n : ℕ} {Z : Ω → Fin n → ℝ} + (hZ : HasGaussianLaw Z P) (m : Fin n → ℝ) (hm : (∫ ω, Z ω ∂P) = m) + (S : Matrix (Fin n) (Fin n) ℝ) + (hcov : ∀ i j, covariance (fun ω => Z ω i) (fun ω => Z ω j) P = S i j) : + S.PosSemidef ∧ P.map (fun ω => toLp 2 (Z ω)) = multivariateGaussian (toLp 2 m) S := by + have : IsProbabilityMeasure P := hZ.isProbabilityMeasure + let e : (Fin n → ℝ) ≃L[ℝ] EuclideanSpace ℝ (Fin n) := + (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin n => ℝ)).symm + have hE : HasGaussianLaw (fun ω => toLp 2 (Z ω)) P := hZ.map_equiv_fun e + have : IsGaussian (P.map (fun ω => toLp 2 (Z ω))) := hE.isGaussian_map + have hZi (i : Fin n) : MemLp (fun ω => Z ω i) 2 P := (hZ.eval i).memLp_two + have hS : S.PosSemidef := by + apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg + · ext i j + simp only [Matrix.conjTranspose_apply, star_trivial] + rw [← hcov i j, ← hcov j i] + exact covariance_comm _ _ + · intro x + have hp := covarianceBilin_self_nonneg + (μ := P.map (fun ω => toLp 2 (Z ω))) (toLp 2 x) + rw [covarianceBilin_apply_pi hZi] at hp + simpa [hcov, dotProduct, Matrix.mulVec, Finset.mul_sum, mul_assoc, mul_left_comm, + mul_comm] using hp + refine ⟨hS, ?_⟩ + apply IsGaussian.ext + · simp only [id_eq] + rw [integral_id_multivariateGaussian] + rw [integral_map hE.aemeasurable (by fun_prop)] + change (∫ ω, e (Z ω) ∂P) = e m + rw [e.integral_comp_comm, hm] + · rw [← ContinuousLinearMap.toBilinForm_inj] + refine LinearMap.BilinForm.ext_basis (EuclideanSpace.basisFun (Fin n) ℝ).toBasis fun i j => ?_ + simp only [ContinuousLinearMap.toBilinForm_apply] + change covarianceBilin (P.map (fun ω => toLp 2 (Z ω))) + (EuclideanSpace.basisFun (Fin n) ℝ i) (EuclideanSpace.basisFun (Fin n) ℝ j) = + covarianceBilin (multivariateGaussian (toLp 2 m) S) + (EuclideanSpace.basisFun (Fin n) ℝ i) (EuclideanSpace.basisFun (Fin n) ℝ j) + rw [covarianceBilin_apply_basisFun hZi, covarianceBilin_multivariateGaussian hS] + simpa using hcov i j + +/-- The actual adjusted-endpoint Schur covariance. -/ +def sourceEndpointSchurAt (L r : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + sourceEndpointCovarianceAt L r - + sourceCrossCovarianceAt L r * (sourcePinCovarianceAt L r)⁻¹ * (sourceCrossCovarianceAt L r)ᵀ + +/-- Gaussianity is inherited from the actual jointly Gaussian pin/endpoint vector. -/ +theorem sourceEndpointResidual_hasGaussianLaw {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + HasGaussianLaw (sourceEndpointResidual L r) (smoothLaw 2 L) := by + have hR := (regression_residual_jointGaussian (sourcePins_adjusted_hasGaussianLaw hL r) + (sourceRegressionMapAt L r) 0 0).snd + unfold sourceEndpointResidual + simpa only [sub_zero] using hR + +/-- A literal posterior sample at a prescribed pin. -/ +def sourceEndpointPosterior (L r : ℝ) (x : Fin 6 → ℝ) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + sourceEndpointResidual L r ω + sourceRegressionMapAt L r x + +theorem sourceEndpointPosterior_hasGaussianLaw {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : HasGaussianLaw (sourceEndpointPosterior L r x) (smoothLaw 2 L) := by + have hR := sourceEndpointResidual_hasGaussianLaw hL r + refine ⟨hR.aemeasurable.add aemeasurable_const, ?_⟩ + have : IsGaussian ((smoothLaw 2 L).map (sourceEndpointResidual L r)) := hR.isGaussian_map + convert (inferInstance : IsGaussian (((smoothLaw 2 L).map (sourceEndpointResidual L r)).map + (fun v => v + sourceRegressionMapAt L r x))) using 1 + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hR.aemeasurable] + rfl + +theorem sourceEndpointPosterior_integral {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : (∫ ω, sourceEndpointPosterior L r x ω ∂smoothLaw 2 L) = + sourceRegressionMapAt L r x := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + unfold sourceEndpointPosterior + rw [integral_add (sourceEndpointResidual_hasGaussianLaw hL r).integrable (integrable_const _), + sourceEndpointResidual_integral_zero hL r, integral_const, zero_add] + simp + +theorem sourceEndpointPosterior_covariance {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (x : Fin 6 → ℝ) (i j : Fin 6) : + covariance (fun ω => sourceEndpointPosterior L r x ω i) + (fun ω => sourceEndpointPosterior L r x ω j) (smoothLaw 2 L) = + sourceEndpointSchurAt L r i j := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + have hR := sourceEndpointResidual_hasGaussianLaw hL r + change covariance (fun ω => sourceEndpointResidual L r ω i + sourceRegressionMapAt L r x i) + (fun ω => sourceEndpointResidual L r ω j + sourceRegressionMapAt L r x j) _ = _ + rw [covariance_add_const_left (hR.eval i).integrable, + covariance_add_const_right (hR.eval j).integrable] + exact sourceEndpointResidual_covariance hL r hG i j + +/-- The canonical all-pin law is the literal posterior sampling map. -/ +theorem sourceEndpointKernel_eq_map_posterior {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : sourceEndpointKernel L r x = + (smoothLaw 2 L).map (sourceEndpointPosterior L r x) := by + rw [sourceEndpointKernel_apply hL r x, + AEMeasurable.map_map_of_aemeasurable (by fun_prop) + (sourceEndpointResidual_hasGaussianLaw hL r).aemeasurable] + rfl + +theorem sourceEndpointKernel_isGaussian {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : IsGaussian (sourceEndpointKernel L r x) := by + rw [sourceEndpointKernel_eq_map_posterior hL r x] + exact (sourceEndpointPosterior_hasGaussianLaw hL r x).isGaussian_map + +/-- Positive semidefiniteness follows from the actual residual covariance, including singular cases. -/ +theorem sourceEndpointSchurAt_posSemidef {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) : (sourceEndpointSchurAt L r).PosSemidef := by + exact (finiteGaussian_law_parameters (sourceEndpointPosterior_hasGaussianLaw hL r 0) + (sourceRegressionMapAt L r 0) (sourceEndpointPosterior_integral hL r 0) + (sourceEndpointSchurAt L r) (sourceEndpointPosterior_covariance hL r hG 0)).1 + +/-- Exact literal multivariate Gaussian law at every prescribed pin, with no residual PSD premise. -/ +theorem sourceEndpointKernel_eq_multivariateGaussian {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (x : Fin 6 → ℝ) : + (sourceEndpointKernel L r x).map (toLp 2) = + multivariateGaussian (toLp 2 (sourceRegressionMapAt L r x)) (sourceEndpointSchurAt L r) := by + rw [sourceEndpointKernel_eq_map_posterior hL r x, + AEMeasurable.map_map_of_aemeasurable (by fun_prop) + (sourceEndpointPosterior_hasGaussianLaw hL r x).aemeasurable] + exact (finiteGaussian_law_parameters (sourceEndpointPosterior_hasGaussianLaw hL r x) + (sourceRegressionMapAt L r x) (sourceEndpointPosterior_integral hL r x) + (sourceEndpointSchurAt L r) (sourceEndpointPosterior_covariance hL r hG x)).2 + +/-- The original Pi-space kernel mean is the actual regression mean. -/ +theorem sourceEndpointKernel_integral {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : (∫ v, v ∂sourceEndpointKernel L r x) = sourceRegressionMapAt L r x := by + rw [sourceEndpointKernel_eq_map_posterior hL r x, + integral_map (sourceEndpointPosterior_hasGaussianLaw hL r x).aemeasurable (by fun_prop)] + exact sourceEndpointPosterior_integral hL r x + +/-- The original Pi-space kernel covariance is the derived actual Schur covariance. -/ +theorem sourceEndpointKernel_covariance {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (x : Fin 6 → ℝ) (i j : Fin 6) : + covariance (fun v => v i) (fun v => v j) (sourceEndpointKernel L r x) = + sourceEndpointSchurAt L r i j := by + rw [sourceEndpointKernel_eq_map_posterior hL r x, + covariance_map_fun (measurable_pi_apply i).aestronglyMeasurable + (measurable_pi_apply j).aestronglyMeasurable + (sourceEndpointPosterior_hasGaussianLaw hL r x).aemeasurable] + exact sourceEndpointPosterior_covariance hL r hG x i j + +end I4SourceEndpointParameters diff --git a/frontiers/source_derivative_gaussian_20261025/SourceEndpointRegression.lean b/frontiers/source_derivative_gaussian_20261025/SourceEndpointRegression.lean new file mode 100644 index 000000000..9e832ed60 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourceEndpointRegression.lean @@ -0,0 +1,206 @@ +import SourceConfluentPins +import ResearchFormalCoreR1.GaussianRegressionKernel +import Mathlib.LinearAlgebra.Matrix.ToLin + +/-! +# Canonical regression kernel for actual adjusted endpoint observations + +The definitions are families in every real separation, so limits can be stated +without dependent proof arguments. Their values at zero are not claimed to be +contact limits. For nonzero period the actual pin/adjusted-endpoint pair is a +linear image of the proved twelve-observation Gaussian vector. + +The only current algebraic premise in the regression theorem is invertibility +of the actual pin covariance. The residual law and the conditional-law conclusion +are derived. Values of the explicit kernel are specified at every pin vector. +See `SOURCE_ENDPOINT_REGRESSION.md` for the frozen source contract. +-/ + +noncomputable section + +open I4SmoothCarrier I4SourceConfluentPins ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Matrix +open scoped BigOperators Matrix + +set_option autoImplicit false + +namespace I4SourceEndpointRegression + +/-- Actual confluent observations as a nondependent family in r. -/ +def sourceConfluentPinsAt (L r : ℝ) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + confluentTransform r (rawPins L r ω) + +@[simp] theorem sourceConfluentPinsAt_eq (L r : ℝ) (hr : r ≠ 0) : + sourceConfluentPinsAt L r = confluentPins L r hr := rfl + +/-- LM006 order `(xxM,xxS,xyM,xyS,yyM,yyS)`, with the confluent xx/xy pins subtracted. -/ +def sourceAdjustedEndpoints (L r : ℝ) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + let x := sourceConfluentPinsAt L r ω + let v := rawPinsWithHessians L r ω + ![(v 6 - x 3)/r, (v 9 - x 3)/r, (v 7 - x 4)/r, (v 10 - x 4)/r, v 8, v 11] + +/-- The actual pin covariance G. -/ +def sourcePinCovarianceAt (L r : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + fun i j => covariance (fun ω => sourceConfluentPinsAt L r ω i) + (fun ω => sourceConfluentPinsAt L r ω j) (smoothLaw 2 L) + +/-- The actual adjusted-endpoint/pin cross covariance C, with endpoint rows. -/ +def sourceCrossCovarianceAt (L r : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + fun i j => covariance (fun ω => sourceAdjustedEndpoints L r ω i) + (fun ω => sourceConfluentPinsAt L r ω j) (smoothLaw 2 L) + +/-- The actual adjusted endpoint covariance, without an assumed limiting model. -/ +def sourceEndpointCovarianceAt (L r : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + fun i j => covariance (fun ω => sourceAdjustedEndpoints L r ω i) + (fun ω => sourceAdjustedEndpoints L r ω j) (smoothLaw 2 L) + +def sourceRegressionMatrixAt (L r : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + sourceCrossCovarianceAt L r * (sourcePinCovarianceAt L r)⁻¹ + +def sourceRegressionMapAt (L r : ℝ) : (Fin 6 → ℝ) →L[ℝ] (Fin 6 → ℝ) := + (Matrix.toLin' (sourceRegressionMatrixAt L r)).toContinuousLinearMap + +/-- Deterministic twelve-observation map: no observation law is built into its definition. -/ +def endpointObservationMap (r : ℝ) : + (Fin 12 → ℝ) →L[ℝ] ((Fin 6 → ℝ) × (Fin 6 → ℝ)) where + toFun v := + let x := confluentTransform r (fun i : Fin 6 => v (Fin.castAdd 6 i)) + (x, ![(v 6 - x 3)/r, (v 9 - x 3)/r, (v 7 - x 4)/r, (v 10 - x 4)/r, v 8, v 11]) + map_add' v w := by + ext i <;> fin_cases i <;> simp [confluentTransform] <;> ring + map_smul' c v := by + ext i <;> fin_cases i <;> simp [confluentTransform] <;> ring + cont := by + unfold confluentTransform + fun_prop + +lemma endpointObservationMap_actual (L r : ℝ) (ω : SmoothSample 2 L) : + endpointObservationMap r (rawPinsWithHessians L r ω) = + (sourceConfluentPinsAt L r ω, sourceAdjustedEndpoints L r ω) := by + have hp : (fun i : Fin 6 => rawPinsWithHessians L r ω (Fin.castAdd 6 i)) = + rawPins L r ω := by + ext i + fin_cases i <;> rfl + change (confluentTransform r _, _) = _ + simp only [hp] + rfl + +/-- The actual joint Gaussian law is a proved image of the actual twelve derivative observations. -/ +theorem sourcePins_adjusted_hasGaussianLaw {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + HasGaussianLaw (fun ω => (sourceConfluentPinsAt L r ω, sourceAdjustedEndpoints L r ω)) + (smoothLaw 2 L) := by + have hG := (rawPinsWithHessians_hasGaussianLaw hL r).map_fun (endpointObservationMap r) + apply hG.congr + filter_upwards with ω + exact endpointObservationMap_actual L r ω + +@[simp] theorem sourceRegressionMapAt_apply (L r : ℝ) (x : Fin 6 → ℝ) : + sourceRegressionMapAt L r x = sourceRegressionMatrixAt L r *ᵥ x := rfl + +@[simp] theorem sourceRegressionMapAt_single (L r : ℝ) (l j : Fin 6) : + sourceRegressionMapAt L r (Pi.single l 1) j = sourceRegressionMatrixAt L r j l := by + simp [sourceRegressionMapAt] + +/-- The matrix inverse derives the regression normal equations for these actual observations. -/ +theorem sourceRegression_normal_equations (L r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) : + ∀ i j, covariance (fun ω => sourceConfluentPinsAt L r ω i) + (fun ω => sourceAdjustedEndpoints L r ω j) (smoothLaw 2 L) = + ∑ l, sourceRegressionMapAt L r (Pi.single l 1) j * + covariance (fun ω => sourceConfluentPinsAt L r ω i) + (fun ω => sourceConfluentPinsAt L r ω l) (smoothLaw 2 L) := by + let _ : Invertible (sourcePinCovarianceAt L r) := hG.invertible + have hprod : sourceCrossCovarianceAt L r * (sourcePinCovarianceAt L r)⁻¹ * + sourcePinCovarianceAt L r = sourceCrossCovarianceAt L r := + Matrix.inv_mul_cancel_right_of_invertible _ _ + intro i j + calc + covariance (fun ω => sourceConfluentPinsAt L r ω i) + (fun ω => sourceAdjustedEndpoints L r ω j) (smoothLaw 2 L) = + sourceCrossCovarianceAt L r j i := covariance_comm _ _ + _ = ∑ l, sourceRegressionMatrixAt L r j l * sourcePinCovarianceAt L r l i := + (congrArg (fun M => M j i) hprod).symm + _ = ∑ l, sourceRegressionMapAt L r (Pi.single l 1) j * + covariance (fun ω => sourceConfluentPinsAt L r ω i) + (fun ω => sourceConfluentPinsAt L r ω l) (smoothLaw 2 L) := by + apply Finset.sum_congr rfl + intro l _ + rw [sourceRegressionMapAt_single] + congr 1 + exact covariance_comm _ _ + +/-- The actual affine regression residual, with no assumed residual law. -/ +def sourceEndpointResidual (L r : ℝ) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + sourceAdjustedEndpoints L r ω - sourceRegressionMapAt L r (sourceConfluentPinsAt L r ω) + +/-- The canonical actual endpoint kernel, specified at every pin value. -/ +def sourceEndpointKernel (L r : ℝ) : Kernel (Fin 6 → ℝ) (Fin 6 → ℝ) := + regressionKernel ((smoothLaw 2 L).map (sourceEndpointResidual L r)) + (sourceRegressionMapAt L r) 0 0 + +theorem sourceEndpointKernel_markov {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + IsMarkovKernel (sourceEndpointKernel L r) := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + unfold sourceEndpointKernel + infer_instance + +theorem sourceEndpointKernel_measurable (L r : ℝ) : Measurable (sourceEndpointKernel L r) := + Kernel.measurable _ + +/-- The all-pin pushforward identity includes every probability-zero prescribed pin vector. -/ +theorem sourceEndpointKernel_apply {L : ℝ} (hL : L ≠ 0) (r : ℝ) (x : Fin 6 → ℝ) : + sourceEndpointKernel L r x = + ((smoothLaw 2 L).map (sourceEndpointResidual L r)).map + (fun v => v + sourceRegressionMapAt L r x) := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + simpa [sourceEndpointKernel] using + regressionKernel_apply ((smoothLaw 2 L).map (sourceEndpointResidual L r)) + (sourceRegressionMapAt L r) 0 0 x + +/-- The actual target's conditional law, with Gaussian residual hypotheses derived rather than assumed. +Current pin-covariance invertibility is the sole algebraic premise. -/ +theorem sourceEndpointKernel_hasCondDistrib {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) : + HasCondDistrib (sourceAdjustedEndpoints L r) (sourceConfluentPinsAt L r) + (sourceEndpointKernel L r) (smoothLaw 2 L) := by + have hcond := gaussianRegression_observed_hasCondDistrib + (sourcePins_adjusted_hasGaussianLaw hL r) (sourceRegressionMapAt L r) 0 0 + (sourceRegression_normal_equations L r hG) + unfold sourceEndpointKernel sourceEndpointResidual + simpa only [sub_zero] using hcond + +/-- Combined canonical law for the actual pins and adjusted endpoint vector. -/ +theorem sourceEndpointKernel_spec {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) : + IsMarkovKernel (sourceEndpointKernel L r) ∧ + Measurable (sourceEndpointKernel L r) ∧ + (∀ x, sourceEndpointKernel L r x = + ((smoothLaw 2 L).map (sourceEndpointResidual L r)).map + (fun v => v + sourceRegressionMapAt L r x)) ∧ + HasCondDistrib (sourceAdjustedEndpoints L r) (sourceConfluentPinsAt L r) + (sourceEndpointKernel L r) (smoothLaw 2 L) := + ⟨sourceEndpointKernel_markov hL r, sourceEndpointKernel_measurable L r, + sourceEndpointKernel_apply hL r, sourceEndpointKernel_hasCondDistrib hL r hG⟩ + +/-- The literal physical normalized endpoint vector in LM006 order. -/ +def sourceNormalizedEndpoints (L r : ℝ) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + let v := rawPinsWithHessians L r ω + ![v 6/r, v 9/r, v 7/r, v 10/r, v 8, v 11] + +def sourcePrescribedPins (r b : ℝ) : Fin 6 → ℝ := ![b-r^3/12,0,0,0,0,2] + +/-- On the two zero xx/xy pin coordinates, adjusted entries are exactly physical normalized entries. -/ +theorem sourceAdjustedEndpoints_on_pin_fiber (L r : ℝ) (ω : SmoothSample 2 L) + (h3 : sourceConfluentPinsAt L r ω 3 = 0) (h4 : sourceConfluentPinsAt L r ω 4 = 0) : + sourceAdjustedEndpoints L r ω = sourceNormalizedEndpoints L r ω := by + simp [sourceAdjustedEndpoints, sourceNormalizedEndpoints, h3, h4] + +/-- Exact deterministic reconstruction at the prescribed pin fiber, without choosing an arbitrary RCD. -/ +theorem sourceAdjustedEndpoints_on_prescribed (L r b : ℝ) (ω : SmoothSample 2 L) + (hp : sourceConfluentPinsAt L r ω = sourcePrescribedPins r b) : + sourceAdjustedEndpoints L r ω = sourceNormalizedEndpoints L r ω := by + apply sourceAdjustedEndpoints_on_pin_fiber + · simpa [sourcePrescribedPins] using congrFun hp 3 + · simpa [sourcePrescribedPins] using congrFun hp 4 + +end I4SourceEndpointRegression diff --git a/frontiers/source_derivative_gaussian_20261025/SourcePhysicalEndpointKernel.lean b/frontiers/source_derivative_gaussian_20261025/SourcePhysicalEndpointKernel.lean new file mode 100644 index 000000000..5b345c2f1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/SourcePhysicalEndpointKernel.lean @@ -0,0 +1,131 @@ +import SourceEndpointParameters + +/-! +# Canonical physical endpoint law at every prescribed null pin + +The physical and adjusted endpoint vectors differ by a literal pin-dependent +linear correction. Both canonical kernels use the same actual residual law. +The correction vanishes at the prescribed vector, proving equality there as +literal measures. This argument does not substitute an arbitrary conditional +law version at a zero-probability fiber. See `SOURCE_PHYSICAL_ENDPOINT_KERNEL.md`. +-/ + +noncomputable section + +open I4SmoothCarrier I4SourceEndpointRegression I4SourceEndpointParameters ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Matrix +open scoped BigOperators Matrix + +set_option autoImplicit false + +namespace I4SourcePhysicalEndpointKernel + +/-- The literal correction from adjusted entries to physical normalized endpoint entries. -/ +def sourcePhysicalCorrection (r : ℝ) : (Fin 6 → ℝ) →L[ℝ] (Fin 6 → ℝ) where + toFun x := ![x 3/r, x 3/r, x 4/r, x 4/r, 0, 0] + map_add' x y := by ext i; fin_cases i <;> simp <;> ring + map_smul' c x := by ext i; fin_cases i <;> simp <;> ring + cont := by fun_prop + +def sourcePhysicalRegressionMap (L r : ℝ) : (Fin 6 → ℝ) →L[ℝ] (Fin 6 → ℝ) := + sourceRegressionMapAt L r + sourcePhysicalCorrection r + +/-- A pointwise identity for the actual source observations, at every separation. -/ +theorem sourceNormalizedEndpoints_eq_adjusted (L r : ℝ) (ω : SmoothSample 2 L) : + sourceNormalizedEndpoints L r ω = + sourceAdjustedEndpoints L r ω + sourcePhysicalCorrection r (sourceConfluentPinsAt L r ω) := by + ext i + fin_cases i <;> simp [sourceNormalizedEndpoints, sourceAdjustedEndpoints, sourcePhysicalCorrection] + all_goals ring + +/-- The physical canonical kernel retains exactly the same actual adjusted residual law. -/ +def sourcePhysicalEndpointKernel (L r : ℝ) : Kernel (Fin 6 → ℝ) (Fin 6 → ℝ) := + regressionKernel ((smoothLaw 2 L).map (sourceEndpointResidual L r)) + (sourcePhysicalRegressionMap L r) 0 0 + +theorem sourcePhysicalEndpointKernel_markov {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + IsMarkovKernel (sourcePhysicalEndpointKernel L r) := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + unfold sourcePhysicalEndpointKernel + infer_instance + +theorem sourcePhysicalEndpointKernel_measurable (L r : ℝ) : + Measurable (sourcePhysicalEndpointKernel L r) := Kernel.measurable _ + +/-- Its all-pin pushforward is literal, rather than an almost-everywhere kernel identity. -/ +theorem sourcePhysicalEndpointKernel_apply {L : ℝ} (hL : L ≠ 0) (r : ℝ) (x : Fin 6 → ℝ) : + sourcePhysicalEndpointKernel L r x = + ((smoothLaw 2 L).map (sourceEndpointResidual L r)).map + (fun v => v + sourceRegressionMapAt L r x + sourcePhysicalCorrection r x) := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + simpa only [sourcePhysicalEndpointKernel, sourcePhysicalRegressionMap, sub_zero, add_zero, + _root_.add_apply, add_assoc] using + regressionKernel_apply ((smoothLaw 2 L).map (sourceEndpointResidual L r)) + (sourcePhysicalRegressionMap L r) 0 0 x + +/-- The physical observations have this canonical conditional law, derived from the actual residual. -/ +theorem sourcePhysicalEndpointKernel_hasCondDistrib {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) : + HasCondDistrib (sourceNormalizedEndpoints L r) (sourceConfluentPinsAt L r) + (sourcePhysicalEndpointKernel L r) (smoothLaw 2 L) := by + have hXV := sourcePins_adjusted_hasGaussianLaw hL r + have hXR : HasGaussianLaw (fun ω => + (sourceConfluentPinsAt L r ω, sourceEndpointResidual L r ω)) (smoothLaw 2 L) := by + unfold sourceEndpointResidual + simpa only [sub_zero] using regression_residual_jointGaussian hXV (sourceRegressionMapAt L r) 0 0 + have hzero : ∀ i j, covariance (fun ω => sourceConfluentPinsAt L r ω i) + (fun ω => sourceEndpointResidual L r ω j) (smoothLaw 2 L) = 0 := by + unfold sourceEndpointResidual + simpa only [sub_zero] using regression_residual_zero_covariance hXV + (sourceRegressionMapAt L r) 0 0 (sourceRegression_normal_equations L r hG) + have hcond := gaussianRegression_hasCondDistrib hXR hzero (sourcePhysicalRegressionMap L r) 0 0 + unfold sourcePhysicalEndpointKernel + convert hcond using 1 + funext ω + rw [sourceNormalizedEndpoints_eq_adjusted] + simp only [add_zero, sub_zero, sourceEndpointResidual, sourcePhysicalRegressionMap, + _root_.add_apply] + abel + +/-- The explicit physical correction is zero at every prescribed pin, for arbitrary height. -/ +theorem sourcePhysicalCorrection_prescribed (r b : ℝ) : + sourcePhysicalCorrection r (sourcePrescribedPins r b) = 0 := by + ext i + fin_cases i <;> simp [sourcePhysicalCorrection, sourcePrescribedPins] + +/-- Equality at the prescribed null pin is equality of the two literal canonical measures. -/ +theorem sourcePhysicalEndpointKernel_at_prescribed {L : ℝ} (hL : L ≠ 0) (r b : ℝ) : + sourcePhysicalEndpointKernel L r (sourcePrescribedPins r b) = + sourceEndpointKernel L r (sourcePrescribedPins r b) := by + rw [sourcePhysicalEndpointKernel_apply hL, sourceEndpointKernel_apply hL, + sourcePhysicalCorrection_prescribed] + simp only [add_zero] + +/-- The physical kernel has the extra pin-dependent correction in its mean away from the prescribed fiber. -/ +theorem sourcePhysicalEndpointKernel_integral {L : ℝ} (hL : L ≠ 0) (r : ℝ) (x : Fin 6 → ℝ) : + (∫ v, v ∂sourcePhysicalEndpointKernel L r x) = + sourceRegressionMapAt L r x + sourcePhysicalCorrection r x := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + have hR := sourceEndpointResidual_hasGaussianLaw hL r + rw [sourcePhysicalEndpointKernel_apply hL r x, + integral_map (by fun_prop) (by fun_prop), integral_map hR.aemeasurable (by fun_prop)] + rw [integral_add (f := fun ω => sourceEndpointResidual L r ω + sourceRegressionMapAt L r x) + (g := fun _ => sourcePhysicalCorrection r x) + (hR.integrable.add (integrable_const _)) (integrable_const _), + integral_add hR.integrable (integrable_const _), sourceEndpointResidual_integral_zero hL r] + simp + +/-- All-point physical kernel specification; no residual hypothesis is assumed. -/ +theorem sourcePhysicalEndpointKernel_spec {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) : + IsMarkovKernel (sourcePhysicalEndpointKernel L r) ∧ + Measurable (sourcePhysicalEndpointKernel L r) ∧ + (∀ x, sourcePhysicalEndpointKernel L r x = + ((smoothLaw 2 L).map (sourceEndpointResidual L r)).map + (fun v => v + sourceRegressionMapAt L r x + sourcePhysicalCorrection r x)) ∧ + HasCondDistrib (sourceNormalizedEndpoints L r) (sourceConfluentPinsAt L r) + (sourcePhysicalEndpointKernel L r) (smoothLaw 2 L) := + ⟨sourcePhysicalEndpointKernel_markov hL r, sourcePhysicalEndpointKernel_measurable L r, + sourcePhysicalEndpointKernel_apply hL r, sourcePhysicalEndpointKernel_hasCondDistrib hL r hG⟩ + +end I4SourcePhysicalEndpointKernel diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/AxiomAudit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/AxiomAudit.lean.txt new file mode 100644 index 000000000..c1d9dca2c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/AxiomAudit.lean.txt @@ -0,0 +1,8 @@ +import ConfluentIntegralIdentities +#print axioms I4ConfluentIntegralIdentities.pin3_integral +#print axioms I4ConfluentIntegralIdentities.pin4_integral +#print axioms I4ConfluentIntegralIdentities.pin5_integral +#print axioms I4ConfluentIntegralIdentities.adjusted_second_left +#print axioms I4ConfluentIntegralIdentities.adjusted_second_right +#print axioms I4ConfluentIntegralIdentities.adjusted_transverse_left +#print axioms I4ConfluentIntegralIdentities.adjusted_transverse_right diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/FalseCubicCoefficient.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/FalseCubicCoefficient.lean.txt new file mode 100644 index 000000000..60143167a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/FalseCubicCoefficient.lean.txt @@ -0,0 +1,5 @@ +import ConfluentIntegralIdentities +open I4ConfluentIntegralIdentities +-- A missing factor of two in the cubic Peano normalization is false. +example : pin5 (fun x : ℝ => x^3) (-2) = 3 := by + norm_num [pin5, deriv_pow_field] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/FalseTransverseDerivative.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/FalseTransverseDerivative.lean.txt new file mode 100644 index 000000000..4b6dda8eb --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/FalseTransverseDerivative.lean.txt @@ -0,0 +1,5 @@ +import ConfluentIntegralIdentities +open I4ConfluentIntegralIdentities +-- Using H first derivatives in the P4 divided difference would give two. +example : pin4 (fun x : ℝ => x^2) 2 = 2 := by + norm_num [pin4] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/HANDOFF.md b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/HANDOFF.md new file mode 100644 index 000000000..52debe405 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/HANDOFF.md @@ -0,0 +1,67 @@ +# Exact confluent integral identities: author-side handoff + +Sole new package source: +`frontiers/source_derivative_gaussian_20261025/ConfluentIntegralIdentities.lean`. + +Source SHA256: +`0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8`. + +Read the entire frozen contract at Math- `d382da3`, +`SOURCE_CONFLUENT_INTEGRALS.md`. All seven literal formulas are proved in +namespace `I4ConfluentIntegralIdentities`: + +- `pin3_integral` +- `pin4_integral` +- `pin5_integral` +- `adjusted_second_left` +- `adjusted_second_right` +- `adjusted_transverse_left` +- `adjusted_transverse_right` + +Definitions `pin3`, `pin4`, `pin5` are the literal divided differences. The +public F theorems assume precisely `ContDiff ℝ 3 F`; H theorems assume precisely +`ContDiff ℝ 2 H`. Every theorem allows signed `r ≠ 0`. No integral identity, +Taylor remainder, convergence or Gaussianity premise is added. + +Generic private derivative-chain helpers are instantiated from those stated +smoothness assumptions. The averages use FTOC, adjusted endpoint formulas use +affine-weight integration by parts, and the Peano identity differentiates the +explicit primitive `(1/4-u²)F''(ru)+(2/r)uF'(ru)-(2/r²)F(ru)` before FTOC. + +Actual author lineage: OpenAI/Codex `/root/dm2_landed_composition` owns the +module. Actual OpenAI/Codex helper `/root/dm2_landed_composition/peano_strategy` +proved the generic Peano scratch lemma at `../PeanoHelper.lean`; that argument +was copied/adapted into the module. No helper import or new package dependency +was introduced. This is author-side proof work, not independent review. + +## Completed new-scope verification + +`qualification_receipt.json` binds exact commands, exits, raw logs and source +hashes. Original compilation was reused for later qualification. + +- Final new-module compile: exit 0, `.olean` generated. Its raw output has one + harmless `unnecessarySeqFocus` linter warning at line 71. +- Named `leanchecker ConfluentIntegralIdentities`: run once, exit 0, empty raw + output. +- All seven public transitive axiom reports: only `propext`, `Classical.choice`, + `Quot.sound`. +- Positive controls: exit 0. Cubic pin and quadratic average at radius minus + two; literal H versus H' distinction; opposite adjusted endpoint signs; and + normalized cubic kernel mass one obtained from the actual Peano theorem. +- False cubic normalization and false H'-based transverse value controls: + each exit 1 at an unsolved `False` proof goal. + +`positive-attempt1.{json,log}` records an initial control-only failure because +`norm_num` already factored the integral, making an extra rewrite obsolete. +Only that new control was corrected/rechecked. No theorem source was changed +after the completed final compile, and no kernel/axiom/old verification was +repeated. + +Existing proof evidence is preserved. Actual differentiated-source-mode +instantiation, justified infinite-sum covariance limit, conditional regression +and Palm event joins remain open. Root alone integrates package targets, +inventory and final evidence. Scientific effect NONE. + +Per `/workspace/AGENTS.md`, completed applicable verification is reusable; do +not repeat these checks merely because the work is handed off or a separate +process/review requirement remains pending. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/PositiveControls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/PositiveControls.lean.txt new file mode 100644 index 000000000..30c71c8c4 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/PositiveControls.lean.txt @@ -0,0 +1,28 @@ +import ConfluentIntegralIdentities +import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas +open I4ConfluentIntegralIdentities + +-- Negative nonunit radius preserves the exact cubic coefficient. +example : pin5 (fun x : ℝ => x^3) (-2) = 6 := by + norm_num [pin5, deriv_pow_field] + +-- The second-derivative average remains correct for signed separation. +example : pin3 (fun x : ℝ => x^2) (-2) = 2 := by + norm_num [pin3, deriv_pow_field] + +-- The transverse pin uses H itself, not its derivative. +example : pin4 (fun x : ℝ => x^2) 2 = 0 := by + norm_num [pin4] + +-- Left and right adjusted transverse values carry opposite endpoint signs. +example : (deriv (fun x : ℝ => x^2) (-(-2)/2) - pin4 (fun x : ℝ => x^2) (-2))/(-2) = -1 := by + norm_num [pin4, deriv_pow_field] +example : (deriv (fun x : ℝ => x^2) ((-2)/2) - pin4 (fun x : ℝ => x^2) (-2))/(-2) = 1 := by + norm_num [pin4, deriv_pow_field] + +-- The actual Peano theorem on a cubic gives normalized kernel mass one. +example : 6 * (∫ u in (-1/2 : ℝ)..(1/2 : ℝ), (1/4-u^2)) = 1 := by + have h := pin5_integral (fun x : ℝ => x^3) (by fun_prop) (-2) (by norm_num) + norm_num [pin5, deriv_pow_field, iteratedDeriv_pow] at h + norm_num at h ⊢ + linarith [h] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt1.json new file mode 100644 index 000000000..5dbffc203 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt1.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/ConfluentIntegralIdentities.olean", + "ConfluentIntegralIdentities.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 1, + "source_sha256": "54cb0b5178f81009b50eed936785ff55e0aeb48a564abafd1e9a9f9782f29503" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt1.log new file mode 100644 index 000000000..9fed01092 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt1.log @@ -0,0 +1,65 @@ +ConfluentIntegralIdentities.lean:36:2: error: Type mismatch: After simplification, term + HasDerivAt.comp u (hg (r * u)) (HasDerivAt.const_mul r (hasDerivAt_id u)) + has type + @HasDerivAt ℝ DenselyNormedField.toNontriviallyNormedField ℝ + DenselyNormedField.toNontriviallyNormedField.toDivisionRing.toAddCommGroup (NormedAlgebra.toNormedSpace ℝ).toModule + PseudoMetricSpace.toUniformSpace.toTopologicalSpace ⋯ (g ∘ HMul.hMul r) (r * g' (r * u)) u +but is expected to have type + @HasDerivAt ℝ DenselyNormedField.toNontriviallyNormedField ℝ Real.instAddCommGroup Semiring.toModule + PseudoMetricSpace.toUniformSpace.toTopologicalSpace ⋯ (fun u => g (r * u)) (r * g' (r * u)) u +ConfluentIntegralIdentities.lean:47:47: error: Application type mismatch: The argument + Continuous.intervalIntegrable hc' +has type + ∀ (a b : ℝ), IntervalIntegrable (fun u => r * g' (r * u)) ?m.148 a b +but is expected to have type + IntervalIntegrable (fun u => r * g' (r * u)) volume (-1 / 2) (1 / 2) +in the application + intervalIntegral.integral_eq_sub_of_hasDerivAt (fun u x => scaled_hasDerivAt g g' hg r u) + (Continuous.intervalIntegrable hc') +ConfluentIntegralIdentities.lean:70:4: error: Application type mismatch: The argument + Continuous.intervalIntegrable continuous_const +has type + ∀ (a b : ℝ), IntervalIntegrable (fun x => ?m.297) ?m.300 a b +but is expected to have type + IntervalIntegrable (fun x => -1) volume (-1 / 2) (1 / 2) +in the application + intervalIntegral.integral_mul_deriv_eq_deriv_mul_of_hasDerivAt ?m.257 + (Continuous.continuousOn (Continuous.comp hgc (Continuous.mul continuous_const continuous_id))) (fun u x => ?m.289) + (fun u x => scaled_hasDerivAt g g' hg r u) (Continuous.intervalIntegrable continuous_const) +ConfluentIntegralIdentities.lean:68:19: error: Type mismatch: After simplification, term + HasDerivAt.sub (hasDerivAt_const u (1 / 2)) (hasDerivAt_id u) + has type + @HasDerivAt ℝ DenselyNormedField.toNontriviallyNormedField ℝ Real.normedAddCommGroup.toAddCommGroup + RCLike.toInnerProductSpaceReal.toModule PseudoMetricSpace.toUniformSpace.toTopologicalSpace ⋯ ((fun x => 2⁻¹) - id) + (-1) u +but is expected to have type + @HasDerivAt ℝ DenselyNormedField.toNontriviallyNormedField ℝ Real.normedCommRing.toAddCommGroup + (NormedAlgebra.toNormedSpace ℝ).toModule PseudoMetricSpace.toUniformSpace.toTopologicalSpace ⋯ (fun u => 2⁻¹ - u) + (-1) u +ConfluentIntegralIdentities.lean:77:2: error: linarith failed to find a contradiction +case h1 +g g' : ℝ → ℝ +hg : ∀ (x : ℝ), HasDerivAt g (g' x) x +hc : Continuous g' +r : ℝ +hr : r ≠ 0 +hgc : Continuous g +hc' : Continuous fun u => r * g' (r * u) +hfun : (fun u => (1 / 2 - u) * (r * g' (r * u))) = fun u => r * ((1 / 2 - u) * g' (r * u)) +hleft : r * (-1 / 2) = -r / 2 +h : + r * ∫ (x : ℝ) in -(1 / 2)..1 / 2, (1 / 2 - x) * g' (r * x) = + -g (-(r * (1 / 2))) + ∫ (x : ℝ) in -(1 / 2)..1 / 2, g (r * x) +a✝ : g (-r / 2) - ∫ (u : ℝ) in -1 / 2..1 / 2, g (r * u) < (-∫ (u : ℝ) in -1 / 2..1 / 2, (1 / 2 - u) * g' (r * u)) * r +⊢ False +failed +ConfluentIntegralIdentities.lean:94:4: error: Application type mismatch: The argument + Continuous.intervalIntegrable continuous_const +has type + ∀ (a b : ℝ), IntervalIntegrable (fun x => ?m.290) ?m.293 a b +but is expected to have type + IntervalIntegrable (fun x => 1) volume (-1 / 2) (1 / 2) +in the application + intervalIntegral.integral_mul_deriv_eq_deriv_mul_of_hasDerivAt ?m.250 + (Continuous.continuousOn (Continuous.comp hgc (Continuous.mul continuous_const continuous_id))) (fun u x => ?m.282) + (fun u x => scaled_hasDerivAt g g' hg r u) (Continuous.intervalIntegrable continuous_const) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt2.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt2.json new file mode 100644 index 000000000..5b909d0c4 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt2.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/ConfluentIntegralIdentities.olean", + "ConfluentIntegralIdentities.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 1, + "source_sha256": "89fdef1ec0ae0983d2eb5ee80bd8fe3cbd2c5d07dd13f91b242487a229c87a8c" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt2.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt2.log new file mode 100644 index 000000000..ce648bafb --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt2.log @@ -0,0 +1,41 @@ +ConfluentIntegralIdentities.lean:70:19: error: Type mismatch: After simplification, term + HasDerivAt.sub (hasDerivAt_const u (1 / 2)) (hasDerivAt_id u) + has type + @HasDerivAt ℝ DenselyNormedField.toNontriviallyNormedField ℝ Real.normedAddCommGroup.toAddCommGroup + RCLike.toInnerProductSpaceReal.toModule PseudoMetricSpace.toUniformSpace.toTopologicalSpace ⋯ ((fun x => 2⁻¹) - id) + (-1) u +but is expected to have type + @HasDerivAt ℝ DenselyNormedField.toNontriviallyNormedField ℝ Real.normedCommRing.toAddCommGroup + (NormedAlgebra.toNormedSpace ℝ).toModule PseudoMetricSpace.toUniformSpace.toTopologicalSpace ⋯ (fun u => 2⁻¹ - u) + (-1) u +ConfluentIntegralIdentities.lean:78:2: error: linarith failed to find a contradiction +case h1 +g g' : ℝ → ℝ +hg : ∀ (x : ℝ), HasDerivAt g (g' x) x +hc : Continuous g' +r : ℝ +hr : r ≠ 0 +hgc : Continuous g +hc' : Continuous fun u => r * g' (r * u) +hfun : (fun u => (1 / 2 - u) * (r * g' (r * u))) = fun u => r * ((1 / 2 - u) * g' (r * u)) +h : + r * ∫ (x : ℝ) in -(1 / 2)..1 / 2, (1 / 2 - x) * g' (r * x) = + -g (-(r * (1 / 2))) + ∫ (x : ℝ) in -(1 / 2)..1 / 2, g (r * x) +a✝ : g (-r / 2) - ∫ (u : ℝ) in -1 / 2..1 / 2, g (r * u) < (-∫ (u : ℝ) in -1 / 2..1 / 2, (1 / 2 - u) * g' (r * u)) * r +⊢ False +failed +ConfluentIntegralIdentities.lean:101:2: error: linarith failed to find a contradiction +case h1 +g g' : ℝ → ℝ +hg : ∀ (x : ℝ), HasDerivAt g (g' x) x +hc : Continuous g' +r : ℝ +hr : r ≠ 0 +hgc : Continuous g +hc' : Continuous fun u => r * g' (r * u) +hfun : (fun u => (u + 1 / 2) * (r * g' (r * u))) = fun u => r * ((u + 1 / 2) * g' (r * u)) +h : + r * ∫ (x : ℝ) in -(1 / 2)..1 / 2, (x + 1 / 2) * g' (r * x) = g (r * (1 / 2)) - ∫ (x : ℝ) in -(1 / 2)..1 / 2, g (r * x) +a✝ : g (r / 2) - ∫ (u : ℝ) in -1 / 2..1 / 2, g (r * u) < (∫ (u : ℝ) in -1 / 2..1 / 2, (u + 1 / 2) * g' (r * u)) * r +⊢ False +failed diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt3.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt3.json new file mode 100644 index 000000000..b7dfe5221 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt3.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/ConfluentIntegralIdentities.olean", + "ConfluentIntegralIdentities.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 1, + "source_sha256": "03ffd76905e0326b5569627f16d55ea6735c33eff63920f69ce5b472b07e7341" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt3.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt3.log new file mode 100644 index 000000000..d7d3b2c87 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/attempt3.log @@ -0,0 +1,32 @@ +ConfluentIntegralIdentities.lean:80:2: error: linarith failed to find a contradiction +case h1 +g g' : ℝ → ℝ +hg : ∀ (x : ℝ), HasDerivAt g (g' x) x +hc : Continuous g' +r : ℝ +hr : r ≠ 0 +hgc : Continuous g +hc' : Continuous fun u => r * g' (r * u) +hfun : (fun u => (1 / 2 - u) * (r * g' (r * u))) = fun u => r * ((1 / 2 - u) * g' (r * u)) +h : + r * ∫ (x : ℝ) in -(1 / 2)..1 / 2, (1 / 2 - x) * g' (r * x) = + -g (-(r * (1 / 2))) + ∫ (x : ℝ) in -(1 / 2)..1 / 2, g (r * x) +a✝ : + g (-r / 2) - ∫ (u : ℝ) in -(1 / 2)..1 / 2, g (r * u) < -((∫ (u : ℝ) in -(1 / 2)..1 / 2, (1 / 2 - u) * g' (r * u)) * r) +⊢ False +failed +ConfluentIntegralIdentities.lean:104:2: error: linarith failed to find a contradiction +case h1 +g g' : ℝ → ℝ +hg : ∀ (x : ℝ), HasDerivAt g (g' x) x +hc : Continuous g' +r : ℝ +hr : r ≠ 0 +hgc : Continuous g +hc' : Continuous fun u => r * g' (r * u) +hfun : (fun u => (u + 1 / 2) * (r * g' (r * u))) = fun u => r * ((u + 1 / 2) * g' (r * u)) +h : + r * ∫ (x : ℝ) in -(1 / 2)..1 / 2, (x + 1 / 2) * g' (r * x) = g (r * (1 / 2)) - ∫ (x : ℝ) in -(1 / 2)..1 / 2, g (r * x) +a✝ : g (r / 2) - ∫ (u : ℝ) in -(1 / 2)..1 / 2, g (r * u) < (∫ (u : ℝ) in -(1 / 2)..1 / 2, (u + 1 / 2) * g' (r * u)) * r +⊢ False +failed diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/axioms.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/axioms.json new file mode 100644 index 000000000..94bfe6f65 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/axioms.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_integrals/AxiomAudit.lean" + ], + "exit_code": 0, + "duration_seconds": 5.193881511688232, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/axioms.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/axioms.log new file mode 100644 index 000000000..c2e3c383a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/axioms.log @@ -0,0 +1,7 @@ +'I4ConfluentIntegralIdentities.pin3_integral' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ConfluentIntegralIdentities.pin4_integral' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ConfluentIntegralIdentities.pin5_integral' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ConfluentIntegralIdentities.adjusted_second_left' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ConfluentIntegralIdentities.adjusted_second_right' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ConfluentIntegralIdentities.adjusted_transverse_left' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ConfluentIntegralIdentities.adjusted_transverse_right' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-cubic.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-cubic.json new file mode 100644 index 000000000..ae7f6efaf --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-cubic.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_integrals/FalseCubicCoefficient.lean" + ], + "exit_code": 1, + "duration_seconds": 5.286576271057129, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-cubic.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-cubic.log new file mode 100644 index 000000000..f73e44499 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-cubic.log @@ -0,0 +1,2 @@ +/workspace/scratch/dm2_landed_composition/confluent_integrals/FalseCubicCoefficient.lean:4:46: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-transverse.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-transverse.json new file mode 100644 index 000000000..fa784e13b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-transverse.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_integrals/FalseTransverseDerivative.lean" + ], + "exit_code": 1, + "duration_seconds": 5.0333075523376465, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-transverse.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-transverse.log new file mode 100644 index 000000000..18c6df15d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/false-transverse.log @@ -0,0 +1,2 @@ +/workspace/scratch/dm2_landed_composition/confluent_integrals/FalseTransverseDerivative.lean:4:43: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/final_compile.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/final_compile.json new file mode 100644 index 000000000..f8e490bdc --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/final_compile.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/ConfluentIntegralIdentities.olean", + "ConfluentIntegralIdentities.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/final_compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/final_compile.log new file mode 100644 index 000000000..a1040aaf9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/final_compile.log @@ -0,0 +1,3 @@ +ConfluentIntegralIdentities.lean:71:60: warning: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/kernel.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/kernel.json new file mode 100644 index 000000000..314a506d4 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/kernel.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "leanchecker", + "ConfluentIntegralIdentities" + ], + "exit_code": 0, + "duration_seconds": 7.803982257843018, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/kernel.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/kernel.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/PeanoHelper.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/PeanoHelper.lean.txt new file mode 100644 index 000000000..686cbcec6 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/PeanoHelper.lean.txt @@ -0,0 +1,51 @@ +import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus +import Mathlib.Analysis.Calculus.Deriv.Pow +import Mathlib.Analysis.Calculus.Deriv.Mul +import Mathlib.Tactic.FieldSimp +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.Ring + +noncomputable section + +open MeasureTheory + +namespace I4ConfluentIntegralsHelper + +/-- The cubic Peano identity follows by an explicitly differentiated primitive. +Only the derivative chain is assumed; no integral remainder is a premise. -/ +theorem peano_generic + (f f1 f2 f3 : ℝ → ℝ) + (h0 : ∀ x, HasDerivAt f (f1 x) x) + (h1 : ∀ x, HasDerivAt f1 (f2 x) x) + (h2 : ∀ x, HasDerivAt f2 (f3 x) x) + (_hc1 : Continuous f1) (_hc2 : Continuous f2) (hc3 : Continuous f3) + (r : ℝ) (hr : r ≠ 0) : + (12 / r ^ 2) * ((f1 (r/2) + f1 (-r/2))/2 - (f (r/2) - f (-r/2))/r) = + 6 * ∫ u in (-1/2 : ℝ)..(1/2 : ℝ), (1/4 - u ^ 2) * f3 (r*u) := by + let A : ℝ → ℝ := fun u => + (1/4-u^2)*f2 (r*u) + (2/r)*u*f1 (r*u) - (2/r^2)*f (r*u) + have hA (u : ℝ) : HasDerivAt A (r*((1/4-u^2)*f3 (r*u))) u := by + have hru : HasDerivAt (fun u : ℝ => r*u) r u := by + simpa [id] using (hasDerivAt_id u).const_mul r + have hK : HasDerivAt (fun u : ℝ => 1/4-u^2) (-2*u) u := by + convert ((hasDerivAt_id u).pow 2).const_sub (1/4 : ℝ) using 1 <;> simp [id] + have hU : HasDerivAt (fun u : ℝ => (2/r)*u) (2/r) u := by + simpa [id] using (hasDerivAt_id u).const_mul (2/r) + have h := ((hK.mul ((h2 (r*u)).comp u hru)).add + (hU.mul ((h1 (r*u)).comp u hru))).sub (((h0 (r*u)).comp u hru).const_mul (2/r^2)) + convert h using 1 + · rfl + · dsimp [Function.comp_def] + field_simp [hr] + ring + have hcont : Continuous (fun u : ℝ => r*((1/4-u^2)*f3 (r*u))) := + (continuous_const.sub (continuous_id.pow 2)).mul + (hc3.comp (continuous_const.mul continuous_id)) |>.const_mul r + have hft := intervalIntegral.integral_eq_sub_of_hasDerivAt + (fun u _ => hA u) (hcont.intervalIntegrable (-1/2 : ℝ) (1/2 : ℝ)) + rw [intervalIntegral.integral_const_mul] at hft + norm_num [A, mul_div_assoc, mul_neg] at hft + field_simp [hr] at hft ⊢ + nlinarith [hft] + +end I4ConfluentIntegralsHelper diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt1.json new file mode 100644 index 000000000..5eece9929 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt1.json @@ -0,0 +1,10 @@ +{ + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/PeanoHelper.lean" + ], + "exit_code": 1, + "source_sha256": "60a0b740f99eb5df5807cb1f07e6f8f1984f5b65dae07f08fbb24b2af2e74c99" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt1.log new file mode 100644 index 000000000..69c4c6ec4 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt1.log @@ -0,0 +1,54 @@ +/workspace/scratch/dm2_landed_composition/PeanoHelper.lean:27:70: error(lean.invalidField): Invalid field `const_mul`: The environment does not contain `HasFDerivAtFilter.const_mul`, so it is not possible to project the field `const_mul` from an expression + hasDerivAt_id u +of type + HasFDerivAtFilter id (ContinuousLinearMap.toSpanSingleton ℝ 1) (nhds u ×ˢ pure u) +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +/workspace/scratch/dm2_landed_composition/PeanoHelper.lean:28:60: error: unsolved goals +case e'_8 +f f1 f2 f3 : ℝ → ℝ +h0 : ∀ (x : ℝ), HasDerivAt f (f1 x) x +h1 : ∀ (x : ℝ), HasDerivAt f1 (f2 x) x +h2 : ∀ (x : ℝ), HasDerivAt f2 (f3 x) x +_hc1 : Continuous f1 +_hc2 : Continuous f2 +hc3 : Continuous f3 +r : ℝ +hr : r ≠ 0 +A : ℝ → ℝ := fun u => (1 / 4 - u ^ 2) * f2 (r * u) + 2 / r * u * f1 (r * u) - 2 / r ^ 2 * f (r * u) +u : ℝ +hru : HasDerivAt (fun u => r * u) r u +⊢ (fun u => 1 / 4 - u ^ 2) = fun x => 1 / 4 - (id ^ 2) x + +case e'_9 +f f1 f2 f3 : ℝ → ℝ +h0 : ∀ (x : ℝ), HasDerivAt f (f1 x) x +h1 : ∀ (x : ℝ), HasDerivAt f1 (f2 x) x +h2 : ∀ (x : ℝ), HasDerivAt f2 (f3 x) x +_hc1 : Continuous f1 +_hc2 : Continuous f2 +hc3 : Continuous f3 +r : ℝ +hr : r ≠ 0 +A : ℝ → ℝ := fun u => (1 / 4 - u ^ 2) * f2 (r * u) + 2 / r * u * f1 (r * u) - 2 / r ^ 2 * f (r * u) +u : ℝ +hru : HasDerivAt (fun u => r * u) r u +⊢ -(u * 2) = -(id u * 2) +/workspace/scratch/dm2_landed_composition/PeanoHelper.lean:31:24: error(lean.invalidField): Invalid field `const_mul`: The environment does not contain `HasFDerivAtFilter.const_mul`, so it is not possible to project the field `const_mul` from an expression + hasDerivAt_id u +of type + HasFDerivAtFilter id (ContinuousLinearMap.toSpanSingleton ℝ 1) (nhds u ×ˢ pure u) +/workspace/scratch/dm2_landed_composition/PeanoHelper.lean:32:19: error(lean.invalidField): Invalid field `mul`: The environment does not contain `HasFDerivAtFilter.mul`, so it is not possible to project the field `mul` from an expression + hK +of type + HasFDerivAtFilter (fun u => 1 / 4 - u ^ 2) (ContinuousLinearMap.toSpanSingleton ℝ (-2 * u)) (nhds u ×ˢ pure u) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt2.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt2.json new file mode 100644 index 000000000..a257aa1fe --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt2.json @@ -0,0 +1,10 @@ +{ + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/PeanoHelper.lean" + ], + "exit_code": 1, + "source_sha256": "9320d72b03fffea8c55915653804b36486d65db6e6cc13707a785dffb10af495" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt2.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt2.log new file mode 100644 index 000000000..d0bfa52bd --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt2.log @@ -0,0 +1,47 @@ +/workspace/scratch/dm2_landed_composition/PeanoHelper.lean:28:52: error: Type mismatch + HasDerivAt.const_mul r (hasDerivAt_id u) +has type + HasDerivAt (fun y => r * id y) (r * 1) u +but is expected to have type + HasDerivAt (fun u => r * u) r u +/workspace/scratch/dm2_landed_composition/PeanoHelper.lean:32:6: error: Type mismatch + HasDerivAt.const_mul (2 / r) (hasDerivAt_id u) +has type + HasDerivAt (fun y => 2 / r * id y) (2 / r * 1) u +but is expected to have type + HasDerivAt (fun u => 2 / r * u) (2 / r) u +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +/workspace/scratch/dm2_landed_composition/PeanoHelper.lean:27:63: error: unsolved goals +case e'_8 +f f1 f2 f3 : ℝ → ℝ +h0 : ∀ (x : ℝ), HasDerivAt f (f1 x) x +h1 : ∀ (x : ℝ), HasDerivAt f1 (f2 x) x +h2 : ∀ (x : ℝ), HasDerivAt f2 (f3 x) x +_hc1 : Continuous f1 +_hc2 : Continuous f2 +hc3 : Continuous f3 +r : ℝ +hr : r ≠ 0 +A : ℝ → ℝ := fun u => (1 / 4 - u ^ 2) * f2 (r * u) + 2 / r * u * f1 (r * u) - 2 / r ^ 2 * f (r * u) +u : ℝ +hru : HasDerivAt (fun u => r * u) r u +hK : HasDerivAt (fun u => 1 / 4 - u ^ 2) (-2 * u) u +hU : HasDerivAt (fun u => 2 / r * u) (2 / r) u +h : + HasDerivAt + ((fun u => 1 / 4 - u ^ 2) * f2 ∘ HMul.hMul r + (fun u => 2 / r * u) * f1 ∘ HMul.hMul r - fun y => + 2 / r ^ 2 * (f ∘ HMul.hMul r) y) + (-2 * u * (f2 ∘ HMul.hMul r) u + (1 / 4 - u ^ 2) * (f3 (r * u) * r) + + (2 / r * (f1 ∘ HMul.hMul r) u + 2 / r * u * (f2 (r * u) * r)) - + 2 / r ^ 2 * (f1 (r * u) * r)) + u +⊢ (fun u => + r * u * f1 (r * u) * r⁻¹ ^ 2 * 2 - r ^ 2 * u ^ 2 * f2 (r * u) * r⁻¹ ^ 2 + r ^ 2 * f2 (r * u) * r⁻¹ ^ 2 * (1 / 4) - + f (r * u) * r⁻¹ ^ 2 * 2) = + (fun u => 1 / 4 - u ^ 2) * f2 ∘ HMul.hMul r + (fun u => r⁻¹ * u * 2) * f1 ∘ HMul.hMul r - fun y => + r⁻¹ ^ 2 * f (r * y) * 2 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt3.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt3.json new file mode 100644 index 000000000..496c39780 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt3.json @@ -0,0 +1,10 @@ +{ + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/PeanoHelper.lean" + ], + "exit_code": 0, + "source_sha256": "bd5eb2469adbc58560b6f412d14350c5a85c0e92368f5508262d98aeaaa18b52" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt3.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt3.log new file mode 100644 index 000000000..a72c5aa1b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_attempt3.log @@ -0,0 +1,6 @@ +/workspace/scratch/dm2_landed_composition/PeanoHelper.lean:31:86: warning: this tactic is never executed + +Note: This linter can be disabled with `set_option linter.unreachableTactic false` +/workspace/scratch/dm2_landed_composition/PeanoHelper.lean:31:86: warning: Unused tactic linter: `ring` does nothing + +Note: This linter can be disabled with `set_option linter.unusedTactic false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_final.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_final.json new file mode 100644 index 000000000..e4f24419b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_final.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/PeanoHelper.lean" + ], + "exit_code": 0, + "source_sha256": "ce78d03bae945c8092053b271cc3ee801af6da07e9be987218cfb1f03f398315", + "scope": "NEW_SCRATCH_PEANO_LEMMA_ONLY", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition/peano_strategy" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_final.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/peano_helper/peano_helper_final.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive-attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive-attempt1.json new file mode 100644 index 000000000..aa2599324 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive-attempt1.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_integrals/PositiveControls.lean" + ], + "exit_code": 1, + "duration_seconds": 5.598034620285034, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive-attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive-attempt1.log new file mode 100644 index 000000000..f876d3832 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive-attempt1.log @@ -0,0 +1,8 @@ +/workspace/scratch/dm2_landed_composition/confluent_integrals/PositiveControls.lean:30:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + fun u => (1 / 4 - u ^ 2) * 6 +in the target expression + (∫ (x : ℝ) in -(1 / 2)..1 / 2, 1 / 4 - x ^ 2) * 6 = 1 + +h : (∫ (x : ℝ) in -(1 / 2)..1 / 2, 1 / 4 - x ^ 2) * 6 = 1 +heq : (fun u => (1 / 4 - u ^ 2) * 6) = fun u => 6 * (1 / 4 - u ^ 2) +⊢ 6 * ∫ (u : ℝ) in -1 / 2..1 / 2, 1 / 4 - u ^ 2 = 1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive.json new file mode 100644 index 000000000..ab57baab3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_integrals/PositiveControls.lean" + ], + "exit_code": 0, + "duration_seconds": 3.1239266395568848, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive.log b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/positive.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/qualification_receipt.json b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/qualification_receipt.json new file mode 100644 index 000000000..0eb80027c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/qualification_receipt.json @@ -0,0 +1,164 @@ +{ + "scope": "NEW_CONFLUENT_INTEGRAL_IDENTITIES_ONLY_LOCAL_QUALIFICATION", + "scientific_effect": "NONE", + "source_path": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025/ConfluentIntegralIdentities.lean", + "source_sha256": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "frozen_contract_commit": "d382da3", + "frozen_contract_path": "frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_INTEGRALS.md", + "author_side_lineage": [ + "OpenAI/Codex /root/dm2_landed_composition", + "OpenAI/Codex /root/dm2_landed_composition/peano_strategy: actual generic Peano scratch helper" + ], + "compile_evidence_reused": { + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/ConfluentIntegralIdentities.olean", + "ConfluentIntegralIdentities.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_integrals/final_compile.log", + "raw_log_sha256": "022f4f8b18c8d5d1fc60362ee2a5a1dcf68599cb92994e4900143d4894238508" + }, + "axiom_audit": { + "I4ConfluentIntegralIdentities.pin3_integral": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4ConfluentIntegralIdentities.pin4_integral": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4ConfluentIntegralIdentities.pin5_integral": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4ConfluentIntegralIdentities.adjusted_second_left": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4ConfluentIntegralIdentities.adjusted_second_right": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4ConfluentIntegralIdentities.adjusted_transverse_left": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4ConfluentIntegralIdentities.adjusted_transverse_right": [ + "propext", + "Classical.choice", + "Quot.sound" + ] + }, + "checks": { + "kernel": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "leanchecker", + "ConfluentIntegralIdentities" + ], + "exit_code": 0, + "duration_seconds": 7.803982257843018, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_integrals/kernel.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855", + "expected_exit_code": 0 + }, + "axioms": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_integrals/AxiomAudit.lean" + ], + "exit_code": 0, + "duration_seconds": 5.193881511688232, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_integrals/axioms.log", + "raw_log_sha256": "0ea71ea95ab6444015bb28170bb6cf70e61de2af3be047daf1a2abe5a63a4d57", + "expected_exit_code": 0 + }, + "positive": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_integrals/PositiveControls.lean" + ], + "exit_code": 0, + "duration_seconds": 3.1239266395568848, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_integrals/positive.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855", + "expected_exit_code": 0 + }, + "false-cubic": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_integrals/FalseCubicCoefficient.lean" + ], + "exit_code": 1, + "duration_seconds": 5.286576271057129, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_integrals/false-cubic.log", + "raw_log_sha256": "2f47cdff53f210cc7b7f6fe9a2594085065a3530d8ad49b59d34183a41360ad2", + "expected_exit_code": 1 + }, + "false-transverse": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_integrals/FalseTransverseDerivative.lean" + ], + "exit_code": 1, + "duration_seconds": 5.0333075523376465, + "source_sha256_before": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "source_sha256_after": "0324d7556d485cd360a513587199354fa1f3863d15cf06b52c86e362de5e56b8", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_integrals/false-transverse.log", + "raw_log_sha256": "64b33fa722fa610db1fca752962ee9abb252c58100c48dd9edc26205159291c2", + "expected_exit_code": 1 + } + }, + "control_source_sha256": { + "AxiomAudit.lean": "362b1e333bb42de7a1dd616d75344e11001ac58a7b77281300464d45b1e55274", + "PositiveControls.lean": "d551c3f96a6f60eaa3d769b3ad12da6e9487f41491f9cff5c63ed8e68f0bb191", + "FalseCubicCoefficient.lean": "b3911ae4f12a23a10fd6d30fffbf81791981c1bce070348815da6e66c3a79c70", + "FalseTransverseDerivative.lean": "2a14a1824d2db7550604896de10b6a2b2ece559441378eab4add8c3cd3adeb14" + }, + "helper_evidence": { + "source": "/workspace/scratch/dm2_landed_composition/PeanoHelper.lean", + "metadata": "/workspace/scratch/dm2_landed_composition/peano_helper_final.json", + "raw_log": "/workspace/scratch/dm2_landed_composition/peano_helper_final.log" + }, + "not_established": [ + "actual source mode instantiation", + "justified covariance infinite sum limit", + "actual conditional regression or Palm law", + "independent scientific review", + "overall campaign completion" + ] +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/run_one.py b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/run_one.py new file mode 100644 index 000000000..eabc7c3af --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ConfluentIntegralIdentities/run_one.py @@ -0,0 +1,17 @@ +import hashlib,json,pathlib,subprocess,sys,time +p=pathlib.Path(__file__).parent +label=sys.argv[1] +source=pathlib.Path("ConfluentIntegralIdentities.lean") +sha=hashlib.sha256(source.read_bytes()).hexdigest() +if label=="kernel": + cmd=["lake","env","leanchecker","ConfluentIntegralIdentities"] +else: + cmd=["lake","env","lean",str(p/{"axioms":"AxiomAudit.lean","positive":"PositiveControls.lean","false-cubic":"FalseCubicCoefficient.lean","false-transverse":"FalseTransverseDerivative.lean"}[label])] +start=time.time() +r=subprocess.run(cmd,stdout=subprocess.PIPE,stderr=subprocess.STDOUT) +(p/(label+".log")).write_bytes(r.stdout) +meta={"cwd":str(pathlib.Path.cwd()),"command":cmd,"exit_code":r.returncode,"duration_seconds":time.time()-start,"source_sha256_before":sha,"source_sha256_after":hashlib.sha256(source.read_bytes()).hexdigest()} +(p/(label+".json")).write_text(json.dumps(meta,indent=2)+"\n") +print(json.dumps(meta)) +print(r.stdout.decode(),end="") +raise SystemExit(r.returncode) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/AxiomAudit-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/AxiomAudit-01.log new file mode 100644 index 000000000..b153886ae --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/AxiomAudit-01.log @@ -0,0 +1,8 @@ +'I4ContactGram.contactWitness_transpose_mulVec_eq_zero' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactGram.contactWitness_transpose_injective' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactGram.contactWitness_amplitude_transpose_injective' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactGram.coefficient_product_summable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactGram.coefficientGram_quadratic_hasSum' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactGram.coefficientGram_posDef_of_separating' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactGram.contact_coefficientGram_posDef' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactGram.contact_schur_pos' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/AxiomAudit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/AxiomAudit.lean.txt new file mode 100644 index 000000000..f183705ec --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/AxiomAudit.lean.txt @@ -0,0 +1,9 @@ +import ContactJetCovariance +#print axioms I4ContactGram.contactWitness_transpose_mulVec_eq_zero +#print axioms I4ContactGram.contactWitness_transpose_injective +#print axioms I4ContactGram.contactWitness_amplitude_transpose_injective +#print axioms I4ContactGram.coefficient_product_summable +#print axioms I4ContactGram.coefficientGram_quadratic_hasSum +#print axioms I4ContactGram.coefficientGram_posDef_of_separating +#print axioms I4ContactGram.contact_coefficientGram_posDef +#print axioms I4ContactGram.contact_schur_pos diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/ContactJetCovariance-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/ContactJetCovariance-01.log new file mode 100644 index 000000000..14db20c03 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/ContactJetCovariance-01.log @@ -0,0 +1,2 @@ +no such file or directory (error code: 2) + file: ContactJetCovariance.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/ContactJetCovariance-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/ContactJetCovariance-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/EVIDENCE.json b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/EVIDENCE.json new file mode 100644 index 000000000..e01fd9cdb --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/EVIDENCE.json @@ -0,0 +1,80 @@ +{ + "source": "ContactJetCovariance.lean", + "source_sha256": "50df0281c580ac7af018713a38c6bdd5dd34b52e3834b573165045bc64acd49d", + "source_contract_commit": "45b0ede", + "author": "OpenAI/Codex /root/regression_kernel with OpenAI/Codex /root/regression_kernel/gaussian_api optional Schur API/proof support", + "scientific_effect": "NONE; author-side verification only; actual coefficient and physical covariance identification separately required", + "new_module_build": { + "command": "/home/agent/.elan/bin/lake env lean -o .lake/build/lib/lean/ContactJetCovariance.olean ContactJetCovariance.lean", + "exit": 0, + "log": "ContactJetCovariance-02.log" + }, + "new_module_kernel_replay": { + "command": "/home/agent/.elan/bin/lake env leanchecker ContactJetCovariance", + "exit": 0, + "log": "KernelReplay-01.log" + }, + "axiom_audit": { + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/AxiomAudit.lean", + "exit": 0, + "targets": 8, + "allowed": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "log": "AxiomAudit-01.log" + }, + "controls": [ + { + "file": "PositiveControls.lean", + "exit": 0, + "log": "PositiveControls-03.log", + "claims": [ + "positive finite Gram c=1 amplitudes=1", + "positive finite Gram c=-1 amplitudes=2", + "actual failure of transpose injectivity for c=0", + "actual failure of transpose injectivity for zero amplitudes" + ] + }, + { + "file": "RejectedZeroFrequency.lean", + "exit": 1, + "log": "RejectedZeroFrequency-01.log", + "reason": "unsolved False side condition c\u22600" + }, + { + "file": "RejectedZeroAmplitude.lean", + "exit": 1, + "log": "RejectedZeroAmplitude-01.log", + "reason": "unsolved False side condition \u2200s a_s\u22600" + } + ], + "logs": { + "ContactJetCovariance-02.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "PositiveControls-03.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "RejectedZeroFrequency-01.log": { + "bytes": 115, + "sha256": "15bbcb32f00f3b6855f8d63d99ce54f84221a07548d69ddc33714842cdaf9e01" + }, + "RejectedZeroAmplitude-01.log": { + "bytes": 125, + "sha256": "be30efec397f160a20ff069eed66d1490704e096c08060cab37d5dc2ec69c929" + }, + "AxiomAudit-01.log": { + "bytes": 868, + "sha256": "d767504ed20e66d9b326686703729954244281675cd41ca15cfce5cc10e832f1" + }, + "KernelReplay-01.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + } + }, + "reuse": "Previously completed regression proofs and source-input checks were not rerun; these checks concern the new contact certificate and its new controls." +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/GramProbe-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/GramProbe-01.log new file mode 100644 index 000000000..517ce7939 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/GramProbe-01.log @@ -0,0 +1,12 @@ +/workspace/scratch/regression-kernel/contact-gram/GramProbe.lean:14:46: error(lean.invalidField): Invalid field `norm`: The environment does not contain `Exists.norm`, so it is not possible to project the field `norm` from an expression + ha +of type `∃ a_1, HasSum a a_1` +/workspace/scratch/regression-kernel/contact-gram/GramProbe.lean:14:54: error(lean.invalidField): Invalid field `norm`: The environment does not contain `Exists.norm`, so it is not possible to project the field `norm` from an expression + hb +of type `∃ a, HasSum b a` +/workspace/scratch/regression-kernel/contact-gram/GramProbe.lean:13:75: error: unsolved goals +ι : Type u_1 +a b : ι → ℝ +ha : Summable a +hb : Summable b +⊢ Summable fun m => a m * b m diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/GramProbe-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/GramProbe-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/GramProbe.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/GramProbe.lean.txt new file mode 100644 index 000000000..7860c7677 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/GramProbe.lean.txt @@ -0,0 +1,54 @@ +import Mathlib.LinearAlgebra.Matrix.PosDef +import Mathlib.Analysis.Normed.Ring.InfiniteSum +import Mathlib.Analysis.Normed.Module.FiniteDimension + +open Matrix +open scoped BigOperators Matrix +set_option autoImplicit false +namespace I4ContactGram + +noncomputable def coefficientGram {ι : Type*} {n : ℕ} (coeff : Fin n → ι → ℝ) : + Matrix (Fin n) (Fin n) ℝ := fun i j => ∑' m, coeff i m * coeff j m + +lemma coefficient_product_summable {ι : Type*} {a b : ι → ℝ} + (ha : Summable a) (hb : Summable b) : Summable (fun m => a m * b m) := by + have hp := summable_mul_of_summable_norm ha.norm hb.norm + simpa only [Function.comp_def] using hp.comp_injective + (show Function.Injective (fun m : ι => (m,m)) from + fun i j h => congrArg Prod.fst h) + +lemma coefficientGram_quadratic_hasSum {ι : Type*} {n : ℕ} (coeff : Fin n → ι → ℝ) + (hc : ∀ i, Summable (coeff i)) (z : Fin n → ℝ) : + HasSum (fun m => (∑ i, z i * coeff i m)^2) + (star z ⬝ᵥ (coefficientGram coeff *ᵥ z)) := by + have hs : HasSum (fun m => ∑ i, ∑ j, z i * (coeff i m * coeff j m) * z j) + (∑ i, ∑ j, z i * coefficientGram coeff i j * z j) := + hasSum_sum (fun i _ => hasSum_sum (fun j _ => + (((coefficient_product_summable (hc i) (hc j)).hasSum).mul_left (z i)).mul_right (z j))) + convert hs using 1 + · funext m + rw [pow_two, Finset.sum_mul_sum] + apply Finset.sum_congr rfl + intro i _ + apply Finset.sum_congr rfl + intro j _ + ring + · simp [Matrix.mulVec, dotProduct, Finset.mul_sum, mul_assoc] + +lemma coefficientGram_posDef_of_separating {ι : Type*} {n : ℕ} + (coeff : Fin n → ι → ℝ) (hc : ∀ i, Summable (coeff i)) + (hsep : ∀ z : Fin n → ℝ, z ≠ 0 → ∃ m, (∑ i, z i * coeff i m) ≠ 0) : + (coefficientGram coeff).PosDef := by + apply Matrix.PosDef.of_dotProduct_mulVec_pos + · ext i j + simp only [coefficientGram, Matrix.conjTranspose_apply, star_trivial] + apply tsum_congr + intro m + ring + · intro z hz + have hs := coefficientGram_quadratic_hasSum coeff hc z + rw [← hs.tsum_eq] + obtain ⟨m, hm⟩ := hsep z hz + exact hs.summable.tsum_pos (fun m => sq_nonneg _) m (sq_pos_of_ne_zero hm) + +end I4ContactGram diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/KernelReplay-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/KernelReplay-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls-01.log new file mode 100644 index 000000000..fb70caf1f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls-01.log @@ -0,0 +1,12 @@ +/workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean:8:44: error: unsolved goals +⊢ ¬?m.30 = 0 +/workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean:6:57: error: unsolved goals +i s : Fin 7 +⊢ contactWitness 1 i s = contactWitness ?m.30 i s + +⊢ ℝ +/workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean:15:44: error: unsolved goals +⊢ ¬?m.52 = 0 +/workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean:33:64: error: unsolved goals +h : Function.Injective (contactWitness 1 * diagonal fun x => 0)ᵀ.mulVec +⊢ col 0 1 = 0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls-02.log new file mode 100644 index 000000000..c4251929e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls-02.log @@ -0,0 +1,4 @@ +/workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean:33:64: error: unsolved goals +h : Function.Injective (contactWitness 1 * diagonal fun x => 0)ᵀ.mulVec +hmat : (contactWitness 1 * diagonal fun x => 0) = 0 +⊢ col 0 1 = 0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls.lean.txt new file mode 100644 index 000000000..1c593e8cb --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/PositiveControls.lean.txt @@ -0,0 +1,41 @@ +import ContactJetCovariance + +open Matrix I4ContactGram +open scoped BigOperators Matrix + +example : (coefficientGram (contactWitness 1)).PosDef := by + apply contact_coefficientGram_posDef (c := 1) (contactWitness 1) + (fun i => (hasSum_fintype _).summable) (by norm_num) id (fun _ => 1) + (fun _ => one_ne_zero) + intro i s + simp + +example : (coefficientGram (fun i s => 2 * contactWitness (-1) i s)).PosDef := by + apply contact_coefficientGram_posDef (c := -1) (fun i s => 2 * contactWitness (-1) i s) + (fun i => (hasSum_fintype _).summable) (by norm_num) id (fun _ => 2) (by norm_num) + intro i s + rfl + +example : ¬ Function.Injective ((contactWitness 0)ᵀ).mulVec := by + intro h + have hz : (contactWitness 0)ᵀ *ᵥ (Pi.single (1 : Fin 7) (1 : ℝ)) = + (contactWitness 0)ᵀ *ᵥ 0 := by + ext s + fin_cases s <;> norm_num [Matrix.mulVec, dotProduct, contactWitness, Fin.sum_univ_succ, + Pi.single_apply] + have heq := congrFun (h hz) (1 : Fin 7) + norm_num at heq + +example : ¬ Function.Injective ((contactWitness 1 * diagonal (fun _ => (0 : ℝ)))ᵀ).mulVec := by + intro h + have hz : (contactWitness 1 * diagonal (fun _ => (0 : ℝ)))ᵀ *ᵥ + (Pi.single (1 : Fin 7) (1 : ℝ)) = + (contactWitness 1 * diagonal (fun _ => (0 : ℝ)))ᵀ *ᵥ 0 := by + have hmat : contactWitness 1 * diagonal (fun _ => (0 : ℝ)) = 0 := by + ext i j + rw [Matrix.mul_diagonal] + simp + rw [hmat] + simp only [Matrix.transpose_zero, Matrix.zero_mulVec] + have heq := congrFun (h hz) (1 : Fin 7) + norm_num at heq diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe-01.log new file mode 100644 index 000000000..20f70209a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe-01.log @@ -0,0 +1 @@ +/workspace/scratch/regression-kernel/contact-gram/RankProbe.lean:1:0: error: object file '/workspace/Math-/frontiers/determinant_moment_20261025/.lake/packages/mathlib/.lake/build/lib/lean/Mathlib/Tactic/Nlinarith.olean' of module Mathlib.Tactic.Nlinarith does not exist diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe-02.log new file mode 100644 index 000000000..cf2effde5 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe-02.log @@ -0,0 +1,55 @@ +/workspace/scratch/regression-kernel/contact-gram/RankProbe.lean:34:63: error: linarith failed to find a contradiction +case h1 +c : ℝ +hc : c ≠ 0 +z : Fin 7 → ℝ +h : (contactWitness c)ᵀ *ᵥ z = 0 +hs : ∀ (s : Fin 7), ((contactWitness c)ᵀ *ᵥ z) s = 0 s +h0 : z 0 = 0 +h1 : c = 0 ∨ z 3 = 0 +h2 : c = 0 ∨ z 6 = 0 +h3 : -(c ^ 2 * z 3) + (-(c ^ 2 * z 4) + -(c ^ 2 * z 6)) = 0 +h4 : c * z 1 + -(c ^ 3 * z 5) = 0 +h5 : 2 * c * z 1 + -(8 * c ^ 3 * z 5) = 0 +h6 : c = 0 ∨ z 2 = 0 +hc2 : c ^ 2 ≠ 0 +hc3 : c ^ 3 ≠ 0 +a✝ : c ^ 2 * z 3 < 0 +⊢ False +failed +/workspace/scratch/regression-kernel/contact-gram/RankProbe.lean:35:63: error: linarith failed to find a contradiction +case h1 +c : ℝ +hc : c ≠ 0 +z : Fin 7 → ℝ +h : (contactWitness c)ᵀ *ᵥ z = 0 +hs : ∀ (s : Fin 7), ((contactWitness c)ᵀ *ᵥ z) s = 0 s +h0 : z 0 = 0 +h1 : c = 0 ∨ z 3 = 0 +h2 : c = 0 ∨ z 6 = 0 +h3 : -(c ^ 2 * z 3) + (-(c ^ 2 * z 4) + -(c ^ 2 * z 6)) = 0 +h4 : c * z 1 + -(c ^ 3 * z 5) = 0 +h5 : 2 * c * z 1 + -(8 * c ^ 3 * z 5) = 0 +h6 : c = 0 ∨ z 2 = 0 +hc2 : c ^ 2 ≠ 0 +hc3 : c ^ 3 ≠ 0 +hz3 : z 3 = 0 +a✝ : c ^ 2 * z 6 < 0 +⊢ False +failed +/workspace/scratch/regression-kernel/contact-gram/RankProbe.lean:39:40: error: Application type mismatch: The argument + h6 +has type + c = 0 ∨ z 2 = 0 +but is expected to have type + ?m.373 * ?m.374 = 0 +in the application + mul_eq_zero.mp h6 +/workspace/scratch/regression-kernel/contact-gram/RankProbe.lean:59:32: error: Application type mismatch: The argument + hs +has type + a s * ((contactWitness c)ᵀ *ᵥ x) s = ((diagonal a * (contactWitness c)ᵀ) *ᵥ y) s +but is expected to have type + a s * ((contactWitness c)ᵀ *ᵥ x) s = a s * ((contactWitness c)ᵀ *ᵥ y) s +in the application + mul_left_cancel₀ (ha s) hs diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe.lean.txt new file mode 100644 index 000000000..f0e7c455d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RankProbe.lean.txt @@ -0,0 +1,61 @@ +import Mathlib.LinearAlgebra.Matrix.PosDef +import Mathlib.LinearAlgebra.Matrix.Notation +import Mathlib.Tactic.FinCases +import Mathlib.Tactic.Linarith + +open Matrix +open scoped BigOperators Matrix +set_option autoImplicit false +namespace I4ContactGram + +def contactWitness (c : ℝ) : Matrix (Fin 7) (Fin 7) ℝ := + !![1,1,1,1,0,0,0; + 0,0,0,0,c,2*c,0; + 0,0,0,0,0,0,c; + 0,-c^2,0,-c^2,0,0,0; + 0,0,0,-c^2,0,0,0; + 0,0,0,0,-c^3,-8*c^3,0; + 0,0,-c^2,-c^2,0,0,0] + +lemma contactWitness_transpose_mulVec_eq_zero {c : ℝ} (hc : c ≠ 0) + (z : Fin 7 → ℝ) (h : (contactWitness c)ᵀ *ᵥ z = 0) : z = 0 := by + have hs (s : Fin 7) := congrFun h s + have h0 : z 0 = 0 := by + simpa [contactWitness, Matrix.mulVec, dotProduct, Fin.sum_univ_succ] using hs 0 + have h1 := hs 1 + have h2 := hs 2 + have h3 := hs 3 + have h4 := hs 4 + have h5 := hs 5 + have h6 := hs 6 + simp [contactWitness, Matrix.mulVec, dotProduct, Fin.sum_univ_succ, h0] at h1 h2 h3 h4 h5 h6 + have hc2 : c^2 ≠ 0 := pow_ne_zero _ hc + have hc3 : c^3 ≠ 0 := pow_ne_zero _ hc + have hz3 : z 3 = 0 := h1.resolve_left hc + have hz6 : z 6 = 0 := h2.resolve_left hc + have hz4 : z 4 = 0 := (mul_eq_zero.mp (show c^2 * z 4 = 0 by nlinarith [h3])).resolve_left hc2 + have hz5 : z 5 = 0 := (mul_eq_zero.mp (show c^3 * z 5 = 0 by nlinarith [h4, h5])).resolve_left hc3 + have hz1 : z 1 = 0 := (mul_eq_zero.mp (show c * z 1 = 0 by nlinarith [h4])).resolve_left hc + have hz2 : z 2 = 0 := h6.resolve_left hc + funext i + fin_cases i <;> assumption + +lemma contactWitness_transpose_injective {c : ℝ} (hc : c ≠ 0) : + Function.Injective ((contactWitness c)ᵀ).mulVec := by + intro x y h + apply sub_eq_zero.mp + apply contactWitness_transpose_mulVec_eq_zero hc + simp [Matrix.mulVec_sub, h] + +lemma contactWitness_amplitude_transpose_injective {c : ℝ} (hc : c ≠ 0) + (a : Fin 7 → ℝ) (ha : ∀ s, a s ≠ 0) : + Function.Injective ((contactWitness c * diagonal a)ᵀ).mulVec := by + intro x y h + apply contactWitness_transpose_injective hc + funext s + have hs := congrFun h s + simp only [Matrix.transpose_mul, Matrix.diagonal_transpose, ← Matrix.mulVec_mulVec, + Matrix.mulVec_diagonal] at hs + exact mul_left_cancel₀ (ha s) hs + +end I4ContactGram diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroAmplitude-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroAmplitude-01.log new file mode 100644 index 000000000..8f2db3f8d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroAmplitude-01.log @@ -0,0 +1,3 @@ +/workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean:3:91: error: unsolved goals +s : Fin 7 +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroAmplitude.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroAmplitude.lean.txt new file mode 100644 index 000000000..18456148e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroAmplitude.lean.txt @@ -0,0 +1,6 @@ +import ContactJetCovariance +open Matrix I4ContactGram +example : Function.Injective ((contactWitness 1 * diagonal (fun _ => (0 : ℝ)))ᵀ).mulVec := by + apply contactWitness_amplitude_transpose_injective (by norm_num) + intro s + norm_num diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroFrequency-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroFrequency-01.log new file mode 100644 index 000000000..1c3f6631c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroFrequency-01.log @@ -0,0 +1,2 @@ +/workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean:3:61: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroFrequency.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroFrequency.lean.txt new file mode 100644 index 000000000..c023377c0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/RejectedZeroFrequency.lean.txt @@ -0,0 +1,5 @@ +import ContactJetCovariance +open Matrix I4ContactGram +example : Function.Injective ((contactWitness 0)ᵀ).mulVec := by + apply contactWitness_transpose_injective + norm_num diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/assemble.py b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/assemble.py new file mode 100644 index 000000000..db3a68e90 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/ContactJetCovariance/assemble.py @@ -0,0 +1,61 @@ +rank=open('/workspace/scratch/regression-kernel/contact-gram/RankProbe.lean').read() +gram=open('/workspace/scratch/regression-kernel/contact-gram/GramProbe.lean').read() +imports='''import Mathlib.LinearAlgebra.Matrix.PosDef +import Mathlib.LinearAlgebra.Matrix.Notation +import Mathlib.Analysis.Normed.Ring.InfiniteSum +import Mathlib.Analysis.Normed.Module.FiniteDimension +import Mathlib.Tactic.FinCases +import Mathlib.Tactic.Linarith + +''' +header='''/-! +# Seven-mode certificate for contact-jet covariance + +The row order is `(f, fx, fy, fxx, fxy, fxxx, fyy)`, with selected columns +`(DC, cos(1,0), cos(0,1), cos(1,1), sin(1,0), sin(2,0), sin(0,1))`. +Nonzero mode amplitudes are retained. The witness forces strict positivity of an +infinite coefficient Gram matrix by its sum-of-squares quadratic identity. + +This is a coefficient certificate. Applying it to the literal smooth source requires +the exact derivative-coefficient identities and physical covariance identification; +these are not premises that may be silently omitted. See `SOURCE_CONTACT_GRAM.md`. +-/ + +open Matrix +open scoped BigOperators Matrix +set_option autoImplicit false + +namespace I4ContactGram + +''' +rankbody=rank[rank.index('def contactWitness'):rank.rindex('end I4ContactGram')] +grambody=gram[gram.index('noncomputable def coefficientGram'):gram.rindex('end I4ContactGram')] +final='''/-- Seven explicit independent mode columns force positivity of the entire infinite Gram sum. +No positive-definiteness or invertibility conclusion is an input premise. -/ +theorem contact_coefficientGram_posDef {ι : Type*} (coeff : Fin 7 → ι → ℝ) + (hcoeff : ∀ i, Summable (coeff i)) {c : ℝ} (hc : c ≠ 0) + (mode : Fin 7 → ι) (a : Fin 7 → ℝ) (ha : ∀ s, a s ≠ 0) + (hcolumns : ∀ i s, coeff i (mode s) = a s * contactWitness c i s) : + (coefficientGram coeff).PosDef := by + apply coefficientGram_posDef_of_separating coeff hcoeff + intro z hz + classical + by_cases hex : ∃ m, (∑ i, z i * coeff i m) ≠ 0 + · exact hex + · exfalso + apply hz + apply contactWitness_amplitude_transpose_injective hc a ha + rw [Matrix.mulVec_zero] + funext s + have hm : (∑ i, z i * coeff i (mode s)) = 0 := + not_ne_iff.mp (not_exists.mp hex (mode s)) + simp_rw [hcolumns] at hm + simpa [Matrix.mulVec, dotProduct, Matrix.mul_diagonal, mul_comm, mul_left_comm, + mul_assoc] using hm + +''' +schur=open('/workspace/scratch/regression-kernel/contact-schur/SchurProbe.lean').read() +schurbody=schur[schur.index('theorem block_posDef'):schur.rindex('end ContactSchurProbe')] +schurbody=schurbody.replace('theorem block_posDef_schur_pos', '''/-- A positive-definite six-plus-one block covariance has strictly positive scalar Schur complement. -/ +theorem contact_schur_pos''',1) +open('ContactJetCovariance.lean','w').write(imports+header+rankbody+grambody+final+schurbody+'\nend I4ContactGram\n') diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe-01.log new file mode 100644 index 000000000..04645840e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe-01.log @@ -0,0 +1,20 @@ +/workspace/scratch/regression-limits/bounds-probe.lean:20:35: error(lean.synthInstanceFailed): failed to synthesize instance of type class + Norm (ι → ℝ) + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/bounds-probe.lean:20:43: error(lean.synthInstanceFailed): failed to synthesize instance of type class + Norm (ι → ℝ) + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/bounds-probe.lean:21:40: error: linarith failed to find a contradiction +ι : Type u_1 +inst✝ : Fintype ι +f : ℝ → ι → ℝ +g : ℝ → Matrix ι ι ℝ +f0 : ι → ℝ +g0 : Matrix ι ι ℝ +hf : Tendsto f (𝓝[>] 0) (𝓝 f0) +hg : Tendsto g (𝓝[>] 0) (𝓝 g0) +a✝ : ?m.112 ≤ ‖?m.98‖ +⊢ False +failed diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe-02.log new file mode 100644 index 000000000..0f8d72aa5 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe-02.log @@ -0,0 +1,21 @@ +/workspace/scratch/regression-limits/bounds-probe.lean:23:35: error(lean.synthInstanceFailed): failed to synthesize instance of type class + Norm (Matrix ι ι ℝ) + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/bounds-probe.lean:23:43: error(lean.synthInstanceFailed): failed to synthesize instance of type class + Norm (Matrix ι ι ℝ) + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/bounds-probe.lean:24:40: error: linarith failed to find a contradiction +ι : Type u_1 +inst✝ : Fintype ι +f : ℝ → ι → ℝ +g : ℝ → Matrix ι ι ℝ +f0 : ι → ℝ +g0 : Matrix ι ι ℝ +hf : Tendsto f (𝓝[>] 0) (𝓝 f0) +hg : Tendsto g (𝓝[>] 0) (𝓝 g0) +hfb : ∀ᶠ (r : ℝ) in 𝓝[>] 0, ‖f r‖ < ‖f0‖ + 1 +a✝ : ?m.163 ≤ ‖?m.149‖ +⊢ False +failed diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe.lean.txt new file mode 100644 index 000000000..0dc6d09ea --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/bounds-probe.lean.txt @@ -0,0 +1,39 @@ +import Mathlib.Analysis.Normed.Group.Real +import Mathlib.Analysis.Normed.Group.Constructions +import Mathlib.Analysis.Normed.Group.Continuity +import Mathlib.Topology.Order.LeftRightNhds +import Mathlib.Topology.Instances.Matrix +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.Positivity + +open Filter Matrix +open scoped Topology +set_option autoImplicit false +namespace ResearchFormalCoreR1 + +theorem bounds_from_limits {ι : Type*} [Fintype ι] + (f : ℝ → (ι → ℝ)) (g : ℝ → Matrix ι ι ℝ) + (f0 : ι → ℝ) (g0 : Matrix ι ι ℝ) + (hf : Tendsto f (𝓝[>] (0 : ℝ)) (𝓝 f0)) + (hg : Tendsto g (𝓝[>] (0 : ℝ)) (𝓝 g0)) : + ∃ B V δ : ℝ, 0 ≤ B ∧ 0 ≤ V ∧ 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → + (∀ i, |f r i| ≤ B) ∧ (∀ i, |g r i i| ≤ V) := by + have hfb : ∀ᶠ r in 𝓝[>] (0 : ℝ), ‖f r‖ < ‖f0‖ + 1 := + hf.norm.eventually (gt_mem_nhds (by linarith)) + have hgg : Tendsto (fun r => fun i j => g r i j) (𝓝[>] (0 : ℝ)) + (𝓝 (fun i j => g0 i j)) := hg + have hgb : ∀ᶠ r in 𝓝[>] (0 : ℝ), ‖fun i j => g r i j‖ < ‖fun i j => g0 i j‖ + 1 := + hgg.norm.eventually (gt_mem_nhds (by linarith)) + obtain ⟨δ, hδ, hbounds⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp (hfb.and hgb) + refine ⟨‖f0‖+1, ‖fun i j => g0 i j‖+1, δ, by positivity, by positivity, hδ, ?_⟩ + intro r hr hrδ + have hb := hbounds ⟨hr, hrδ⟩ + constructor + · intro i + exact (show |f r i| ≤ ‖f r‖ by simpa only [Real.norm_eq_abs] using norm_le_pi_norm (f r) i).trans hb.1.le + · intro i + exact (show |g r i i| ≤ ‖fun i j => g r i j‖ by + exact (show |g r i i| ≤ ‖g r i‖ by simpa only [Real.norm_eq_abs] using norm_le_pi_norm (g r i) i).trans + (norm_le_pi_norm (fun i j => g r i j) i)).trans hb.2.le +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe-01.log new file mode 100644 index 000000000..46d076605 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe-01.log @@ -0,0 +1,25 @@ +/workspace/scratch/regression-limits/inverse-probe.lean:13:5: error(lean.synthInstanceFailed): failed to synthesize instance of type class + IsTopologicalRing ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/inverse-probe.lean:19:47: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousInv₀ ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/inverse-probe.lean:20:11: error(lean.synthInstanceFailed): failed to synthesize instance of type class + IsTopologicalRing ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/inverse-probe.lean:10:73: error: unsolved goals +case refine_1 +p : Type u_1 +a : Type u_2 +inst✝¹ : Fintype p +inst✝ : DecidableEq p +u : Filter a +G : a → Matrix p p ℝ +G0 : Matrix p p ℝ +hG : Tendsto G u (𝓝 G0) +hG0 : IsUnit G0 +hdet0 : G0.det ≠ 0 +⊢ IsTopologicalRing ℝ diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe-02.log new file mode 100644 index 000000000..5844eb14e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe-02.log @@ -0,0 +1,25 @@ +/workspace/scratch/regression-limits/inverse-probe.lean:14:5: error(lean.synthInstanceFailed): failed to synthesize instance of type class + IsTopologicalRing ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/inverse-probe.lean:20:47: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousInv₀ ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/inverse-probe.lean:21:11: error(lean.synthInstanceFailed): failed to synthesize instance of type class + IsTopologicalRing ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/inverse-probe.lean:11:73: error: unsolved goals +case refine_1 +p : Type u_1 +a : Type u_2 +inst✝¹ : Fintype p +inst✝ : DecidableEq p +u : Filter a +G : a → Matrix p p ℝ +G0 : Matrix p p ℝ +hG : Tendsto G u (𝓝 G0) +hG0 : IsUnit G0 +hdet0 : G0.det ≠ 0 +⊢ IsTopologicalRing ℝ diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe.lean.txt new file mode 100644 index 000000000..a27dd0be5 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/inverse-probe.lean.txt @@ -0,0 +1,23 @@ +import Mathlib.Topology.Instances.Matrix +import Mathlib.Topology.Algebra.GroupWithZero +import Mathlib.Analysis.Normed.Group.Real +import Mathlib.Analysis.Normed.Field.Basic +import Mathlib.Topology.Algebra.Ring.Real +open Matrix Filter +open scoped Topology +set_option autoImplicit false +example {p a : Type*} [Fintype p] [DecidableEq p] + (u : Filter a) (G : a → Matrix p p ℝ) (G0 : Matrix p p ℝ) + (hG : Tendsto G u (𝓝 G0)) (hG0 : IsUnit G0) : + (∀ᶠ r in u, IsUnit (G r)) ∧ Tendsto (fun r => (G r)⁻¹) u (𝓝 G0⁻¹) := by + have hdet0 : G0.det ≠ 0 := ((Matrix.isUnit_iff_isUnit_det G0).mp hG0).ne_zero + have hdet : Tendsto (fun r => (G r).det) u (𝓝 G0.det) := + (continuous_id.matrix_det.tendsto G0).comp hG + have hunit : ∀ᶠ r in u, IsUnit (G r) := by + filter_upwards [hdet.eventually_ne hdet0] with r hr + exact (Matrix.isUnit_iff_isUnit_det (G r)).mpr (isUnit_iff_ne_zero.mpr hr) + have hInv : Tendsto (fun r => (G r)⁻¹) u (𝓝 G0⁻¹) := by + have hc : ContinuousAt (Ring.inverse : ℝ → ℝ) G0.det := by + simpa only [Ring.inverse_eq_inv'] using (continuousAt_inv₀ hdet0) + exact (continuousAt_matrix_inv G0 hc).tendsto.comp hG + exact ⟨hunit, hInv⟩ diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-01.log new file mode 100644 index 000000000..51de76f51 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-01.log @@ -0,0 +1,16 @@ +/workspace/scratch/regression-limits/matrix-probe.lean:24:50: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousMul ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/matrix-probe.lean:24:8: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousAdd ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/matrix-probe.lean:38:50: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousMul ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/matrix-probe.lean:38:8: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousAdd ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-02.log new file mode 100644 index 000000000..1c631ceb1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-02.log @@ -0,0 +1 @@ +/workspace/scratch/regression-limits/matrix-probe.lean:1:0: error: object file '/workspace/Math-/frontiers/determinant_moment_20261025/.lake/packages/mathlib/.lake/build/lib/lean/Mathlib/Analysis/Normed/Ring/Real.olean' of module Mathlib.Analysis.Normed.Ring.Real does not exist diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-03.log new file mode 100644 index 000000000..973ac0ba3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-03.log @@ -0,0 +1,16 @@ +/workspace/scratch/regression-limits/matrix-probe.lean:25:50: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousMul ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/matrix-probe.lean:25:8: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousAdd ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/matrix-probe.lean:39:50: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousMul ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +/workspace/scratch/regression-limits/matrix-probe.lean:39:8: error(lean.synthInstanceFailed): failed to synthesize instance of type class + ContinuousAdd ℝ + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-04.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe-04.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe.lean.txt new file mode 100644 index 000000000..74fabdef0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/matrix-probe.lean.txt @@ -0,0 +1,41 @@ +import Mathlib.Topology.Instances.Real.Lemmas +import Mathlib.Analysis.Normed.Field.Basic +import Mathlib.Topology.Instances.Matrix +import Mathlib.Analysis.Normed.Group.Real + +open Filter Matrix +open scoped Topology +set_option autoImplicit false +namespace ResearchFormalCoreR1 + +theorem tendsto_matrix_mul_explicit {α p t q : Type*} [Fintype t] + {u : Filter α} {A : α → Matrix p t ℝ} {A0 : Matrix p t ℝ} + {B : α → Matrix t q ℝ} {B0 : Matrix t q ℝ} + (hA : Tendsto A u (𝓝 A0)) (hB : Tendsto B u (𝓝 B0)) : + Tendsto (fun r => A r * B r) u (𝓝 (A0 * B0)) := by + have hAij (i : p) (j : t) : Tendsto (fun r => A r i j) u (𝓝 (A0 i j)) := + ((continuous_apply j).tendsto (A0 i)).comp + (((continuous_apply i).tendsto A0).comp hA) + have hBij (i : t) (j : q) : Tendsto (fun r => B r i j) u (𝓝 (B0 i j)) := + ((continuous_apply j).tendsto (B0 i)).comp + (((continuous_apply i).tendsto B0).comp hB) + apply tendsto_pi_nhds.mpr + intro i + apply tendsto_pi_nhds.mpr + intro j + exact tendsto_finsetSum Finset.univ (fun k _ => (hAij i k).mul (hBij k j)) + +theorem tendsto_matrix_mulVec_explicit {α p t : Type*} [Fintype t] + {u : Filter α} {A : α → Matrix p t ℝ} {A0 : Matrix p t ℝ} + {b : α → (t → ℝ)} {b0 : t → ℝ} + (hA : Tendsto A u (𝓝 A0)) (hb : Tendsto b u (𝓝 b0)) : + Tendsto (fun r => A r *ᵥ b r) u (𝓝 (A0 *ᵥ b0)) := by + have hAij (i : p) (j : t) : Tendsto (fun r => A r i j) u (𝓝 (A0 i j)) := + ((continuous_apply j).tendsto (A0 i)).comp + (((continuous_apply i).tendsto A0).comp hA) + have hbi (i : t) : Tendsto (fun r => b r i) u (𝓝 (b0 i)) := + ((continuous_apply i).tendsto b0).comp hb + apply tendsto_pi_nhds.mpr + intro i + exact tendsto_finsetSum Finset.univ (fun j _ => (hAij i j).mul (hbi j)) +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-01.log new file mode 100644 index 000000000..395c7d5df --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-01.log @@ -0,0 +1 @@ +GaussianRegressionLimits.lean:1:0: error: object file '/workspace/Math-/frontiers/determinant_moment_20261025/.lake/packages/mathlib/.lake/build/lib/lean/Mathlib/Analysis/Normed/Group/Pi.olean' of module Mathlib.Analysis.Normed.Group.Pi does not exist diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-02.log new file mode 100644 index 000000000..b523a575c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-02.log @@ -0,0 +1,6 @@ +GaussianRegressionLimits.lean:36:0: error: (deterministic) timeout at `whnf`, maximum number of heartbeats (200000) has been reached + +Note: Use `set_option maxHeartbeats ` to set the limit. + +Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command. +GaussianRegressionLimits.lean:106:8: error: (kernel) unknown constant 'I4GaussianRegressionLimits.regression_parameter_limits' diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls-01.log new file mode 100644 index 000000000..25388b29c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls-01.log @@ -0,0 +1,38 @@ +/workspace/scratch/regression-limits/regression-limits-controls.lean:14:41: error: typeclass instance problem is stuck + OfNat (Matrix (?m.68 M V δ r) (Fin 1) ℝ) 1 + +Note: Lean will not try to resolve this typeclass instance problem because the first type argument to `OfNat` contains metavariables. This argument must be fully determined before Lean will try to resolve the typeclass. + +Hint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck. +numerals are polymorphic in Lean, but the numeral `1` cannot be used in a context where the expected type is + Matrix (?m.68 M V δ r) (Fin 1) ℝ +due to the absence of the instance above +I4GaussianRegressionLimits.regressionMean.{u_1, u_2} {p : Type u_1} {t : Type u_2} [Fintype p] [DecidableEq p] + (G : Matrix p p ℝ) (C : Matrix t p ℝ) (d : p → ℝ) : t → ℝ +I4GaussianRegressionLimits.regressionCovariance.{u_1, u_2} {p : Type u_1} {t : Type u_2} [Fintype p] [DecidableEq p] + (G : Matrix p p ℝ) (C : Matrix t p ℝ) (B : Matrix t t ℝ) : Matrix t t ℝ +I4GaussianRegressionLimits.regression_parameter_limits.{u_1, u_2, u_3} {p : Type u_1} {t : Type u_2} {a : Type u_3} + [Fintype p] [DecidableEq p] [Fintype t] (u : Filter a) (G : a → Matrix p p ℝ) (C : a → Matrix t p ℝ) + (B : a → Matrix t t ℝ) (d : a → p → ℝ) (G0 : Matrix p p ℝ) (C0 : Matrix t p ℝ) (B0 : Matrix t t ℝ) (d0 : p → ℝ) + (hG : Tendsto G u (𝓝 G0)) (hC : Tendsto C u (𝓝 C0)) (hB : Tendsto B u (𝓝 B0)) (hd : Tendsto d u (𝓝 d0)) + (hG0 : IsUnit G0) : + (∀ᶠ (r : a) in u, IsUnit (G r)) ∧ + Tendsto (fun r => (G r)⁻¹) u (𝓝 G0⁻¹) ∧ + Tendsto (fun r => regressionMean (G r) (C r) (d r)) u (𝓝 (regressionMean G0 C0 d0)) ∧ + Tendsto (fun r => regressionCovariance (G r) (C r) (B r)) u (𝓝 (regressionCovariance G0 C0 B0)) +I4GaussianRegressionLimits.regression_uniform_bounds.{u_1, u_2} {p : Type u_1} {t : Type u_2} [Fintype p] + [DecidableEq p] [Fintype t] (G : ℝ → Matrix p p ℝ) (C : ℝ → Matrix t p ℝ) (B : ℝ → Matrix t t ℝ) (d : ℝ → p → ℝ) + (G0 : Matrix p p ℝ) (C0 : Matrix t p ℝ) (B0 : Matrix t t ℝ) (d0 : p → ℝ) (hG : Tendsto G (𝓝[>] 0) (𝓝 G0)) + (hC : Tendsto C (𝓝[>] 0) (𝓝 C0)) (hB : Tendsto B (𝓝[>] 0) (𝓝 B0)) (hd : Tendsto d (𝓝[>] 0) (𝓝 d0)) (hG0 : IsUnit G0) : + ∃ M V δ, + 0 ≤ M ∧ + 0 ≤ V ∧ + 0 < δ ∧ + ∀ (r : ℝ), + 0 < r → + r < δ → + IsUnit (G r) ∧ + (∀ (i : t), |regressionMean (G r) (C r) (d r) i| ≤ M) ∧ + ∀ (i : t), |regressionCovariance (G r) (C r) (B r) i i| ≤ V +'I4GaussianRegressionLimits.regression_parameter_limits' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4GaussianRegressionLimits.regression_uniform_bounds' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls-before.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls-before.lean.txt new file mode 100644 index 000000000..5c3b570ef --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls-before.lean.txt @@ -0,0 +1,50 @@ +import GaussianRegressionLimits +import Mathlib.Tactic.FinCases +import Mathlib.Tactic.NormNum +namespace I4GaussianRegressionLimits +open Matrix Filter +open scoped Topology + +private def controlG (r : ℝ) : Matrix (Fin 1) (Fin 1) ℝ := fun _ _ => 1+r^2 +private def controlD (r : ℝ) : Fin 1 → ℝ := fun _ => 3+r + +-- Changing pin covariance and nonzero targets; the limiting target covariance is singular. +example : ∃ M V δ : ℝ, 0 ≤ M ∧ 0 ≤ V ∧ 0 < δ ∧ ∀ r : ℝ, 0 < r → r < δ → + IsUnit (controlG r) ∧ + (∀ i, |regressionMean (controlG r) 1 (controlD r) i| ≤ M) ∧ + (∀ i, |regressionCovariance (controlG r) 1 1 i i| ≤ V) := by + have hG : Tendsto controlG (𝓝[>] (0:ℝ)) (𝓝 (controlG 0)) := by + have hc : Continuous controlG := by unfold controlG; fun_prop + exact hc.continuousWithinAt + have hd : Tendsto controlD (𝓝[>] (0:ℝ)) (𝓝 (controlD 0)) := by + have hc : Continuous controlD := by unfold controlD; fun_prop + exact hc.continuousWithinAt + have hunit : IsUnit (controlG 0) := by + have heq : controlG 0 = 1 := by + ext i j + fin_cases i + fin_cases j + norm_num [controlG] + rw [heq] + exact isUnit_one + exact regression_uniform_bounds controlG (fun _r => 1) (fun _r => 1) controlD + (controlG 0) 1 1 (controlD 0) hG tendsto_const_nhds tendsto_const_nhds hd hunit + +-- Both finite sizes may be empty; no Nonempty assumption is hidden in the inverse API. +example : ∃ M V δ : ℝ, 0 ≤ M ∧ 0 ≤ V ∧ 0 < δ ∧ ∀ r : ℝ, 0 < r → r < δ → + IsUnit (1 : Matrix (Fin 0) (Fin 0) ℝ) ∧ + (∀ i : Fin 0, |regressionMean (1 : Matrix (Fin 0) (Fin 0) ℝ) + (0 : Matrix (Fin 0) (Fin 0) ℝ) (0 : Fin 0 → ℝ) i| ≤ M) ∧ + (∀ i : Fin 0, |regressionCovariance (1 : Matrix (Fin 0) (Fin 0) ℝ) + (0 : Matrix (Fin 0) (Fin 0) ℝ) 0 i i| ≤ V) := by + exact regression_uniform_bounds (fun _r => (1 : Matrix (Fin 0) (Fin 0) ℝ)) + (fun _r => (0 : Matrix (Fin 0) (Fin 0) ℝ)) (fun _r => 0) (fun _r => 0) + 1 0 0 0 tendsto_const_nhds tendsto_const_nhds tendsto_const_nhds tendsto_const_nhds isUnit_one + +#check regressionMean +#check regressionCovariance +#check regression_parameter_limits +#check regression_uniform_bounds +#print axioms regression_parameter_limits +#print axioms regression_uniform_bounds +end I4GaussianRegressionLimits diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls.lean.txt new file mode 100644 index 000000000..0041575a2 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-controls.lean.txt @@ -0,0 +1,44 @@ +import GaussianRegressionLimits +import Mathlib.Tactic.FinCases +import Mathlib.Tactic.NormNum +namespace I4GaussianRegressionLimits +open Matrix Filter +open scoped Topology + +private def controlG (r : ℝ) : Matrix (Fin 1) (Fin 1) ℝ := fun _ _ => 1+r^2 +private def controlD (r : ℝ) : Fin 1 → ℝ := fun _ => 3+r + +-- Changing pin covariance and nonzero targets; the limiting target covariance is singular. +example : ∃ M V δ : ℝ, 0 ≤ M ∧ 0 ≤ V ∧ 0 < δ ∧ ∀ r : ℝ, 0 < r → r < δ → + IsUnit (controlG r) ∧ + (∀ i : Fin 1, |regressionMean (controlG r) 1 (controlD r) i| ≤ M) ∧ + (∀ i : Fin 1, |regressionCovariance (controlG r) 1 1 i i| ≤ V) := by + have hG : Tendsto controlG (𝓝[>] (0:ℝ)) (𝓝 (controlG 0)) := by + have hc : Continuous controlG := by unfold controlG; fun_prop + exact hc.continuousWithinAt + have hd : Tendsto controlD (𝓝[>] (0:ℝ)) (𝓝 (controlD 0)) := by + have hc : Continuous controlD := by unfold controlD; fun_prop + exact hc.continuousWithinAt + have hunit : IsUnit (controlG 0) := by + have heq : controlG 0 = 1 := by + ext i j + fin_cases i + fin_cases j + norm_num [controlG] + rw [heq] + exact isUnit_one + exact regression_uniform_bounds controlG (fun _r => 1) (fun _r => 1) controlD + (controlG 0) 1 1 (controlD 0) hG tendsto_const_nhds tendsto_const_nhds hd hunit + +-- Both finite sizes may be empty; no Nonempty assumption is hidden in the inverse API. +example : ∃ M V δ : ℝ, 0 ≤ M ∧ 0 ≤ V ∧ 0 < δ ∧ ∀ r : ℝ, 0 < r → r < δ → + IsUnit (1 : Matrix (Fin 0) (Fin 0) ℝ) ∧ + (∀ i : Fin 0, |regressionMean (1 : Matrix (Fin 0) (Fin 0) ℝ) + (0 : Matrix (Fin 0) (Fin 0) ℝ) (0 : Fin 0 → ℝ) i| ≤ M) ∧ + (∀ i : Fin 0, |regressionCovariance (1 : Matrix (Fin 0) (Fin 0) ℝ) + (0 : Matrix (Fin 0) (Fin 0) ℝ) 0 i i| ≤ V) := by + exact regression_uniform_bounds (fun _r => (1 : Matrix (Fin 0) (Fin 0) ℝ)) + (fun _r => (0 : Matrix (Fin 0) (Fin 0) ℝ)) (fun _r => 0) (fun _r => 0) + 1 0 0 0 tendsto_const_nhds tendsto_const_nhds tendsto_const_nhds tendsto_const_nhds isUnit_one + +end I4GaussianRegressionLimits diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-leanchecker.log b/frontiers/source_derivative_gaussian_20261025/evidence/GaussianRegressionLimits/regression-limits-leanchecker.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/00-initial-api.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/00-initial-api.command new file mode 100644 index 000000000..2451db71b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/00-initial-api.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean /workspace/scratch/source-confluent-limits/initial_api.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/00-initial-api.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/00-initial-api.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/00-initial-api.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/00-initial-api.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/00-initial-api.log new file mode 100644 index 000000000..3ef91e0ec --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/00-initial-api.log @@ -0,0 +1,5 @@ +/workspace/scratch/source-confluent-limits/initial_api.lean:4:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceConfluentLimits.sourceConfluentPinsAt_series` +/workspace/scratch/source-confluent-limits/initial_api.lean:5:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceConfluentLimits.sourceAdjustedEndpoints_series` +/workspace/scratch/source-confluent-limits/initial_api.lean:6:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceConfluentLimits.sourcePinCovarianceAt_coefficient` +/workspace/scratch/source-confluent-limits/initial_api.lean:7:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceConfluentLimits.sourceCrossCovarianceAt_coefficient` +/workspace/scratch/source-confluent-limits/initial_api.lean:8:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceConfluentLimits.sourceEndpointCovarianceAt_coefficient` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/01-coefficient-representation-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/01-coefficient-representation-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/01-coefficient-representation-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/01-coefficient-representation-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/01-coefficient-representation-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/01-coefficient-representation-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/01-coefficient-representation-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/01-coefficient-representation-compile.log new file mode 100644 index 000000000..0b9edef77 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/01-coefficient-representation-compile.log @@ -0,0 +1,29 @@ +SourceConfluentLimits.lean:79:8: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + fun i => a i m * ω m +in the target expression + T (fun i => a i m) j * ω m = (⇑(↑T).toAddMonoidHom ∘ fun m i => a i m * ω m) m j + +case e'_5 +ι : Type u_1 +n q : ℕ +a : Fin n → ι → ℝ +ω : ι → ℝ +ha : ∀ (i : Fin n), Summable fun m => a i m * ω m +T : (Fin n → ℝ) →L[ℝ] Fin q → ℝ +hs : HasSum (fun m i => a i m * ω m) fun i => scalarSeries (a i) ω +ht : HasSum (⇑(↑T).toAddMonoidHom ∘ fun m i => a i m * ω m) ((↑T).toAddMonoidHom fun i => scalarSeries (a i) ω) +j : Fin q +m : ι +hm : (fun i => a i m * ω m) = ω m • fun i => a i m +⊢ T (fun i => a i m) j * ω m = (⇑(↑T).toAddMonoidHom ∘ fun m i => a i m * ω m) m j +SourceConfluentLimits.lean:96:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + scalarSeries (derivativeCoefficient 2 L ?k ?x ?u) (sampleInclusion 2 L ω) +in the target expression + rawPinsWithHessians L r ω i = scalarSeries (rawDerivativeCoefficient L r i) (sampleInclusion 2 L ω) + +L : ℝ +hL : L ≠ 0 +r : ℝ +ω : SmoothSample 2 L +i : Fin 12 +⊢ rawPinsWithHessians L r ω i = scalarSeries (rawDerivativeCoefficient L r i) (sampleInclusion 2 L ω) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/02-coefficient-joins-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/02-coefficient-joins-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/02-coefficient-joins-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/02-coefficient-joins-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/02-coefficient-joins-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/02-coefficient-joins-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/02-coefficient-joins-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/02-coefficient-joins-compile.log new file mode 100644 index 000000000..102980b67 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/02-coefficient-joins-compile.log @@ -0,0 +1,28 @@ +SourceConfluentLimits.lean:73:47: error: unsolved goals +case e'_6 +ι : Type u_1 +n q : ℕ +a : Fin n → ι → ℝ +ω : ι → ℝ +ha : ∀ (i : Fin n), Summable fun m => a i m * ω m +T : (Fin n → ℝ) →L[ℝ] Fin q → ℝ +hs : HasSum (fun m i => a i m * ω m) fun i => scalarSeries (a i) ω +ht : HasSum (⇑(↑T).toAddMonoidHom ∘ fun m i => a i m * ω m) ((↑T).toAddMonoidHom fun i => scalarSeries (a i) ω) +j : Fin q +⊢ T (fun i => scalarSeries (a i) ω) j = (↑T).toAddMonoidHom (fun i => scalarSeries (a i) ω) j +SourceConfluentLimits.lean:105:15: error: No applicable extensionality theorem found for type + Type + +Note: Extensionality theorems can be registered by marking them with the `[ext]` attribute +SourceConfluentLimits.lean:105:15: error: No applicable extensionality theorem found for type + Type + +Note: Extensionality theorems can be registered by marking them with the `[ext]` attribute +SourceConfluentLimits.lean:105:15: error: No applicable extensionality theorem found for type + Type + +Note: Extensionality theorems can be registered by marking them with the `[ext]` attribute +SourceConfluentLimits.lean:105:15: error: No applicable extensionality theorem found for type + Type + +Note: Extensionality theorems can be registered by marking them with the `[ext]` attribute diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/03-exact-coefficient-representations-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/03-exact-coefficient-representations-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/03-exact-coefficient-representations-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/03-exact-coefficient-representations-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/03-exact-coefficient-representations-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/03-exact-coefficient-representations-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/03-exact-coefficient-representations-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/03-exact-coefficient-representations-compile.log new file mode 100644 index 000000000..c549499b8 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/03-exact-coefficient-representations-compile.log @@ -0,0 +1,80 @@ +SourceConfluentLimits.lean:109:6: error: Tactic `apply` failed: could not unify the conclusion of `congrArg + ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 +with the goal + (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionX = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) (rawDerivativeDirections 2) + +Note: The full type of `congrArg ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` is + ?m.1388 = ?m.1389 → + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 + +case «2» +L : ℝ +hL : L ≠ 0 +r : ℝ +ω : SmoothSample 2 L +⊢ (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionX = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) (rawDerivativeDirections 2) +SourceConfluentLimits.lean:109:6: error: Tactic `apply` failed: could not unify the conclusion of `congrArg + ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 +with the goal + (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionX = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) (rawDerivativeDirections 3) + +Note: The full type of `congrArg ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` is + ?m.1388 = ?m.1389 → + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 + +case «3» +L : ℝ +hL : L ≠ 0 +r : ℝ +ω : SmoothSample 2 L +⊢ (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionX = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) (rawDerivativeDirections 3) +SourceConfluentLimits.lean:109:6: error: Tactic `apply` failed: could not unify the conclusion of `congrArg + ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 +with the goal + (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionY = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) (rawDerivativeDirections 4) + +Note: The full type of `congrArg ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` is + ?m.1388 = ?m.1389 → + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 + +case «4» +L : ℝ +hL : L ≠ 0 +r : ℝ +ω : SmoothSample 2 L +⊢ (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionY = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) (rawDerivativeDirections 4) +SourceConfluentLimits.lean:109:6: error: Tactic `apply` failed: could not unify the conclusion of `congrArg + ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 +with the goal + (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionY = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) (rawDerivativeDirections 5) + +Note: The full type of `congrArg ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` is + ?m.1388 = ?m.1389 → + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 + +case «5» +L : ℝ +hL : L ≠ 0 +r : ℝ +ω : SmoothSample 2 L +⊢ (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionY = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) (rawDerivativeDirections 5) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/04-raw-direction-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/04-raw-direction-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/04-raw-direction-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/04-raw-direction-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/04-raw-direction-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/04-raw-direction-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/04-raw-direction-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/04-raw-direction-compile.log new file mode 100644 index 000000000..be99f27bd --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/04-raw-direction-compile.log @@ -0,0 +1,80 @@ +SourceConfluentLimits.lean:110:6: error: Tactic `apply` failed: could not unify the conclusion of `congrArg + ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 +with the goal + (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionX = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) (rawDerivativeDirections 2) + +Note: The full type of `congrArg ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` is + ?m.1388 = ?m.1389 → + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 + +case «2» +L : ℝ +hL : L ≠ 0 +r : ℝ +ω : SmoothSample 2 L +⊢ (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionX = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) (rawDerivativeDirections 2) +SourceConfluentLimits.lean:110:6: error: Tactic `apply` failed: could not unify the conclusion of `congrArg + ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 +with the goal + (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionX = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) (rawDerivativeDirections 3) + +Note: The full type of `congrArg ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` is + ?m.1388 = ?m.1389 → + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 + +case «3» +L : ℝ +hL : L ≠ 0 +r : ℝ +ω : SmoothSample 2 L +⊢ (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionX = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) (rawDerivativeDirections 3) +SourceConfluentLimits.lean:110:6: error: Tactic `apply` failed: could not unify the conclusion of `congrArg + ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 +with the goal + (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionY = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) (rawDerivativeDirections 4) + +Note: The full type of `congrArg ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` is + ?m.1388 = ?m.1389 → + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 + +case «4» +L : ℝ +hL : L ≠ 0 +r : ℝ +ω : SmoothSample 2 L +⊢ (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionY = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) (rawDerivativeDirections 4) +SourceConfluentLimits.lean:110:6: error: Tactic `apply` failed: could not unify the conclusion of `congrArg + ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 +with the goal + (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionY = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) (rawDerivativeDirections 5) + +Note: The full type of `congrArg ⇑(iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r))` is + ?m.1388 = ?m.1389 → + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1388 = + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) (endpointS r)) ?m.1389 + +case «5» +L : ℝ +hL : L ≠ 0 +r : ℝ +ω : SmoothSample 2 L +⊢ (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionY = + (iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) (rawDerivativeDirections 5) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/05-raw-one-derivative-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/05-raw-one-derivative-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/05-raw-one-derivative-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/05-raw-one-derivative-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/05-raw-one-derivative-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/05-raw-one-derivative-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/05-raw-one-derivative-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/05-raw-one-derivative-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/06-mode-qualification-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/06-mode-qualification-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/06-mode-qualification-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/06-mode-qualification-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/06-mode-qualification-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/06-mode-qualification-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/06-mode-qualification-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/06-mode-qualification-compile.log new file mode 100644 index 000000000..5591829f3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/06-mode-qualification-compile.log @@ -0,0 +1,79 @@ +SourceConfluentLimits.lean:321:12: error: `simp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 0 + has type + Fin (rawDerivativeOrder 0) → Fin 2 → ℝ + but is expected to have type + Fin 0 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 0 (endpointM r) (rawDerivativeDirections 0) +SourceConfluentLimits.lean:321:12: error: `simp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 1 + has type + Fin (rawDerivativeOrder 1) → Fin 2 → ℝ + but is expected to have type + Fin 0 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 0 (endpointS r) (rawDerivativeDirections 1) +SourceConfluentLimits.lean:321:12: error: `simp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 4 + has type + Fin (rawDerivativeOrder 4) → Fin 2 → ℝ + but is expected to have type + Fin 1 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 1 (endpointM r) (rawDerivativeDirections 4) +SourceConfluentLimits.lean:321:12: error: `simp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 5 + has type + Fin (rawDerivativeOrder 5) → Fin 2 → ℝ + but is expected to have type + Fin 1 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 1 (endpointS r) (rawDerivativeDirections 5) +SourceConfluentLimits.lean:321:12: error: `simp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 8 + has type + Fin (rawDerivativeOrder 8) → Fin 2 → ℝ + but is expected to have type + Fin 2 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 2 (endpointM r) (rawDerivativeDirections 8) +SourceConfluentLimits.lean:321:12: error: `simp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 11 + has type + Fin (rawDerivativeOrder 11) → Fin 2 → ℝ + but is expected to have type + Fin 2 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 2 (endpointS r) (rawDerivativeDirections 11) +SourceConfluentLimits.lean:316:8: error: (kernel) declaration has metavariables 'I4SourceConfluentLimits.rawDerivativeCoefficient_explicit' +SourceConfluentLimits.lean:334:63: error: unsolved goals +case «5» +L r : ℝ +m : SourceMode 2 +⊢ deriv (modeF L m) (-r / 2) + deriv (modeF L m) (r / 2) = deriv (modeF L m) (r / 2) + deriv (modeF L m) (-r / 2) ∨ + r = 0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/07-explicit-mode-values-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/07-explicit-mode-values-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/07-explicit-mode-values-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/07-explicit-mode-values-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/07-explicit-mode-values-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/07-explicit-mode-values-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/07-explicit-mode-values-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/07-explicit-mode-values-compile.log new file mode 100644 index 000000000..a700b286d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/07-explicit-mode-values-compile.log @@ -0,0 +1,103 @@ +SourceConfluentLimits.lean:323:12: error: `dsimp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 0 + has type + Fin (rawDerivativeOrder 0) → Fin 2 → ℝ + but is expected to have type + Fin 0 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 0 (endpointM r) (rawDerivativeDirections 0) +SourceConfluentLimits.lean:323:12: error: `dsimp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 1 + has type + Fin (rawDerivativeOrder 1) → Fin 2 → ℝ + but is expected to have type + Fin 0 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 0 (endpointS r) (rawDerivativeDirections 1) +SourceConfluentLimits.lean:323:12: error: `dsimp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 2 + has type + Fin (rawDerivativeOrder 2) → Fin 2 → ℝ + but is expected to have type + Fin 1 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 1 (endpointM r) (rawDerivativeDirections 2) +SourceConfluentLimits.lean:323:12: error: `dsimp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 3 + has type + Fin (rawDerivativeOrder 3) → Fin 2 → ℝ + but is expected to have type + Fin 1 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 1 (endpointS r) (rawDerivativeDirections 3) +SourceConfluentLimits.lean:323:12: error: `dsimp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 6 + has type + Fin (rawDerivativeOrder 6) → Fin 2 → ℝ + but is expected to have type + Fin 2 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 2 (endpointM r) (rawDerivativeDirections 6) +SourceConfluentLimits.lean:323:12: error: `dsimp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 7 + has type + Fin (rawDerivativeOrder 7) → Fin 2 → ℝ + but is expected to have type + Fin 2 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 2 (endpointM r) (rawDerivativeDirections 7) +SourceConfluentLimits.lean:323:12: error: `dsimp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 9 + has type + Fin (rawDerivativeOrder 9) → Fin 2 → ℝ + but is expected to have type + Fin 2 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 2 (endpointS r) (rawDerivativeDirections 9) +SourceConfluentLimits.lean:323:12: error: `dsimp` made no progress + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + Application type mismatch: The argument + rawDerivativeDirections 10 + has type + Fin (rawDerivativeOrder 10) → Fin 2 → ℝ + but is expected to have type + Fin 2 → Fin 2 → ℝ + in the application + derivativeCoefficient 2 L 2 (endpointS r) (rawDerivativeDirections 10) +SourceConfluentLimits.lean:317:73: error: unsolved goals +case «11» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L 2 (endpointS r) (rawDerivativeDirections 11) m = + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionY, directionY] m +SourceConfluentLimits.lean:316:8: error: (kernel) declaration has metavariables 'I4SourceConfluentLimits.rawDerivativeCoefficient_explicit' diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/08-canonical-direction-data-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/08-canonical-direction-data-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/08-canonical-direction-data-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/08-canonical-direction-data-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/08-canonical-direction-data-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/08-canonical-direction-data-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/08-canonical-direction-data-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/08-canonical-direction-data-compile.log new file mode 100644 index 000000000..c863ef745 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/08-canonical-direction-data-compile.log @@ -0,0 +1,392 @@ +SourceConfluentLimits.lean:318:14: error: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` +SourceConfluentLimits.lean:322:14: error: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` +SourceConfluentLimits.lean:326:14: error: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` +SourceConfluentLimits.lean:330:14: error: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨2, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨2, ⋯⟩) + (rawDerivativeDirections ⟨2, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨2, ⋯⟩ + +case «2» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨2, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨2, ⋯⟩) + (rawDerivativeDirections ⟨2, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨2, ⋯⟩ +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨3, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨3, ⋯⟩) + (rawDerivativeDirections ⟨3, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨3, ⋯⟩ + +case «3» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨3, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨3, ⋯⟩) + (rawDerivativeDirections ⟨3, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨3, ⋯⟩ +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨4, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨4, ⋯⟩) + (rawDerivativeDirections ⟨4, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨4, ⋯⟩ + +case «4» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨4, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨4, ⋯⟩) + (rawDerivativeDirections ⟨4, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨4, ⋯⟩ +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨5, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨5, ⋯⟩) + (rawDerivativeDirections ⟨5, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨5, ⋯⟩ + +case «5» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨5, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨5, ⋯⟩) + (rawDerivativeDirections ⟨5, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨5, ⋯⟩ +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨6, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨6, ⋯⟩) + (rawDerivativeDirections ⟨6, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨6, ⋯⟩ + +case «6» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨6, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨6, ⋯⟩) + (rawDerivativeDirections ⟨6, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨6, ⋯⟩ +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨7, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨7, ⋯⟩) + (rawDerivativeDirections ⟨7, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨7, ⋯⟩ + +case «7» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨7, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨7, ⋯⟩) + (rawDerivativeDirections ⟨7, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨7, ⋯⟩ +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨8, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨8, ⋯⟩) + (rawDerivativeDirections ⟨8, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨8, ⋯⟩ + +case «8» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨8, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨8, ⋯⟩) + (rawDerivativeDirections ⟨8, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨8, ⋯⟩ +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨9, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨9, ⋯⟩) + (rawDerivativeDirections ⟨9, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨9, ⋯⟩ + +case «9» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨9, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨9, ⋯⟩) + (rawDerivativeDirections ⟨9, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨9, ⋯⟩ +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨10, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨10, ⋯⟩) + (rawDerivativeDirections ⟨10, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨10, ⋯⟩ + +case «10» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨10, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨10, ⋯⟩) + (rawDerivativeDirections ⟨10, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨10, ⋯⟩ +SourceConfluentLimits.lean:365:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨11, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨11, ⋯⟩) + (rawDerivativeDirections ⟨11, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨11, ⋯⟩ + +case «11» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨11, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨11, ⋯⟩) + (rawDerivativeDirections ⟨11, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨11, ⋯⟩ +SourceConfluentLimits.lean:362:23: error: This simp argument is unused: + rawDirections2 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:362:39: error: This simp argument is unused: + rawDirections3 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections4, rawDirections5, rawDirections6, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:362:55: error: This simp argument is unused: + rawDirections4 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections5, rawDirections6, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:362:71: error: This simp argument is unused: + rawDirections5 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections6, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:362:87: error: This simp argument is unused: + rawDirections6 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:362:103: error: This simp argument is unused: + rawDirections7 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:362:119: error: This simp argument is unused: + rawDirections8 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:362:135: error: This simp argument is unused: + rawDirections9 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, + rawDirections8, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:362:151: error: This simp argument is unused: + rawDirections10 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, + rawDirections8, rawDirections9, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:362:168: error: This simp argument is unused: + rawDirections11 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, + rawDirections8, rawDirections9, rawDirections10] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/09-reducible-order-table-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/09-reducible-order-table-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/09-reducible-order-table-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/09-reducible-order-table-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/09-reducible-order-table-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/09-reducible-order-table-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/09-reducible-order-table-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/09-reducible-order-table-compile.log new file mode 100644 index 000000000..c39f362b2 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/09-reducible-order-table-compile.log @@ -0,0 +1,380 @@ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨2, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨2, ⋯⟩) + (rawDerivativeDirections ⟨2, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨2, ⋯⟩ + +case «2» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨2, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨2, ⋯⟩) + (rawDerivativeDirections ⟨2, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨2, ⋯⟩ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨3, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨3, ⋯⟩) + (rawDerivativeDirections ⟨3, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨3, ⋯⟩ + +case «3» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨3, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨3, ⋯⟩) + (rawDerivativeDirections ⟨3, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨3, ⋯⟩ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨4, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨4, ⋯⟩) + (rawDerivativeDirections ⟨4, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨4, ⋯⟩ + +case «4» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨4, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨4, ⋯⟩) + (rawDerivativeDirections ⟨4, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨4, ⋯⟩ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨5, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨5, ⋯⟩) + (rawDerivativeDirections ⟨5, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨5, ⋯⟩ + +case «5» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨5, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨5, ⋯⟩) + (rawDerivativeDirections ⟨5, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨5, ⋯⟩ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨6, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨6, ⋯⟩) + (rawDerivativeDirections ⟨6, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨6, ⋯⟩ + +case «6» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨6, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨6, ⋯⟩) + (rawDerivativeDirections ⟨6, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨6, ⋯⟩ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨7, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨7, ⋯⟩) + (rawDerivativeDirections ⟨7, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨7, ⋯⟩ + +case «7» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨7, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨7, ⋯⟩) + (rawDerivativeDirections ⟨7, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨7, ⋯⟩ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨8, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨8, ⋯⟩) + (rawDerivativeDirections ⟨8, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨8, ⋯⟩ + +case «8» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨8, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨8, ⋯⟩) + (rawDerivativeDirections ⟨8, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨8, ⋯⟩ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨9, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨9, ⋯⟩) + (rawDerivativeDirections ⟨9, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨9, ⋯⟩ + +case «9» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨9, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨9, ⋯⟩) + (rawDerivativeDirections ⟨9, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨9, ⋯⟩ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨10, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨10, ⋯⟩) + (rawDerivativeDirections ⟨10, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨10, ⋯⟩ + +case «10» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨10, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨10, ⋯⟩) + (rawDerivativeDirections ⟨10, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨10, ⋯⟩ +SourceConfluentLimits.lean:369:12: error: Tactic `rfl` failed: The left-hand side + derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨11, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨11, ⋯⟩) + (rawDerivativeDirections ⟨11, ⋯⟩) m +is not definitionally equal to the right-hand side + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨11, ⋯⟩ + +case «11» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (![0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ⟨11, ⋯⟩) + (![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointM r, + endpointM r, endpointS r, endpointS r, endpointS r] + ⟨11, ⋯⟩) + (rawDerivativeDirections ⟨11, ⋯⟩) m = + ![modeF L m (-r / 2), modeF L m (r / 2), derivativeCoefficient 2 L 1 ![-r / 2, 0] ![directionX] m, + derivativeCoefficient 2 L 1 ![r / 2, 0] ![directionX] m, modeH L m (-r / 2), modeH L m (r / 2), + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![-r / 2, 0] ![directionX, directionY] m, modeYY L m (-r / 2), + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionX] m, + derivativeCoefficient 2 L 2 ![r / 2, 0] ![directionX, directionY] m, modeYY L m (r / 2)] + ⟨11, ⋯⟩ +SourceConfluentLimits.lean:366:23: error: This simp argument is unused: + rawDirections2 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:366:39: error: This simp argument is unused: + rawDirections3 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections4, rawDirections5, rawDirections6, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:366:55: error: This simp argument is unused: + rawDirections4 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections5, rawDirections6, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:366:71: error: This simp argument is unused: + rawDirections5 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections6, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:366:87: error: This simp argument is unused: + rawDirections6 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections7, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:366:103: error: This simp argument is unused: + rawDirections7 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections8, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:366:119: error: This simp argument is unused: + rawDirections8 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, + rawDirections9, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:366:135: error: This simp argument is unused: + rawDirections9 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, + rawDirections8, rawDirections10, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:366:151: error: This simp argument is unused: + rawDirections10 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, + rawDirections8, rawDirections9, rawDirections11] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentLimits.lean:366:168: error: This simp argument is unused: + rawDirections11 + +Hint: Omit it from the simp argument list. + [apply] simp only [rawDirections2, rawDirections3, rawDirections4, rawDirections5, rawDirections6, rawDirections7, + rawDirections8, rawDirections9, rawDirections10] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/10-direction-rewrite-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/10-direction-rewrite-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/10-direction-rewrite-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/10-direction-rewrite-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/10-direction-rewrite-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/10-direction-rewrite-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/10-direction-rewrite-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/10-direction-rewrite-compile.log new file mode 100644 index 000000000..c910ee583 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/10-direction-rewrite-compile.log @@ -0,0 +1,130 @@ +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨2, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨2, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨2, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨2, ⋯⟩) + +case «2» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨2, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨2, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨2, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨2, ⋯⟩) +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨3, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨3, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨3, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨3, ⋯⟩) + +case «3» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨3, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨3, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨3, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨3, ⋯⟩) +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨4, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨4, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨4, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨4, ⋯⟩) + +case «4» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨4, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨4, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨4, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨4, ⋯⟩) +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨5, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨5, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨5, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨5, ⋯⟩) + +case «5» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨5, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨5, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨5, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨5, ⋯⟩) +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨6, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨6, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨6, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨6, ⋯⟩) + +case «6» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨6, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨6, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨6, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨6, ⋯⟩) +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨7, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨7, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨7, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨7, ⋯⟩) + +case «7» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨7, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨7, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨7, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨7, ⋯⟩) +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨8, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨8, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨8, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨8, ⋯⟩) + +case «8» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨8, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨8, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨8, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨8, ⋯⟩) +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨9, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨9, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨9, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨9, ⋯⟩) + +case «9» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨9, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨9, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨9, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨9, ⋯⟩) +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨10, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨10, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨10, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨10, ⋯⟩) + +case «10» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨10, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨10, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨10, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨10, ⋯⟩) +SourceConfluentLimits.lean:376:10: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + rawDerivativeDirections 11 +in the target expression + derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨11, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨11, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨11, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨11, ⋯⟩) + +case «11» +L r : ℝ +m : SourceMode 2 +⊢ derivativeCoefficient 2 L (rawDerivativeOrder ((fun i => i) ⟨11, ⋯⟩)) (rawDerivativePoint r ((fun i => i) ⟨11, ⋯⟩)) + (rawDerivativeDirections ((fun i => i) ⟨11, ⋯⟩)) m = + modeRawValues L r m ((fun i => i) ⟨11, ⋯⟩) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/11-explicit-row-indices-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/11-explicit-row-indices-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/11-explicit-row-indices-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/11-explicit-row-indices-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/11-explicit-row-indices-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/11-explicit-row-indices-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/11-explicit-row-indices-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/11-explicit-row-indices-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/12-peano-extension-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/12-peano-extension-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/12-peano-extension-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/12-peano-extension-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/12-peano-extension-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/12-peano-extension-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/12-peano-extension-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/12-peano-extension-compile.log new file mode 100644 index 000000000..38a4ddb67 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/12-peano-extension-compile.log @@ -0,0 +1,26 @@ +SourceConfluentLimits.lean:529:60: error: unsolved goals +hi : IntervalIntegrable (fun t => t) ℙ (-1 / 2) (1 / 2) +hp : IntervalIntegrable (fun t => t ^ 2) ℙ (-1 / 2) (1 / 2) +⊢ ∫ (t : ℝ) in -(1 / 2)..1 / 2, 1 / 2 - t = 1 / 2 +SourceConfluentLimits.lean:568:4: error: Tactic `rfl` failed: The left-hand side + ∫ (u : ℝ) in -1 / 2..1 / 2, derivativeCoefficient 2 L 2 ![r * u, 0] ![directionX, directionX] m +is not definitionally equal to the right-hand side + regularizedPinCoefficient L r ⟨3, ⋯⟩ m + +case «3» +L r : ℝ +hr : r ≠ 0 +m : SourceMode 2 +⊢ ∫ (u : ℝ) in -1 / 2..1 / 2, derivativeCoefficient 2 L 2 ![r * u, 0] ![directionX, directionX] m = + regularizedPinCoefficient L r ⟨3, ⋯⟩ m +SourceConfluentLimits.lean:572:4: error: Tactic `rfl` failed: The left-hand side + ∫ (u : ℝ) in -1 / 2..1 / 2, derivativeCoefficient 2 L 2 ![r * u, 0] ![directionX, directionY] m +is not definitionally equal to the right-hand side + regularizedPinCoefficient L r ⟨4, ⋯⟩ m + +case «4» +L r : ℝ +hr : r ≠ 0 +m : SourceMode 2 +⊢ ∫ (u : ℝ) in -1 / 2..1 / 2, derivativeCoefficient 2 L 2 ![r * u, 0] ![directionX, directionY] m = + regularizedPinCoefficient L r ⟨4, ⋯⟩ m diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/13-peano-masses-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/13-peano-masses-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/13-peano-masses-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/13-peano-masses-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/13-peano-masses-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/13-peano-masses-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/13-peano-masses-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/13-peano-masses-compile.log new file mode 100644 index 000000000..a92fd38c5 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/13-peano-masses-compile.log @@ -0,0 +1,6 @@ +SourceConfluentLimits.lean:526:7: error: Variable name `t` is not explicitly referenced. + +Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning: + [apply] _t + +Note: This linter can be disabled with `set_option linter.unusedVariables false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/14-peano-extension-clean-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/14-peano-extension-clean-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/14-peano-extension-clean-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/14-peano-extension-clean-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/14-peano-extension-clean-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/14-peano-extension-clean-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/14-peano-extension-clean-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/14-peano-extension-clean-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/15-uniform-majorant-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/15-uniform-majorant-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/15-uniform-majorant-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/15-uniform-majorant-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/15-uniform-majorant-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/15-uniform-majorant-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/15-uniform-majorant-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/15-uniform-majorant-compile.log new file mode 100644 index 000000000..cc2f19461 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/15-uniform-majorant-compile.log @@ -0,0 +1,78 @@ +SourceConfluentLimits.lean:782:17: error(lean.unknownIdentifier): Unknown identifier `norm_ofNat` +SourceConfluentLimits.lean:784:2: error: linarith failed to find a contradiction +a b M : ℝ +ha : ‖a‖ ≤ M +hb : ‖b‖ ≤ M +h : ‖a + b‖ ≤ M + M +a✝ : M < ‖a + b‖ / ‖2‖ +⊢ False +failed +SourceConfluentLimits.lean:803:80: error: unexpected token '!['; expected ')', ',' or ':' +SourceConfluentLimits.lean:803:14: error: Type mismatch: After simplification, term + weightedDerivativeIntegral_norm_le hL 2 + has type + ∀ (u : Fin 2 → Fin 2 → ℝ) (m : SourceMode 2) (w : ℝ → ℝ) (K : ℝ), + 0 ≤ K → + (∀ t ∈ Ι (-1 / 2) (1 / 2), |w t| ≤ K) → + ∀ (r : ℝ), ‖weightedDerivativeIntegral L 2 u m w r‖ ≤ K * derivativeMajorant 2 L 2 u m +but is expected to have type + ‖regularizedPinCoefficient L r ⟨3, ⋯⟩ m‖ ≤ confluentModeMajorant L m +SourceConfluentLimits.lean:793:71: error: unsolved goals +case «4» +L : ℝ +hL : L ≠ 0 +r : ℝ +m : SourceMode 2 +⊢ ‖regularizedPinCoefficient L r ((fun i => i) ⟨4, ⋯⟩) m‖ ≤ confluentModeMajorant L m + +case «5» +L : ℝ +hL : L ≠ 0 +r : ℝ +m : SourceMode 2 +⊢ ‖regularizedPinCoefficient L r ((fun i => i) ⟨5, ⋯⟩) m‖ ≤ confluentModeMajorant L m +SourceConfluentLimits.lean:823:80: error: unexpected token '!['; expected ')', ',' or ':' +SourceConfluentLimits.lean:823:14: error: Type mismatch: After simplification, term + weightedDerivativeIntegral_norm_le hL 3 + has type + ∀ (u : Fin 3 → Fin 2 → ℝ) (m : SourceMode 2) (w : ℝ → ℝ) (K : ℝ), + 0 ≤ K → + (∀ t ∈ Ι (-1 / 2) (1 / 2), |w t| ≤ K) → + ∀ (r : ℝ), ‖weightedDerivativeIntegral L 3 u m w r‖ ≤ K * derivativeMajorant 2 L 3 u m +but is expected to have type + ‖weightedDerivativeIntegral L 3 ![directionX, directionX, directionX] m peanoLeft r‖ ≤ confluentModeMajorant L m +SourceConfluentLimits.lean:819:76: error: unsolved goals +case «1» +L : ℝ +hL : L ≠ 0 +r : ℝ +m : SourceMode 2 +⊢ ‖regularizedAdjustedCoefficient L r ((fun i => i) ⟨1, ⋯⟩) m‖ ≤ confluentModeMajorant L m + +case «2» +L : ℝ +hL : L ≠ 0 +r : ℝ +m : SourceMode 2 +⊢ ‖regularizedAdjustedCoefficient L r ((fun i => i) ⟨2, ⋯⟩) m‖ ≤ confluentModeMajorant L m + +case «3» +L : ℝ +hL : L ≠ 0 +r : ℝ +m : SourceMode 2 +⊢ ‖regularizedAdjustedCoefficient L r ((fun i => i) ⟨3, ⋯⟩) m‖ ≤ confluentModeMajorant L m + +case «4» +L : ℝ +hL : L ≠ 0 +r : ℝ +m : SourceMode 2 +⊢ ‖regularizedAdjustedCoefficient L r ((fun i => i) ⟨4, ⋯⟩) m‖ ≤ confluentModeMajorant L m + +case «5» +L : ℝ +hL : L ≠ 0 +r : ℝ +m : SourceMode 2 +⊢ ‖regularizedAdjustedCoefficient L r ((fun i => i) ⟨5, ⋯⟩) m‖ ≤ confluentModeMajorant L m diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/16-majorant-bound-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/16-majorant-bound-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/16-majorant-bound-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/16-majorant-bound-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/16-majorant-bound-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/16-majorant-bound-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/16-majorant-bound-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/16-majorant-bound-compile.log new file mode 100644 index 000000000..19a982902 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/16-majorant-bound-compile.log @@ -0,0 +1,8 @@ +SourceConfluentLimits.lean:821:41: error: Type mismatch: After simplification, term + h + has type + @LE.le ℝ Real.instLE ‖weightedDerivativeIntegral L 3 ![directionX, directionX, directionX] m peanoCubic r‖ + (derivativeMajorant 2 L 3 ![directionX, directionX, directionX] m) +but is expected to have type + @LE.le ℝ Real.instPreorder.toLE ‖weightedDerivativeIntegral L 3 ![directionX, directionX, directionX] m peanoCubic r‖ + (modeComponentMajorant L m 5) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/17-global-peano-bounds-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/17-global-peano-bounds-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/17-global-peano-bounds-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/17-global-peano-bounds-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/17-global-peano-bounds-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/17-global-peano-bounds-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/17-global-peano-bounds-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/17-global-peano-bounds-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/18-actual-block-limits-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/18-actual-block-limits-compile.command new file mode 100644 index 000000000..545d2e272 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/18-actual-block-limits-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceConfluentLimits.olean SourceConfluentLimits.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/18-actual-block-limits-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/18-actual-block-limits-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/18-actual-block-limits-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/18-actual-block-limits-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/18-actual-block-limits-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.command new file mode 100644 index 000000000..b7211c139 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/source-confluent-limits/controls.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.json new file mode 100644 index 000000000..b1a402882 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.json @@ -0,0 +1,14 @@ +{ + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/source-confluent-limits/controls.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exitcode": 0, + "source_sha256_before": "1635a05bbd5b56608bc0a1a04eb4404c2df4601f1255f5846c1342a597bf5b0b", + "source_sha256_after": "1635a05bbd5b56608bc0a1a04eb4404c2df4601f1255f5846c1342a597bf5b0b", + "log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/19-controls.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.command new file mode 100644 index 000000000..79b356a4b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/source-confluent-limits/axiom_audit.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.json new file mode 100644 index 000000000..16a2b41eb --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.json @@ -0,0 +1,14 @@ +{ + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "lean", + "-DwarningAsError=true", + "/workspace/scratch/source-confluent-limits/axiom_audit.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exitcode": 0, + "source_sha256_before": "1635a05bbd5b56608bc0a1a04eb4404c2df4601f1255f5846c1342a597bf5b0b", + "source_sha256_after": "1635a05bbd5b56608bc0a1a04eb4404c2df4601f1255f5846c1342a597bf5b0b", + "log_sha256": "4063c4cd1d47da1f737e29bfc71f1a54764255b658a6cfa14e9b906e72a41ebe" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.log new file mode 100644 index 000000000..14c86168b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/20-axiom-audit.log @@ -0,0 +1,87 @@ +'I4SourceConfluentLimits.rawPinsWithHessians_series' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.pinCoefficient_summable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.adjustedCoefficient_summable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.sourceConfluentPinsAt_series' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.sourceAdjustedEndpoints_series' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.sourcePinCovarianceAt_coefficient' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.sourceCrossCovarianceAt_coefficient' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.sourceEndpointCovarianceAt_coefficient' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4ModeCurveChains.mode_smooth' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.axis_iteratedDeriv' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.axis_transverse_smooth' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.axis_transverse_iteratedDeriv' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.modeH_eq' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.modeF_C3' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.modeH_C2' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.modeF_deriv' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.modeF_second' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.modeF_third' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.modeH_deriv' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ModeCurveChains.modeH_second' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.rawDerivativeCoefficient_explicit' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.pinCoefficient_explicit' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.adjustedCoefficient_explicit' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4WeightedModeIntegrals.weighted_integrand_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4WeightedModeIntegrals.weightedDerivativeIntegral_continuous' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4WeightedModeIntegrals.weightedDerivativeIntegral_norm_le' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4WeightedModeIntegrals.weightedDerivativeIntegral_zero' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.peano_kernel_masses' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.pinCoefficient_eq_regularized' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.adjustedCoefficient_eq_regularized' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactCoefficientLimits.contactPinCoefficient_summable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactCoefficientLimits.contactAdjustedCoefficient_summable' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4ContactCoefficientLimits.sourcePinVector_eq_contactSeries' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4ContactCoefficientLimits.sourceContactEndpointVector_eq_contactSeries' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4ContactCoefficientLimits.sourcePinCovariance_eq_contactCoefficientGram' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4ContactCoefficientLimits.sourceContactEndpointCrossCovariance_eq_contactCoefficientGram' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4ContactCoefficientLimits.sourceContactEndpointCovariance_eq_contactCoefficientGram' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceConfluentLimits.confluentModeMajorant_nonneg' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.confluentModeMajorant_summable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.regularizedPinCoefficient_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.regularizedAdjustedCoefficient_bound' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceConfluentLimits.regularizedPinCoefficient_continuous' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceConfluentLimits.regularizedAdjustedCoefficient_continuous' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceConfluentLimits.regularizedPinCoefficient_zero' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.regularizedAdjustedCoefficient_zero' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.regularizedPinCovariance_continuous' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.regularizedCrossCovariance_continuous' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceConfluentLimits.regularizedEndpointCovariance_continuous' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceConfluentLimits.sourcePinCovarianceAt_eq_regularized' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceConfluentLimits.sourceCrossCovarianceAt_eq_regularized' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceConfluentLimits.sourceEndpointCovarianceAt_eq_regularized' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceConfluentLimits.regularizedPinCovariance_zero' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.regularizedCrossCovariance_zero' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.regularizedEndpointCovariance_zero' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.sourcePinCovarianceAt_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.sourceCrossCovarianceAt_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentLimits.sourceEndpointCovarianceAt_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.command new file mode 100644 index 000000000..1226874f0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env leanchecker SourceConfluentLimits diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.json new file mode 100644 index 000000000..b854fc889 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.json @@ -0,0 +1,15 @@ +{ + "command": [ + "/home/agent/.elan/bin/lake", + "env", + "leanchecker", + "SourceConfluentLimits" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exitcode": 0, + "elapsed_seconds": 18.978482509999594, + "source_sha256_before": "1635a05bbd5b56608bc0a1a04eb4404c2df4601f1255f5846c1342a597bf5b0b", + "source_sha256_after": "1635a05bbd5b56608bc0a1a04eb4404c2df4601f1255f5846c1342a597bf5b0b", + "toolchain": "leanprover/lean4:v4.34.1", + "log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/21-kernel-replay.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/ContactCoefficientLimits.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/ContactCoefficientLimits.lean.txt new file mode 100644 index 000000000..1f9c259a6 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/ContactCoefficientLimits.lean.txt @@ -0,0 +1,124 @@ +import SourceContactEndpointParameters +import SourceConfluentPins + +noncomputable section +open I4Foundation I4Source I4SourceGaussian I4SmoothCarrier I4SourceDerivativeGaussian +open I4SourceContact I4SourceConfluentPins MeasureTheory ProbabilityTheory +open scoped BigOperators +set_option autoImplicit false + +namespace I4ContactCoefficientLimits + +def contactPinCoefficient (L : ℝ) (i : Fin 6) (m : SourceMode 2) : ℝ := + sourceContactCoefficient L i.castSucc m + +def contactAdjustedCoefficient (L : ℝ) (i : Fin 6) (m : SourceMode 2) : ℝ := + ![-derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m / 2, + derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m / 2, + -derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionY] m / 2, + derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionY] m / 2, + derivativeCoefficient 2 L 2 0 ![directionY,directionY] m, + derivativeCoefficient 2 L 2 0 ![directionY,directionY] m] i + +theorem contactPinCoefficient_summable {L : ℝ} (hL : L ≠ 0) (i : Fin 6) : + Summable (contactPinCoefficient L i) := + sourceContactCoefficient_summable hL i.castSucc + +theorem contactAdjustedCoefficient_summable {L : ℝ} (hL : L ≠ 0) (i : Fin 6) : + Summable (contactAdjustedCoefficient L i) := by + change Summable (fun m => contactAdjustedCoefficient L i m) + have hx := derivativeCoefficient_summable 2 hL 3 0 ![directionX,directionX,directionX] + have ha := derivativeCoefficient_summable 2 hL 3 0 ![directionX,directionX,directionY] + have hq := derivativeCoefficient_summable 2 hL 2 0 ![directionY,directionY] + fin_cases i + · simpa [contactAdjustedCoefficient, div_eq_mul_inv, mul_comm] using + hx.neg.mul_left ((2 : ℝ)⁻¹) + · simpa [contactAdjustedCoefficient, div_eq_mul_inv, mul_comm] using + hx.mul_left ((2 : ℝ)⁻¹) + · simpa [contactAdjustedCoefficient, div_eq_mul_inv, mul_comm] using + ha.neg.mul_left ((2 : ℝ)⁻¹) + · simpa [contactAdjustedCoefficient, div_eq_mul_inv, mul_comm] using + ha.mul_left ((2 : ℝ)⁻¹) + · exact hq + · exact hq + +theorem sourcePinVector_eq_contactSeries {L : ℝ} (hL : L ≠ 0) + (ω : SmoothSample 2 L) (i : Fin 6) : + sourcePinVector L ω i = + scalarSeries (contactPinCoefficient L i) (sampleInclusion 2 L ω) := + smoothSourceField_derivative_series 2 hL ω (contactOrder i.castSucc) 0 + (contactDirections i.castSucc) + +private theorem scalarSeries_div {ι : Type*} (a : ι → ℝ) (ω : ι → ℝ) (c : ℝ) : + scalarSeries (fun m => a m / c) ω = scalarSeries a ω / c := by + simp [scalarSeries, div_mul_eq_mul_div, tsum_div_const] + +private theorem scalarSeries_neg {ι : Type*} (a : ι → ℝ) (ω : ι → ℝ) : + scalarSeries (fun m => -a m) ω = -scalarSeries a ω := by + simp [scalarSeries, neg_mul, tsum_neg] + +theorem sourceContactEndpointVector_eq_contactSeries {L : ℝ} (hL : L ≠ 0) + (ω : SmoothSample 2 L) (i : Fin 6) : + sourceContactEndpointVector L ω i = + scalarSeries (contactAdjustedCoefficient L i) (sampleInclusion 2 L ω) := by + have hx := smoothSourceField_derivative_series 2 hL ω 3 0 + ![directionX,directionX,directionX] + have ha := smoothSourceField_derivative_series 2 hL ω 3 0 + ![directionX,directionX,directionY] + have hq := smoothSourceField_derivative_series 2 hL ω 2 0 ![directionY,directionY] + fin_cases i + · change -iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0 + ![directionX,directionX,directionX] / 2 = + scalarSeries (fun m => -derivativeCoefficient 2 L 3 0 + ![directionX,directionX,directionX] m / 2) (sampleInclusion 2 L ω) + rw [scalarSeries_div, scalarSeries_neg, hx] + · change iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0 + ![directionX,directionX,directionX] / 2 = + scalarSeries (fun m => derivativeCoefficient 2 L 3 0 + ![directionX,directionX,directionX] m / 2) (sampleInclusion 2 L ω) + rw [scalarSeries_div, hx] + · change -iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0 + ![directionX,directionX,directionY] / 2 = + scalarSeries (fun m => -derivativeCoefficient 2 L 3 0 + ![directionX,directionX,directionY] m / 2) (sampleInclusion 2 L ω) + rw [scalarSeries_div, scalarSeries_neg, ha] + · change iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0 + ![directionX,directionX,directionY] / 2 = + scalarSeries (fun m => derivativeCoefficient 2 L 3 0 + ![directionX,directionX,directionY] m / 2) (sampleInclusion 2 L ω) + rw [scalarSeries_div, ha] + · exact hq + · exact hq + +theorem sourcePinCovariance_eq_contactCoefficientGram {L : ℝ} (hL : L ≠ 0) + (i j : Fin 6) : sourcePinCovariance L i j = + ∑' m, contactPinCoefficient L i m * contactPinCoefficient L j m := by + change covariance (fun ω => sourcePinVector L ω i) (fun ω => sourcePinVector L ω j) + (smoothLaw 2 L) = _ + simp_rw [sourcePinVector_eq_contactSeries hL] + rw [covariance_comp_inclusion 2 hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (contactPinCoefficient_summable hL i) (contactPinCoefficient_summable hL j) + +theorem sourceContactEndpointCrossCovariance_eq_contactCoefficientGram {L : ℝ} + (hL : L ≠ 0) (i j : Fin 6) : sourceContactEndpointCrossCovariance L i j = + ∑' m, contactAdjustedCoefficient L i m * contactPinCoefficient L j m := by + unfold sourceContactEndpointCrossCovariance + simp_rw [sourceContactEndpointVector_eq_contactSeries hL, sourcePinVector_eq_contactSeries hL] + rw [covariance_comp_inclusion 2 hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (contactAdjustedCoefficient_summable hL i) (contactPinCoefficient_summable hL j) + +theorem sourceContactEndpointCovariance_eq_contactCoefficientGram {L : ℝ} + (hL : L ≠ 0) (i j : Fin 6) : sourceContactEndpointCovariance L i j = + ∑' m, contactAdjustedCoefficient L i m * contactAdjustedCoefficient L j m := by + unfold sourceContactEndpointCovariance + simp_rw [sourceContactEndpointVector_eq_contactSeries hL] + rw [covariance_comp_inclusion 2 hL _ _ (scalarSeries_measurable _) + (scalarSeries_measurable _)] + exact I4SourceCovariance.scalarSeries_covariance _ _ + (contactAdjustedCoefficient_summable hL i) (contactAdjustedCoefficient_summable hL j) + +end I4ContactCoefficientLimits diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/RegularizedZero.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/RegularizedZero.lean.txt new file mode 100644 index 000000000..ed0e16af3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/RegularizedZero.lean.txt @@ -0,0 +1,64 @@ +import SourceConfluentLimits + +noncomputable section +open I4Source I4SourceDerivativeGaussian I4SourceConfluentPins I4SourceContact +open I4SourceConfluentLimits I4ModeCurveChains I4WeightedModeIntegrals I4ContactCoefficientLimits +open MeasureTheory +open scoped BigOperators Interval +set_option autoImplicit false + +namespace I4SourceConfluentLimits + +theorem regularizedPinCoefficient_zero (L : ℝ) (i : Fin 6) (m : SourceMode 2) : + regularizedPinCoefficient L 0 i m = contactPinCoefficient L i m := by + have hpoint : (![0,0] : Fin 2 → ℝ) = 0 := by ext j; fin_cases j <;> rfl + have hm := peano_kernel_masses + fin_cases i + · change (modeF L m (-0/2) + modeF L m (0/2))/2 = + derivativeCoefficient 2 L 0 0 ![] m + simp [modeF, derivativeCoefficient, iteratedFDeriv_zero_apply, hpoint] + · change (deriv (modeF L m) (-0/2) + deriv (modeF L m) (0/2))/2 = + derivativeCoefficient 2 L 1 0 ![directionX] m + simp only [neg_zero, zero_div, modeF_deriv, hpoint] + ring + · change (modeH L m (-0/2) + modeH L m (0/2))/2 = + derivativeCoefficient 2 L 1 0 ![directionY] m + simp only [neg_zero, zero_div, modeH, hpoint] + ring + · change weightedDerivativeIntegral L 2 ![directionX,directionX] m (fun _ => 1) 0 = + derivativeCoefficient 2 L 2 0 ![directionX,directionX] m + rw [weightedDerivativeIntegral_zero, hm.1, one_mul] + · change weightedDerivativeIntegral L 2 ![directionX,directionY] m (fun _ => 1) 0 = + derivativeCoefficient 2 L 2 0 ![directionX,directionY] m + rw [weightedDerivativeIntegral_zero, hm.1, one_mul] + · change 6 * weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] + m peanoCubic 0 = derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m + rw [weightedDerivativeIntegral_zero, ← mul_assoc, hm.2.2.2, one_mul] + +theorem regularizedAdjustedCoefficient_zero (L : ℝ) (i : Fin 6) (m : SourceMode 2) : + regularizedAdjustedCoefficient L 0 i m = contactAdjustedCoefficient L i m := by + have hpoint : (![0,0] : Fin 2 → ℝ) = 0 := by ext j; fin_cases j <;> rfl + have hm := peano_kernel_masses + fin_cases i + · change -weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] + m peanoLeft 0 = -derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m / 2 + rw [weightedDerivativeIntegral_zero, hm.2.1] + ring + · change weightedDerivativeIntegral L 3 ![directionX,directionX,directionX] + m peanoRight 0 = derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionX] m / 2 + rw [weightedDerivativeIntegral_zero, hm.2.2.1] + ring + · change -weightedDerivativeIntegral L 3 ![directionX,directionX,directionY] + m peanoLeft 0 = -derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionY] m / 2 + rw [weightedDerivativeIntegral_zero, hm.2.1] + ring + · change weightedDerivativeIntegral L 3 ![directionX,directionX,directionY] + m peanoRight 0 = derivativeCoefficient 2 L 3 0 ![directionX,directionX,directionY] m / 2 + rw [weightedDerivativeIntegral_zero, hm.2.2.1] + ring + · change modeYY L m (-0/2) = derivativeCoefficient 2 L 2 0 ![directionY,directionY] m + simp only [neg_zero, zero_div, modeYY, hpoint] + · change modeYY L m (0/2) = derivativeCoefficient 2 L 2 0 ![directionY,directionY] m + simp only [zero_div, modeYY, hpoint] + +end I4SourceConfluentLimits diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/contact-coefficients-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/contact-coefficients-01.log new file mode 100644 index 000000000..5e1b6f8ee --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/contact-coefficients-01.log @@ -0,0 +1,135 @@ +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:33:4: error: Type mismatch: After simplification, term + Summable.mul_left 2⁻¹ (Summable.neg hx) + has type + Summable fun i => -(2⁻¹ * derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionX] i) +but is expected to have type + Summable (contactAdjustedCoefficient L 0) +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:35:4: error: Type mismatch: After simplification, term + Summable.mul_left 2⁻¹ hx + has type + Summable fun i => 2⁻¹ * derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionX] i +but is expected to have type + Summable (contactAdjustedCoefficient L 1) +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:37:4: error: Type mismatch: After simplification, term + Summable.mul_left 2⁻¹ (Summable.neg ha) + has type + Summable fun i => -(2⁻¹ * derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionY] i) +but is expected to have type + Summable (contactAdjustedCoefficient L 2) +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:39:4: error: Type mismatch: After simplification, term + Summable.mul_left 2⁻¹ ha + has type + Summable fun i => 2⁻¹ * derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionY] i +but is expected to have type + Summable (contactAdjustedCoefficient L 3) +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:71:2: error: unsolved goals +case «0» +L : ℝ +hL : L ≠ 0 +ω : SmoothSample 2 L +hx : + (iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0) ![directionX, directionX, directionX] = + scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionX]) (sampleInclusion 2 L ω) +ha : + (iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0) ![directionX, directionX, directionY] = + scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionY]) (sampleInclusion 2 L ω) +hq : + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) 0) ![directionY, directionY] = + scalarSeries (derivativeCoefficient 2 L 2 0 ![directionY, directionY]) (sampleInclusion 2 L ω) +⊢ -scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionX]) (sampleInclusion 2 L ω) / 2 = + scalarSeries (contactAdjustedCoefficient L ⟨0, ⋯⟩) (sampleInclusion 2 L ω) +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:74:2: error: unsolved goals +case «1» +L : ℝ +hL : L ≠ 0 +ω : SmoothSample 2 L +hx : + (iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0) ![directionX, directionX, directionX] = + scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionX]) (sampleInclusion 2 L ω) +ha : + (iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0) ![directionX, directionX, directionY] = + scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionY]) (sampleInclusion 2 L ω) +hq : + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) 0) ![directionY, directionY] = + scalarSeries (derivativeCoefficient 2 L 2 0 ![directionY, directionY]) (sampleInclusion 2 L ω) +⊢ scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionX]) (sampleInclusion 2 L ω) / 2 = + scalarSeries (contactAdjustedCoefficient L ⟨1, ⋯⟩) (sampleInclusion 2 L ω) +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:77:2: error: unsolved goals +case «2» +L : ℝ +hL : L ≠ 0 +ω : SmoothSample 2 L +hx : + (iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0) ![directionX, directionX, directionX] = + scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionX]) (sampleInclusion 2 L ω) +ha : + (iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0) ![directionX, directionX, directionY] = + scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionY]) (sampleInclusion 2 L ω) +hq : + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) 0) ![directionY, directionY] = + scalarSeries (derivativeCoefficient 2 L 2 0 ![directionY, directionY]) (sampleInclusion 2 L ω) +⊢ -scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionY]) (sampleInclusion 2 L ω) / 2 = + scalarSeries (contactAdjustedCoefficient L ⟨2, ⋯⟩) (sampleInclusion 2 L ω) +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:80:2: error: unsolved goals +case «3» +L : ℝ +hL : L ≠ 0 +ω : SmoothSample 2 L +hx : + (iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0) ![directionX, directionX, directionX] = + scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionX]) (sampleInclusion 2 L ω) +ha : + (iteratedFDeriv ℝ 3 (smoothSourceField 2 L ω) 0) ![directionX, directionX, directionY] = + scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionY]) (sampleInclusion 2 L ω) +hq : + (iteratedFDeriv ℝ 2 (smoothSourceField 2 L ω) 0) ![directionY, directionY] = + scalarSeries (derivativeCoefficient 2 L 2 0 ![directionY, directionY]) (sampleInclusion 2 L ω) +⊢ scalarSeries (derivativeCoefficient 2 L 3 0 ![directionX, directionX, directionY]) (sampleInclusion 2 L ω) / 2 = + scalarSeries (contactAdjustedCoefficient L ⟨3, ⋯⟩) (sampleInclusion 2 L ω) +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:69:23: warning: This simp argument is unused: + contactAdjustedCoefficient + +Hint: Omit it from the simp argument list. + [apply] simp only [Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val, scalarSeries_div, scalarSeries_neg] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:69:51: warning: This simp argument is unused: + Matrix.cons_val_zero + +Hint: Omit it from the simp argument list. + [apply] simp only [contactAdjustedCoefficient, Matrix.cons_val_one, Matrix.cons_val, scalarSeries_div, + scalarSeries_neg] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:69:73: warning: This simp argument is unused: + Matrix.cons_val_one + +Hint: Omit it from the simp argument list. + [apply] simp only [contactAdjustedCoefficient, Matrix.cons_val_zero, Matrix.cons_val, scalarSeries_div, + scalarSeries_neg] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:70:4: warning: This simp argument is unused: + Matrix.cons_val + +Hint: Omit it from the simp argument list. + [apply] simp only [contactAdjustedCoefficient, Matrix.cons_val_zero, Matrix.cons_val_one, scalarSeries_div, + scalarSeries_neg] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:70:21: warning: This simp argument is unused: + scalarSeries_div + +Hint: Omit it from the simp argument list. + [apply] simp only [contactAdjustedCoefficient, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val, + scalarSeries_neg] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +/workspace/scratch/contact-coefficient-limits/ContactCoefficientLimits.lean:70:39: warning: This simp argument is unused: + scalarSeries_neg + +Hint: Omit it from the simp argument list. + [apply] simp only [contactAdjustedCoefficient, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val, + scalarSeries_div] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/contact-coefficients-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/contact-coefficients-02.log new file mode 100644 index 000000000..e69de29bb diff --git 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"292ba3ce2d7ba18bdf099a1a21dab9a94c55f19661ddec58303bfa3b0be58b07", + "source_sha256_after": "292ba3ce2d7ba18bdf099a1a21dab9a94c55f19661ddec58303bfa3b0be58b07", + "log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/regularized-zero-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_contact_coefficient_helpers/regularized-zero-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/ModeCurveChains.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/ModeCurveChains.lean.txt new file mode 100644 index 000000000..bbcb046df --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/ModeCurveChains.lean.txt @@ -0,0 +1,120 @@ +import SourceConfluentPins +import ConfluentIntegralIdentities +import Mathlib.Analysis.Calculus.ContDiff.Comp +import Mathlib.Tactic.FinCases + +noncomputable section + +open I4Source I4Weights I4SourceDerivativeGaussian I4SourceConfluentPins + +namespace I4ModeCurveChains + +def axisMap : ℝ →L[ℝ] (Fin 2 → ℝ) where + toFun t := ![t,0] + map_add' s t := by ext i; fin_cases i <;> simp + map_smul' c t := by ext i; fin_cases i <;> simp + cont := by fun_prop + +theorem mode_smooth (L : ℝ) (m : SourceMode 2) : + ContDiff ℝ (⊤ : ℕ∞) (modeCoefficient 2 L m) := by + cases m with + | none => + change ContDiff ℝ (⊤ : ℕ∞) (fun _ : Fin 2 → ℝ => Real.sqrt (1 / sourceZ 2 L)) + fun_prop + | some p => + rcases p with ⟨k,b⟩ + change ContDiff ℝ (⊤ : ℕ∞) (fun x : Fin 2 → ℝ => + Real.sqrt (2 * sourceWeight 2 L k / sourceZ 2 L) * + if b then Real.sin (phase 2 L k x) else Real.cos (phase 2 L k x)) + cases b <;> simp only [Bool.false_eq_true, ite_false, ite_true] <;> + unfold phase <;> fun_prop + +theorem axis_iteratedDeriv (f : (Fin 2 → ℝ) → ℝ) + (hf : ContDiff ℝ (⊤ : ℕ∞) f) (k : ℕ) (t : ℝ) : + iteratedDeriv k (fun s => f ![s,0]) t = + iteratedFDeriv ℝ k f ![t,0] (fun _ => directionX) := by + rw [iteratedDeriv_eq_iteratedFDeriv] + have h := axisMap.iteratedFDeriv_comp_right hf t (i := k) (by simp) + have hh := congrArg (fun A => A (fun _ => (1 : ℝ))) h + simpa [Function.comp_def, axisMap, directionX, + ContinuousMultilinearMap.compContinuousLinearMap_apply] using hh + +theorem axis_transverse_smooth (f : (Fin 2 → ℝ) → ℝ) + (hf : ContDiff ℝ (⊤ : ℕ∞) f) : + ContDiff ℝ (⊤ : ℕ∞) (fun t => fderiv ℝ f ![t,0] directionY) := by + have hdf : ContDiff ℝ (⊤ : ℕ∞) (fderiv ℝ f) := hf.fderiv_right (by simp) + exact (hdf.clm_apply contDiff_const).comp axisMap.contDiff + +theorem axis_transverse_iteratedDeriv (f : (Fin 2 → ℝ) → ℝ) + (hf : ContDiff ℝ (⊤ : ℕ∞) f) (k : ℕ) (t : ℝ) : + iteratedDeriv k (fun s => fderiv ℝ f ![s,0] directionY) t = + iteratedFDeriv ℝ (k+1) f ![t,0] (Fin.snoc (fun _ => directionX) directionY) := by + have hdf : ContDiff ℝ (⊤ : ℕ∞) (fderiv ℝ f) := hf.fderiv_right (by simp) + have hfun : ContDiff ℝ (⊤ : ℕ∞) (fun x => fderiv ℝ f x directionY) := + hdf.clm_apply contDiff_const + calc + _ = iteratedFDeriv ℝ k (fun x => fderiv ℝ f x directionY) ![t,0] + (fun _ => directionX) := axis_iteratedDeriv _ hfun k t + _ = iteratedFDeriv ℝ k (fderiv ℝ f) ![t,0] (fun _ => directionX) directionY := + iteratedFDeriv_clm_apply_const_apply hdf (by simp) + _ = _ := by + rw [iteratedFDeriv_succ_apply_right, Fin.init_snoc, Fin.snoc_last] + +def modeF (L : ℝ) (m : SourceMode 2) (t : ℝ) : ℝ := modeCoefficient 2 L m ![t,0] +def modeH (L : ℝ) (m : SourceMode 2) (t : ℝ) : ℝ := + derivativeCoefficient 2 L 1 ![t,0] ![directionY] m + +theorem modeH_eq (L : ℝ) (m : SourceMode 2) : + modeH L m = fun t => fderiv ℝ (modeCoefficient 2 L m) ![t,0] directionY := by + funext t + exact iteratedFDeriv_one_apply _ + +theorem modeF_C3 (L : ℝ) (m : SourceMode 2) : ContDiff ℝ 3 (modeF L m) := + ((mode_smooth L m).comp axisMap.contDiff).of_le (by simp) + +theorem modeH_C2 (L : ℝ) (m : SourceMode 2) : ContDiff ℝ 2 (modeH L m) := by + rw [modeH_eq] + exact (axis_transverse_smooth _ (mode_smooth L m)).of_le (by simp) + +theorem modeF_deriv (L : ℝ) (m : SourceMode 2) (t : ℝ) : + deriv (modeF L m) t = derivativeCoefficient 2 L 1 ![t,0] ![directionX] m := by + change deriv (fun s => modeCoefficient 2 L m ![s,0]) t = _ + have h := axis_iteratedDeriv _ (mode_smooth L m) 1 t + simpa [derivativeCoefficient, iteratedDeriv_one] using h + +theorem modeF_second (L : ℝ) (m : SourceMode 2) (t : ℝ) : + iteratedDeriv 2 (modeF L m) t = + derivativeCoefficient 2 L 2 ![t,0] ![directionX,directionX] m := by + change iteratedDeriv 2 (fun s => modeCoefficient 2 L m ![s,0]) t = _ + have h := axis_iteratedDeriv _ (mode_smooth L m) 2 t + have hd : (fun _ : Fin 2 => directionX) = ![directionX,directionX] := by + ext i; fin_cases i <;> rfl + simpa only [derivativeCoefficient, hd] using h + +theorem modeF_third (L : ℝ) (m : SourceMode 2) (t : ℝ) : + iteratedDeriv 3 (modeF L m) t = + derivativeCoefficient 2 L 3 ![t,0] ![directionX,directionX,directionX] m := by + change iteratedDeriv 3 (fun s => modeCoefficient 2 L m ![s,0]) t = _ + have h := axis_iteratedDeriv _ (mode_smooth L m) 3 t + have hd : (fun _ : Fin 3 => directionX) = ![directionX,directionX,directionX] := by + ext i; fin_cases i <;> rfl + simpa only [derivativeCoefficient, hd] using h + +theorem modeH_deriv (L : ℝ) (m : SourceMode 2) (t : ℝ) : + deriv (modeH L m) t = derivativeCoefficient 2 L 2 ![t,0] ![directionX,directionY] m := by + rw [modeH_eq] + have h := axis_transverse_iteratedDeriv _ (mode_smooth L m) 1 t + have hd : Fin.snoc (fun _ : Fin 1 => directionX) directionY = + ![directionX,directionY] := by ext i; fin_cases i <;> rfl + simpa only [iteratedDeriv_one, derivativeCoefficient, hd] using h + +theorem modeH_second (L : ℝ) (m : SourceMode 2) (t : ℝ) : + iteratedDeriv 2 (modeH L m) t = + derivativeCoefficient 2 L 3 ![t,0] ![directionX,directionX,directionY] m := by + rw [modeH_eq] + have h := axis_transverse_iteratedDeriv _ (mode_smooth L m) 2 t + have hd : Fin.snoc (fun _ : Fin 2 => directionX) directionY = + ![directionX,directionX,directionY] := by ext i; fin_cases i <;> rfl + simpa only [derivativeCoefficient, hd] using h + +end I4ModeCurveChains diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/RegularizedCoefficientContinuity.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/RegularizedCoefficientContinuity.lean.txt new file mode 100644 index 000000000..ec7bae23b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/RegularizedCoefficientContinuity.lean.txt @@ -0,0 +1,75 @@ +import SourceConfluentLimits + +noncomputable section + +open I4Source I4SourceDerivativeGaussian I4SourceDerivativeContinuity +open I4SourceConfluentPins I4ModeCurveChains I4WeightedModeIntegrals +open MeasureTheory Set +open scoped Interval + +namespace I4SourceConfluentLimits + +private theorem helper_left_bound (t : ℝ) + (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |peanoLeft t| ≤ 1 := by + rw [Set.uIoc_of_le (by norm_num)] at ht + unfold peanoLeft + exact abs_le.mpr ⟨by linarith [ht.2], by linarith [ht.1]⟩ + +private theorem helper_right_bound (t : ℝ) + (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |peanoRight t| ≤ 1 := by + rw [Set.uIoc_of_le (by norm_num)] at ht + unfold peanoRight + exact abs_le.mpr ⟨by linarith [ht.1], by linarith [ht.2]⟩ + +private theorem helper_cubic_bound (t : ℝ) + (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : |peanoCubic t| ≤ 1 := by + rw [Set.uIoc_of_le (by norm_num)] at ht + have habs : |t| ≤ (1/2 : ℝ) := abs_le.mpr ⟨by linarith [ht.1], ht.2⟩ + have hsq := (sq_le_sq₀ (abs_nonneg t) (by norm_num : (0 : ℝ) ≤ 1/2)).2 habs + rw [sq_abs] at hsq + unfold peanoCubic + exact abs_le.mpr ⟨by nlinarith, by nlinarith [sq_nonneg t]⟩ + +theorem regularizedPinCoefficient_continuous {L : ℝ} (hL : L ≠ 0) + (i : Fin 6) (m : SourceMode 2) : + Continuous (fun r => regularizedPinCoefficient L r i m) := by + have hF := (modeF_C3 L m).continuous + have hH := (modeH_C2 L m).continuous + have hF' : Continuous (deriv (modeF L m)) := by + simpa only [iteratedDeriv_one] using + (modeF_C3 L m).continuous_iteratedDeriv 1 (by norm_num) + have hpoly : Continuous peanoCubic := by unfold peanoCubic; fun_prop + fin_cases i + · change Continuous (fun r : ℝ => (modeF L m (-r/2) + modeF L m (r/2))/2) + exact ((hF.comp (by fun_prop)).add (hF.comp (by fun_prop))).div_const 2 + · change Continuous (fun r : ℝ => (deriv (modeF L m) (-r/2) + deriv (modeF L m) (r/2))/2) + exact ((hF'.comp (by fun_prop)).add (hF'.comp (by fun_prop))).div_const 2 + · change Continuous (fun r : ℝ => (modeH L m (-r/2) + modeH L m (r/2))/2) + exact ((hH.comp (by fun_prop)).add (hH.comp (by fun_prop))).div_const 2 + · exact weightedDerivativeIntegral_continuous hL 2 ![directionX,directionX] m + (fun _ => 1) continuous_const 1 (by norm_num) (by intro t ht; norm_num) + · exact weightedDerivativeIntegral_continuous hL 2 ![directionX,directionY] m + (fun _ => 1) continuous_const 1 (by norm_num) (by intro t ht; norm_num) + · exact (weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionX] m + peanoCubic hpoly 1 (by norm_num) helper_cubic_bound).const_mul 6 + +theorem regularizedAdjustedCoefficient_continuous {L : ℝ} (hL : L ≠ 0) + (i : Fin 6) (m : SourceMode 2) : + Continuous (fun r => regularizedAdjustedCoefficient L r i m) := by + have hleft : Continuous peanoLeft := by unfold peanoLeft; fun_prop + have hright : Continuous peanoRight := by unfold peanoRight; fun_prop + have hYY : Continuous (modeYY L m) := + (derivativeCoefficient_continuous 2 L 2 ![directionY,directionY] m).comp (by fun_prop) + fin_cases i + · exact (weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionX] m + peanoLeft hleft 1 (by norm_num) helper_left_bound).neg + · exact weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionX] m + peanoRight hright 1 (by norm_num) helper_right_bound + · exact (weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionY] m + peanoLeft hleft 1 (by norm_num) helper_left_bound).neg + · exact weightedDerivativeIntegral_continuous hL 3 ![directionX,directionX,directionY] m + peanoRight hright 1 (by norm_num) helper_right_bound + · exact hYY.comp (by fun_prop) + · exact hYY.comp (by fun_prop) + +end I4SourceConfluentLimits diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/WeightedModeIntegrals.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/WeightedModeIntegrals.lean.txt new file mode 100644 index 000000000..b1bef35b2 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/WeightedModeIntegrals.lean.txt @@ -0,0 +1,66 @@ +import SourceDerivativeContinuity +import Mathlib.MeasureTheory.Integral.DominatedConvergence +import Mathlib.Tactic.NormNum + +noncomputable section + +open I4Source I4SourceDerivativeGaussian I4SourceDerivativeContinuity +open MeasureTheory Set +open scoped Interval + +namespace I4WeightedModeIntegrals + +def weightedDerivativeIntegral (L : ℝ) (k : ℕ) (u : Fin k → (Fin 2 → ℝ)) + (m : SourceMode 2) (w : ℝ → ℝ) (r : ℝ) : ℝ := + ∫ t in (-1/2 : ℝ)..(1/2 : ℝ), w t * derivativeCoefficient 2 L k ![r*t,0] u m + +theorem weighted_integrand_bound {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) (m : SourceMode 2) (w : ℝ → ℝ) + (K : ℝ) (hK : 0 ≤ K) + (hwbound : ∀ t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ), |w t| ≤ K) + (r t : ℝ) (ht : t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ)) : + ‖w t * derivativeCoefficient 2 L k ![r*t,0] u m‖ ≤ + K * derivativeMajorant 2 L k u m := by + rw [norm_mul, Real.norm_eq_abs] + exact mul_le_mul (hwbound t ht) (derivativeCoefficient_bound 2 hL k u m ![r*t,0]) + (norm_nonneg _) hK + +theorem weightedDerivativeIntegral_continuous {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) (m : SourceMode 2) (w : ℝ → ℝ) (hw : Continuous w) + (K : ℝ) (hK : 0 ≤ K) + (hwbound : ∀ t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ), |w t| ≤ K) : + Continuous (weightedDerivativeIntegral L k u m w) := by + unfold weightedDerivativeIntegral + apply intervalIntegral.continuous_of_dominated_interval + (bound := fun _ => K * derivativeMajorant 2 L k u m) + · intro r + have hc : Continuous (fun t => w t * derivativeCoefficient 2 L k ![r*t,0] u m) := + hw.mul ((derivativeCoefficient_continuous 2 L k u m).comp (by fun_prop)) + exact hc.aestronglyMeasurable + · intro r + exact Filter.Eventually.of_forall (fun t ht => + weighted_integrand_bound hL k u m w K hK hwbound r t ht) + · exact continuous_const.intervalIntegrable _ _ + · exact Filter.Eventually.of_forall (fun t _ => + continuous_const.mul ((derivativeCoefficient_continuous 2 L k u m).comp (by fun_prop))) + +theorem weightedDerivativeIntegral_norm_le {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) (m : SourceMode 2) (w : ℝ → ℝ) + (K : ℝ) (hK : 0 ≤ K) + (hwbound : ∀ t ∈ Set.uIoc (-1/2 : ℝ) (1/2 : ℝ), |w t| ≤ K) (r : ℝ) : + ‖weightedDerivativeIntegral L k u m w r‖ ≤ K * derivativeMajorant 2 L k u m := by + have h := intervalIntegral.norm_integral_le_of_norm_le_const + (fun t ht => weighted_integrand_bound hL k u m w K hK hwbound r t ht) + norm_num at h + simpa only [weightedDerivativeIntegral, Real.norm_eq_abs, neg_div] using h + +theorem weightedDerivativeIntegral_zero (L : ℝ) (k : ℕ) + (u : Fin k → (Fin 2 → ℝ)) (m : SourceMode 2) (w : ℝ → ℝ) : + weightedDerivativeIntegral L k u m w 0 = + (∫ t in (-1/2 : ℝ)..(1/2 : ℝ), w t) * derivativeCoefficient 2 L k 0 u m := by + unfold weightedDerivativeIntegral + have hzero : (![0,0] : Fin 2 → ℝ) = 0 := by ext i; fin_cases i <;> rfl + simp_rw [zero_mul, hzero] + exact intervalIntegral.integral_mul_const _ _ + +end I4WeightedModeIntegrals diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_attempt1.json new file mode 100644 index 000000000..b6280607d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_attempt1.json @@ -0,0 +1,12 @@ +{ + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/ModeCurveChains.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 1, + "source_sha256": "fce4eff5e0add8496f2876234ecede8080b43e4c896c0bc7738e4a767c8bbaa4", + "scope": "NEW_SCRATCH_ACTUAL_MODE_CHAINS_ONLY" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_attempt1.log new file mode 100644 index 000000000..e9352eb60 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_attempt1.log @@ -0,0 +1,21 @@ +/workspace/scratch/dm2_landed_composition/ModeCurveChains.lean:82:2: error: Type mismatch: After simplification, term + h + has type + @Eq ℝ (deriv (fun s => modeCoefficient 2 L m ![s, 0]) t) ((fderiv ℝ (modeCoefficient 2 L m) ![t, 0]) directionX) +but is expected to have type + @Eq ℝ (deriv (modeF L m) t) ((fderiv ℝ (modeCoefficient 2 L m) ![t, 0]) directionX) +/workspace/scratch/dm2_landed_composition/ModeCurveChains.lean:90:2: error: Type mismatch: After simplification, term + h + has type + @Eq ℝ (iteratedDeriv 2 (fun s => modeCoefficient 2 L m ![s, 0]) t) + ((iteratedFDeriv ℝ 2 (modeCoefficient 2 L m) ![t, 0]) ![directionX, directionX]) +but is expected to have type + @Eq ℝ (iteratedDeriv 2 (modeF L m) t) ((iteratedFDeriv ℝ 2 (modeCoefficient 2 L m) ![t, 0]) ![directionX, directionX]) +/workspace/scratch/dm2_landed_composition/ModeCurveChains.lean:98:2: error: Type mismatch: After simplification, term + h + has type + @Eq ℝ (iteratedDeriv 3 (fun s => modeCoefficient 2 L m ![s, 0]) t) + ((iteratedFDeriv ℝ 3 (modeCoefficient 2 L m) ![t, 0]) ![directionX, directionX, directionX]) +but is expected to have type + @Eq ℝ (iteratedDeriv 3 (modeF L m) t) + ((iteratedFDeriv ℝ 3 (modeCoefficient 2 L m) ![t, 0]) ![directionX, directionX, directionX]) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_final.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_final.json new file mode 100644 index 000000000..7fb9c2a58 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_final.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/ModeCurveChains.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256": "6f51b0d187a5e07c2cee93da8238ac2d6f3250990fc3d3b5ed5ebab5607d6724", + "scope": "NEW_SCRATCH_ACTUAL_MODE_CHAINS_ONLY", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_final.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/mode_chains_final.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/regularized_continuity_attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/regularized_continuity_attempt1.json new file mode 100644 index 000000000..3a051d6fa --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/regularized_continuity_attempt1.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/RegularizedCoefficientContinuity.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256": "323dd93689bd95cfd050078aa170a81c31b3f3480887fb7a438a535711bf650c", + "scope": "NEW_SCRATCH_REGULARIZED_CONTINUITY_ONLY", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/regularized_continuity_attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/regularized_continuity_attempt1.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_attempt1.json new file mode 100644 index 000000000..bdd5a9271 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_attempt1.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/WeightedModeIntegrals.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 1, + "source_sha256": "ff25e6292d58a9649858b6a2b03add8f0f9eb7ca7a57d7c5124560195c4a179e", + "scope": "NEW_SCRATCH_WEIGHTED_MODE_INTEGRALS_ONLY", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_attempt1.log new file mode 100644 index 000000000..5482d0639 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_attempt1.log @@ -0,0 +1,6 @@ +/workspace/scratch/dm2_landed_composition/WeightedModeIntegrals.lean:55:2: error: Type mismatch + h +has type + |∫ (x : ℝ) in -(1 / 2)..1 / 2, w x * derivativeCoefficient 2 L k ![r * x, 0] u m| ≤ K * derivativeMajorant 2 L k u m +but is expected to have type + ‖weightedDerivativeIntegral L k u m w r‖ ≤ K * derivativeMajorant 2 L k u m diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_final.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_final.json new file mode 100644 index 000000000..09510e881 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_final.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/WeightedModeIntegrals.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256": "9d1aa74e3207394da92b8a0c2d467d47bc3d972aa6ff44faaa5391dbaca67075", + "scope": "NEW_SCRATCH_WEIGHTED_MODE_INTEGRALS_ONLY", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_final.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/actual_dm2_helpers/weighted_mode_final.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/axiom_audit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/axiom_audit.lean.txt new file mode 100644 index 000000000..9f5f7090a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/axiom_audit.lean.txt @@ -0,0 +1,59 @@ +import SourceConfluentLimits + +#print axioms I4SourceConfluentLimits.rawPinsWithHessians_series +#print axioms I4SourceConfluentLimits.pinCoefficient_summable +#print axioms I4SourceConfluentLimits.adjustedCoefficient_summable +#print axioms I4SourceConfluentLimits.sourceConfluentPinsAt_series +#print axioms I4SourceConfluentLimits.sourceAdjustedEndpoints_series +#print axioms I4SourceConfluentLimits.sourcePinCovarianceAt_coefficient +#print axioms I4SourceConfluentLimits.sourceCrossCovarianceAt_coefficient +#print axioms I4SourceConfluentLimits.sourceEndpointCovarianceAt_coefficient +#print axioms I4ModeCurveChains.mode_smooth +#print axioms I4ModeCurveChains.axis_iteratedDeriv +#print axioms I4ModeCurveChains.axis_transverse_smooth +#print axioms I4ModeCurveChains.axis_transverse_iteratedDeriv +#print axioms I4ModeCurveChains.modeH_eq +#print axioms I4ModeCurveChains.modeF_C3 +#print axioms I4ModeCurveChains.modeH_C2 +#print axioms I4ModeCurveChains.modeF_deriv +#print axioms I4ModeCurveChains.modeF_second +#print axioms I4ModeCurveChains.modeF_third +#print axioms I4ModeCurveChains.modeH_deriv +#print axioms I4ModeCurveChains.modeH_second +#print axioms I4SourceConfluentLimits.rawDerivativeCoefficient_explicit +#print axioms I4SourceConfluentLimits.pinCoefficient_explicit +#print axioms I4SourceConfluentLimits.adjustedCoefficient_explicit +#print axioms I4WeightedModeIntegrals.weighted_integrand_bound +#print axioms I4WeightedModeIntegrals.weightedDerivativeIntegral_continuous +#print axioms I4WeightedModeIntegrals.weightedDerivativeIntegral_norm_le +#print axioms I4WeightedModeIntegrals.weightedDerivativeIntegral_zero +#print axioms I4SourceConfluentLimits.peano_kernel_masses +#print axioms I4SourceConfluentLimits.pinCoefficient_eq_regularized +#print axioms I4SourceConfluentLimits.adjustedCoefficient_eq_regularized +#print axioms I4ContactCoefficientLimits.contactPinCoefficient_summable +#print axioms I4ContactCoefficientLimits.contactAdjustedCoefficient_summable +#print axioms I4ContactCoefficientLimits.sourcePinVector_eq_contactSeries +#print axioms I4ContactCoefficientLimits.sourceContactEndpointVector_eq_contactSeries +#print axioms I4ContactCoefficientLimits.sourcePinCovariance_eq_contactCoefficientGram +#print axioms I4ContactCoefficientLimits.sourceContactEndpointCrossCovariance_eq_contactCoefficientGram +#print axioms I4ContactCoefficientLimits.sourceContactEndpointCovariance_eq_contactCoefficientGram +#print axioms I4SourceConfluentLimits.confluentModeMajorant_nonneg +#print axioms I4SourceConfluentLimits.confluentModeMajorant_summable +#print axioms I4SourceConfluentLimits.regularizedPinCoefficient_bound +#print axioms I4SourceConfluentLimits.regularizedAdjustedCoefficient_bound +#print axioms I4SourceConfluentLimits.regularizedPinCoefficient_continuous +#print axioms I4SourceConfluentLimits.regularizedAdjustedCoefficient_continuous +#print axioms I4SourceConfluentLimits.regularizedPinCoefficient_zero +#print axioms I4SourceConfluentLimits.regularizedAdjustedCoefficient_zero +#print axioms I4SourceConfluentLimits.regularizedPinCovariance_continuous +#print axioms I4SourceConfluentLimits.regularizedCrossCovariance_continuous +#print axioms I4SourceConfluentLimits.regularizedEndpointCovariance_continuous +#print axioms I4SourceConfluentLimits.sourcePinCovarianceAt_eq_regularized +#print axioms I4SourceConfluentLimits.sourceCrossCovarianceAt_eq_regularized +#print axioms I4SourceConfluentLimits.sourceEndpointCovarianceAt_eq_regularized +#print axioms I4SourceConfluentLimits.regularizedPinCovariance_zero +#print axioms I4SourceConfluentLimits.regularizedCrossCovariance_zero +#print axioms I4SourceConfluentLimits.regularizedEndpointCovariance_zero +#print axioms I4SourceConfluentLimits.sourcePinCovarianceAt_tendsto +#print axioms I4SourceConfluentLimits.sourceCrossCovarianceAt_tendsto +#print axioms I4SourceConfluentLimits.sourceEndpointCovarianceAt_tendsto diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/controls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/controls.lean.txt new file mode 100644 index 000000000..7e3da41fe --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/controls.lean.txt @@ -0,0 +1,42 @@ +import SourceConfluentLimits + +open I4Source I4Weights I4SmoothCarrier I4SourceConfluentPins I4SourceContact +open I4SourceEndpointRegression I4SourceConfluentLimits I4ContactCoefficientLimits +open MeasureTheory ProbabilityTheory Filter +open scoped Topology + +example (L : ℝ) (ω : SmoothSample 2 L) : sourceConfluentPinsAt L 0 ω 5 = 0 := by + simp [sourceConfluentPinsAt, confluentTransform] + +example (L : ℝ) (m : SourceMode 2) : pinCoefficient L 0 5 m = 0 := by + simp [pinCoefficient, pinObservationMap, endpointObservationMap, confluentTransform] + +example {L : ℝ} (hL : L ≠ 0) : + regularizedPinCoefficient L 0 5 (sourceContactMode 5) ≠ 0 := by + rw [regularizedPinCoefficient_zero] + change sourceContactCoefficient L 5 (sourceContactMode 5) ≠ 0 + have hcol : sourceContactCoefficient L 5 (sourceContactMode 5) = + sourceContactAmplitude L 5 * (-8 * sourceC L ^ 3) := by + simpa [I4ContactGram.contactWitness] using sourceContact_selected_columns L 5 5 + rw [hcol] + have hc : sourceC L ≠ 0 := div_ne_zero + (mul_ne_zero (by norm_num) (ne_of_gt Real.pi_pos)) hL + exact mul_ne_zero (sourceContactAmplitude_pos hL 5).ne' + (mul_ne_zero (by norm_num) (pow_ne_zero _ hc)) + +example (L : ℝ) (m : SourceMode 2) (i : Fin 6) : + pinCoefficient L (-2) i m = regularizedPinCoefficient L (-2) i m := + pinCoefficient_eq_regularized L (-2) (by norm_num) i m + +example (L : ℝ) (m : SourceMode 2) (i : Fin 6) : + adjustedCoefficient L (-2) i m = regularizedAdjustedCoefficient L (-2) i m := + adjustedCoefficient_eq_regularized L (-2) (by norm_num) i m + +example : Tendsto (sourcePinCovarianceAt 24) (𝓝[>] (0 : ℝ)) + (𝓝 (sourcePinCovariance 24)) := sourcePinCovarianceAt_tendsto (by norm_num) + +example : Tendsto (sourceCrossCovarianceAt 24) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactEndpointCrossCovariance 24)) := sourceCrossCovarianceAt_tendsto (by norm_num) + +example : Tendsto (sourceEndpointCovarianceAt (-24)) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactEndpointCovariance (-24))) := sourceEndpointCovarianceAt_tendsto (by norm_num) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/initial_api.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/initial_api.lean.txt new file mode 100644 index 000000000..7682d6c23 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/initial_api.lean.txt @@ -0,0 +1,8 @@ +import SourceEndpointRegression +import SourceDerivativeContinuity + +#check I4SourceConfluentLimits.sourceConfluentPinsAt_series +#check I4SourceConfluentLimits.sourceAdjustedEndpoints_series +#check I4SourceConfluentLimits.sourcePinCovarianceAt_coefficient +#check I4SourceConfluentLimits.sourceCrossCovarianceAt_coefficient +#check I4SourceConfluentLimits.sourceEndpointCovarianceAt_coefficient diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/targets.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/targets.json new file mode 100644 index 000000000..a1355cd5f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentLimits/targets.json @@ -0,0 +1,59 @@ +[ + "I4SourceConfluentLimits.rawPinsWithHessians_series", + "I4SourceConfluentLimits.pinCoefficient_summable", + "I4SourceConfluentLimits.adjustedCoefficient_summable", + "I4SourceConfluentLimits.sourceConfluentPinsAt_series", + "I4SourceConfluentLimits.sourceAdjustedEndpoints_series", + "I4SourceConfluentLimits.sourcePinCovarianceAt_coefficient", + "I4SourceConfluentLimits.sourceCrossCovarianceAt_coefficient", + "I4SourceConfluentLimits.sourceEndpointCovarianceAt_coefficient", + "I4ModeCurveChains.mode_smooth", + "I4ModeCurveChains.axis_iteratedDeriv", + "I4ModeCurveChains.axis_transverse_smooth", + "I4ModeCurveChains.axis_transverse_iteratedDeriv", + "I4ModeCurveChains.modeH_eq", + "I4ModeCurveChains.modeF_C3", + "I4ModeCurveChains.modeH_C2", + "I4ModeCurveChains.modeF_deriv", + "I4ModeCurveChains.modeF_second", + "I4ModeCurveChains.modeF_third", + "I4ModeCurveChains.modeH_deriv", + "I4ModeCurveChains.modeH_second", + "I4SourceConfluentLimits.rawDerivativeCoefficient_explicit", + "I4SourceConfluentLimits.pinCoefficient_explicit", + "I4SourceConfluentLimits.adjustedCoefficient_explicit", + "I4WeightedModeIntegrals.weighted_integrand_bound", + "I4WeightedModeIntegrals.weightedDerivativeIntegral_continuous", + "I4WeightedModeIntegrals.weightedDerivativeIntegral_norm_le", + "I4WeightedModeIntegrals.weightedDerivativeIntegral_zero", + "I4SourceConfluentLimits.peano_kernel_masses", + "I4SourceConfluentLimits.pinCoefficient_eq_regularized", + "I4SourceConfluentLimits.adjustedCoefficient_eq_regularized", + "I4ContactCoefficientLimits.contactPinCoefficient_summable", + "I4ContactCoefficientLimits.contactAdjustedCoefficient_summable", + "I4ContactCoefficientLimits.sourcePinVector_eq_contactSeries", + "I4ContactCoefficientLimits.sourceContactEndpointVector_eq_contactSeries", + "I4ContactCoefficientLimits.sourcePinCovariance_eq_contactCoefficientGram", + "I4ContactCoefficientLimits.sourceContactEndpointCrossCovariance_eq_contactCoefficientGram", + "I4ContactCoefficientLimits.sourceContactEndpointCovariance_eq_contactCoefficientGram", + "I4SourceConfluentLimits.confluentModeMajorant_nonneg", + "I4SourceConfluentLimits.confluentModeMajorant_summable", + "I4SourceConfluentLimits.regularizedPinCoefficient_bound", + "I4SourceConfluentLimits.regularizedAdjustedCoefficient_bound", + "I4SourceConfluentLimits.regularizedPinCoefficient_continuous", + "I4SourceConfluentLimits.regularizedAdjustedCoefficient_continuous", + "I4SourceConfluentLimits.regularizedPinCoefficient_zero", + "I4SourceConfluentLimits.regularizedAdjustedCoefficient_zero", + "I4SourceConfluentLimits.regularizedPinCovariance_continuous", + "I4SourceConfluentLimits.regularizedCrossCovariance_continuous", + "I4SourceConfluentLimits.regularizedEndpointCovariance_continuous", + "I4SourceConfluentLimits.sourcePinCovarianceAt_eq_regularized", + "I4SourceConfluentLimits.sourceCrossCovarianceAt_eq_regularized", + "I4SourceConfluentLimits.sourceEndpointCovarianceAt_eq_regularized", + "I4SourceConfluentLimits.regularizedPinCovariance_zero", + "I4SourceConfluentLimits.regularizedCrossCovariance_zero", + "I4SourceConfluentLimits.regularizedEndpointCovariance_zero", + "I4SourceConfluentLimits.sourcePinCovarianceAt_tendsto", + "I4SourceConfluentLimits.sourceCrossCovarianceAt_tendsto", + "I4SourceConfluentLimits.sourceEndpointCovarianceAt_tendsto" +] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/AxiomAudit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/AxiomAudit.lean.txt new file mode 100644 index 000000000..ad525f9b1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/AxiomAudit.lean.txt @@ -0,0 +1,9 @@ +import SourceConfluentPins +#print axioms I4SourceConfluentPins.rawTransform_confluentTransform +#print axioms I4SourceConfluentPins.confluentTransform_rawTransform +#print axioms I4SourceConfluentPins.confluentEquiv_apply +#print axioms I4SourceConfluentPins.confluentEquiv_symm_apply +#print axioms I4SourceConfluentPins.prescribed_target +#print axioms I4SourceConfluentPins.rawPins_hasGaussianLaw +#print axioms I4SourceConfluentPins.confluentPins_hasGaussianLaw +#print axioms I4SourceConfluentPins.rawPinsWithHessians_hasGaussianLaw diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/FalseZeroCubicPin.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/FalseZeroCubicPin.lean.txt new file mode 100644 index 000000000..6ba0d481f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/FalseZeroCubicPin.lean.txt @@ -0,0 +1,5 @@ +import SourceConfluentPins +open I4SourceConfluentPins +-- Dropping r != 0 would incorrectly retain the prescribed cubic pin two. +example : confluentTransform 0 ![7, 7, 0, 0, 0, 0] 5 = 2 := by + norm_num [confluentTransform] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/FalseZeroInverse.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/FalseZeroInverse.lean.txt new file mode 100644 index 000000000..286d05869 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/FalseZeroInverse.lean.txt @@ -0,0 +1,5 @@ +import SourceConfluentPins +open I4SourceConfluentPins +-- Dropping r != 0 would incorrectly recover the first raw pin as zero. +example : rawTransform 0 (confluentTransform 0 ![0, 2, 0, 0, 0, 0]) 0 = 0 := by + norm_num [rawTransform, confluentTransform] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/PositiveControls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/PositiveControls.lean.txt new file mode 100644 index 000000000..00b8aecf0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/PositiveControls.lean.txt @@ -0,0 +1,25 @@ +import SourceConfluentPins +open I4SourceConfluentPins + +-- The prescribed arbitrary height and nonunit radius give a nontrivial target. +example : confluentTransform 2 ![7, 17/3, 0, 0, 0, 0] = + ![19/3, 0, 0, 0, 0, 2] := by + have h := prescribed_target (2 : ℝ) 7 (by norm_num) + norm_num [confluentEquiv_apply] at h + exact h + +-- Nonzero signed separation is allowed, and the cubic height correction changes sign. +example : confluentTransform (-2) ![7, 25/3, 0, 0, 0, 0] = + ![23/3, 0, 0, 0, 0, 2] := by + have h := prescribed_target (-2 : ℝ) 7 (by norm_num) + norm_num [confluentEquiv_apply] at h + exact h + +-- Zero radius collapses two unequal value pins to their common average. +example : rawTransform 0 (confluentTransform 0 ![0, 2, 0, 0, 0, 0]) = + ![1, 1, 0, 0, 0, 0] := by + norm_num [rawTransform, confluentTransform] + +-- Zero radius cannot produce the required nonzero cubic confluent pin. +example : confluentTransform 0 ![7, 7, 0, 0, 0, 0] 5 = 0 := by + norm_num [confluentTransform] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/README.md b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/README.md new file mode 100644 index 000000000..d232ec38d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/README.md @@ -0,0 +1,40 @@ +# New-module qualification only + +The completed SourceConfluentPins compilation was reused. This run covers the +previously uncovered named kernel replay, eight public target axiom audit and +new meaningful controls. Existing inputs and previously qualified modules were +not separately rechecked. + +`qualification_receipt.json` records exact subprocess commands, working +directories, exit codes, raw-log paths and digests, and the unchanged source +SHA256. Every subprocess's untouched combined stdout/stderr is retained in +its `.log`; metadata is retained in its `.json`. + +Results: + +- `lake env leanchecker SourceConfluentPins`: exit 0, empty raw output. +- Eight public-target transitive axiom reports: only `propext`, + `Classical.choice`, `Quot.sound`. +- Positive controls: exit 0. Height seven/radius two gives the literal target + `(19/3,0,0,0,0,2)`; radius minus two gives `(23/3,0,0,0,0,2)`. + Zero-radius controls prove the noninverse collapse to `(1,1,0,0,0,0)` and + zero cubic coordinate. +- The false zero-radius inverse and false cubic-coordinate-two controls each + exit 1 at an unsolved `False` proof goal, with no import or syntax failure. + +Source SHA256 throughout: +`41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f`. + +A preliminary CLI discovery call `lake env leanchecker --help` returned exit 1 +with `uncaught exception: Could not find any oleans for: Source_smooth_realization`. +That output is reconstructed here from the actual tool response; it was not +redirected. It did not replay any old module. The actual requested named +`SourceConfluentPins` replay was then run exactly once, with its raw log retained. + +Actual author-side lineage is OpenAI/Codex `/root/dm2_landed_composition`. +No helper was created. Qualification is local mathematical evidence, separate +from independent scientific review. Actual-source covariance limits, +conditioning/regression joins and overall campaign completion remain open. + +Per `/workspace/AGENTS.md`, reuse these completed applicable checks; a later +handoff or process obligation does not invalidate the mathematics checked here. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/axioms.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/axioms.json new file mode 100644 index 000000000..e2c37422b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/axioms.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_qualification/AxiomAudit.lean" + ], + "exit_code": 0, + "duration_seconds": 5.699351787567139, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/axioms.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/axioms.log new file mode 100644 index 000000000..c2bb67fd8 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/axioms.log @@ -0,0 +1,8 @@ +'I4SourceConfluentPins.rawTransform_confluentTransform' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentPins.confluentTransform_rawTransform' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentPins.confluentEquiv_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentPins.confluentEquiv_symm_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentPins.prescribed_target' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentPins.rawPins_hasGaussianLaw' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentPins.confluentPins_hasGaussianLaw' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceConfluentPins.rawPinsWithHessians_hasGaussianLaw' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt1.log new file mode 100644 index 000000000..1d13a3581 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt1.log @@ -0,0 +1,75 @@ +SourceConfluentPins.lean:43:78: warning: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` +SourceConfluentPins.lean:93:52: error: unsolved goals +case «0» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 0 (smoothSourceField 2 L ω) (endpointM r)) fun x => directionX) = + smoothSourceField 2 L ω (endpointM r) + +case «1» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 0 (smoothSourceField 2 L ω) (endpointS r)) fun x => directionX) = + smoothSourceField 2 L ω (endpointS r) + +case «2» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) fun x => directionX) = + (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionX + +case «3» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) fun x => directionX) = + (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionX + +case «4» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) fun x => directionY) = + (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionY + +case «5» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) fun x => directionY) = + (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionY diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt2.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt2.log new file mode 100644 index 000000000..8995bb881 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt2.log @@ -0,0 +1,89 @@ +SourceConfluentPins.lean:43:78: warning: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` +SourceConfluentPins.lean:93:52: error: unsolved goals +case «0» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 0 (smoothSourceField 2 L ω) (endpointM r)) fun x => directionX) = + smoothSourceField 2 L ω (endpointM r) + +case «1» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 0 (smoothSourceField 2 L ω) (endpointS r)) fun x => directionX) = + smoothSourceField 2 L ω (endpointS r) + +case «2» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) fun x => directionX) = + (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionX + +case «3» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) fun x => directionX) = + (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionX + +case «4» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointM r)) fun x => directionY) = + (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionY + +case «5» +L : ℝ +hL : L ≠ 0 +r : ℝ +k : Fin 6 → ℕ := ![0, 0, 1, 1, 1, 1] +x : Fin 6 → Fin 2 → ℝ := ![endpointM r, endpointS r, endpointM r, endpointS r, endpointM r, endpointS r] +u : (j : Fin 6) → Fin (k j) → Fin 2 → ℝ := fun j x => if ↑j < 4 then directionX else directionY +hG : HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField 2 L ω) (x j)) (u j)) (smoothLaw 2 L) +ω : SmoothSample 2 L +⊢ ((iteratedFDeriv ℝ 1 (smoothSourceField 2 L ω) (endpointS r)) fun x => directionY) = + (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionY +SourceConfluentPins.lean:103:42: warning: This simp argument is unused: + iteratedFDeriv_zero_apply + +Hint: Omit it from the simp argument list. + [apply] simp [rawPins, k, x, u, iteratedFDeriv_one_apply] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceConfluentPins.lean:104:4: warning: This simp argument is unused: + iteratedFDeriv_one_apply + +Hint: Omit it from the simp argument list. + [apply] simp [rawPins, k, x, u, iteratedFDeriv_zero_apply] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt2.meta b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt2.meta new file mode 100644 index 000000000..9e9a5a20a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt2.meta @@ -0,0 +1,3 @@ +cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025 +command=['lake', 'env', 'lean', '-o', '.lake/build/lib/lean/SourceConfluentPins.olean', 'SourceConfluentPins.lean'] +exit_code=1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt3.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt3.log new file mode 100644 index 000000000..33f60a1cc --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt3.log @@ -0,0 +1,3 @@ +SourceConfluentPins.lean:43:78: warning: Used `tac1 <;> tac2` where `(tac1; tac2)` would suffice + +Note: This linter can be disabled with `set_option linter.unnecessarySeqFocus false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt3.meta b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt3.meta new file mode 100644 index 000000000..3fd733dd1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt3.meta @@ -0,0 +1,3 @@ +cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025 +command=['lake', 'env', 'lean', '-o', '.lake/build/lib/lean/SourceConfluentPins.olean', 'SourceConfluentPins.lean'] +exit_code=0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt4.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt4.log new file mode 100644 index 000000000..6c0dfd0a5 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt4.log @@ -0,0 +1,2 @@ +SourceConfluentPins.lean:43:80: error: No goals to be solved +SourceConfluentPins.lean:43:80: error: No goals to be solved diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt4.meta b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt4.meta new file mode 100644 index 000000000..9e9a5a20a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_attempt4.meta @@ -0,0 +1,3 @@ +cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025 +command=['lake', 'env', 'lean', '-o', '.lake/build/lib/lean/SourceConfluentPins.olean', 'SourceConfluentPins.lean'] +exit_code=1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_final.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_final.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_final.meta b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_final.meta new file mode 100644 index 000000000..3fd733dd1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_final.meta @@ -0,0 +1,3 @@ +cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025 +command=['lake', 'env', 'lean', '-o', '.lake/build/lib/lean/SourceConfluentPins.olean', 'SourceConfluentPins.lean'] +exit_code=0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_handoff.md b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_handoff.md new file mode 100644 index 000000000..2af78661e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/earlier_compile/source_confluent_handoff.md @@ -0,0 +1,67 @@ +# SourceConfluentPins author-side handoff + +Source file: +`/workspace/Math-/frontiers/source_derivative_gaussian_20261025/SourceConfluentPins.lean` + +Frozen source contract read in full from Math- commit `93c9a30`, +`frontiers/source_derivative_gaussian_20261025/SOURCE_CONFLUENT_PINS.md`. + +Author: actual OpenAI/Codex agent `/root/dm2_landed_composition`, delegated by +`/root`. No helper agent was created. This is author-side proof work, not +independent scientific review or provider-distinct alignment evidence. + +## New proved declarations + +Namespace `I4SourceConfluentPins`: + +- `rawTransform_confluentTransform` +- `confluentTransform_rawTransform` +- `confluentEquiv_apply` +- `confluentEquiv_symm_apply` +- `prescribed_target` +- `rawPins_hasGaussianLaw` +- `confluentPins_hasGaussianLaw` +- `rawPinsWithHessians_hasGaussianLaw` + +The actual `confluentEquiv r hr` continuous linear equivalence contains proved +linearity, continuity and both inverse laws, using the literal stated formulas. +The prescribed target retains arbitrary real height `b` and nonzero radius. + +`rawPins` are actual field values and fderiv observations at the two endpoints. +`rawPinsWithHessians` contains the raw six pins followed by actual iterated +second derivatives in order `(fxxM,fxyM,fyyM,fxxS,fxyS,fyyS)`. Their joint +Gaussianity follows from one 12-entry heterogeneous derivative theorem. + +No covariance nondegeneracy, conditional field construction, regression block, +Palm event or scientific acceptance is inferred. + +## Actual final scoped verification + +The completed final command was: + +```sh +lake env lean -o .lake/build/lib/lean/SourceConfluentPins.olean SourceConfluentPins.lean +``` + +Working directory: +`/workspace/Math-/frontiers/source_derivative_gaussian_20261025` + +Exit code: 0. Raw merged stdout/stderr: empty. +The `.olean` was generated in the package's existing `.lake/build/lib/lean`. + +Direct subprocess-captured evidence: + +- `/workspace/scratch/dm2_landed_composition/source_confluent_final.log` +- `/workspace/scratch/dm2_landed_composition/source_confluent_final.meta` + +Intermediate attempts are retained as `source_confluent_attempt1.log`, +`source_confluent_attempt2.{log,meta}`, `source_confluent_attempt3.{log,meta}` +and `source_confluent_attempt4.{log,meta}`. Attempt 3 completed the mandatory +scope. Attempt 4 added the optional joint derivative vector and exposed a +tactic sequencing issue in an attempted linter cleanup; the final version +uses `all_goals` and compiled cleanly. + +Existing source inputs and `SourceDerivativeGaussian` verification were +reused, not rerun. Per `/workspace/AGENTS.md`, do not repeat completed applicable +checks merely because of a handoff. Root owns package integration, inventory, +final gate receipt and the separate scientific-review process. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-cubic.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-cubic.json new file mode 100644 index 000000000..e93ceb7c8 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-cubic.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_qualification/FalseZeroCubicPin.lean" + ], + "exit_code": 1, + "duration_seconds": 7.52158784866333, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-cubic.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-cubic.log new file mode 100644 index 000000000..8f8e34e89 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-cubic.log @@ -0,0 +1,2 @@ +/workspace/scratch/dm2_landed_composition/confluent_qualification/FalseZeroCubicPin.lean:4:60: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-inverse.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-inverse.json new file mode 100644 index 000000000..0c3e00ba9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-inverse.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_qualification/FalseZeroInverse.lean" + ], + "exit_code": 1, + "duration_seconds": 7.989002466201782, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-inverse.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-inverse.log new file mode 100644 index 000000000..d5d2b39b3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/false-zero-inverse.log @@ -0,0 +1,2 @@ +/workspace/scratch/dm2_landed_composition/confluent_qualification/FalseZeroInverse.lean:4:77: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/kernel.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/kernel.json new file mode 100644 index 000000000..e1aef2ea3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/kernel.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "leanchecker", + "SourceConfluentPins" + ], + "exit_code": 0, + "duration_seconds": 11.284632682800293, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/kernel.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/kernel.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/positive.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/positive.json new file mode 100644 index 000000000..d116dbd17 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/positive.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_qualification/PositiveControls.lean" + ], + "exit_code": 0, + "duration_seconds": 7.29991602897644, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/positive.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/positive.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/qualification_receipt.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/qualification_receipt.json new file mode 100644 index 000000000..17532f6f2 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/qualification_receipt.json @@ -0,0 +1,157 @@ +{ + "scope": "NEW_SOURCE_CONFLUENT_PINS_ONLY_LOCAL_QUALIFICATION", + "scientific_effect": "NONE", + "author_side_agent": "OpenAI/Codex /root/dm2_landed_composition", + "source_path": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025/SourceConfluentPins.lean", + "source_sha256": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "compile_evidence_reused": { + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/SourceConfluentPins.olean", + "SourceConfluentPins.lean" + ], + "exit_code": 0, + "raw_log": "/workspace/scratch/dm2_landed_composition/source_confluent_final.log", + "metadata": "/workspace/scratch/dm2_landed_composition/source_confluent_final.meta" + }, + "axiom_audit": { + "I4SourceConfluentPins.rawTransform_confluentTransform": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceConfluentPins.confluentTransform_rawTransform": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceConfluentPins.confluentEquiv_apply": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceConfluentPins.confluentEquiv_symm_apply": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceConfluentPins.prescribed_target": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceConfluentPins.rawPins_hasGaussianLaw": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceConfluentPins.confluentPins_hasGaussianLaw": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceConfluentPins.rawPinsWithHessians_hasGaussianLaw": [ + "propext", + "Classical.choice", + "Quot.sound" + ] + }, + "checks": { + "kernel": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "leanchecker", + "SourceConfluentPins" + ], + "exit_code": 0, + "duration_seconds": 11.284632682800293, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_qualification/kernel.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855", + "expected_exit_code": 0 + }, + "axioms": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_qualification/AxiomAudit.lean" + ], + "exit_code": 0, + "duration_seconds": 5.699351787567139, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_qualification/axioms.log", + "raw_log_sha256": "88fab8464a91fa4d5c239f06458cff95566fdb5e64228ebada870581771a3a20", + "expected_exit_code": 0 + }, + "positive": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_qualification/PositiveControls.lean" + ], + "exit_code": 0, + "duration_seconds": 7.29991602897644, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_qualification/positive.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855", + "expected_exit_code": 0 + }, + "false-zero-inverse": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_qualification/FalseZeroInverse.lean" + ], + "exit_code": 1, + "duration_seconds": 7.989002466201782, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_qualification/false-zero-inverse.log", + "raw_log_sha256": "abe6a0ad3e7510758b2ce683a3fd3fac8bea7fa5f10a876456d085c0253e7434", + "expected_exit_code": 1 + }, + "false-zero-cubic": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/confluent_qualification/FalseZeroCubicPin.lean" + ], + "exit_code": 1, + "duration_seconds": 7.52158784866333, + "source_sha256_before": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "source_sha256_after": "41e1ea1ef4f9960307552a8271a5c04d2b72d94736b98eb601b51669ed603c3f", + "raw_log": "/workspace/scratch/dm2_landed_composition/confluent_qualification/false-zero-cubic.log", + "raw_log_sha256": "57fa6b8b5bdd65fdd1bd7b77e555773d00a054e1a137f9f368038559afe662e7", + "expected_exit_code": 1 + } + }, + "control_sources": { + "AxiomAudit.lean": "1f14cbf5cbc43c71e5f05ed837fcde4b80f82a27d02cfa6885d77b3cbd2f41c0", + "PositiveControls.lean": "c8954c8a2a67c38b855c3665f8b0738f1ed9488af589ef73ecd95ef2a8fd1d39", + "FalseZeroInverse.lean": "5105e7581dfd422360e48f7399130d830cfaaa27ca9d9f5a9bf1fdf530276799", + "FalseZeroCubicPin.lean": "7d7ada04833212dd49ab73ca02f9af6cbc93a8d18e6efe741dd6eb0ace394c2e" + }, + "not_established": [ + "covariance limits or invertibility", + "actual conditional regression blocks", + "Palm event identification", + "independent scientific review", + "overall campaign completion" + ] +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/run_one.py b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/run_one.py new file mode 100644 index 000000000..6f486db9a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceConfluentPins/run_one.py @@ -0,0 +1,17 @@ +import hashlib,json,pathlib,subprocess,sys,time +p=pathlib.Path(__file__).parent +label=sys.argv[1] +source=pathlib.Path("SourceConfluentPins.lean") +sha=hashlib.sha256(source.read_bytes()).hexdigest() +if label=="kernel": + cmd=["lake","env","leanchecker","SourceConfluentPins"] +else: + cmd=["lake","env","lean",str(p/{"axioms":"AxiomAudit.lean","positive":"PositiveControls.lean","false-zero-inverse":"FalseZeroInverse.lean","false-zero-cubic":"FalseZeroCubicPin.lean"}[label])] +start=time.time() +r=subprocess.run(cmd,stdout=subprocess.PIPE,stderr=subprocess.STDOUT) +(p/(label+".log")).write_bytes(r.stdout) +meta={"cwd":str(pathlib.Path.cwd()),"command":cmd,"exit_code":r.returncode,"duration_seconds":time.time()-start,"source_sha256_before":sha,"source_sha256_after":hashlib.sha256(source.read_bytes()).hexdigest()} +(p/(label+".json")).write_text(json.dumps(meta,indent=2)+"\n") +print(json.dumps(meta)) +print(r.stdout.decode(),end="") +raise SystemExit(r.returncode) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/00-initial-api.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/00-initial-api.command new file mode 100644 index 000000000..1957775c3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/00-initial-api.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean /workspace/scratch/source-contact-covariance/initial_api.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/00-initial-api.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/00-initial-api.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/00-initial-api.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/00-initial-api.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/00-initial-api.log new file mode 100644 index 000000000..375a00c92 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/00-initial-api.log @@ -0,0 +1,4 @@ +/workspace/scratch/source-contact-covariance/initial_api.lean:4:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceContact.sourceContactVector_hasGaussianLaw` +/workspace/scratch/source-contact-covariance/initial_api.lean:5:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceContact.sourceContactCovariance_eq_coefficientGram` +/workspace/scratch/source-contact-covariance/initial_api.lean:6:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceContact.sourceContact_selected_columns` +/workspace/scratch/source-contact-covariance/initial_api.lean:7:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceContact.sourceContactCovariance_posDef` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/01-first-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/01-first-compile.command new file mode 100644 index 000000000..0bcaad56f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/01-first-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceContactCovariance.olean SourceContactCovariance.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/01-first-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/01-first-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/01-first-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/01-first-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/01-first-compile.log new file mode 100644 index 000000000..9f982c77e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/01-first-compile.log @@ -0,0 +1,312 @@ +SourceContactCovariance.lean:25:2: error(nested.lean.propRecLargeElim): Tactic `cases` failed with a nested error: +Tactic `induction` failed: recursor `List.Mem.casesOn` can only eliminate into `Prop` + +i : Fin 7 +this✝ : i ∈ Fintype.elems +a✝ : List (Fin 7) +this : List.Mem i a✝ +⊢ List.finRange 7 = a✝ → this✝ ≍ this → Fin (contactOrder i) → Fin 2 → ℝ +SourceContactCovariance.lean:26:4: error: Type mismatch + ![] +has type + Fin 0 → ?m.8 +but is expected to have type + Fin (contactOrder i) → Fin 2 → ℝ +SourceContactCovariance.lean:27:2: error: No goals to be solved +SourceContactCovariance.lean:109:11: error: unsolved goals +case succ +L : ℝ +k : ℕ +u : Fin (k + 1) → Fin 2 → ℝ +⊢ (iteratedFDeriv ℝ (k + 1) (modeCoefficient 2 L none) 0) u = 0 +SourceContactCovariance.lean:109:43: error: This simp argument is unused: + modeCoefficient + +Hint: Omit it from the simp argument list. + [apply] simp [derivativeCoefficient, iteratedFDeriv_succ_const] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceContactCovariance.lean:109:60: error: This simp argument is unused: + iteratedFDeriv_succ_const + +Hint: Omit it from the simp argument list. + [apply] simp [derivativeCoefficient, modeCoefficient] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceContactCovariance.lean:120:84: error: `fun_prop` was unable to prove `ContDiff ℝ (↑n) (if false = true then Real.sin else Real.cos)` + +Issues: + No theorems found for `ite` in order to prove `ContDiff ℝ ↑n fun a => (if false = true then Real.sin else Real.cos) a` +SourceContactCovariance.lean:128:13: error: Ambiguous term + smul_apply +Possible interpretations: + _root_.smul_apply : ∀ (f : ?m.260) (r : ?m.259) (x : ?m.261), (r • f) x = r • f x + + Matrix.smul_apply : ∀ (r : ?m.262) (A : Matrix ?m.259 ?m.260 ?m.261) (i : ?m.259) (j : ?m.260), + (r • A) i j = r • A i j +SourceContactCovariance.lean:130:2: error: Tactic `rfl` failed: The left-hand side + (A • (iteratedFDeriv ℝ n g 0).compContinuousLinearMap fun x => ell) u +is not definitionally equal to the right-hand side + √(2 * sourceWeight 2 L ↑k / sourceZ 2 L) * + ((∏ j, (phaseCLM 2 L ↑k) (u j)) * iteratedDeriv n (if b = true then Real.sin else Real.cos) 0) + +L : ℝ +k : ↑(halfLattice 2) +b : Bool +n : ℕ +u : Fin n → Fin 2 → ℝ +A : ℝ := √(2 * sourceWeight 2 L ↑k / sourceZ 2 L) +g : ℝ → ℝ := if b = true then Real.sin else Real.cos +ell : (Fin 2 → ℝ) →L[ℝ] ℝ := phaseCLM 2 L ↑k +hg : ContDiff ℝ (↑n) g +hcomp : ContDiff ℝ (↑n) (g ∘ ⇑ell) +heq : modeCoefficient 2 L (some (k, b)) = A • g ∘ ⇑ell +⊢ (A • (iteratedFDeriv ℝ n g 0).compContinuousLinearMap fun x => ell) u = + √(2 * sourceWeight 2 L ↑k / sourceZ 2 L) * + ((∏ j, (phaseCLM 2 L ↑k) (u j)) * iteratedDeriv n (if b = true then Real.sin else Real.cos) 0) +SourceContactCovariance.lean:120:69: error: This simp argument is unused: + ite_false + +Hint: Omit it from the simp argument list. + [apply] simp only [g, ite_true] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceContactCovariance.lean:128:25: error: This simp argument is unused: + ContinuousMultilinearMap.compContinuousLinearMap_apply + +Hint: Omit it from the simp argument list. + [apply] simp only [smul_apply, map_zero, iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod, smul_eq_mul] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceContactCovariance.lean:129:14: error: This simp argument is unused: + iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod + +Hint: Omit it from the simp argument list. + [apply] simp only [smul_apply, ContinuousMultilinearMap.compContinuousLinearMap_apply, map_zero, smul_eq_mul] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceContactCovariance.lean:129:62: error: This simp argument is unused: + smul_eq_mul + +Hint: Omit it from the simp argument list. + [apply] simp only [smul_apply, ContinuousMultilinearMap.compContinuousLinearMap_apply, map_zero, + iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +SourceContactCovariance.lean:136:69: error: unsolved goals +case «0».«1» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) * ∏ x, sourceC L * sorry x 0 = + √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) + +case «0».«2» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) * ∏ x, sourceC L * sorry x 1 = + √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) + +case «0».«3» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) * ∏ x, (sourceC L * sorry x 0 + sourceC L * sorry x 1) = + √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) + +case «1».«4» +L : ℝ +⊢ ∏ x, sourceC L * sorry x 0 = sourceC L ∨ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «1».«5» +L : ℝ +⊢ ∏ x, sourceC L * sorry x 0 * 2 = sourceC L * 2 ∨ √(sourceWeight 2 L ![2, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «1».«6» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ ∏ x, sourceC L * sorry x 1 = 0 + +case «2».«4» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ ∏ x, sourceC L * sorry x 0 = 0 + +case «2».«5» +L : ℝ +⊢ √(sourceWeight 2 L ![2, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ ∏ x, sourceC L * sorry x 0 * 2 = 0 + +case «2».«6» +L : ℝ +⊢ ∏ x, sourceC L * sorry x 1 = sourceC L ∨ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «3».«1» +L : ℝ +⊢ ∏ x, sourceC L * sorry x 0 = sourceC L ^ 2 ∨ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «3».«2» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ ∏ x, sourceC L * sorry x 1 = 0 + +case «3».«3» +L : ℝ +⊢ ∏ x, (sourceC L * sorry x 0 + sourceC L * sorry x 1) = sourceC L ^ 2 ∨ + √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «4».«1» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ ∏ x, sourceC L * sorry x 0 = 0 + +case «4».«2» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ ∏ x, sourceC L * sorry x 1 = 0 + +case «4».«3» +L : ℝ +⊢ ∏ x, (sourceC L * sorry x 0 + sourceC L * sorry x 1) = sourceC L ^ 2 ∨ + √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «5».«4» +L : ℝ +⊢ ∏ x, sourceC L * sorry x 0 = sourceC L ^ 3 ∨ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «5».«5» +L : ℝ +⊢ ∏ x, sourceC L * sorry x 0 * 2 = sourceC L ^ 3 * 8 ∨ √(sourceWeight 2 L ![2, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «5».«6» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ ∏ x, sourceC L * sorry x 1 = 0 + +case «6».«1» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ ∏ x, sourceC L * sorry x 0 = 0 + +case «6».«2» +L : ℝ +⊢ ∏ x, sourceC L * sorry x 1 = sourceC L ^ 2 ∨ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «6».«3» +L : ℝ +⊢ ∏ x, (sourceC L * sorry x 0 + sourceC L * sorry x 1) = sourceC L ^ 2 ∨ + √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 +SourceContactCovariance.lean:139:41: error: declaration uses `sorry` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/02-dependent-directions-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/02-dependent-directions-compile.command new file mode 100644 index 000000000..0bcaad56f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/02-dependent-directions-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceContactCovariance.olean SourceContactCovariance.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/02-dependent-directions-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/02-dependent-directions-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/02-dependent-directions-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/02-dependent-directions-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/02-dependent-directions-compile.log new file mode 100644 index 000000000..ac33bfca3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/02-dependent-directions-compile.log @@ -0,0 +1,36 @@ +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +SourceContactCovariance.lean:136:0: error: (deterministic) timeout at `whnf`, maximum number of heartbeats (200000) has been reached + +Note: Use `set_option maxHeartbeats ` to set the limit. + +Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command. +SourceContactCovariance.lean:151:8: error: (kernel) unknown constant 'I4SourceContact.sourceContact_selected_columns' diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/03-column-reduction-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/03-column-reduction-compile.command new file mode 100644 index 000000000..0bcaad56f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/03-column-reduction-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceContactCovariance.olean SourceContactCovariance.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/03-column-reduction-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/03-column-reduction-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/03-column-reduction-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/03-column-reduction-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/03-column-reduction-compile.log new file mode 100644 index 000000000..106efdeb2 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/03-column-reduction-compile.log @@ -0,0 +1,867 @@ +SourceContactCovariance.lean:137:80: error: unexpected token 'set_option'; expected 'lemma' +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +SourceContactCovariance.lean:141:69: error: unsolved goals +case «0».«1» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) * + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨0, ⋯⟩ x 0 = + √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) + +case «0».«2» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) * + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨0, ⋯⟩ x 1 = + √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) + +case «0».«3» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) * + ∏ x, + (sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨0, ⋯⟩ x 0 + + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨0, ⋯⟩ x 1) = + √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) + +case «1».«4» +L : ℝ +⊢ ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨1, ⋯⟩ x 0 = + sourceC L ∨ + √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «1».«5» +L : ℝ +⊢ ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨1, ⋯⟩ x 0 * + 2 = + sourceC L * 2 ∨ + √(sourceWeight 2 L ![2, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «1».«6» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨1, ⋯⟩ x 1 = + 0 + +case «2».«4» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨2, ⋯⟩ x 0 = + 0 + +case «2».«5» +L : ℝ +⊢ √(sourceWeight 2 L ![2, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨2, ⋯⟩ x 0 * + 2 = + 0 + +case «2».«6» +L : ℝ +⊢ ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨2, ⋯⟩ x 1 = + sourceC L ∨ + √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «3».«1» +L : ℝ +⊢ ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨3, ⋯⟩ x 0 = + sourceC L ^ 2 ∨ + √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «3».«2» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨3, ⋯⟩ x 1 = + 0 + +case «3».«3» +L : ℝ +⊢ ∏ x, + (sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨3, ⋯⟩ x 0 + + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨3, ⋯⟩ x 1) = + sourceC L ^ 2 ∨ + √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «4».«1» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨4, ⋯⟩ x 0 = + 0 + +case «4».«2» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨4, ⋯⟩ x 1 = + 0 + +case «4».«3» +L : ℝ +⊢ ∏ x, + (sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨4, ⋯⟩ x 0 + + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨4, ⋯⟩ x 1) = + sourceC L ^ 2 ∨ + √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «5».«4» +L : ℝ +⊢ ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨5, ⋯⟩ x 0 = + sourceC L ^ 3 ∨ + √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «5».«5» +L : ℝ +⊢ ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨5, ⋯⟩ x 0 * + 2 = + sourceC L ^ 3 * 8 ∨ + √(sourceWeight 2 L ![2, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «5».«6» +L : ℝ +⊢ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨5, ⋯⟩ x 1 = + 0 + +case «6».«1» +L : ℝ +⊢ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 ∨ + ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨6, ⋯⟩ x 0 = + 0 + +case «6».«2» +L : ℝ +⊢ ∏ x, + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) ![![0, 1], ![0, 1]] + (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨6, ⋯⟩ x 1 = + sourceC L ^ 2 ∨ + √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «6».«3» +L : ℝ +⊢ ∏ x, + (sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨6, ⋯⟩ x 0 + + sourceC L * + Fin.cases (motive := fun x => Fin (contactOrder x) → Fin 2 → ℝ) ![] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ) → Fin 2 → ℝ) ![![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ) → Fin 2 → ℝ) ![![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![0, 1]] + (fun i => + Fin.cases (motive := fun x => Fin (contactOrder x.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![1, 0], ![1, 0], ![1, 0]] + (fun i => + Fin.cases (motive := fun x => + Fin (contactOrder x.succ.succ.succ.succ.succ.succ) → Fin 2 → ℝ) + ![![0, 1], ![0, 1]] (fun i => i.elim0) i) + i) + i) + i) + i) + i) + ⟨6, ⋯⟩ x 1) = + sourceC L ^ 2 ∨ + √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/04-normalized-row-types-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/04-normalized-row-types-compile.command new file mode 100644 index 000000000..0bcaad56f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/04-normalized-row-types-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceContactCovariance.olean SourceContactCovariance.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/04-normalized-row-types-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/04-normalized-row-types-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/04-normalized-row-types-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/04-normalized-row-types-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/04-normalized-row-types-compile.log new file mode 100644 index 000000000..fac0ba312 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/04-normalized-row-types-compile.log @@ -0,0 +1,80 @@ +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +Try this: + [apply] ring_nf + + The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form. + + Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead. +SourceContactCovariance.lean:141:69: error: unsolved goals +case «1».«5» +L : ℝ +⊢ True ∨ √(sourceWeight 2 L ![2, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «3».«1» +L : ℝ +⊢ True ∨ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «3».«3» +L : ℝ +⊢ True ∨ √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «4».«3» +L : ℝ +⊢ True ∨ √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «5».«4» +L : ℝ +⊢ True ∨ √(sourceWeight 2 L ![1, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «5».«5» +L : ℝ +⊢ True ∨ √(sourceWeight 2 L ![2, 0] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «6».«2» +L : ℝ +⊢ True ∨ √(sourceWeight 2 L ![0, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 + +case «6».«3» +L : ℝ +⊢ True ∨ √(sourceWeight 2 L ![1, 1] * (sourceZ 2 L)⁻¹ * 2) = 0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/05-column-identities-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/05-column-identities-compile.command new file mode 100644 index 000000000..0bcaad56f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/05-column-identities-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceContactCovariance.olean SourceContactCovariance.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/05-column-identities-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/05-column-identities-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/05-column-identities-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/05-column-identities-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/05-column-identities-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/06-controls.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/06-controls.command new file mode 100644 index 000000000..3ef2ec895 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/06-controls.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/source-contact-covariance/controls.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/06-controls.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/06-controls.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/06-controls.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/06-controls.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/06-controls.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/07-axiom-audit.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/07-axiom-audit.command new file mode 100644 index 000000000..0ad187151 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/07-axiom-audit.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/source-contact-covariance/axiom_audit.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/07-axiom-audit.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/07-axiom-audit.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/07-axiom-audit.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/07-axiom-audit.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/07-axiom-audit.log new file mode 100644 index 000000000..47db98052 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/07-axiom-audit.log @@ -0,0 +1,7 @@ +'I4SourceContact.sourceContactVector_hasGaussianLaw' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactCoefficient_summable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactCovariance_eq_coefficientGram' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactMode_nonzero_frequency' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactAmplitude_pos' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContact_selected_columns' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactCovariance_posDef' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/08-kernel-replay.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/08-kernel-replay.command new file mode 100644 index 000000000..31f2d8f31 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/08-kernel-replay.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env leanchecker SourceContactCovariance diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/08-kernel-replay.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/08-kernel-replay.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/08-kernel-replay.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/08-kernel-replay.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/08-kernel-replay.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/axiom_audit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/axiom_audit.lean.txt new file mode 100644 index 000000000..f8be15f4b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/axiom_audit.lean.txt @@ -0,0 +1,9 @@ +import SourceContactCovariance + +#print axioms I4SourceContact.sourceContactVector_hasGaussianLaw +#print axioms I4SourceContact.sourceContactCoefficient_summable +#print axioms I4SourceContact.sourceContactCovariance_eq_coefficientGram +#print axioms I4SourceContact.sourceContactMode_nonzero_frequency +#print axioms I4SourceContact.sourceContactAmplitude_pos +#print axioms I4SourceContact.sourceContact_selected_columns +#print axioms I4SourceContact.sourceContactCovariance_posDef diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/controls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/controls.lean.txt new file mode 100644 index 000000000..2806fd46b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/controls.lean.txt @@ -0,0 +1,40 @@ +import SourceContactCovariance + +open I4Source I4Weights I4SmoothCarrier I4ContactGram I4SourceContact +open MeasureTheory ProbabilityTheory + +example : (sourceContactMode 5).map (fun p => ((p.1 : Fin 2 → ℤ), p.2)) = + some (![2,0], true) := rfl + +example : (sourceContactMode 3).map (fun p => ((p.1 : Fin 2 → ℤ), p.2)) = + some (![1,1], false) := rfl + +example (L : ℝ) : sourceContactCoefficient L 5 (sourceContactMode 5) = + sourceContactAmplitude L 5 * (-8 * sourceC L ^ 3) := by + simpa [contactWitness] using sourceContact_selected_columns L 5 5 + +example (L : ℝ) : sourceContactCoefficient L 4 (sourceContactMode 3) = + sourceContactAmplitude L 3 * (-sourceC L ^ 2) := by + simpa [contactWitness] using sourceContact_selected_columns L 4 3 + +example {L : ℝ} (hL : L ≠ 0) : sourceContactCoefficient L 5 (sourceContactMode 5) ≠ + sourceContactAmplitude L 5 * (-2 * sourceC L ^ 3) := by + have hcol : sourceContactCoefficient L 5 (sourceContactMode 5) = + sourceContactAmplitude L 5 * (-8 * sourceC L ^ 3) := by + simpa [contactWitness] using sourceContact_selected_columns L 5 5 + rw [hcol] + intro h + have hc : sourceC L ≠ 0 := div_ne_zero + (mul_ne_zero (by norm_num) (ne_of_gt Real.pi_pos)) hL + have hp : sourceContactAmplitude L 5 * sourceC L ^ 3 ≠ 0 := + mul_ne_zero (sourceContactAmplitude_pos hL 5).ne' (pow_ne_zero _ hc) + apply hp + nlinarith [h] + +example : (sourceContactCovariance 24).PosDef := sourceContactCovariance_posDef (by norm_num) + +example : (sourceContactCovariance (-24)).PosDef := sourceContactCovariance_posDef (by norm_num) + +example {L : ℝ} (hL : L ≠ 0) : + HasGaussianLaw (sourceContactVector L) (smoothLaw 2 L) := + sourceContactVector_hasGaussianLaw hL diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/initial_api.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/initial_api.lean.txt new file mode 100644 index 000000000..595f50e91 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactCovariance/initial_api.lean.txt @@ -0,0 +1,7 @@ +import SourceDerivativeGaussian +import ContactJetCovariance + +#check I4SourceContact.sourceContactVector_hasGaussianLaw +#check I4SourceContact.sourceContactCovariance_eq_coefficientGram +#check I4SourceContact.sourceContact_selected_columns +#check I4SourceContact.sourceContactCovariance_posDef diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-01.log new file mode 100644 index 000000000..cc49e0268 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-01.log @@ -0,0 +1,28 @@ +SourceContactEndpointParameters.lean:74:39: error(lean.invalidField): Invalid field `div_const`: The environment does not contain `WithTop.LT.div_const`, so it is not possible to project the field `div_const` from an expression + MemLp.neg (sourcePin_memLp hL 5) +of type + WithTop.LT (eLpNorm (-fun ω => sourcePinVector L ω 5) 2 (smoothLaw 2 L)) ⊤ +SourceContactEndpointParameters.lean:75:33: error(lean.invalidField): Invalid field `div_const`: The environment does not contain `WithTop.LT.div_const`, so it is not possible to project the field `div_const` from an expression + sourcePin_memLp hL 5 +of type + WithTop.LT (eLpNorm (fun ω => sourcePinVector L ω 5) 2 (smoothLaw 2 L)) ⊤ +SourceContactEndpointParameters.lean:76:40: error(lean.invalidField): Invalid field `div_const`: The environment does not contain `WithTop.LT.div_const`, so it is not possible to project the field `div_const` from an expression + MemLp.neg (sourceContactA_memLp hL) +of type + WithTop.LT (eLpNorm (-sourceContactA L) 2 (smoothLaw 2 L)) ⊤ +SourceContactEndpointParameters.lean:77:36: error(lean.invalidField): Invalid field `div_const`: The environment does not contain `WithTop.LT.div_const`, so it is not possible to project the field `div_const` from an expression + sourceContactA_memLp hL +of type + WithTop.LT (eLpNorm (sourceContactA L) 2 (smoothLaw 2 L)) ⊤ +SourceContactEndpointParameters.lean:116:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + (?M * ?N) *ᵥ ?v +in the target expression + (sourceContactEndpointRegression L *ᵥ x) 0 = -x 5 / 2 ∧ (sourceContactEndpointRegression L *ᵥ x) 1 = x 5 / 2 + +L : ℝ +hL : L ≠ 0 +x : Fin 6 → ℝ +x✝ : Invertible (sourcePinCovariance L) := ⋯.invertible +hcancel : sourcePinCovariance L *ᵥ (sourcePinCovariance L)⁻¹ *ᵥ x = x +hc : (sourcePinCovariance L *ᵥ (sourcePinCovariance L)⁻¹ *ᵥ x) 5 = x 5 +⊢ (sourceContactEndpointRegression L *ᵥ x) 0 = -x 5 / 2 ∧ (sourceContactEndpointRegression L *ᵥ x) 1 = x 5 / 2 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-02.log new file mode 100644 index 000000000..67507a8cc --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-02.log @@ -0,0 +1,40 @@ +SourceContactEndpointParameters.lean:127:14: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ∑ i ∈ ?s, ?f i / ?a +in the target expression + ∑ x_1, -sourcePinCovariance L 5 x_1 / 2 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1 = + (-∑ i, sourcePinCovariance L 5 i * ∑ i_1, (sourcePinCovariance L)⁻¹ i i_1 * x i_1) / 2 + +case left +L : ℝ +hL : L ≠ 0 +x : Fin 6 → ℝ +x✝ : Invertible (sourcePinCovariance L) := ⋯.invertible +hcancel : sourcePinCovariance L *ᵥ (sourcePinCovariance L)⁻¹ *ᵥ x = x +hc : ∑ i, sourcePinCovariance L 5 i * ∑ i_1, (sourcePinCovariance L)⁻¹ i i_1 * x i_1 = x 5 +⊢ ∑ x_1, -sourcePinCovariance L 5 x_1 / 2 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1 = + (-∑ i, sourcePinCovariance L 5 i * ∑ i_1, (sourcePinCovariance L)⁻¹ i i_1 * x i_1) / 2 + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + function expected + sourcePinCovariance L 5 +SourceContactEndpointParameters.lean:131:14: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ∑ i ∈ ?s, ?f i / ?a +in the target expression + ∑ x_1, sourcePinCovariance L 5 x_1 / 2 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1 = + (∑ i, sourcePinCovariance L 5 i * ∑ i_1, (sourcePinCovariance L)⁻¹ i i_1 * x i_1) / 2 + +case right +L : ℝ +hL : L ≠ 0 +x : Fin 6 → ℝ +x✝ : Invertible (sourcePinCovariance L) := ⋯.invertible +hcancel : sourcePinCovariance L *ᵥ (sourcePinCovariance L)⁻¹ *ᵥ x = x +hc : ∑ i, sourcePinCovariance L 5 i * ∑ i_1, (sourcePinCovariance L)⁻¹ i i_1 * x i_1 = x 5 +⊢ ∑ x_1, sourcePinCovariance L 5 x_1 / 2 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1 = + (∑ i, sourcePinCovariance L 5 i * ∑ i_1, (sourcePinCovariance L)⁻¹ i i_1 * x i_1) / 2 + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + function expected + sourcePinCovariance L 5 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-03.log new file mode 100644 index 000000000..633f05d17 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-03.log @@ -0,0 +1,41 @@ +SourceContactEndpointParameters.lean:198:40: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ∑ i ∈ ?s, ?f i / ?a +in the target expression + ∑ x_1, -sourcePinCovariance L 5 x_1 / 2 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1 = + (∑ x_1, -(sourcePinCovariance L 5 x_1 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1)) / 2 + +case left +L : ℝ +hL : L ≠ 0 +x : Fin 6 → ℝ +x✝ : Invertible (sourcePinCovariance L) := ⋯.invertible +hcancel : sourcePinCovariance L *ᵥ (sourcePinCovariance L)⁻¹ *ᵥ x = x +hc : ∑ j, sourcePinCovariance L 5 j * ((sourcePinCovariance L)⁻¹ *ᵥ x) j = x 5 +⊢ ∑ x_1, -sourcePinCovariance L 5 x_1 / 2 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1 = + (∑ x_1, -(sourcePinCovariance L 5 x_1 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1)) / 2 + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + function expected + sourcePinCovariance L 5 +SourceContactEndpointParameters.lean:202:14: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ∑ i ∈ ?s, ?f i / ?a +in the target expression + ∑ x_1, sourcePinCovariance L 5 x_1 / 2 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1 = + (∑ j, sourcePinCovariance L 5 j * ((sourcePinCovariance L)⁻¹ *ᵥ x) j) / 2 + +case right +L : ℝ +hL : L ≠ 0 +x : Fin 6 → ℝ +x✝ : Invertible (sourcePinCovariance L) := ⋯.invertible +hcancel : sourcePinCovariance L *ᵥ (sourcePinCovariance L)⁻¹ *ᵥ x = x +hc : ∑ j, sourcePinCovariance L 5 j * ((sourcePinCovariance L)⁻¹ *ᵥ x) j = x 5 +⊢ ∑ x_1, sourcePinCovariance L 5 x_1 / 2 * ((sourcePinCovariance L)⁻¹ *ᵥ x) x_1 = + (∑ j, sourcePinCovariance L 5 j * ((sourcePinCovariance L)⁻¹ *ᵥ x) j) / 2 + +Note: The target expression is not type-correct under the `implicit` transparency level, which may have triggered the failure. This is usually caused by unfolding of semireducible definitions in prior tactic steps. Use `set_option linter.tacticCheckInstances true` to investigate the source of the issue. +Full error: + function expected + sourcePinCovariance L 5 +SourceContactEndpointParameters.lean:241:64: error(lean.unknownIdentifier): Unknown constant `Matrix.empty_val` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-04.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-04.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-05.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-05.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-compile-author-history.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-compile-author-history.json new file mode 100644 index 000000000..a04445dd5 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-compile-author-history.json @@ -0,0 +1,19 @@ +{ + "provenance": "author tool-history report recorded after command; not a retrospective machine receipt", + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/SourceContactEndpointParameters.olean", + "SourceContactEndpointParameters.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "observed_exit_code": 0, + "source_sha256": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "source_unchanged_since_final_successful_compile": true, + "log": "/workspace/scratch/contact-endpoint-parameters/contact-parameters-05.log", + "log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855", + "new_kernel_receipt": "contact-parameters-leanchecker.json", + "new_controls_and_axiom_receipt": "contact-parameters-controls-01.json" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-controls-01.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-controls-01.json new file mode 100644 index 000000000..ab8e55591 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-controls-01.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/contact-endpoint-parameters/contact-parameters-controls.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256_before": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "source_sha256_after": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "log_sha256": "3eb74647a2d1698732992034708b9c24d542250b15358cac3db4130130c3183a" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-controls-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-controls-01.log new file mode 100644 index 000000000..c9946b94d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-controls-01.log @@ -0,0 +1,27 @@ +I4SourceContact.sourceContactEndpointVector (L : ℝ) (ω : I4SmoothCarrier.SmoothSample 2 L) : Fin 6 → ℝ +I4SourceContact.sourceContactEndpointCrossCovariance (L : ℝ) : Matrix (Fin 6) (Fin 6) ℝ +I4SourceContact.sourceContactEndpointCovariance (L : ℝ) : Matrix (Fin 6) (Fin 6) ℝ +I4SourceContact.sourceContactEndpointMean_contact {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + sourceContactEndpointMean L b 0 = -1 ∧ + sourceContactEndpointMean L b 1 = 1 ∧ + sourceContactEndpointMean L b 2 + sourceContactEndpointMean L b 3 = 0 ∧ + sourceContactEndpointMean L b 4 - sourceContactEndpointMean L b 5 = 0 +I4SourceContact.sourceContactEndpointSchur_covariance {L : ℝ} (hL : L ≠ 0) : + (∀ (i j : Fin 6), + ProbabilityTheory.covariance (fun ω => sourceContactEndpointResidual L ω i) + (fun ω => sourceContactEndpointResidual L ω j) (I4SmoothCarrier.smoothLaw 2 L) = + sourceContactEndpointSchur L i j) ∧ + (sourceContactEndpointSchur L).PosSemidef +I4SourceContact.sourceContactEndpointSchur_contact {L : ℝ} (hL : L ≠ 0) : + sourceContactEndpointSchur L 0 0 = 0 ∧ + sourceContactEndpointSchur L 1 1 = 0 ∧ + sourceContactEndpointSchur L 2 2 + 2 * sourceContactEndpointSchur L 2 3 + sourceContactEndpointSchur L 3 3 = 0 ∧ + sourceContactEndpointSchur L 4 4 - 2 * sourceContactEndpointSchur L 4 5 + sourceContactEndpointSchur L 5 5 = 0 +I4SourceContact.sourceContactEndpointSchur_referenceVariance_pos {L : ℝ} (hL : L ≠ 0) : + 0 < sourceContactEndpointSchur L 4 4 +'I4SourceContact.sourceContactEndpointMean_contact' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactEndpointSchur_covariance' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactEndpointSchur_contact' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactEndpointSchur_referenceVariance_pos' depends on axioms: [propext, + Classical.choice, + Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-controls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-controls.lean.txt new file mode 100644 index 000000000..22c4404af --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-controls.lean.txt @@ -0,0 +1,33 @@ +import SourceContactEndpointParameters + +open I4SourceContact Matrix +noncomputable section + +-- Actual torus source, arbitrary conditioning level, and a singular endpoint covariance. +example (b : ℝ) : + sourceContactEndpointMean 1 b 0 = -1 ∧ + sourceContactEndpointMean 1 b 1 = 1 ∧ + (sourceContactEndpointSchur 1).PosSemidef ∧ + ¬ (sourceContactEndpointSchur 1).PosDef ∧ + 0 < sourceContactEndpointSchur 1 4 4 := by + have hL : (1 : ℝ) ≠ 0 := by norm_num + have hm := sourceContactEndpointMean_contact hL b + have hz := (sourceContactEndpointSchur_contact hL).1 + refine ⟨hm.1, hm.2.1, sourceContactEndpointSchur_posSemidef hL, ?_, + sourceContactEndpointSchur_referenceVariance_pos hL⟩ + intro hp + have hdiag : 0 < sourceContactEndpointSchur 1 0 0 := hp.diag_pos + rw [hz] at hdiag + exact (lt_irrefl 0) hdiag + +#check sourceContactEndpointVector +#check sourceContactEndpointCrossCovariance +#check sourceContactEndpointCovariance +#check sourceContactEndpointMean_contact +#check sourceContactEndpointSchur_covariance +#check sourceContactEndpointSchur_contact +#check sourceContactEndpointSchur_referenceVariance_pos +#print axioms sourceContactEndpointMean_contact +#print axioms sourceContactEndpointSchur_covariance +#print axioms sourceContactEndpointSchur_contact +#print axioms sourceContactEndpointSchur_referenceVariance_pos diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-leanchecker.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-leanchecker.json new file mode 100644 index 000000000..0be514ba7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-leanchecker.json @@ -0,0 +1,13 @@ +{ + "command": [ + "lake", + "env", + "leanchecker", + "SourceContactEndpointParameters" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256_before": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "source_sha256_after": "da5337a7814a008eaa8b8a98c0df4daf3efb1bc347789a2f75efc63c9c258d01", + "log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-leanchecker.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/contact-parameters-leanchecker.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/generic/residual-probe-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/generic/residual-probe-01.log new file mode 100644 index 000000000..c3f3341a1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/generic/residual-probe-01.log @@ -0,0 +1,41 @@ +/workspace/scratch/contact-endpoint-parameters/generic/residual-probe.lean:52:88: error: unsolved goals +Ω : Type u_1 +ι : Type u_2 +κ : Type u_3 +inst✝⁵ : MeasurableSpace Ω +inst✝⁴ : Fintype ι +inst✝³ : DecidableEq ι +inst✝² : Fintype κ +inst✝¹ : DecidableEq κ +P : Measure Ω +inst✝ : IsFiniteMeasure P +J : Ω → ι → ℝ +T : Ω → κ → ℝ +hJ : ∀ (i : ι), MemLp (fun ω => J ω i) 2 P +hT : ∀ (i : κ), MemLp (fun ω => T ω i) 2 P +hG : IsUnit (of fun i j => cov[fun ω => J ω i, fun ω => J ω j; P]) +G : Matrix ι ι ℝ := fun i j => cov[fun ω => J ω i, fun ω => J ω j; P] +C : Matrix κ ι ℝ := fun i j => cov[fun ω => T ω i, fun ω => J ω j; P] +B : Matrix κ κ ℝ := fun i j => cov[fun ω => T ω i, fun ω => T ω j; P] +D : Matrix κ ι ℝ := C * G⁻¹ +R : Ω → κ → ℝ := fun ω => T ω - D *ᵥ J ω +hG' : IsUnit G +x✝ : Invertible G := hG'.invertible +U : Ω → κ → ℝ := fun ω => D *ᵥ J ω +hU : ∀ (i : κ), MemLp (fun ω => U ω i) 2 P +hR : ∀ (i : κ), MemLp (fun ω => R ω i) 2 P +hnormal : D * G = C +hUJ : ∀ (i : κ) (j : ι), cov[fun ω => U ω i, fun ω => J ω j; P] = C i j +hJR : ∀ (i : ι) (j : κ), cov[fun ω => J ω i, fun ω => R ω j; P] = 0 +hUR : ∀ (i j : κ), cov[fun ω => U ω i, fun ω => R ω j; P] = 0 +i j : κ +a : ι +a✝ : a ∈ Finset.univ +⊢ D i a * cov[fun ω => T ω j, fun ω => J ω a; P] = D i a * (fun i j => cov[fun ω => T ω i, fun ω => J ω j; P])ᵀ a j +/workspace/scratch/contact-endpoint-parameters/generic/residual-probe.lean:56:33: warning: This simp argument is unused: + Matrix.transpose_apply + +Hint: Omit it from the simp argument list. + [apply] simp only [Matrix.mul_apply, C] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/generic/residual-probe-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/generic/residual-probe-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/generic/residual-probe.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/generic/residual-probe.lean.txt new file mode 100644 index 000000000..b13d2f98c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactEndpointParameters/generic/residual-probe.lean.txt @@ -0,0 +1,78 @@ +import SourceEndpointParameters +import SourceContactSchur + +noncomputable section +open MeasureTheory ProbabilityTheory Matrix WithLp +open scoped BigOperators RealInnerProductSpace +set_option autoImplicit false +namespace I4SourceEndpointParameters + +theorem generic_residual_covariance {Ω ι κ : Type*} [MeasurableSpace Ω] + [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] + {P : Measure Ω} [IsFiniteMeasure P] + (J : Ω → ι → ℝ) (T : Ω → κ → ℝ) + (hJ : ∀ i, MemLp (fun ω => J ω i) 2 P) + (hT : ∀ i, MemLp (fun ω => T ω i) 2 P) + (hG : IsUnit (Matrix.of (fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P) : Matrix ι ι ℝ)) : + let G : Matrix ι ι ℝ := fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P + let C : Matrix κ ι ℝ := fun i j => covariance (fun ω => T ω i) (fun ω => J ω j) P + let B : Matrix κ κ ℝ := fun i j => covariance (fun ω => T ω i) (fun ω => T ω j) P + let D := C * G⁻¹ + let R := fun ω => T ω - D *ᵥ J ω + (∀ i j, covariance (fun ω => R ω i) (fun ω => R ω j) P = (B - C * G⁻¹ * Cᵀ) i j) ∧ + (B - C * G⁻¹ * Cᵀ).PosSemidef := by + classical + intro G C B D R + have hG' : IsUnit G := hG + let _ : Invertible G := hG'.invertible + let U := fun ω => D *ᵥ J ω + have hU (i : κ) : MemLp (fun ω => U ω i) 2 P := + memLp_finsetSum Finset.univ (fun a _ => (hJ a).const_mul (D i a)) + have hR (i : κ) : MemLp (fun ω => R ω i) 2 P := (hT i).sub (hU i) + have hnormal : D * G = C := by + dsimp [D] + rw [Matrix.mul_assoc, Matrix.inv_mul_of_invertible, Matrix.mul_one] + have hUJ (i : κ) (j : ι) : covariance (fun ω => U ω i) (fun ω => J ω j) P = C i j := by + change covariance (fun ω => ∑ a, D i a * J ω a) (fun ω => J ω j) P = C i j + rw [covariance_fun_sum_left (fun a => (hJ a).const_mul _) (hJ j)] + simp_rw [covariance_const_mul_left] + exact congrFun (congrFun hnormal i) j + have hJR (i : ι) (j : κ) : covariance (fun ω => J ω i) (fun ω => R ω j) P = 0 := by + change covariance (fun ω => J ω i) (fun ω => T ω j - U ω j) P = 0 + rw [covariance_fun_sub_right (hJ i) (hT j) (hU j), + covariance_comm (fun ω => J ω i) (fun ω => U ω j), hUJ] + change covariance (fun ω => J ω i) (fun ω => T ω j) P - + covariance (fun ω => T ω j) (fun ω => J ω i) P = 0 + rw [covariance_comm, sub_self] + have hUR (i j : κ) : covariance (fun ω => U ω i) (fun ω => R ω j) P = 0 := by + change covariance (fun ω => ∑ a, D i a * J ω a) (fun ω => R ω j) P = 0 + rw [covariance_fun_sum_left (fun a => (hJ a).const_mul _) (hR j)] + simp_rw [covariance_const_mul_left, hJR, mul_zero] + exact Finset.sum_const_zero + have hUT (i j : κ) : covariance (fun ω => U ω i) (fun ω => T ω j) P = (D * Cᵀ) i j := by + change covariance (fun ω => ∑ a, D i a * J ω a) (fun ω => T ω j) P = _ + rw [covariance_fun_sum_left (fun a => (hJ a).const_mul _) (hT j)] + simp_rw [covariance_const_mul_left] + simp only [Matrix.mul_apply, C] + apply Finset.sum_congr rfl + intro a _ + rw [covariance_comm (fun ω => J ω a)] + rfl + have hcov (i j : κ) : covariance (fun ω => R ω i) (fun ω => R ω j) P = (B - C * G⁻¹ * Cᵀ) i j := by + change covariance (fun ω => R ω i) (fun ω => T ω j - U ω j) P = _ + rw [covariance_fun_sub_right (hR i) (hT j) (hU j), + covariance_comm (fun ω => R ω i) (fun ω => U ω j), hUR, sub_zero] + change covariance (fun ω => T ω i - U ω i) (fun ω => T ω j) P = _ + rw [covariance_fun_sub_left (hT i) (hU i) (hT j), hUT] + rfl + refine ⟨hcov, ?_⟩ + apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg + · ext i j + simp only [Matrix.conjTranspose_apply, star_trivial] + rw [← hcov i j, ← hcov j i] + exact covariance_comm _ _ + · intro x + have hp := covarianceBilin_self_nonneg (μ := P.map (fun ω => toLp 2 (R ω))) (toLp 2 x) + rw [covarianceBilin_apply_pi hR] at hp + simpa [hcov, dotProduct, Matrix.mulVec, Finset.mul_sum, mul_assoc, mul_left_comm, mul_comm] using hp +end I4SourceEndpointParameters diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/AxiomAudit-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/AxiomAudit-01.log new file mode 100644 index 000000000..65ea54362 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/AxiomAudit-01.log @@ -0,0 +1,13 @@ +'I4SourceContact.sourceContactBlockEquiv_inl' depends on axioms: [propext, Quot.sound] +'I4SourceContact.sourceContactBlockEquiv_inr' depends on axioms: [propext, Quot.sound] +'I4SourceContact.sourceContactCovariance_block_identity' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourcePinCovariance_posDef' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourcePinCovariance_isUnit' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourcePinCovarianceInvertible' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactQVariance_eq_variance' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactSchur_pos' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.contact_residual_variance_of_isUnit_covariance' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceContact.sourceContactResidual_variance' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceContact.sourceContactResidual_variance_pos' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/AxiomAudit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/AxiomAudit.lean.txt new file mode 100644 index 000000000..6389c8778 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/AxiomAudit.lean.txt @@ -0,0 +1,12 @@ +import SourceContactSchur +#print axioms I4SourceContact.sourceContactBlockEquiv_inl +#print axioms I4SourceContact.sourceContactBlockEquiv_inr +#print axioms I4SourceContact.sourceContactCovariance_block_identity +#print axioms I4SourceContact.sourcePinCovariance_posDef +#print axioms I4SourceContact.sourcePinCovariance_isUnit +#print axioms I4SourceContact.sourcePinCovarianceInvertible +#print axioms I4SourceContact.sourceContactQVariance_eq_variance +#print axioms I4SourceContact.sourceContactSchur_pos +#print axioms I4SourceContact.contact_residual_variance_of_isUnit_covariance +#print axioms I4SourceContact.sourceContactResidual_variance +#print axioms I4SourceContact.sourceContactResidual_variance_pos diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/EVIDENCE.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/EVIDENCE.json new file mode 100644 index 000000000..e477f5f28 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/EVIDENCE.json @@ -0,0 +1,124 @@ +{ + "source": "SourceContactSchur.lean", + "source_sha256": "2fe122a50ec1889b579558d80086251d87ed0cec2420d268ab1d04f6bfff4409", + "source_contract_commit": "3bb99c9", + "author": "OpenAI/Codex /root/regression_kernel with OpenAI/Codex /root/regression_kernel/gaussian_api generic L2 residual-variance proof support", + "scientific_effect": "NONE; author-side evidence; different-family scientific review remains separate", + "build": { + "command": "/home/agent/.elan/bin/lake env lean -o .lake/build/lib/lean/SourceContactSchur.olean SourceContactSchur.lean", + "exit": 0, + "log": "SourceContactSchur-04.log" + }, + "kernel": { + "command": "/home/agent/.elan/bin/lake env leanchecker SourceContactSchur", + "exit": 0, + "log": "KernelReplay-01.log" + }, + "axioms": { + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/AxiomAudit.lean", + "exit": 0, + "targets": { + "I4SourceContact.sourceContactBlockEquiv_inl": [ + "Quot.sound", + "propext" + ], + "I4SourceContact.sourceContactBlockEquiv_inr": [ + "Quot.sound", + "propext" + ], + "I4SourceContact.sourceContactCovariance_block_identity": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceContact.sourcePinCovariance_posDef": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceContact.sourcePinCovariance_isUnit": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceContact.sourcePinCovarianceInvertible": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceContact.sourceContactQVariance_eq_variance": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceContact.sourceContactSchur_pos": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceContact.contact_residual_variance_of_isUnit_covariance": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceContact.sourceContactResidual_variance": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceContact.sourceContactResidual_variance_pos": [ + "Classical.choice", + "Quot.sound", + "propext" + ] + }, + "log": "AxiomAudit-01.log" + }, + "controls": [ + { + "file": "PositiveControls.lean", + "exit": 0, + "log": "PositiveControls-01.log", + "checks": "literal G positive/invertible at \u00b11; actual Schur positive at \u00b11; actual residual variance positive; exact block indices/roundtrip" + }, + { + "file": "RejectedZeroPeriod.lean", + "exit": 1, + "log": "RejectedZeroPeriod-01.log", + "reason": "False nonzero-period side condition" + }, + { + "file": "RejectedWrongQIndex.lean", + "exit": 1, + "log": "RejectedWrongQIndex-01.log", + "reason": "False equality of Q index6 with index0" + } + ], + "logs": { + "SourceContactSchur-04.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "PositiveControls-01.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "RejectedZeroPeriod-01.log": { + "bytes": 120, + "sha256": "a32530bb6df4ff65058d541b0bdd131f54e1cf4c879b308012e30a65fe6a4140" + }, + "RejectedWrongQIndex-01.log": { + "bytes": 121, + "sha256": "fb8ff7ec25f05d726bb11363c9948bfa76eb723c6ed26798d177eac9504675b9" + }, + "AxiomAudit-01.log": { + "bytes": 1163, + "sha256": "f1e25d18acc7628a1300de08d5807a8667fe7f377248e303b9f769f999e621f0" + }, + "KernelReplay-01.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + } + }, + "reuse": "Completed Gaussian regression, contact Gram and literal source covariance evidence reused; only this new module and its controls audited." +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/KernelReplay-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/KernelReplay-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/PositiveControls-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/PositiveControls-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/PositiveControls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/PositiveControls.lean.txt new file mode 100644 index 000000000..40af49867 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/PositiveControls.lean.txt @@ -0,0 +1,14 @@ +import SourceContactSchur +open I4SourceContact I4SmoothCarrier ProbabilityTheory + +example : (sourcePinCovariance 1).PosDef := sourcePinCovariance_posDef (by norm_num) +example : IsUnit (sourcePinCovariance (-1)) := sourcePinCovariance_isUnit (by norm_num) +example : 0 < sourceContactSchur 1 := sourceContactSchur_pos (by norm_num) +example : 0 < sourceContactSchur (-1) := sourceContactSchur_pos (by norm_num) +example : 0 < Var[sourceContactResidual 1; smoothLaw 2 1] := + sourceContactResidual_variance_pos (by norm_num) +example (i : Fin 6) : sourceContactBlockEquiv (Sum.inl i) = i.castSucc := by simp +example : sourceContactBlockEquiv (Sum.inr ()) = (6 : Fin 7) := by simp +example : sourceContactBlockEquiv.symm (6 : Fin 7) = Sum.inr () := by + apply sourceContactBlockEquiv.injective + simp diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedWrongQIndex-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedWrongQIndex-01.log new file mode 100644 index 000000000..79cbdcc04 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedWrongQIndex-01.log @@ -0,0 +1,2 @@ +/workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean:3:64: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedWrongQIndex.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedWrongQIndex.lean.txt new file mode 100644 index 000000000..6c65399f8 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedWrongQIndex.lean.txt @@ -0,0 +1,4 @@ +import SourceContactSchur +open I4SourceContact +example : sourceContactBlockEquiv (Sum.inr ()) = (0 : Fin 7) := by + norm_num diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedZeroPeriod-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedZeroPeriod-01.log new file mode 100644 index 000000000..de936efc7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedZeroPeriod-01.log @@ -0,0 +1,2 @@ +/workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean:3:38: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedZeroPeriod.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedZeroPeriod.lean.txt new file mode 100644 index 000000000..0726152b4 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/RejectedZeroPeriod.lean.txt @@ -0,0 +1,5 @@ +import SourceContactSchur +open I4SourceContact +example : 0 < sourceContactSchur 0 := by + apply sourceContactSchur_pos + norm_num diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-01.log new file mode 100644 index 000000000..8b55c7e82 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-01.log @@ -0,0 +1,24 @@ +SourceContactSchur.lean:79:6: error: Type mismatch + covariance_comm +has type + ∀ (X Y : ?m.168 → ℝ) {μ : Measure ?m.168}, cov[X, Y; μ] = cov[Y, X; μ] +but is expected to have type + cov[fun ω => sourceContactVector L ω 6, fun ω => sourceContactVector L ω j.castSucc; smoothLaw 2 L] = + cov[fun ω => sourceContactVector L ω j.castSucc, fun ω => sourceContactVector L ω 6; smoothLaw 2 L] +SourceContactSchur.lean:84:48: error: Application type mismatch: The argument + Fin.castSucc_injective +has type + ∀ (n : ℕ), Function.Injective Fin.castSucc +but is expected to have type + Function.Injective Fin.castSucc +in the application + PosDef.submatrix (sourceContactCovariance_posDef hL) Fin.castSucc_injective +SourceContactSchur.lean:90:0: warning: Definition `I4SourceContact.sourcePinCovarianceInvertible` of class type is semireducible. Most type class instances should be instance-reducible, so consider marking this +definition with `@[instance_reducible]`. If it is intentionally semireducible, this warning can be disabled with `set_option warn.classDefReducibility false`. +SourceContactSchur.lean:109:2: error: Type mismatch: After simplification, term + hs + has type + ∑ j, (∑ j_1, sourceContactCross L j_1 * (sourcePinCovariance L)⁻¹ j_1 j) * sourceContactCross L j < + sourceContactQVariance L +but is expected to have type + ∑ x, ∑ i, sourceContactCross L x * ((sourcePinCovariance L)⁻¹ x i * sourceContactCross L i) < sourceContactQVariance L diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-02.log new file mode 100644 index 000000000..1adc5caa3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-02.log @@ -0,0 +1,22 @@ +SourceContactSchur.lean:118:4: error: Tactic `apply` failed: could not unify the conclusion of `Finset.sum_congr rfl` + Finset.sum ?m.105 ?m.102 = Finset.sum ?m.105 ?m.103 +with the goal + ∑ x, sourceContactCross L i * (sourcePinCovariance L)⁻¹ i x * sourceContactCross L x = + sourceContactCross L i * ∑ x, (sourcePinCovariance L)⁻¹ i x * sourceContactCross L x + +Note: The full type of `Finset.sum_congr rfl` is + (∀ x ∈ ?m.105, ?m.102 x = ?m.103 x) → Finset.sum ?m.105 ?m.102 = Finset.sum ?m.105 ?m.103 + +L : ℝ +hL : L ≠ 0 +hblock : + (fromBlocks (sourcePinCovariance L) (sourceContactCrossMatrix L) (sourceContactCrossMatrix L)ᴴ + (sourceContactQMatrix L)).PosDef +hs : + 0 < + (sourceContactQMatrix L - (sourceContactCrossMatrix L)ᴴ * (sourcePinCovariance L)⁻¹ * sourceContactCrossMatrix L) () + () +i : Fin 6 +a✝ : i ∈ Finset.univ +⊢ ∑ x, sourceContactCross L i * (sourcePinCovariance L)⁻¹ i x * sourceContactCross L x = + sourceContactCross L i * ∑ x, (sourcePinCovariance L)⁻¹ i x * sourceContactCross L x diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-03.log new file mode 100644 index 000000000..918987da0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-03.log @@ -0,0 +1,8 @@ +SourceContactSchur.lean:189:56: error: unsolved goals +L : ℝ +hL : L ≠ 0 +this : IsProbabilityMeasure (smoothLaw 2 L) +hJ : ∀ (i : Fin 6), MemLp (fun ω => sourcePinVector L ω i) 2 (smoothLaw 2 L) +hQ : MemLp (sourceContactQ L) 2 (smoothLaw 2 L) +⊢ (of fun i j => cov[fun ω => sourcePinVector L ω i, fun ω => sourcePinVector L ω j; smoothLaw 2 L]) = + sourcePinCovariance L diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-04.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceContactSchur/SourceContactSchur-04.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/00-initial-api.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/00-initial-api.command new file mode 100644 index 000000000..98a76b2ab --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/00-initial-api.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean /workspace/scratch/source-derivative-continuity/initial_api.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/00-initial-api.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/00-initial-api.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/00-initial-api.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/00-initial-api.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/00-initial-api.log new file mode 100644 index 000000000..dbbf8307f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/00-initial-api.log @@ -0,0 +1,3 @@ +/workspace/scratch/source-derivative-continuity/initial_api.lean:3:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceDerivativeContinuity.derivativeCoefficient_continuous` +/workspace/scratch/source-derivative-continuity/initial_api.lean:4:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceDerivativeContinuity.smoothSourceField_derivative_covariance_continuous` +/workspace/scratch/source-derivative-continuity/initial_api.lean:5:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceDerivativeContinuity.smoothSourceField_derivative_integral_zero` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/01-first-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/01-first-compile.command new file mode 100644 index 000000000..a78d7bb03 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/01-first-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceDerivativeContinuity.olean SourceDerivativeContinuity.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/01-first-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/01-first-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/01-first-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/01-first-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/01-first-compile.log new file mode 100644 index 000000000..456006a6a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/01-first-compile.log @@ -0,0 +1,14 @@ +SourceDerivativeContinuity.lean:52:9: error(lean.unknownIdentifier): Unknown constant `ContinuousMultilinearMap.continuous_eval_const` +SourceDerivativeContinuity.lean:93:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + sourceLaw d +in the target expression + ∫ (ω : SourceMode d → ℝ), scalarSeries (derivativeCoefficient d L k x u) ω ∂coefficientLaw (SourceMode d) = 0 + +d : ℕ +L : ℝ +hL : L ≠ 0 +k : ℕ +x : Fin d → ℝ +u : Fin k → Fin d → ℝ +h : ∫ (ω : SourceMode d → ℝ), scalarSeries (derivativeCoefficient d L k x u) ω ∂coefficientLaw (SourceMode d) = 0 +⊢ ∫ (ω : SmoothSample d L), scalarSeries (derivativeCoefficient d L k x u) (sampleInclusion d L ω) ∂smoothLaw d L = 0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/02-evaluation-and-law-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/02-evaluation-and-law-compile.command new file mode 100644 index 000000000..a78d7bb03 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/02-evaluation-and-law-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceDerivativeContinuity.olean SourceDerivativeContinuity.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/02-evaluation-and-law-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/02-evaluation-and-law-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/02-evaluation-and-law-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/02-evaluation-and-law-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/02-evaluation-and-law-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/03-controls.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/03-controls.command new file mode 100644 index 000000000..2e113f5c7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/03-controls.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/source-derivative-continuity/controls.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/03-controls.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/03-controls.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/03-controls.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/03-controls.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/03-controls.log new file mode 100644 index 000000000..22199f103 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/03-controls.log @@ -0,0 +1,3 @@ +/workspace/scratch/source-derivative-continuity/controls.lean:39:77: error: `fun_prop` was unable to prove `Continuous ?m.43` + +Issues: diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/04-axiom-audit.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/04-axiom-audit.command new file mode 100644 index 000000000..116394815 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/04-axiom-audit.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/source-derivative-continuity/axiom_audit.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/04-axiom-audit.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/04-axiom-audit.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/04-axiom-audit.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/04-axiom-audit.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/04-axiom-audit.log new file mode 100644 index 000000000..efbd2a059 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/04-axiom-audit.log @@ -0,0 +1,12 @@ +'I4SourceDerivativeContinuity.derivativeMajorant_nonneg' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceDerivativeContinuity.derivativeMajorant_summable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceDerivativeContinuity.derivativeCoefficient_bound' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceDerivativeContinuity.derivativeCoefficient_continuous' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceDerivativeContinuity.smoothSourceField_derivative_covariance_continuous' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceDerivativeContinuity.smoothSourceField_derivative_integral_zero' depends on axioms: [propext, + Classical.choice, + Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/05-controls-explicit-path.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/05-controls-explicit-path.command new file mode 100644 index 000000000..2e113f5c7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/05-controls-explicit-path.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/source-derivative-continuity/controls.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/05-controls-explicit-path.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/05-controls-explicit-path.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/05-controls-explicit-path.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/05-controls-explicit-path.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/05-controls-explicit-path.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/06-kernel-replay.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/06-kernel-replay.command new file mode 100644 index 000000000..a6bc4acc3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/06-kernel-replay.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env leanchecker SourceDerivativeContinuity diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/06-kernel-replay.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/06-kernel-replay.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/06-kernel-replay.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/06-kernel-replay.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/06-kernel-replay.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/axiom_audit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/axiom_audit.lean.txt new file mode 100644 index 000000000..e0a584a8d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/axiom_audit.lean.txt @@ -0,0 +1,8 @@ +import SourceDerivativeContinuity + +#print axioms I4SourceDerivativeContinuity.derivativeMajorant_nonneg +#print axioms I4SourceDerivativeContinuity.derivativeMajorant_summable +#print axioms I4SourceDerivativeContinuity.derivativeCoefficient_bound +#print axioms I4SourceDerivativeContinuity.derivativeCoefficient_continuous +#print axioms I4SourceDerivativeContinuity.smoothSourceField_derivative_covariance_continuous +#print axioms I4SourceDerivativeContinuity.smoothSourceField_derivative_integral_zero diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/controls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/controls.lean.txt new file mode 100644 index 000000000..d115e381a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/controls.lean.txt @@ -0,0 +1,40 @@ +import SourceDerivativeContinuity + +open I4SmoothCarrier I4SourceDerivativeContinuity MeasureTheory ProbabilityTheory + +example (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + Continuous (fun xy : (Fin d → ℝ) × (Fin d → ℝ) => + cov[fun ω => smoothSourceField d L ω xy.1, + fun ω => smoothSourceField d L ω xy.2; smoothLaw d L]) := by + simpa only [iteratedFDeriv_zero_apply] using + smoothSourceField_derivative_covariance_continuous d hL 0 0 + (fun i => Fin.elim0 i) (fun i => Fin.elim0 i) + +example (d : ℕ) {L : ℝ} (hL : L ≠ 0) (k : ℕ) + (u : Fin k → (Fin d → ℝ)) : + Continuous (fun x : Fin d → ℝ => + cov[fun ω => iteratedFDeriv ℝ k (smoothSourceField d L ω) x u, + fun ω => iteratedFDeriv ℝ k (smoothSourceField d L ω) x u; smoothLaw d L]) := + (smoothSourceField_derivative_covariance_continuous d hL k k u u).comp + (continuous_id.prodMk continuous_id) + +example {L : ℝ} (hL : L ≠ 0) (k l : ℕ) + (u : Fin k → (Fin 0 → ℝ)) (v : Fin l → (Fin 0 → ℝ)) : + Continuous (fun xy : (Fin 0 → ℝ) × (Fin 0 → ℝ) => + cov[fun ω => iteratedFDeriv ℝ k (smoothSourceField 0 L ω) xy.1 u, + fun ω => iteratedFDeriv ℝ l (smoothSourceField 0 L ω) xy.2 v; smoothLaw 0 L]) := + smoothSourceField_derivative_covariance_continuous 0 hL k l u v + +example (d : ℕ) {L : ℝ} (hL : L ≠ 0) (x : Fin d → ℝ) : + (∫ ω, smoothSourceField d L ω x ∂smoothLaw d L) = 0 := by + simpa only [iteratedFDeriv_zero_apply] using + smoothSourceField_derivative_integral_zero d hL 0 x (fun i => Fin.elim0 i) + +example (d : ℕ) {L : ℝ} (hL : L ≠ 0) (k l : ℕ) + (u : Fin k → (Fin d → ℝ)) (v : Fin l → (Fin d → ℝ)) + (a b : Fin d → ℝ) : + Continuous (fun r : ℝ => + cov[fun ω => iteratedFDeriv ℝ k (smoothSourceField d L ω) (r • a) u, + fun ω => iteratedFDeriv ℝ l (smoothSourceField d L ω) (r • b) v; smoothLaw d L]) := + (smoothSourceField_derivative_covariance_continuous d hL k l u v).comp + (show Continuous (fun r : ℝ => (r • a, r • b)) from by fun_prop) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/initial_api.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/initial_api.lean.txt new file mode 100644 index 000000000..83d1ecfa4 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeContinuity/initial_api.lean.txt @@ -0,0 +1,5 @@ +import SourceDerivativeGaussian + +#check I4SourceDerivativeContinuity.derivativeCoefficient_continuous +#check I4SourceDerivativeContinuity.smoothSourceField_derivative_covariance_continuous +#check I4SourceDerivativeContinuity.smoothSourceField_derivative_integral_zero diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/00-required-input-compilation.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/00-required-input-compilation.command new file mode 100644 index 000000000..7a7061393 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/00-required-input-compilation.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake build SourceDerivativeInputs diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/00-required-input-compilation.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/00-required-input-compilation.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/00-required-input-compilation.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/00-required-input-compilation.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/00-required-input-compilation.log new file mode 100644 index 000000000..2a69a6d2c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/00-required-input-compilation.log @@ -0,0 +1,17 @@ +✔ [3354/3366] Built GaussianSeriesFoundation (3.0s) +✔ [3355/3366] Built SpectralWeights (2.0s) +✔ [3356/3366] Built SourceFourierSeries (2.9s) +✔ [3357/3366] Built SourceC4 (5.4s) +✔ [3358/3366] Built SourceLattice (5.4s) +✔ [3359/3366] Built SourceGaussianLaw (5.7s) +⚠ [3360/3366] Built SourceSmooth (5.3s) +warning: inputs/SourceSmooth.lean:48:15: try 'simp' instead of 'simpa' + +Note: This linter can be disabled with `set_option linter.unnecessarySimpa false` +✔ [3361/3366] Built SourceKernel (23s) +✔ [3362/3366] Built SourceCovariance (2.9s) +✔ [3363/3366] Built SourceTorus (2.3s) +✔ [3364/3366] Built SourceTorusC4 (2.7s) +⚠ [3365/3366] Built SourceSmoothCarrier (2.6s) +warning: inputs/SourceSmoothCarrier.lean:67:24: `Set.mem_setOf_eq` has been deprecated: Use `Set.mem_ofPred_eq` instead +Build completed successfully (3366 jobs). diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/01-initial-api.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/01-initial-api.command new file mode 100644 index 000000000..56b7d1d4f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/01-initial-api.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean /workspace/scratch/source-derivative-gaussian/initial_api.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/01-initial-api.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/01-initial-api.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/01-initial-api.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/01-initial-api.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/01-initial-api.log new file mode 100644 index 000000000..13ddad5e3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/01-initial-api.log @@ -0,0 +1,4 @@ +/workspace/scratch/source-derivative-gaussian/initial_api.lean:3:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceDerivativeGaussian.derivativeCoefficient_summable` +/workspace/scratch/source-derivative-gaussian/initial_api.lean:4:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceDerivativeGaussian.smoothSourceField_derivative_series` +/workspace/scratch/source-derivative-gaussian/initial_api.lean:5:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceDerivativeGaussian.scalarSeries_finiteVector_hasGaussianLaw` +/workspace/scratch/source-derivative-gaussian/initial_api.lean:6:7: error(lean.unknownIdentifier): Unknown identifier `I4SourceDerivativeGaussian.smoothSourceField_derivativeVector_hasGaussianLaw` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/02-first-module-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/02-first-module-compile.command new file mode 100644 index 000000000..a41c82332 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/02-first-module-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceDerivativeGaussian.olean SourceDerivativeGaussian.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/02-first-module-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/02-first-module-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/02-first-module-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/02-first-module-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/02-first-module-compile.log new file mode 100644 index 000000000..78d7badda --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/02-first-module-compile.log @@ -0,0 +1,30 @@ +SourceDerivativeGaussian.lean:37:41: error: Application type mismatch: The argument + le_top +has type + ?m.47 ≤ ⊤ +but is expected to have type + ↑k ≤ ↑⊤ +in the application + ContDiff.of_le (modeCoefficient_smooth d L m) le_top +SourceDerivativeGaussian.lean:81:42: error: Application type mismatch: The argument + le_top +has type + ?m.55 ≤ ⊤ +but is expected to have type + ↑↑k ≤ ↑⊤ +in the application + ContDiff.of_le (modeCoefficient_smooth d L m) le_top +SourceDerivativeGaussian.lean:131:56: error: unsolved goals +ι : Type u_1 +inst✝ : Countable ι +n : ℕ +a : Fin n → ι → ℝ +ha : ∀ (j : Fin n), Summable (a j) +hm : Measurable fun ω j => scalarSeries (a j) ω +ell : (Fin n → ℝ) →L[ℝ] ℝ +c : Fin n → ℝ := fun j => ell (Pi.single j 1) +z : Fin n → ℝ +j : Fin n +a✝ : j ∈ Finset.univ +hs : Pi.single j (z j) = z j • Pi.single j 1 +⊢ z j * ell (Pi.single j 1) = (∑ i, Pi.single i (z i)) j * ell (Pi.single j 1) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/03-order-coercions-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/03-order-coercions-compile.command new file mode 100644 index 000000000..a41c82332 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/03-order-coercions-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceDerivativeGaussian.olean SourceDerivativeGaussian.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/03-order-coercions-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/03-order-coercions-compile.exitcode new file mode 100644 index 000000000..d00491fd7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/03-order-coercions-compile.exitcode @@ -0,0 +1 @@ +1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/03-order-coercions-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/03-order-coercions-compile.log new file mode 100644 index 000000000..fe1adef10 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/03-order-coercions-compile.log @@ -0,0 +1,42 @@ +SourceDerivativeGaussian.lean:97:41: error: `ContinuousMultilinearMap.smul_apply` has been deprecated: Use `smul_apply` instead + +Note: The updated constant is in a different namespace. Dot notation may need to be changed (e.g., from `x.smul_apply` to `smul_apply x`). +SourceDerivativeGaussian.lean:76:80: error: unsolved goals +d : ℕ +L : ℝ +hL : L ≠ 0 +ω : SmoothSample d L +k : ℕ +x : Fin d → ℝ +u : Fin k → Fin d → ℝ +f : SourceMode d → (Fin d → ℝ) → ℝ := fun m y => modeCoefficient d L m y * sampleInclusion d L ω m +major : ℕ → SourceMode d → ℝ := fun j m => orderModeBound j d L m * |sampleInclusion d L ω m| +hf : ∀ (m : SourceMode d), ContDiff ℝ (↑↑k) (f m) +hmajor : ∀ (j : ℕ), ↑j ≤ ↑k → Summable (major j) +hbound : ∀ (j : ℕ) (m : SourceMode d) (y : Fin d → ℝ), ↑j ≤ ↑k → ‖iteratedFDeriv ℝ j (f m) y‖ ≤ major j m +hsum : Summable fun m => iteratedFDeriv ℝ k (f m) x +hdiff : iteratedFDeriv ℝ k (fun y => ∑' (n : SourceMode d), f n y) x = ∑' (n : SourceMode d), iteratedFDeriv ℝ k (f n) x +m : SourceMode d +⊢ (iteratedFDeriv ℝ k (fun y => sampleInclusion d L ω m * modeCoefficient d L m y) x) u = + sampleInclusion d L ω m * (iteratedFDeriv ℝ k (modeCoefficient d L m) x) u +SourceDerivativeGaussian.lean:97:16: error: This simp argument is unused: + weighted_iteratedFDeriv + +Hint: Omit it from the simp argument list. + [apply] simp only [f, ContinuousMultilinearMap.smul_apply, smul_eq_mul, derivativeCoefficient, mul_comm] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceDerivativeGaussian.lean:97:41: error: This simp argument is unused: + ContinuousMultilinearMap.smul_apply + +Hint: Omit it from the simp argument list. + [apply] simp only [f, weighted_iteratedFDeriv, smul_eq_mul, derivativeCoefficient, mul_comm] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +SourceDerivativeGaussian.lean:98:4: error: This simp argument is unused: + smul_eq_mul + +Hint: Omit it from the simp argument list. + [apply] simp only [f, weighted_iteratedFDeriv, ContinuousMultilinearMap.smul_apply, derivativeCoefficient, mul_comm] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/04-core-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/04-core-compile.command new file mode 100644 index 000000000..a41c82332 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/04-core-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceDerivativeGaussian.olean SourceDerivativeGaussian.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/04-core-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/04-core-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/04-core-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/04-core-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/04-core-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/05-covariance-compile.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/05-covariance-compile.command new file mode 100644 index 000000000..a41c82332 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/05-covariance-compile.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true -o .lake/build/lib/lean/SourceDerivativeGaussian.olean SourceDerivativeGaussian.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/05-covariance-compile.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/05-covariance-compile.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/05-covariance-compile.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/05-covariance-compile.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/05-covariance-compile.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/06-controls.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/06-controls.command new file mode 100644 index 000000000..9e434ed5a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/06-controls.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/source-derivative-gaussian/controls.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/06-controls.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/06-controls.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/06-controls.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/06-controls.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/06-controls.log new file mode 100644 index 000000000..39bda87d8 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/06-controls.log @@ -0,0 +1,19 @@ +I4SourceDerivativeGaussian.derivativeCoefficient (d : ℕ) (L : ℝ) (k : ℕ) (x : Fin d → ℝ) (u : Fin k → Fin d → ℝ) + (m : I4Source.SourceMode d) : ℝ +I4SourceDerivativeGaussian.derivativeCoefficient_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) (k : ℕ) (x : Fin d → ℝ) + (u : Fin k → Fin d → ℝ) : Summable (derivativeCoefficient d L k x u) +I4SourceDerivativeGaussian.smoothSourceField_derivative_series (d : ℕ) {L : ℝ} (hL : L ≠ 0) (ω : SmoothSample d L) + (k : ℕ) (x : Fin d → ℝ) (u : Fin k → Fin d → ℝ) : + (iteratedFDeriv ℝ k (smoothSourceField d L ω) x) u = + scalarSeries (derivativeCoefficient d L k x u) (sampleInclusion d L ω) +I4SourceDerivativeGaussian.scalarSeries_finiteVector_hasGaussianLaw.{u_1} {ι : Type u_1} [Countable ι] (n : ℕ) + (a : Fin n → ι → ℝ) (ha : ∀ (j : Fin n), Summable (a j)) : + HasGaussianLaw (fun ω j => scalarSeries (a j) ω) (coefficientLaw ι) +I4SourceDerivativeGaussian.smoothSourceField_derivativeVector_hasGaussianLaw (d : ℕ) {L : ℝ} (hL : L ≠ 0) (n : ℕ) + (k : Fin n → ℕ) (x : Fin n → Fin d → ℝ) (u : (j : Fin n) → Fin (k j) → Fin d → ℝ) : + HasGaussianLaw (fun ω j => (iteratedFDeriv ℝ (k j) (smoothSourceField d L ω) (x j)) (u j)) (smoothLaw d L) +I4SourceDerivativeGaussian.smoothSourceField_derivative_covariance (d : ℕ) {L : ℝ} (hL : L ≠ 0) (k l : ℕ) + (x y : Fin d → ℝ) (u : Fin k → Fin d → ℝ) (v : Fin l → Fin d → ℝ) : + cov[fun ω => (iteratedFDeriv ℝ k (smoothSourceField d L ω) x) u, fun ω => + (iteratedFDeriv ℝ l (smoothSourceField d L ω) y) v; smoothLaw d L] = + ∑' (m : I4Source.SourceMode d), derivativeCoefficient d L k x u m * derivativeCoefficient d L l y v m diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/07-axiom-audit.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/07-axiom-audit.command new file mode 100644 index 000000000..21826483f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/07-axiom-audit.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env lean -DwarningAsError=true /workspace/scratch/source-derivative-gaussian/axiom_audit.lean diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/07-axiom-audit.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/07-axiom-audit.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/07-axiom-audit.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/07-axiom-audit.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/07-axiom-audit.log new file mode 100644 index 000000000..dd3545a92 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/07-axiom-audit.log @@ -0,0 +1,13 @@ +'I4SourceDerivativeGaussian.derivativeCoefficient_summable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceDerivativeGaussian.smoothSourceField_derivative_series' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceDerivativeGaussian.scalarSeries_finiteVector_hasGaussianLaw' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceDerivativeGaussian.smoothSourceField_derivativeVector_hasGaussianLaw' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceDerivativeGaussian.smoothSourceField_derivative_covariance' depends on axioms: [propext, + Classical.choice, + Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/08-kernel-replay.command b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/08-kernel-replay.command new file mode 100644 index 000000000..70f837e86 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/08-kernel-replay.command @@ -0,0 +1 @@ +/home/agent/.elan/bin/lake env leanchecker SourceDerivativeGaussian diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/08-kernel-replay.exitcode b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/08-kernel-replay.exitcode new file mode 100644 index 000000000..573541ac9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/08-kernel-replay.exitcode @@ -0,0 +1 @@ +0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/08-kernel-replay.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/08-kernel-replay.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/axiom_audit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/axiom_audit.lean.txt new file mode 100644 index 000000000..f9cfc2db9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/axiom_audit.lean.txt @@ -0,0 +1,7 @@ +import SourceDerivativeGaussian + +#print axioms I4SourceDerivativeGaussian.derivativeCoefficient_summable +#print axioms I4SourceDerivativeGaussian.smoothSourceField_derivative_series +#print axioms I4SourceDerivativeGaussian.scalarSeries_finiteVector_hasGaussianLaw +#print axioms I4SourceDerivativeGaussian.smoothSourceField_derivativeVector_hasGaussianLaw +#print axioms I4SourceDerivativeGaussian.smoothSourceField_derivative_covariance diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/controls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/controls.lean.txt new file mode 100644 index 000000000..79ef1662f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/controls.lean.txt @@ -0,0 +1,41 @@ +import SourceDerivativeGaussian + +open MeasureTheory ProbabilityTheory I4SmoothCarrier I4SourceDerivativeGaussian +open I4Foundation +open scoped NNReal + +#check derivativeCoefficient +#check derivativeCoefficient_summable +#check smoothSourceField_derivative_series +#check scalarSeries_finiteVector_hasGaussianLaw +#check smoothSourceField_derivativeVector_hasGaussianLaw +#check smoothSourceField_derivative_covariance + +example (d : ℕ) {L : ℝ} (hL : L ≠ 0) (ω : SmoothSample d L) (x : Fin d → ℝ) : + scalarSeries (derivativeCoefficient d L 0 x (fun i => Fin.elim0 i)) + (sampleInclusion d L ω) = smoothSourceField d L ω x := by + simpa only [iteratedFDeriv_zero_apply] using + (smoothSourceField_derivative_series d hL ω 0 x (fun i => Fin.elim0 i)).symm + +example (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + HasGaussianLaw (fun ω (_j : Fin 0) => smoothSourceField d L ω 0) (smoothLaw d L) := by + simpa only [iteratedFDeriv_zero_apply] using + smoothSourceField_derivativeVector_hasGaussianLaw d hL 0 + (fun _ => 0) (fun _ => 0) (fun _ _ => 0) + +example {L : ℝ} (hL : L ≠ 0) (n : ℕ) (k : Fin n → ℕ) + (u : ∀ j, Fin (k j) → (Fin 0 → ℝ)) : + HasGaussianLaw (fun ω j => + iteratedFDeriv ℝ (k j) (smoothSourceField 0 L ω) 0 (u j)) (smoothLaw 0 L) := + smoothSourceField_derivativeVector_hasGaussianLaw 0 hL n k (fun _ => 0) u + +example (d : ℕ) {L : ℝ} (hL : L ≠ 0) (x : Fin d → ℝ) + (u : Fin 2 → (Fin d → ℝ)) : + cov[fun ω => iteratedFDeriv ℝ 2 (smoothSourceField d L ω) x u, + fun ω => iteratedFDeriv ℝ 2 (smoothSourceField d L ω) x u; smoothLaw d L] = + ∑' m, derivativeCoefficient d L 2 x u m ^ 2 := by + simpa only [pow_two] using smoothSourceField_derivative_covariance d hL 2 2 x x u u + +example (d : ℕ) (L : ℝ) (ω : SmoothSample d L) (x : Fin d → ℝ) : + iteratedFDeriv ℝ 2 (smoothSourceField d L ω) x ![0,0] = 0 := by + simp [iteratedFDeriv_two_apply] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/initial_api.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/initial_api.lean.txt new file mode 100644 index 000000000..33836459e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceDerivativeGaussian/initial_api.lean.txt @@ -0,0 +1,6 @@ +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic + +#check I4SourceDerivativeGaussian.derivativeCoefficient_summable +#check I4SourceDerivativeGaussian.smoothSourceField_derivative_series +#check I4SourceDerivativeGaussian.scalarSeries_finiteVector_hasGaussianLaw +#check I4SourceDerivativeGaussian.smoothSourceField_derivativeVector_hasGaussianLaw diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/ActualAssembly.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/ActualAssembly.lean.txt new file mode 100644 index 000000000..e8e5af951 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/ActualAssembly.lean.txt @@ -0,0 +1,104 @@ +/- DRAFT ONLY: not compiled and not a completed mathematical claim. +The three actual SourceConfluentLimits exports referenced below are still +owned by /root/uniform_jet_moment_sources. No desired conclusion is assumed. +Move into production only when those actual proofs/oleans are ready. -/ + +namespace I4SourceEndpointEstimates + +open I4SourceContact I4SourceConfluentLimits I4GaussianRegressionLimits + +theorem sourceEndpoint_parameter_limits {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + (∀ᶠ r in 𝓝[>] (0 : ℝ), IsUnit (sourcePinCovarianceAt L r)) ∧ + Tendsto (fun r => toLp 2 (sourceEndpointMean L r b)) (𝓝[>] (0 : ℝ)) + (𝓝 (toLp 2 (sourceContactEndpointMean L b))) ∧ + Tendsto (sourceEndpointSchurAt L) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactEndpointSchur L)) := by + have hd : Tendsto (fun r => sourcePrescribedPins r b) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactPinValue b)) := by + simpa only [sourceContactPinValue] using sourcePrescribedPins_tendsto b + have hp := regression_parameter_limits (𝓝[>] (0 : ℝ)) + (sourcePinCovarianceAt L) (sourceCrossCovarianceAt L) (sourceEndpointCovarianceAt L) + (fun r => sourcePrescribedPins r b) + (sourcePinCovariance L) (sourceContactEndpointCrossCovariance L) + (sourceContactEndpointCovariance L) (sourceContactPinValue b) + (sourcePinCovarianceAt_tendsto hL) (sourceCrossCovarianceAt_tendsto hL) + (sourceEndpointCovarianceAt_tendsto hL) hd (sourcePinCovariance_isUnit hL) + refine ⟨hp.1, ?_, ?_⟩ + · have hm : Tendsto (fun r => sourceEndpointMean L r b) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactEndpointMean L b)) := by + simpa only [regressionMean, sourceEndpointMean, sourceRegressionMapAt_apply, + sourceRegressionMatrixAt, sourceContactEndpointMean, sourceContactEndpointRegression] + using hp.2.2.1 + exact ((PiLp.continuous_toLp 2 (fun _ : Fin 6 => ℝ)).tendsto _).comp hm + · simpa only [regressionCovariance, sourceEndpointSchurAt, sourceContactEndpointSchur] + using hp.2.2.2 + +theorem sourceEndpoint_uniform_parameter_bounds {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + ∃ M V δ : ℝ, 0 ≤ M ∧ 0 ≤ V ∧ 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → IsUnit (sourcePinCovarianceAt L r) ∧ + (∀ i, |sourceEndpointMean L r b i| ≤ M) ∧ + (∀ i, |sourceEndpointSchurAt L r i i| ≤ V) := by + have hd : Tendsto (fun r => sourcePrescribedPins r b) (𝓝[>] (0 : ℝ)) + (𝓝 (sourceContactPinValue b)) := by + simpa only [sourceContactPinValue] using sourcePrescribedPins_tendsto b + obtain ⟨M,V,δ,hM,hV,hδ,hbounds⟩ := regression_uniform_bounds + (sourcePinCovarianceAt L) (sourceCrossCovarianceAt L) (sourceEndpointCovarianceAt L) + (fun r => sourcePrescribedPins r b) + (sourcePinCovariance L) (sourceContactEndpointCrossCovariance L) + (sourceContactEndpointCovariance L) (sourceContactPinValue b) + (sourcePinCovarianceAt_tendsto hL) (sourceCrossCovarianceAt_tendsto hL) + (sourceEndpointCovarianceAt_tendsto hL) hd (sourcePinCovariance_isUnit hL) + refine ⟨M,V,δ,hM,hV,hδ,?_⟩ + intro r hr hrδ + obtain ⟨hunit,hm,hcov⟩ := hbounds r hr hrδ + refine ⟨hunit,?_,?_⟩ + · simpa only [regressionMean, sourceEndpointMean, sourceRegressionMapAt_apply, + sourceRegressionMatrixAt] using hm + · simpa only [regressionCovariance, sourceEndpointSchurAt] using hcov + +/- Insert sourceEndpoint_normalizer_floor only after its actual-limit proof +is completed. The following final composition consumes that proved result. -/ + +theorem sourceEndpoint_estimates {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + ∃ C c δ : ℝ, 0 ≤ C ∧ 0 < c ∧ 0 < δ ∧ δ ≤ 1 ∧ + ∀ r : ℝ, 0 < r → r < δ → + Integrable (fun v => sourceEndpointWeight r v ^ 2) (sourceEndpointLaw L r b) ∧ + (∫ v, sourceEndpointWeight r v ^ 2 ∂sourceEndpointLaw L r b) ≤ C*r^4 ∧ + c*r^2 ≤ ∫ v, sourceEndpointWeight r v ∂sourceEndpointLaw L r b := by + obtain ⟨M,V,δM,hM,hV,hδM,hbounds⟩ := sourceEndpoint_uniform_parameter_bounds hL b + obtain ⟨c,hc,δF,hδF,hfloor⟩ := sourceEndpoint_normalizer_floor hL b + refine ⟨16 * gm8FiniteM8 6 M V,c,min δM (min δF 1), + mul_nonneg (by norm_num) (gm8FiniteM8_nonneg 6 M V),hc, + lt_min hδM (lt_min hδF (by norm_num)), + (min_le_right _ _).trans (min_le_right _ _),?_⟩ + intro r hr hrδ + have hrM : r < δM := lt_of_lt_of_le hrδ (min_le_left _ _) + have hrF : r < δF := lt_of_lt_of_le hrδ ((min_le_right _ _).trans (min_le_left _ _)) + have hr1 : r ≤ 1 := (le_of_lt hrδ).trans ((min_le_right _ _).trans (min_le_right _ _)) + obtain ⟨hunit,hm,hv⟩ := hbounds r hr hrM + obtain ⟨hsq,hbound⟩ := sourceEndpointLaw_second_moment_of_parameter_bounds hL r b M V + hunit hM hV hr hr1 hm hv + exact ⟨hsq,hbound,hfloor r hr hrF⟩ + +theorem sourceEndpoint_estimates_and_weighted_events {L : ℝ} (hL : L ≠ 0) (b : ℝ) : + ∃ C c δ : ℝ, 0 ≤ C ∧ 0 < c ∧ 0 < δ ∧ δ ≤ 1 ∧ + ∀ r : ℝ, 0 < r → r < δ → + Integrable (fun v => sourceEndpointWeight r v ^ 2) (sourceEndpointLaw L r b) ∧ + (∫ v, sourceEndpointWeight r v ^ 2 ∂sourceEndpointLaw L r b) ≤ C*r^4 ∧ + c*r^2 ≤ (∫ v, sourceEndpointWeight r v ∂sourceEndpointLaw L r b) ∧ + IsProbabilityMeasure (sourceEndpointWeightedProbability L r b) ∧ + ∀ E : Set (Fin 6 → ℝ), MeasurableSet E → + (sourceEndpointWeightedProbability L r b).real E ≤ + (Real.sqrt C/c) * Real.sqrt ((sourceEndpointLaw L r b).real E) := by + obtain ⟨C,c,δ,hC,hc,hδ,hδ1,he⟩ := sourceEndpoint_estimates hL b + refine ⟨C,c,δ,hC,hc,hδ,hδ1,?_⟩ + intro r hr hrδ + obtain ⟨hsq,hm,hf⟩ := he r hr hrδ + have hp := sourceEndpoint_weighted_event_bound_of_estimates hL r b C c hC hc hr + hsq hm hf Set.univ MeasurableSet.univ + refine ⟨hsq,hm,hf,hp.1,?_⟩ + intro E hE + exact (sourceEndpoint_weighted_event_bound_of_estimates hL r b C c hC hc hr + hsq hm hf E hE).2 + +end I4SourceEndpointEstimates diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/AxiomAudit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/AxiomAudit.lean.txt new file mode 100644 index 000000000..4d62a6caf --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/AxiomAudit.lean.txt @@ -0,0 +1,26 @@ +import SourceEndpointEstimates +#print axioms I4SourceEndpointEstimates.sourcePhysicalCoordinates_apply +#print axioms I4SourceEndpointEstimates.sourcePhysicalCoordinates_continuous +#print axioms I4SourceEndpointEstimates.sourcePhysicalCoordinates_inverse +#print axioms I4SourceEndpointEstimates.sourceEndpointWeight_measurable +#print axioms I4SourceEndpointEstimates.sourceEndpointWeight_nonneg +#print axioms I4SourceEndpointEstimates.sourceEndpointWeight_eq_continuous +#print axioms I4SourceEndpointEstimates.sourceEndpointLaw_isProbabilityMeasure +#print axioms I4SourceEndpointEstimates.sourceEndpointLaw_euclidean_hasLaw +#print axioms I4SourceEndpointEstimates.sourceEndpointLaw_physical_marginal +#print axioms I4SourceEndpointEstimates.sourceEndpointLaw_weight_integral +#print axioms I4SourceEndpointEstimates.sourceEndpointLaw_second_moment_of_parameter_bounds +#print axioms I4SourceEndpointEstimates.sourceEndpointLaw_weight_square_integrable +#print axioms I4SourceEndpointEstimates.sourceEndpointLaw_weight_integrable +#print axioms I4SourceEndpointEstimates.sourcePrescribedPins_tendsto +#print axioms I4SourceEndpointEstimates.sourceEndpoint_weighted_event_bound_of_estimates +#print axioms I4SourceEndpointEstimates.sourceEndpoint_parameter_limits +#print axioms I4SourceEndpointEstimates.sourceEndpoint_uniform_parameter_bounds +#print axioms I4SourceEndpointEstimates.sourceEndpoint_normalizer_floor +#print axioms I4SourceEndpointEstimates.sourceEndpoint_estimates +#print axioms I4SourceEndpointEstimates.sourceEndpoint_estimates_and_weighted_events +#print axioms I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_eq +#print axioms I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_isProbabilityMeasure +#print axioms I4SourceEndpointEstimates.sourcePhysicalEndpointKernel_eventually_hasCondDistrib +#print axioms I4SourceEndpointEstimates.sourcePhysicalEndpointWeightedProbability_eq +#print axioms I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/FalseOmitPermutation.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/FalseOmitPermutation.lean.txt new file mode 100644 index 000000000..bf952d0d6 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/FalseOmitPermutation.lean.txt @@ -0,0 +1,7 @@ +import SourceEndpointEstimates +open I4SourceEndpointEstimates ResearchFormalCoreR1 +-- Treating the LM006 vector itself as physical order gives a different weight. +example : sourceEndpointWeight 2 ![-1,1,0,0,-1,-1] = + egm_weight 2 ![-1,1,0,0,-1,-1] := by + norm_num [sourceEndpointWeight, sourcePhysicalCoordinates, egm_weight, egm_HM, + egm_HS, dm2_negative_iff, Matrix.det_fin_two] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/FalseRadiusPower.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/FalseRadiusPower.lean.txt new file mode 100644 index 000000000..432f4bd5d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/FalseRadiusPower.lean.txt @@ -0,0 +1,6 @@ +import SourceEndpointEstimates +open I4SourceEndpointEstimates ResearchFormalCoreR1 +-- Replacing the physical r-squared determinant factor by r is false. +example : sourceEndpointWeight 2 ![-1,1,0,0,-1,-1] = 2 := by + norm_num [sourceEndpointWeight, sourcePhysicalCoordinates, egm_weight, egm_HM, + egm_HS, dm2_negative_iff, Matrix.det_fin_two] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/HANDOFF.md b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/HANDOFF.md new file mode 100644 index 000000000..484bdd25e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/HANDOFF.md @@ -0,0 +1,70 @@ +# Actual physical endpoint composition: completed local proof + +Sole production source: +`/workspace/Math-/frontiers/source_derivative_gaussian_20261025/SourceEndpointEstimates.lean`. + +Source SHA256: +`d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598`. + +Final theorem: +`I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events`. + +Its only research inputs are `L ≠ 0` and an arbitrary fixed real height `b`. +It derives constants `C ≥ 0`, `c > 0` and `0 < δ ≤ 1` such that every +`0 < r < δ` has literal physical weight square integrability, second moment +at most `C*r^4`, normalizer at least `c*r^2`, the explicit W-weighted actual +physical endpoint probability, and the `sqrt(C)/c * sqrt(K_r.real E)` bound +for every measurable endpoint event. + +The final physical law is the canonical `sourcePhysicalEndpointKernel` at the +literal prescribed pin. Identification with the adjusted kernel uses the +proved literal measure equality from the common-residual affine-kernel +construction in `SourcePhysicalEndpointKernel`. A null-fiber identity alone +is not used to identify RCD versions. The module also derives eventual +`HasCondDistrib` for the actual physical normalized endpoint vector using +actual small-radius covariance invertibility. + +No final covariance limit, Gaussian law, invertibility, uniform parameter +bound, moment or positive normalizer is added as a premise. + +## Evidence + +`qualification_receipt.json` binds the source, exact commands, exits, raw logs, +25 public targets, axiom reports, control hashes and remaining boundaries. + +- Final compile `physical_assembly_attempt1`: exit 0, empty output, `.olean` + generated. This completed compile was reused during qualification. +- Named `leanchecker SourceEndpointEstimates`: run once, exit 0, empty output. +- All 25 public transitive axiom reports: only `propext`, `Classical.choice`, + `Quot.sound`. +- Elaborated final types: exit 0; final theorem has only `hL` and `b` inputs. +- Positive physical-order/radius-squared/strict-zero-boundary controls: exit 0. +- False omitted-permutation and false radius-power controls: each exit 1 at an + intended unsolved `False` goal. + +The earlier `stage*` logs and explicitly uncompiled assembly drafts remain as +history. The final source and qualification receipt supersede the pending +checkpoint; drafts are not final proof evidence. + +## Lineage and scope + +Actual OpenAI/Codex `/root/dm2_landed_composition` is the sole implementer. +Actual OpenAI/Codex helper `/root/dm2_landed_composition/peano_strategy` supplied +the normalizer-floor assembly draft incorporated into the module. This is +author-side work, not independent scientific review. + +Completed actual covariance-limit, contact-parameter and canonical physical +kernel proofs are consumed directly. The 23 bound inputs, including the +byte-identical 11-module 66-target packet, were reused without rebuild or +re-audit. Earlier mode-chain, weighted-integral and regularized-continuity +scratch contributions to `SourceConfluentLimits` have separately preserved +evidence in the parent scratch directory. + +Uniformity over unbounded height, whole-field geometric Kac-Rice/Palm +identification, arbitrary field-dependent events, GT5 and later parents remain +outside this theorem. Scientific effect NONE; different-family scientific +review remains separate. No commits or external posts were made. + +Per `/workspace/AGENTS.md`, reuse this completed applicable verification. Do +not repeat it because of a handoff, new session or pending process requirement. +Root owns final inventory, Lakefile, records and publication integration. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/PositiveControls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/PositiveControls.lean.txt new file mode 100644 index 000000000..19077414b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/PositiveControls.lean.txt @@ -0,0 +1,16 @@ +import SourceEndpointEstimates +open I4SourceEndpointEstimates ResearchFormalCoreR1 + +-- Actual LM006 order must be converted to the physical maximum/saddle order. +example : sourcePhysicalCoordinates ![-1,1,0,0,-1,-1] = ![-1,0,-1,1,0,-1] := by + rfl + +-- The literal physical determinant product carries the radius-squared factor. +example : sourceEndpointWeight 2 ![-1,1,0,0,-1,-1] = 4 := by + norm_num [sourceEndpointWeight, sourcePhysicalCoordinates, egm_weight, egm_HM, + egm_HS, dm2_negative_iff, Matrix.det_fin_two] + +-- At the zero maximum-determinant boundary the literal strict type weight vanishes. +example : sourceEndpointWeight 2 ![-1,1,0,0,0,-1] = 0 := by + norm_num [sourceEndpointWeight, sourcePhysicalCoordinates, egm_weight, egm_HM, + egm_HS, dm2_negative_iff, Matrix.det_fin_two] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/STATUS.md b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/STATUS.md new file mode 100644 index 000000000..ebd2390bb --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/STATUS.md @@ -0,0 +1,63 @@ +# Historical endpoint composition checkpoint — superseded + +The final physical theorem is now proved and qualified. This earlier pending +checkpoint is retained only as history. Read `HANDOFF.md` and +`qualification_receipt.json` for the final source, claims and reusable evidence. + +Production owner: actual OpenAI/Codex `/root/dm2_landed_composition`. +Sole production file: `SourceEndpointEstimates.lean`. +Frozen contract read in full: Math- `dffeea6`, `SOURCE_ENDPOINT_ESTIMATES.md`. + +Current production stage 2 compiled cleanly and generated its `.olean`. +Exact command, exit 0, source SHA and empty raw output are preserved in +`stage2_final.{json,log}`. No old modules were rebuilt or rechecked. Stage 2 +has actual coordinate permutation, literal physical weight, measurability and +nonnegativity, inverse normalizer bridge, actual every-pin kernel Gaussian +HasLaw, actual physical marginal Gaussian laws, exact kernel/Gaussian weight +integral transfer, per-radius moment composition, actual W²/W integrability +without supplied parameter bounds, prescribed-pin right limit, and explicit +endpoint weighted measure/event glue. + +The final actual-source small-radius theorem is **not yet proved or claimed**. +It awaits precisely three actual-source covariance-limit exports owned by +`/root/uniform_jet_moment_sources` in `SourceConfluentLimits.lean`: + +- `sourcePinCovarianceAt_tendsto hL` +- `sourceCrossCovarianceAt_tendsto hL` +- `sourceEndpointCovarianceAt_tendsto hL` + +Their target contact matrices are in `I4SourceContact`; definitions and +positive/contact parameter proofs are complete in `SourceContactEndpointParameters`. +No limit, Gaussian law, invertibility, uniform parameter bound, normalizer or +moment conclusion has been added as an assumption of the final theorem. + +Concrete uncompiled assembly drafts (clearly labeled): + +- `ActualAssembly.lean.txt`: actual regression parameter limits, uniform bounds, + constants/moment/floor combination and all measurable endpoint-event bound. +- `../SourceEndpointNormalizerFloorDraft.lean.txt`: actual normalizer floor, + using the proposed `sourceEndpoint_parameter_limits` helper plus proved + contact relations and exact kernel integral transfer. + +The normalizer draft's actual helper lineage is OpenAI/Codex +`/root/dm2_landed_composition/peano_strategy`. That helper performed no checks +and made no production edits for this final-assembly draft. + +Additional scratch support delivered to covariance-limit owner, each compiled +cleanly under its new scope and with exact command/exit/SHA/raw logs: + +- `../ModeCurveChains.lean` and `../mode_chains_final.{json,log}`. +- `../WeightedModeIntegrals.lean` and `../weighted_mode_final.{json,log}`. +- `../RegularizedCoefficientContinuity.lean` and + `../regularized_continuity_attempt1.{json,log}`. + +The last two regularized coefficient continuity lemmas let that owner finish +the actual contact extensions/majorants/dominated Fourier-pairing limit. + +The final assembly should import the completed actual limit/contact modules, +insert its drafts inside `I4SourceEndpointEstimates`, run only the new affected +module check, and then perform its still-uncovered named kernel replay, +transitive public-target axiom audit and meaningful wiring controls once. +Do not repeat existing verification due to a handoff; `/workspace/AGENTS.md` +requires reuse of completed applicable evidence. Independent scientific review +remains separate. No commits, posts or historical source changes were made. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/Types.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/Types.lean.txt new file mode 100644 index 000000000..06a99cdd9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/Types.lean.txt @@ -0,0 +1,7 @@ +import SourceEndpointEstimates +set_option pp.explicit true +#check I4SourceEndpointEstimates.sourceEndpoint_parameter_limits +#check I4SourceEndpointEstimates.sourceEndpoint_uniform_parameter_bounds +#check I4SourceEndpointEstimates.sourceEndpoint_normalizer_floor +#check I4SourceEndpointEstimates.sourcePhysicalEndpointKernel_eventually_hasCondDistrib +#check I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/actual_assembly_attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/actual_assembly_attempt1.json new file mode 100644 index 000000000..3796b614e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/actual_assembly_attempt1.json @@ -0,0 +1,15 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/SourceEndpointEstimates.olean", + "SourceEndpointEstimates.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 1, + "source_sha256": "4192cf503993c9a51822deba5cb8a8a593a28f0707eba38a47837b3c32b82bc0", + "scope": "NEW_SOURCE_ENDPOINT_ESTIMATES_ACTUAL_HL_B_ONLY_ASSEMBLY", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition; normalizer draft helper /root/dm2_landed_composition/peano_strategy" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/actual_assembly_attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/actual_assembly_attempt1.log new file mode 100644 index 000000000..d3017d83b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/actual_assembly_attempt1.log @@ -0,0 +1,16 @@ +SourceEndpointEstimates.lean:214:4: error: Type mismatch: After simplification, term + hp.right.right.right + has type + @Tendsto ℝ (Matrix (Fin 6) (Fin 6) ℝ) + (fun r => + sourceEndpointCovarianceAt L r - + sourceCrossCovarianceAt L r * (sourcePinCovarianceAt L r)⁻¹ * (sourceCrossCovarianceAt L r)ᵀ) + (𝓝[>] 0) + (𝓝 + (sourceContactEndpointCovariance L - + sourceContactEndpointCrossCovariance L * (sourcePinCovariance L)⁻¹ * (sourceContactEndpointCrossCovariance L)ᵀ)) +but is expected to have type + @Tendsto ℝ (Matrix (Fin 6) (Fin 6) ℝ) (sourceEndpointSchurAt L) (𝓝[>] 0) + (𝓝 + (sourceContactEndpointCovariance L - + sourceContactEndpointCrossCovariance L * (sourcePinCovariance L)⁻¹ * (sourceContactEndpointCrossCovariance L)ᵀ)) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/axioms.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/axioms.json new file mode 100644 index 000000000..f6ca4effb --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/axioms.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/AxiomAudit.lean" + ], + "exit_code": 0, + "duration_seconds": 4.68341064453125, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/axioms.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/axioms.log new file mode 100644 index 000000000..ba9c5f94b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/axioms.log @@ -0,0 +1,55 @@ +'I4SourceEndpointEstimates.sourcePhysicalCoordinates_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourcePhysicalCoordinates_continuous' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourcePhysicalCoordinates_inverse' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointWeight_measurable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointWeight_nonneg' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointWeight_eq_continuous' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointLaw_isProbabilityMeasure' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointLaw_euclidean_hasLaw' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointLaw_physical_marginal' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointLaw_weight_integral' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointLaw_second_moment_of_parameter_bounds' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointLaw_weight_square_integrable' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourceEndpointLaw_weight_integrable' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourcePrescribedPins_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourceEndpoint_weighted_event_bound_of_estimates' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourceEndpoint_parameter_limits' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourceEndpoint_uniform_parameter_bounds' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourceEndpoint_normalizer_floor' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourceEndpoint_estimates' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourceEndpoint_estimates_and_weighted_events' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_eq' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_isProbabilityMeasure' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourcePhysicalEndpointKernel_eventually_hasCondDistrib' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourcePhysicalEndpointWeightedProbability_eq' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events' depends on axioms: [propext, + Classical.choice, + Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-permutation.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-permutation.json new file mode 100644 index 000000000..bccfc4091 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-permutation.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/FalseOmitPermutation.lean" + ], + "exit_code": 1, + "duration_seconds": 4.770277976989746, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-permutation.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-permutation.log new file mode 100644 index 000000000..aef84e063 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-permutation.log @@ -0,0 +1,2 @@ +/workspace/scratch/dm2_landed_composition/endpoint_estimates/FalseOmitPermutation.lean:5:38: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-radius.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-radius.json new file mode 100644 index 000000000..5fe196bde --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-radius.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/FalseRadiusPower.lean" + ], + "exit_code": 1, + "duration_seconds": 4.121073007583618, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-radius.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-radius.log new file mode 100644 index 000000000..470318e0a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/false-radius.log @@ -0,0 +1,2 @@ +/workspace/scratch/dm2_landed_composition/endpoint_estimates/FalseRadiusPower.lean:4:58: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/kernel.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/kernel.json new file mode 100644 index 000000000..20fe406ed --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/kernel.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "leanchecker", + "SourceEndpointEstimates" + ], + "exit_code": 0, + "duration_seconds": 8.686837196350098, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/kernel.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/kernel.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/physical_assembly_attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/physical_assembly_attempt1.json new file mode 100644 index 000000000..b9c90d0bc --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/physical_assembly_attempt1.json @@ -0,0 +1,15 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/SourceEndpointEstimates.olean", + "SourceEndpointEstimates.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "scope": "NEW_SOURCE_ENDPOINT_ESTIMATES_ACTUAL_PHYSICAL_KERNEL_COMPOSITION", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition; normalizer draft helper /root/dm2_landed_composition/peano_strategy" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/physical_assembly_attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/physical_assembly_attempt1.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/positive.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/positive.json new file mode 100644 index 000000000..84c67a61b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/positive.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/PositiveControls.lean" + ], + "exit_code": 0, + "duration_seconds": 6.003469705581665, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/positive.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/positive.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/qualification_receipt.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/qualification_receipt.json new file mode 100644 index 000000000..93144c187 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/qualification_receipt.json @@ -0,0 +1,325 @@ +{ + "scope": "NEW_ACTUAL_PHYSICAL_ENDPOINT_ESTIMATES_ONLY", + "scientific_effect": "NONE", + "source_path": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025/SourceEndpointEstimates.lean", + "source_sha256": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "frozen_contracts": [ + { + "commit": "dffeea6", + "path": "frontiers/source_derivative_gaussian_20261025/SOURCE_ENDPOINT_ESTIMATES.md" + }, + { + "commit": "1819e38", + "path": "frontiers/source_derivative_gaussian_20261025/SOURCE_PHYSICAL_ENDPOINT_KERNEL.md", + "role": "actual canonical physical law join consumed" + } + ], + "author_side_lineage": [ + "OpenAI/Codex /root/dm2_landed_composition: sole production implementer", + "OpenAI/Codex /root/dm2_landed_composition/peano_strategy: actual normalizer-floor assembly draft helper" + ], + "final_theorem": "I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events", + "final_research_inputs": [ + "L != 0", + "fixed arbitrary real height b" + ], + "derived_conclusions": [ + "current pin covariance invertibility for every sufficiently small positive radius", + "actual regression means and Schur covariance limits and uniform parameter bounds", + "literal physical weight square integrability and integral <= C*r^4", + "positive normalizer integral >= c*r^2", + "explicit weighted actual physical endpoint probability", + "all measurable endpoint events satisfy sqrt(C)/c * sqrt(base endpoint event probability) bound" + ], + "actual_source_proof_dependencies": [ + "SourceConfluentLimits: three actual covariance block limits", + "SourceContactEndpointParameters: derived contact means/zero variances/PSD/positive reference variance", + "SourceEndpointParameters: actual canonical every-pin Gaussian law", + "SourcePhysicalEndpointKernel: same-residual canonical physical kernel and literal prescribed-pin equality", + "GaussianRegressionLimits: derived inverse/mean/covariance limits and uniform bounds" + ], + "verified_input_reuse": "23 byte-bound inputs, including 11 ResearchFormalCoreR1 modules from the completed 66-target packet at 16f81b973a02783f8be83081aa7f198724cf5767, reused without rebuild or re-audit", + "compile": { + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/SourceEndpointEstimates.olean", + "SourceEndpointEstimates.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "scope": "NEW_SOURCE_ENDPOINT_ESTIMATES_ACTUAL_PHYSICAL_KERNEL_COMPOSITION", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition; normalizer draft helper /root/dm2_landed_composition/peano_strategy", + "raw_log": "/workspace/scratch/dm2_landed_composition/endpoint_estimates/physical_assembly_attempt1.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "public_targets": [ + "I4SourceEndpointEstimates.sourcePhysicalCoordinates_apply", + "I4SourceEndpointEstimates.sourcePhysicalCoordinates_continuous", + "I4SourceEndpointEstimates.sourcePhysicalCoordinates_inverse", + "I4SourceEndpointEstimates.sourceEndpointWeight_measurable", + "I4SourceEndpointEstimates.sourceEndpointWeight_nonneg", + "I4SourceEndpointEstimates.sourceEndpointWeight_eq_continuous", + "I4SourceEndpointEstimates.sourceEndpointLaw_isProbabilityMeasure", + "I4SourceEndpointEstimates.sourceEndpointLaw_euclidean_hasLaw", + "I4SourceEndpointEstimates.sourceEndpointLaw_physical_marginal", + "I4SourceEndpointEstimates.sourceEndpointLaw_weight_integral", + "I4SourceEndpointEstimates.sourceEndpointLaw_second_moment_of_parameter_bounds", + "I4SourceEndpointEstimates.sourceEndpointLaw_weight_square_integrable", + "I4SourceEndpointEstimates.sourceEndpointLaw_weight_integrable", + "I4SourceEndpointEstimates.sourcePrescribedPins_tendsto", + "I4SourceEndpointEstimates.sourceEndpoint_weighted_event_bound_of_estimates", + "I4SourceEndpointEstimates.sourceEndpoint_parameter_limits", + "I4SourceEndpointEstimates.sourceEndpoint_uniform_parameter_bounds", + "I4SourceEndpointEstimates.sourceEndpoint_normalizer_floor", + "I4SourceEndpointEstimates.sourceEndpoint_estimates", + "I4SourceEndpointEstimates.sourceEndpoint_estimates_and_weighted_events", + "I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_eq", + "I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_isProbabilityMeasure", + "I4SourceEndpointEstimates.sourcePhysicalEndpointKernel_eventually_hasCondDistrib", + "I4SourceEndpointEstimates.sourcePhysicalEndpointWeightedProbability_eq", + "I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events" + ], + "axioms": { + "I4SourceEndpointEstimates.sourcePhysicalCoordinates_apply": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourcePhysicalCoordinates_continuous": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourcePhysicalCoordinates_inverse": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointWeight_measurable": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointWeight_nonneg": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointWeight_eq_continuous": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointLaw_isProbabilityMeasure": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointLaw_euclidean_hasLaw": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointLaw_physical_marginal": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointLaw_weight_integral": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointLaw_second_moment_of_parameter_bounds": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointLaw_weight_square_integrable": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpointLaw_weight_integrable": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourcePrescribedPins_tendsto": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpoint_weighted_event_bound_of_estimates": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpoint_parameter_limits": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpoint_uniform_parameter_bounds": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpoint_normalizer_floor": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpoint_estimates": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourceEndpoint_estimates_and_weighted_events": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_eq": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_isProbabilityMeasure": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourcePhysicalEndpointKernel_eventually_hasCondDistrib": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourcePhysicalEndpointWeightedProbability_eq": [ + "propext", + "Classical.choice", + "Quot.sound" + ], + "I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events": [ + "propext", + "Classical.choice", + "Quot.sound" + ] + }, + "checks": { + "kernel": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "leanchecker", + "SourceEndpointEstimates" + ], + "exit_code": 0, + "duration_seconds": 8.686837196350098, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "expected_exit_code": 0, + "raw_log": "/workspace/scratch/dm2_landed_composition/endpoint_estimates/kernel.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "axioms": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/AxiomAudit.lean" + ], + "exit_code": 0, + "duration_seconds": 4.68341064453125, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "expected_exit_code": 0, + "raw_log": "/workspace/scratch/dm2_landed_composition/endpoint_estimates/axioms.log", + "raw_log_sha256": "ded91623f2200300ff7b8ab366f5aa64564a157dd98166cb3eedaa10c884caf1" + }, + "types": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/Types.lean" + ], + "exit_code": 0, + "duration_seconds": 4.792372465133667, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "expected_exit_code": 0, + "raw_log": "/workspace/scratch/dm2_landed_composition/endpoint_estimates/types.log", + "raw_log_sha256": "a0aabbe4809916815304e74fc2cb96ab47d193d6874509e6bdf97d86e62a3f1b" + }, + "positive": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/PositiveControls.lean" + ], + "exit_code": 0, + "duration_seconds": 6.003469705581665, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "expected_exit_code": 0, + "raw_log": "/workspace/scratch/dm2_landed_composition/endpoint_estimates/positive.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "false-permutation": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/FalseOmitPermutation.lean" + ], + "exit_code": 1, + "duration_seconds": 4.770277976989746, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "expected_exit_code": 1, + "raw_log": "/workspace/scratch/dm2_landed_composition/endpoint_estimates/false-permutation.log", + "raw_log_sha256": "f262c0491c0806778fbd7fc56a2fe3668366dac8a386f9c658b323ee3bdd1955" + }, + "false-radius": { + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/FalseRadiusPower.lean" + ], + "exit_code": 1, + "duration_seconds": 4.121073007583618, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "expected_exit_code": 1, + "raw_log": "/workspace/scratch/dm2_landed_composition/endpoint_estimates/false-radius.log", + "raw_log_sha256": "1d211ab1466ecfd781950cf38588cdf7c42f009185c70ad467dcd76037271e47" + } + }, + "control_source_sha256": { + "AxiomAudit.lean": "f6d9f1bf0523abe2bc34e77d0b587e981a82a7c0000589ca9508b25cd6ab0572", + "Types.lean": "c6537a1488393fa76ffbe14c673d9315484d441b87aaf89ee45d77dc43b3d514", + "PositiveControls.lean": "4a3f87cd8e34c7d43d34660fcfcb9531c686aa02e88664a7ad6b660fc48e1209", + "FalseOmitPermutation.lean": "258067b5e2594810e3359a7e8c068a616516472735e6030b3a5519b1d537389e", + "FalseRadiusPower.lean": "030c275aa43471ebce6110688f54d39ca76a57610a2c0e3de56223f5df19a73e" + }, + "not_established": [ + "uniformity over unbounded b", + "whole-field geometric Kac-Rice/Palm identification", + "arbitrary field-dependent event transfer", + "GT5 or further parent theorems", + "independent scientific review or organizational independence credit" + ] +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/run_one.py b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/run_one.py new file mode 100644 index 000000000..a562a2421 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/run_one.py @@ -0,0 +1,12 @@ +import hashlib,json,pathlib,subprocess,sys,time +p=pathlib.Path(__file__).parent +label=sys.argv[1] +source=pathlib.Path("SourceEndpointEstimates.lean") +sha=hashlib.sha256(source.read_bytes()).hexdigest() +if label=="kernel": cmd=["lake","env","leanchecker","SourceEndpointEstimates"] +else: cmd=["lake","env","lean",str(p/{"axioms":"AxiomAudit.lean","types":"Types.lean","positive":"PositiveControls.lean","false-permutation":"FalseOmitPermutation.lean","false-radius":"FalseRadiusPower.lean"}[label])] +start=time.time();r=subprocess.run(cmd,stdout=subprocess.PIPE,stderr=subprocess.STDOUT) +(p/(label+".log")).write_bytes(r.stdout) +meta={"cwd":str(pathlib.Path.cwd()),"command":cmd,"exit_code":r.returncode,"duration_seconds":time.time()-start,"source_sha256_before":sha,"source_sha256_after":hashlib.sha256(source.read_bytes()).hexdigest()} +(p/(label+".json")).write_text(json.dumps(meta,indent=2)+"\n") +print(json.dumps(meta));print(r.stdout.decode(),end="");raise SystemExit(r.returncode) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_attempt1.json new file mode 100644 index 000000000..e94bc1284 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_attempt1.json @@ -0,0 +1,15 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/SourceEndpointEstimates.olean", + "SourceEndpointEstimates.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 1, + "source_sha256": "338535195f9ac8b40003549862de12ab778936bba2368cbb4379fecf677efc84", + "scope": "NEW_SOURCE_ENDPOINT_ESTIMATES_COMPOSITION_ONLY", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_attempt1.log new file mode 100644 index 000000000..8d47690d3 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_attempt1.log @@ -0,0 +1,45 @@ +SourceEndpointEstimates.lean:72:2: warning: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` +SourceEndpointEstimates.lean:73:8: error(lean.synthInstanceFailed): failed to synthesize instance of type class + IsProbabilityMeasure (sourceEndpointLaw L r b) + +Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command. +SourceEndpointEstimates.lean:110:2: error: Type mismatch: After simplification, term + h + has type + @Eq ℝ + (∫ (x : Fin 6 → ℝ), + ((fun x => egm_weight r ![x.ofLp 0, x.ofLp 2, x.ofLp 4, x.ofLp 1, x.ofLp 3, x.ofLp 5]) ∘ toLp 2) + x ∂sourceEndpointLaw L r b) + (∫ (x : WithLp 2 (Fin 6 → ℝ)), + egm_weight r + ![x.ofLp 0, x.ofLp 2, x.ofLp 4, x.ofLp 1, x.ofLp 3, + x.ofLp 5] ∂multivariateGaussian (toLp 2 (sourceEndpointMean L r b)) (sourceEndpointSchurAt L r)) +but is expected to have type + @Eq ℝ (∫ (v : Fin 6 → ℝ), egm_weight r ![v 0, v 2, v 4, v 1, v 3, v 5] ∂sourceEndpointLaw L r b) + (∫ (x : EuclideanSpace ℝ (Fin 6)), + egm_weight r + ![x.ofLp 0, x.ofLp 2, x.ofLp 4, x.ofLp 1, x.ofLp 3, + x.ofLp 5] ∂multivariateGaussian (toLp 2 (sourceEndpointMean L r b)) (sourceEndpointSchurAt L r)) +SourceEndpointEstimates.lean:122:2: warning: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` +SourceEndpointEstimates.lean:150:2: warning: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_final.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_final.json new file mode 100644 index 000000000..d2504fc36 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_final.json @@ -0,0 +1,15 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/SourceEndpointEstimates.olean", + "SourceEndpointEstimates.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256": "dcd3dbf25a199e2b3461bc2f7bcc935acac74483dd525a01d06b3720fa863958", + "scope": "NEW_SOURCE_ENDPOINT_ESTIMATES_COMPOSITION_ONLY", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_final.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_final.log new file mode 100644 index 000000000..f7d124242 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage1_final.log @@ -0,0 +1,24 @@ +SourceEndpointEstimates.lean:73:2: warning: Try this: + haveI̵ + +The goal is a proposition, so `have` is preferred over `haveI`. +The difference between `have` and `haveI` is that `haveI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` +SourceEndpointEstimates.lean:124:2: warning: Try this: + haveI̵ + +The goal is a proposition, so `have` is preferred over `haveI`. +The difference between `have` and `haveI` is that `haveI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` +SourceEndpointEstimates.lean:152:2: warning: Try this: + haveI̵ + +The goal is a proposition, so `have` is preferred over `haveI`. +The difference between `have` and `haveI` is that `haveI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_attempt1.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_attempt1.json new file mode 100644 index 000000000..2a5f24721 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_attempt1.json @@ -0,0 +1,15 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/SourceEndpointEstimates.olean", + "SourceEndpointEstimates.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 1, + "source_sha256": "2c2e6d3cc9af6e061da59ff6e00e7625632b774579f2d91bdf1a2b5e9f32d400", + "scope": "NEW_SOURCE_ENDPOINT_ESTIMATES_INTEGRABILITY_AND_PIN_LIMIT", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_attempt1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_attempt1.log new file mode 100644 index 000000000..f2ad1cd08 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_attempt1.log @@ -0,0 +1,6 @@ +SourceEndpointEstimates.lean:165:2: error: Type mismatch: After simplification, term + Tendsto.mono_left (ContinuousAt.tendsto (Continuous.continuousAt hc)) nhdsWithin_le_nhds + has type + Tendsto (fun r => ![b - r ^ 3 / 12, 0, 0, 0, 0, 2]) (𝓝[?m.131] ?m.130) (𝓝 ![b - ?m.130 ^ 3 / 12, 0, 0, 0, 0, 2]) +but is expected to have type + Tendsto (fun r => ![b - r ^ 3 / 12, 0, 0, 0, 0, 2]) (𝓝[>] 0) (𝓝 ![b, 0, 0, 0, 0, 2]) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_final.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_final.json new file mode 100644 index 000000000..6b985ab45 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_final.json @@ -0,0 +1,15 @@ +{ + "command": [ + "lake", + "env", + "lean", + "-o", + ".lake/build/lib/lean/SourceEndpointEstimates.olean", + "SourceEndpointEstimates.lean" + ], + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "exit_code": 0, + "source_sha256": "d41e6afb5b57ef05e90bae703d863b501517ef36fa2a98be5e2310cfbd3ca245", + "scope": "NEW_SOURCE_ENDPOINT_ESTIMATES_INTEGRABILITY_AND_PIN_LIMIT", + "author_side_lineage": "OpenAI/Codex /root/dm2_landed_composition" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_final.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/stage2_final.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/targets.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/targets.json new file mode 100644 index 000000000..cf7fdf24a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/targets.json @@ -0,0 +1,27 @@ +[ + "I4SourceEndpointEstimates.sourcePhysicalCoordinates_apply", + "I4SourceEndpointEstimates.sourcePhysicalCoordinates_continuous", + "I4SourceEndpointEstimates.sourcePhysicalCoordinates_inverse", + "I4SourceEndpointEstimates.sourceEndpointWeight_measurable", + "I4SourceEndpointEstimates.sourceEndpointWeight_nonneg", + "I4SourceEndpointEstimates.sourceEndpointWeight_eq_continuous", + "I4SourceEndpointEstimates.sourceEndpointLaw_isProbabilityMeasure", + "I4SourceEndpointEstimates.sourceEndpointLaw_euclidean_hasLaw", + "I4SourceEndpointEstimates.sourceEndpointLaw_physical_marginal", + "I4SourceEndpointEstimates.sourceEndpointLaw_weight_integral", + "I4SourceEndpointEstimates.sourceEndpointLaw_second_moment_of_parameter_bounds", + "I4SourceEndpointEstimates.sourceEndpointLaw_weight_square_integrable", + "I4SourceEndpointEstimates.sourceEndpointLaw_weight_integrable", + "I4SourceEndpointEstimates.sourcePrescribedPins_tendsto", + "I4SourceEndpointEstimates.sourceEndpoint_weighted_event_bound_of_estimates", + "I4SourceEndpointEstimates.sourceEndpoint_parameter_limits", + "I4SourceEndpointEstimates.sourceEndpoint_uniform_parameter_bounds", + "I4SourceEndpointEstimates.sourceEndpoint_normalizer_floor", + "I4SourceEndpointEstimates.sourceEndpoint_estimates", + "I4SourceEndpointEstimates.sourceEndpoint_estimates_and_weighted_events", + "I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_eq", + "I4SourceEndpointEstimates.sourcePhysicalEndpointLaw_isProbabilityMeasure", + "I4SourceEndpointEstimates.sourcePhysicalEndpointKernel_eventually_hasCondDistrib", + "I4SourceEndpointEstimates.sourcePhysicalEndpointWeightedProbability_eq", + "I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events" +] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/types.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/types.json new file mode 100644 index 000000000..81b188ba9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/types.json @@ -0,0 +1,13 @@ +{ + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "command": [ + "lake", + "env", + "lean", + "/workspace/scratch/dm2_landed_composition/endpoint_estimates/Types.lean" + ], + "exit_code": 0, + "duration_seconds": 4.792372465133667, + "source_sha256_before": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598", + "source_sha256_after": "d1f08d80d06d1ac08afbdd700370059fd8eaf0d4e483916e742efcdd1f664598" +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/types.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/types.log new file mode 100644 index 000000000..d2899508b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointEstimates/types.log @@ -0,0 +1,253 @@ +I4SourceEndpointEstimates.sourceEndpoint_parameter_limits {L : Real} + (hL : @Ne Real L (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) (b : Real) : + And + (@Filter.Eventually Real + (fun r => + @IsUnit + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (@Semiring.toMonoid + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (@Matrix.semiring (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real Real.semiring + (Fin.fintype (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (instDecidableEqFin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))) + (I4SourceEndpointRegression.sourcePinCovarianceAt L r)) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))) + (And + (@Filter.Tendsto Real + (WithLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + ⋯)) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real)) + (fun r => + @WithLp.toLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + ⋯)) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (I4SourceEndpointEstimates.sourceEndpointMean L r b)) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) + (@nhds + (WithLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + ⋯)) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real)) + (@PiLp.topologicalSpace + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + ⋯)) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun i => + @UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@WithLp.toLp + (@OfNat.ofNat ENNReal (nat_lit 2) + (@instOfNatAtLeastTwo ENNReal (nat_lit 2) + (@AddMonoidWithOne.toNatCast ENNReal + (@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal ENNReal.instAddCommMonoidWithOne)) + ⋯)) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (I4SourceContact.sourceContactEndpointMean L b)))) + (@Filter.Tendsto Real + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (I4SourceEndpointParameters.sourceEndpointSchurAt L) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) + (@nhds + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (@instTopologicalSpaceMatrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace))) + (I4SourceContact.sourceContactEndpointSchur L)))) +I4SourceEndpointEstimates.sourceEndpoint_uniform_parameter_bounds {L : Real} + (hL : @Ne Real L (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) (b : Real) : + ∃ M V δ, + And (@LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) M) + (And (@LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) V) + (And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) δ) + (∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LT.lt Real Real.instLT r δ → + And + (@IsUnit + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (@Semiring.toMonoid + (Matrix (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real) + (@Matrix.semiring (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) Real + Real.semiring (Fin.fintype (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (instDecidableEqFin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))) + (I4SourceEndpointRegression.sourcePinCovarianceAt L r)) + (And + (∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))), + @LE.le Real Real.instLE + (@abs Real Real.lattice Real.instAddGroup + (I4SourceEndpointEstimates.sourceEndpointMean L r b i)) + M) + (∀ (i : Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))), + @LE.le Real Real.instLE + (@abs Real Real.lattice Real.instAddGroup + (I4SourceEndpointParameters.sourceEndpointSchurAt L r i i)) + V))))) +I4SourceEndpointEstimates.sourceEndpoint_normalizer_floor {L : Real} + (hL : @Ne Real L (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) (b : Real) : + ∃ c, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) c) + (∃ δ, + And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) δ) + (∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LT.lt Real Real.instLT r δ → + @LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@MeasureTheory.integral (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) Real + Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) + fun a => Real.measurableSpace) + (I4SourceEndpointEstimates.sourceEndpointLaw L r b) fun v => + I4SourceEndpointEstimates.sourceEndpointWeight r v))) +I4SourceEndpointEstimates.sourcePhysicalEndpointKernel_eventually_hasCondDistrib {L : Real} + (hL : @Ne Real L (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) : + @Filter.Eventually Real + (fun r => + @ProbabilityTheory.HasCondDistrib + (I4SmoothCarrier.SmoothSample (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) L) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (@Subtype.instMeasurableSpace + (I4Source.SourceMode (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real) + (fun x => + @Membership.mem (I4Source.SourceMode (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real) + (Set (I4Source.SourceMode (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real)) + (@Set.instMembership + (I4Source.SourceMode (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → Real)) + (I4SmoothCarrier.goodSet (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) L) x) + (@MeasurableSpace.pi (I4Source.SourceMode (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (fun a => Real) fun a => Real.measurableSpace)) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun a => + Real.measurableSpace) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) fun a => + Real.measurableSpace) + (I4SourceEndpointRegression.sourceNormalizedEndpoints L r) + (I4SourceEndpointRegression.sourceConfluentPinsAt L r) + (I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel L r) + (I4SmoothCarrier.smoothLaw (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) L)) + (@nhdsWithin Real + (@UniformSpace.toTopologicalSpace Real (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) + (@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) +I4SourceEndpointEstimates.sourcePhysicalEndpoint_estimates_and_weighted_events {L : Real} + (hL : @Ne Real L (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) (b : Real) : + ∃ C c δ, + And (@LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) C) + (And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) c) + (And (@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) δ) + (And (@LE.le Real Real.instLE δ (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) + (∀ (r : Real), + @LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) r → + @LT.lt Real Real.instLT r δ → + And + (@MeasureTheory.Integrable Real + (@UniformSpace.toTopologicalSpace Real + (@PseudoMetricSpace.toUniformSpace Real Real.pseudoMetricSpace)) + (@SeminormedAddGroup.toContinuousENorm Real + (@SeminormedAddCommGroup.toSeminormedAddGroup Real + (@NonUnitalSeminormedRing.toSeminormedAddCommGroup Real + (@NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing Real + (@SeminormedCommRing.toNonUnitalSeminormedCommRing Real + (@NormedCommRing.toSeminormedCommRing Real Real.normedCommRing)))))) + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (fun a => Real) fun a => Real.measurableSpace) + (fun v => + @HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (I4SourceEndpointEstimates.sourceEndpointWeight r v) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (I4SourceEndpointEstimates.sourcePhysicalEndpointLaw L r b)) + (And + (@LE.le Real Real.instLE + (@MeasureTheory.integral (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + Real Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (fun a => Real) fun a => Real.measurableSpace) + (I4SourceEndpointEstimates.sourcePhysicalEndpointLaw L r b) fun v => + @HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) + (I4SourceEndpointEstimates.sourceEndpointWeight r v) + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) C + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))) + (And + (@LE.le Real Real.instLE + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c + (@HPow.hPow Real Nat Real + (@instHPow Real Nat (@NPow.toPow Real (@Monoid.toNPow Real Real.instMonoid))) r + (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) + (@MeasureTheory.integral + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) Real + Real.normedAddCommGroup + (@InnerProductSpace.toNormedSpace Real Real Real.instRCLike + (@NormedAddCommGroup.toSeminormedAddCommGroup Real Real.normedAddCommGroup) + (@RCLike.toInnerProductSpaceReal Real Real.instRCLike)) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (fun a => Real) fun a => Real.measurableSpace) + (I4SourceEndpointEstimates.sourcePhysicalEndpointLaw L r b) fun v => + I4SourceEndpointEstimates.sourceEndpointWeight r v)) + (And + (@MeasureTheory.IsProbabilityMeasure + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (fun a => Real) fun a => Real.measurableSpace) + (I4SourceEndpointEstimates.sourcePhysicalEndpointWeightedProbability L r b)) + (∀ (E : Set (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real)), + @MeasurableSet (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (fun a => Real) fun a => Real.measurableSpace) + E → + @LE.le Real Real.instLE + (@MeasureTheory.Measure.real + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (@MeasurableSpace.pi (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) + (fun a => Real) fun a => Real.measurableSpace) + (I4SourceEndpointEstimates.sourcePhysicalEndpointWeightedProbability L r b) E) + (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) + (@HDiv.hDiv Real Real Real + (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) C.sqrt c) + (@MeasureTheory.Measure.real + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))) → Real) + (@MeasurableSpace.pi + (Fin (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))) (fun a => Real) + fun a => Real.measurableSpace) + (I4SourceEndpointEstimates.sourcePhysicalEndpointLaw L r b) E).sqrt))))))))) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/AxiomAudit-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/AxiomAudit-01.log new file mode 100644 index 000000000..3a4ab4e74 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/AxiomAudit-01.log @@ -0,0 +1,45 @@ +'I4SourceEndpointParameters.smoothSourceField_derivative_integral_zero' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.rawPinsWithHessians_integral_zero' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.rawPinsWithHessians_integral_vector_zero' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourcePins_adjusted_integral_zero' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourceConfluentPinsAt_integral_zero' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourceAdjustedEndpoints_integral_zero' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourceEndpointResidual_integral_zero' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourceEndpointResidual_covariance' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.finiteGaussian_law_parameters' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointParameters.sourceEndpointResidual_hasGaussianLaw' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourceEndpointPosterior_hasGaussianLaw' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourceEndpointPosterior_integral' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointParameters.sourceEndpointPosterior_covariance' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourceEndpointKernel_eq_map_posterior' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourceEndpointKernel_isGaussian' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointParameters.sourceEndpointSchurAt_posSemidef' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointParameters.sourceEndpointKernel_eq_multivariateGaussian' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointParameters.sourceEndpointKernel_integral' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointParameters.sourceEndpointKernel_covariance' depends on axioms: [propext, Classical.choice, Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/AxiomAudit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/AxiomAudit.lean.txt new file mode 100644 index 000000000..bfd292262 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/AxiomAudit.lean.txt @@ -0,0 +1,20 @@ +import SourceEndpointParameters +#print axioms I4SourceEndpointParameters.smoothSourceField_derivative_integral_zero +#print axioms I4SourceEndpointParameters.rawPinsWithHessians_integral_zero +#print axioms I4SourceEndpointParameters.rawPinsWithHessians_integral_vector_zero +#print axioms I4SourceEndpointParameters.sourcePins_adjusted_integral_zero +#print axioms I4SourceEndpointParameters.sourceConfluentPinsAt_integral_zero +#print axioms I4SourceEndpointParameters.sourceAdjustedEndpoints_integral_zero +#print axioms I4SourceEndpointParameters.sourceEndpointResidual_integral_zero +#print axioms I4SourceEndpointParameters.sourceEndpointResidual_covariance +#print axioms I4SourceEndpointParameters.finiteGaussian_law_parameters +#print axioms I4SourceEndpointParameters.sourceEndpointResidual_hasGaussianLaw +#print axioms I4SourceEndpointParameters.sourceEndpointPosterior_hasGaussianLaw +#print axioms I4SourceEndpointParameters.sourceEndpointPosterior_integral +#print axioms I4SourceEndpointParameters.sourceEndpointPosterior_covariance +#print axioms I4SourceEndpointParameters.sourceEndpointKernel_eq_map_posterior +#print axioms I4SourceEndpointParameters.sourceEndpointKernel_isGaussian +#print axioms I4SourceEndpointParameters.sourceEndpointSchurAt_posSemidef +#print axioms I4SourceEndpointParameters.sourceEndpointKernel_eq_multivariateGaussian +#print axioms I4SourceEndpointParameters.sourceEndpointKernel_integral +#print axioms I4SourceEndpointParameters.sourceEndpointKernel_covariance diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/EVIDENCE.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/EVIDENCE.json new file mode 100644 index 000000000..bd34d0537 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/EVIDENCE.json @@ -0,0 +1,173 @@ +{ + "source": "SourceEndpointParameters.lean", + "source_sha256": "efc88ac59217ab1960d5518abcd9f16a71b42080a2810f1e337eeb2f686c1e3a", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "source_contract_commit": "20e4634", + "author": "OpenAI/Codex /root/regression_kernel with OpenAI/Codex /root/regression_kernel/gaussian_api residual covariance support and OpenAI/Codex /root/regression_kernel/endpoint_means actual zero-mean proofs", + "scientific_effect": "NONE; author-side evidence; source current G invertibility remains explicit pending limits; different-family review separate", + "build": { + "command": "/home/agent/.elan/bin/lake env lean -o .lake/build/lib/lean/SourceEndpointParameters.olean SourceEndpointParameters.lean", + "exit": 0, + "log": "SourceEndpointParameters-02.log" + }, + "kernel": { + "command": "/home/agent/.elan/bin/lake env leanchecker SourceEndpointParameters", + "exit": 0, + "log": "KernelReplay-01.log" + }, + "axioms": { + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-parameters/AxiomAudit.lean", + "exit": 0, + "targets": { + "I4SourceEndpointParameters.smoothSourceField_derivative_integral_zero": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.rawPinsWithHessians_integral_zero": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.rawPinsWithHessians_integral_vector_zero": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourcePins_adjusted_integral_zero": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceConfluentPinsAt_integral_zero": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceAdjustedEndpoints_integral_zero": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointResidual_integral_zero": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointResidual_covariance": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.finiteGaussian_law_parameters": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointResidual_hasGaussianLaw": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointPosterior_hasGaussianLaw": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointPosterior_integral": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointPosterior_covariance": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointKernel_eq_map_posterior": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointKernel_isGaussian": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointSchurAt_posSemidef": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointKernel_eq_multivariateGaussian": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointKernel_integral": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointParameters.sourceEndpointKernel_covariance": [ + "Classical.choice", + "Quot.sound", + "propext" + ] + }, + "log": "AxiomAudit-01.log" + }, + "controls": [ + { + "file": "PositiveControls.lean", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-parameters/PositiveControls.lean", + "capture_command": "set -o pipefail\n/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-parameters/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/endpoint-parameters/PositiveControls-01.log", + "exit": 0, + "log": "PositiveControls-01.log", + "reason": "actual prescribed-pin Gaussian identity; derived PSD at negative period; singular noncentered Gaussian and empty dimension literal identities" + }, + { + "file": "RejectedCenteredPosterior.lean", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-parameters/RejectedCenteredPosterior.lean", + "capture_command": "set -o pipefail\n/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-parameters/RejectedCenteredPosterior.lean 2>&1 | tee /workspace/scratch/regression-kernel/endpoint-parameters/RejectedCenteredPosterior-01.log", + "exit": 1, + "log": "RejectedCenteredPosterior-01.log", + "reason": "noncentered singular Gaussian has mean3 not0" + }, + { + "file": "RejectedNegativeCovariance.lean", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-parameters/RejectedNegativeCovariance.lean", + "capture_command": "set -o pipefail\n/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-parameters/RejectedNegativeCovariance.lean 2>&1 | tee /workspace/scratch/regression-kernel/endpoint-parameters/RejectedNegativeCovariance-01.log", + "exit": 1, + "log": "RejectedNegativeCovariance-01.log", + "reason": "constant actual Gaussian cannot satisfy negative covariance assumption needed to derive PSD" + } + ], + "logs": { + "SourceEndpointParameters-02.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "PositiveControls-01.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "RejectedCenteredPosterior-01.log": { + "bytes": 126, + "sha256": "fe4eff4da5aab333d2725cc877ee3807f516211d6619001baf2a54125fa824af" + }, + "RejectedNegativeCovariance-01.log": { + "bytes": 196, + "sha256": "79e558f68db2219a662cc7e395064d9b7757b4d20e21e94dc46a9175ba2cb569" + }, + "AxiomAudit-01.log": { + "bytes": 2380, + "sha256": "2316c6b6d0d8ca6ca724a78137145a4aa57ce9d8b8498fcd7a5f86f73da25a49" + }, + "KernelReplay-01.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + } + }, + "reuse": "All completed source/kernel/regression checks reused; only this new parameter module and new controls checked." +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-01.log new file mode 100644 index 000000000..8ddfdd2ec --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-01.log @@ -0,0 +1,56 @@ +/workspace/scratch/regression-kernel/endpoint-parameters/Identification.lean:35:8: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ∫ (x : EuclideanSpace ℝ ?m.277), x ∂multivariateGaussian ?m.280 ?m.281 +in the target expression + ∫ (x : WithLp 2 (Fin n → ℝ)), id x ∂Measure.map (fun ω => toLp 2 (Z ω)) P = + ∫ (x : WithLp 2 (Fin n → ℝ)), id x ∂multivariateGaussian (toLp 2 m) S + +case hm +Ω : Type u_1 +inst✝ : MeasurableSpace Ω +P : Measure Ω +n : ℕ +Z : Ω → Fin n → ℝ +hZ : HasGaussianLaw Z P +m : Fin n → ℝ +hm : ∫ (ω : Ω), Z ω ∂P = m +S : Matrix (Fin n) (Fin n) ℝ +hcov : ∀ (i j : Fin n), cov[fun ω => Z ω i, fun ω => Z ω j; P] = S i j +this✝ : IsProbabilityMeasure P +e : (Fin n → ℝ) ≃L[ℝ] EuclideanSpace ℝ (Fin n) := ⋯ +hE : HasGaussianLaw (fun ω => toLp 2 (Z ω)) P +this : IsGaussian (Measure.map (fun ω => toLp 2 (Z ω)) P) +hZi : ∀ (i : Fin n), MemLp (fun ω => Z ω i) 2 P +hS : S.PosSemidef +⊢ ∫ (x : WithLp 2 (Fin n → ℝ)), id x ∂Measure.map (fun ω => toLp 2 (Z ω)) P = + ∫ (x : WithLp 2 (Fin n → ℝ)), id x ∂multivariateGaussian (toLp 2 m) S +/workspace/scratch/regression-kernel/endpoint-parameters/Identification.lean:42:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ((covarianceBilin (Measure.map (fun ω => toLp 2 fun x => Z ω x) P)) ((EuclideanSpace.basisFun (Fin n) ℝ) ?i)) + ((EuclideanSpace.basisFun (Fin n) ℝ) ?j) +in the target expression + ((covarianceBilin (Measure.map (fun ω => toLp 2 (Z ω)) P)) ((EuclideanSpace.basisFun (Fin n) ℝ).toBasis i)) + ((EuclideanSpace.basisFun (Fin n) ℝ).toBasis j) = + ((covarianceBilin (multivariateGaussian (toLp 2 m) S)) ((EuclideanSpace.basisFun (Fin n) ℝ).toBasis i)) + ((EuclideanSpace.basisFun (Fin n) ℝ).toBasis j) + +case hv +Ω : Type u_1 +inst✝ : MeasurableSpace Ω +P : Measure Ω +n : ℕ +Z : Ω → Fin n → ℝ +hZ : HasGaussianLaw Z P +m : Fin n → ℝ +hm : ∫ (ω : Ω), Z ω ∂P = m +S : Matrix (Fin n) (Fin n) ℝ +hcov : ∀ (i j : Fin n), cov[fun ω => Z ω i, fun ω => Z ω j; P] = S i j +this✝ : IsProbabilityMeasure P +e : (Fin n → ℝ) ≃L[ℝ] EuclideanSpace ℝ (Fin n) := (PiLp.continuousLinearEquiv 2 ℝ fun x => ℝ).symm +hE : HasGaussianLaw (fun ω => toLp 2 (Z ω)) P +this : IsGaussian (Measure.map (fun ω => toLp 2 (Z ω)) P) +hZi : ∀ (i : Fin n), MemLp (fun ω => Z ω i) 2 P +hS : S.PosSemidef +i j : Fin n +⊢ ((covarianceBilin (Measure.map (fun ω => toLp 2 (Z ω)) P)) ((EuclideanSpace.basisFun (Fin n) ℝ).toBasis i)) + ((EuclideanSpace.basisFun (Fin n) ℝ).toBasis j) = + ((covarianceBilin (multivariateGaussian (toLp 2 m) S)) ((EuclideanSpace.basisFun (Fin n) ℝ).toBasis i)) + ((EuclideanSpace.basisFun (Fin n) ℝ).toBasis j) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-02.log new file mode 100644 index 000000000..7dbc9b9fb --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-02.log @@ -0,0 +1,23 @@ +/workspace/scratch/regression-kernel/endpoint-parameters/Identification.lean:39:8: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ∫ (x : Ω), ↑e (Z x) ∂P +in the target expression + ∫ (ω : Ω), e (Z ω) ∂P = e m + +case hm +Ω : Type u_1 +inst✝ : MeasurableSpace Ω +P : Measure Ω +n : ℕ +Z : Ω → Fin n → ℝ +hZ : HasGaussianLaw Z P +m : Fin n → ℝ +hm : ∫ (ω : Ω), Z ω ∂P = m +S : Matrix (Fin n) (Fin n) ℝ +hcov : ∀ (i j : Fin n), cov[fun ω => Z ω i, fun ω => Z ω j; P] = S i j +this✝ : IsProbabilityMeasure P +e : (Fin n → ℝ) ≃L[ℝ] EuclideanSpace ℝ (Fin n) := (PiLp.continuousLinearEquiv 2 ℝ fun x => ℝ).symm +hE : HasGaussianLaw (fun ω => toLp 2 (Z ω)) P +this : IsGaussian (Measure.map (fun ω => toLp 2 (Z ω)) P) +hZi : ∀ (i : Fin n), MemLp (fun ω => Z ω i) 2 P +hS : S.PosSemidef +⊢ ∫ (ω : Ω), e (Z ω) ∂P = e m diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-03.log new file mode 100644 index 000000000..a9f8e29c8 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-03.log @@ -0,0 +1,6 @@ +/workspace/scratch/regression-kernel/endpoint-parameters/Identification.lean:39:4: error: Type mismatch: After simplification, term + ContinuousLinearMap.integral_comp_comm (↑e) (HasGaussianLaw.integrable hZ) + has type + @Eq (EuclideanSpace ℝ (Fin n)) (∫ (x : Ω), ↑e (Z x) ∂P) (↑e m) +but is expected to have type + @Eq (EuclideanSpace ℝ (Fin n)) (∫ (ω : Ω), e (Z ω) ∂P) (e m) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-04.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification-04.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification.lean.txt new file mode 100644 index 000000000..f1bc056b1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/Identification.lean.txt @@ -0,0 +1,50 @@ +import SourceEndpointRegression +import Mathlib.Probability.Distributions.Gaussian.Multivariate + +open MeasureTheory ProbabilityTheory Matrix WithLp +open scoped BigOperators Matrix RealInnerProductSpace +set_option autoImplicit false +namespace I4SourceEndpointParameters + +lemma finiteGaussian_law_parameters {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n : ℕ} {Z : Ω → Fin n → ℝ} + (hZ : HasGaussianLaw Z P) (m : Fin n → ℝ) (hm : (∫ ω, Z ω ∂P) = m) + (S : Matrix (Fin n) (Fin n) ℝ) + (hcov : ∀ i j, covariance (fun ω => Z ω i) (fun ω => Z ω j) P = S i j) : + S.PosSemidef ∧ P.map (fun ω => toLp 2 (Z ω)) = multivariateGaussian (toLp 2 m) S := by + have : IsProbabilityMeasure P := hZ.isProbabilityMeasure + let e : (Fin n → ℝ) ≃L[ℝ] EuclideanSpace ℝ (Fin n) := + (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin n => ℝ)).symm + have hE : HasGaussianLaw (fun ω => toLp 2 (Z ω)) P := hZ.map_equiv_fun e + have : IsGaussian (P.map (fun ω => toLp 2 (Z ω))) := hE.isGaussian_map + have hZi (i : Fin n) : MemLp (fun ω => Z ω i) 2 P := (hZ.eval i).memLp_two + have hS : S.PosSemidef := by + apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg + · ext i j + simp only [Matrix.conjTranspose_apply, star_trivial] + rw [← hcov i j, ← hcov j i] + exact covariance_comm _ _ + · intro x + have hp := covarianceBilin_self_nonneg + (μ := P.map (fun ω => toLp 2 (Z ω))) (toLp 2 x) + rw [covarianceBilin_apply_pi hZi] at hp + simpa [hcov, dotProduct, Matrix.mulVec, Finset.mul_sum, mul_assoc, mul_left_comm, + mul_comm] using hp + refine ⟨hS, ?_⟩ + apply IsGaussian.ext + · simp only [id_eq] + rw [integral_id_multivariateGaussian] + rw [integral_map hE.aemeasurable (by fun_prop)] + change (∫ ω, e (Z ω) ∂P) = e m + rw [e.integral_comp_comm, hm] + · rw [← ContinuousLinearMap.toBilinForm_inj] + refine LinearMap.BilinForm.ext_basis (EuclideanSpace.basisFun (Fin n) ℝ).toBasis fun i j => ?_ + simp only [ContinuousLinearMap.toBilinForm_apply] + change covarianceBilin (P.map (fun ω => toLp 2 (Z ω))) + (EuclideanSpace.basisFun (Fin n) ℝ i) (EuclideanSpace.basisFun (Fin n) ℝ j) = + covarianceBilin (multivariateGaussian (toLp 2 m) S) + (EuclideanSpace.basisFun (Fin n) ℝ i) (EuclideanSpace.basisFun (Fin n) ℝ j) + rw [covarianceBilin_apply_basisFun hZi, covarianceBilin_multivariateGaussian hS] + simpa using hcov i j + +end I4SourceEndpointParameters diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/KernelReplay-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/KernelReplay-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/PositiveControls-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/PositiveControls-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/PositiveControls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/PositiveControls.lean.txt new file mode 100644 index 000000000..3fe127d10 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/PositiveControls.lean.txt @@ -0,0 +1,34 @@ +import SourceEndpointParameters +open I4SourceEndpointRegression I4SourceEndpointParameters I4SmoothCarrier +open MeasureTheory ProbabilityTheory Matrix WithLp +open scoped BigOperators Matrix + +example (r b : ℝ) (hG : IsUnit (sourcePinCovarianceAt 1 r)) : + (sourceEndpointKernel 1 r (sourcePrescribedPins r b)).map (toLp 2) = + multivariateGaussian (toLp 2 (sourceRegressionMapAt 1 r (sourcePrescribedPins r b))) + (sourceEndpointSchurAt 1 r) := + sourceEndpointKernel_eq_multivariateGaussian (by norm_num) r hG _ + +example (r : ℝ) (hG : IsUnit (sourcePinCovarianceAt (-1) r)) : + (sourceEndpointSchurAt (-1) r).PosSemidef := + sourceEndpointSchurAt_posSemidef (by norm_num) r hG + +example : (Measure.dirac (toLp 2 (fun _ : Fin 1 => (3 : ℝ)))) = + multivariateGaussian (toLp 2 (fun _ : Fin 1 => (3 : ℝ))) (0 : Matrix (Fin 1) (Fin 1) ℝ) := by + have hZ : HasGaussianLaw (fun _ : Unit => (fun _ : Fin 1 => (3 : ℝ))) (Measure.dirac ()) := by + refine ⟨measurable_const.aemeasurable, ?_⟩ + rw [Measure.map_dirac' measurable_const] + infer_instance + have h := finiteGaussian_law_parameters hZ (fun _ => 3) (by simp) + (0 : Matrix (Fin 1) (Fin 1) ℝ) (by intro i j; simp) + simpa using h.2 + +example : (Measure.dirac (toLp 2 (0 : Fin 0 → ℝ))) = + multivariateGaussian (toLp 2 (0 : Fin 0 → ℝ)) (0 : Matrix (Fin 0) (Fin 0) ℝ) := by + have hZ : HasGaussianLaw (fun _ : Unit => (0 : Fin 0 → ℝ)) (Measure.dirac ()) := by + refine ⟨measurable_const.aemeasurable, ?_⟩ + rw [Measure.map_dirac' measurable_const] + infer_instance + have h := finiteGaussian_law_parameters hZ 0 (by simp) + (0 : Matrix (Fin 0) (Fin 0) ℝ) (by intro i; exact i.elim0) + simpa using h.2 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedCenteredPosterior-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedCenteredPosterior-01.log new file mode 100644 index 000000000..4e8c585da --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedCenteredPosterior-01.log @@ -0,0 +1,2 @@ +/workspace/scratch/regression-kernel/endpoint-parameters/RejectedCenteredPosterior.lean:4:55: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedCenteredPosterior.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedCenteredPosterior.lean.txt new file mode 100644 index 000000000..599d4a34f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedCenteredPosterior.lean.txt @@ -0,0 +1,5 @@ +import SourceEndpointParameters +open MeasureTheory ProbabilityTheory Matrix WithLp +example : (∫ v, v ∂multivariateGaussian (toLp 2 (fun _ : Fin 1 => (3 : ℝ))) + (0 : Matrix (Fin 1) (Fin 1) ℝ)) (0 : Fin 1) = 0 := by + norm_num diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedNegativeCovariance-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedNegativeCovariance-01.log new file mode 100644 index 000000000..0b2c1792d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedNegativeCovariance-01.log @@ -0,0 +1,4 @@ +/workspace/scratch/regression-kernel/endpoint-parameters/RejectedNegativeCovariance.lean:8:60: error: unsolved goals +case «0».«0» +hZ : HasGaussianLaw (fun x => 0) (Measure.dirac ()) +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedNegativeCovariance.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedNegativeCovariance.lean.txt new file mode 100644 index 000000000..14ac26f9d --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/RejectedNegativeCovariance.lean.txt @@ -0,0 +1,12 @@ +import SourceEndpointParameters +open I4SourceEndpointParameters MeasureTheory ProbabilityTheory Matrix WithLp +example : (- (1 : Matrix (Fin 1) (Fin 1) ℝ)).PosSemidef := by + have hZ : HasGaussianLaw (fun _ : Unit => (0 : Fin 1 → ℝ)) (Measure.dirac ()) := by + refine ⟨measurable_const.aemeasurable, ?_⟩ + rw [Measure.map_dirac' measurable_const] + infer_instance + exact (finiteGaussian_law_parameters hZ 0 (by simp) (-1) (by + intro i j + fin_cases i + fin_cases j + norm_num)).1 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/SourceEndpointParameters-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/SourceEndpointParameters-01.log new file mode 100644 index 000000000..1bef7bed8 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/SourceEndpointParameters-01.log @@ -0,0 +1,13 @@ +SourceEndpointParameters.lean:220:37: error: unsolved goals +L : ℝ +hL : L ≠ 0 +r : ℝ +x : Fin 6 → ℝ +this : IsProbabilityMeasure (smoothLaw 2 L) +⊢ (smoothLaw 2 L).real Set.univ • (sourceRegressionMapAt L r) x = (sourceRegressionMapAt L r) x +SourceEndpointParameters.lean:285:27: error: `fun_prop` was unable to prove `AEStronglyMeasurable (fun v => v i) (Measure.map (sourceEndpointPosterior L r x) (smoothLaw 2 L))` + +Issues: +SourceEndpointParameters.lean:285:41: error: `fun_prop` was unable to prove `AEStronglyMeasurable (fun v => v j) (Measure.map (sourceEndpointPosterior L r x) (smoothLaw 2 L))` + +Issues: diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/SourceEndpointParameters-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/SourceEndpointParameters-02.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/assemble.py b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/assemble.py new file mode 100644 index 000000000..359718502 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/assemble.py @@ -0,0 +1,136 @@ +means=open('/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean').read() +mbody=means[means.index('theorem smoothSourceField_derivative_integral_zero'):means.rindex('end EndpointMeans')] +cov=open('/workspace/scratch/regression-kernel/endpoint-parameters/covariance/ResidualCovariance.lean').read() +cbody=cov[cov.index('theorem sourceEndpointResidual_covariance'):cov.rindex('end EndpointParametersCovarianceProbe')] +ident=open('/workspace/scratch/regression-kernel/endpoint-parameters/Identification.lean').read() +ibody=ident[ident.index('lemma finiteGaussian_law_parameters'):ident.rindex('end I4SourceEndpointParameters')] +header='''import SourceEndpointRegression +import Mathlib.Probability.Distributions.Gaussian.Multivariate + +/-! +# Actual canonical endpoint-kernel Gaussian parameters + +Unconditional means are derived from the literal differentiated source series. +The residual covariance is derived from bilinearity and the actual regression +normal equations. Its Schur matrix is positive semidefinite because it is an +actual Gaussian covariance; no positivity premise is added. + +At every pin the canonical kernel, transported from the Pi space to Euclidean +space, equals the literal `multivariateGaussian` with mean `C G⁻¹ x` and Schur +covariance. Singular residual covariance is retained. Current pin covariance +invertibility is explicit pending the separate covariance-limit join. +See `SOURCE_ENDPOINT_PARAMETERS.md` for the frozen source contract. +-/ + +noncomputable section + +open I4Foundation I4Source I4SourceGaussian I4SmoothCarrier I4SourceDerivativeGaussian +open I4SourceConfluentPins I4SourceEndpointRegression ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Matrix WithLp +open scoped BigOperators Matrix RealInnerProductSpace + +set_option autoImplicit false + +namespace I4SourceEndpointParameters + +''' +actual='''/-- The actual adjusted-endpoint Schur covariance. -/ +def sourceEndpointSchurAt (L r : ℝ) : Matrix (Fin 6) (Fin 6) ℝ := + sourceEndpointCovarianceAt L r - + sourceCrossCovarianceAt L r * (sourcePinCovarianceAt L r)⁻¹ * (sourceCrossCovarianceAt L r)ᵀ + +/-- Gaussianity is inherited from the actual jointly Gaussian pin/endpoint vector. -/ +theorem sourceEndpointResidual_hasGaussianLaw {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + HasGaussianLaw (sourceEndpointResidual L r) (smoothLaw 2 L) := by + have hR := (regression_residual_jointGaussian (sourcePins_adjusted_hasGaussianLaw hL r) + (sourceRegressionMapAt L r) 0 0).snd + unfold sourceEndpointResidual + simpa only [sub_zero] using hR + +/-- A literal posterior sample at a prescribed pin. -/ +def sourceEndpointPosterior (L r : ℝ) (x : Fin 6 → ℝ) (ω : SmoothSample 2 L) : Fin 6 → ℝ := + sourceEndpointResidual L r ω + sourceRegressionMapAt L r x + +theorem sourceEndpointPosterior_hasGaussianLaw {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : HasGaussianLaw (sourceEndpointPosterior L r x) (smoothLaw 2 L) := by + have hR := sourceEndpointResidual_hasGaussianLaw hL r + refine ⟨hR.aemeasurable.add aemeasurable_const, ?_⟩ + have : IsGaussian ((smoothLaw 2 L).map (sourceEndpointResidual L r)) := hR.isGaussian_map + convert (inferInstance : IsGaussian (((smoothLaw 2 L).map (sourceEndpointResidual L r)).map + (fun v => v + sourceRegressionMapAt L r x))) using 1 + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hR.aemeasurable] + rfl + +theorem sourceEndpointPosterior_integral {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : (∫ ω, sourceEndpointPosterior L r x ω ∂smoothLaw 2 L) = + sourceRegressionMapAt L r x := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + unfold sourceEndpointPosterior + rw [integral_add (sourceEndpointResidual_hasGaussianLaw hL r).integrable (integrable_const _), + sourceEndpointResidual_integral_zero hL r, integral_const, zero_add] + +theorem sourceEndpointPosterior_covariance {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (x : Fin 6 → ℝ) (i j : Fin 6) : + covariance (fun ω => sourceEndpointPosterior L r x ω i) + (fun ω => sourceEndpointPosterior L r x ω j) (smoothLaw 2 L) = + sourceEndpointSchurAt L r i j := by + have : IsProbabilityMeasure (smoothLaw 2 L) := smoothLaw_isProbabilityMeasure 2 hL + have hR := sourceEndpointResidual_hasGaussianLaw hL r + change covariance (fun ω => sourceEndpointResidual L r ω i + sourceRegressionMapAt L r x i) + (fun ω => sourceEndpointResidual L r ω j + sourceRegressionMapAt L r x j) _ = _ + rw [covariance_add_const_left (hR.eval i).integrable, + covariance_add_const_right (hR.eval j).integrable] + exact sourceEndpointResidual_covariance hL r hG i j + +/-- The canonical all-pin law is the literal posterior sampling map. -/ +theorem sourceEndpointKernel_eq_map_posterior {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : sourceEndpointKernel L r x = + (smoothLaw 2 L).map (sourceEndpointPosterior L r x) := by + rw [sourceEndpointKernel_apply hL r x, + AEMeasurable.map_map_of_aemeasurable (by fun_prop) + (sourceEndpointResidual_hasGaussianLaw hL r).aemeasurable] + rfl + +theorem sourceEndpointKernel_isGaussian {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : IsGaussian (sourceEndpointKernel L r x) := by + rw [sourceEndpointKernel_eq_map_posterior hL r x] + exact (sourceEndpointPosterior_hasGaussianLaw hL r x).isGaussian_map + +/-- Positive semidefiniteness follows from the actual residual covariance, including singular cases. -/ +theorem sourceEndpointSchurAt_posSemidef {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) : (sourceEndpointSchurAt L r).PosSemidef := by + exact (finiteGaussian_law_parameters (sourceEndpointPosterior_hasGaussianLaw hL r 0) + (sourceRegressionMapAt L r 0) (sourceEndpointPosterior_integral hL r 0) + (sourceEndpointSchurAt L r) (sourceEndpointPosterior_covariance hL r hG 0)).1 + +/-- Exact literal multivariate Gaussian law at every prescribed pin, with no residual PSD premise. -/ +theorem sourceEndpointKernel_eq_multivariateGaussian {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (x : Fin 6 → ℝ) : + (sourceEndpointKernel L r x).map (toLp 2) = + multivariateGaussian (toLp 2 (sourceRegressionMapAt L r x)) (sourceEndpointSchurAt L r) := by + rw [sourceEndpointKernel_eq_map_posterior hL r x, + AEMeasurable.map_map_of_aemeasurable (by fun_prop) + (sourceEndpointPosterior_hasGaussianLaw hL r x).aemeasurable] + exact (finiteGaussian_law_parameters (sourceEndpointPosterior_hasGaussianLaw hL r x) + (sourceRegressionMapAt L r x) (sourceEndpointPosterior_integral hL r x) + (sourceEndpointSchurAt L r) (sourceEndpointPosterior_covariance hL r hG x)).2 + +/-- The original Pi-space kernel mean is the actual regression mean. -/ +theorem sourceEndpointKernel_integral {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (x : Fin 6 → ℝ) : (∫ v, v ∂sourceEndpointKernel L r x) = sourceRegressionMapAt L r x := by + rw [sourceEndpointKernel_eq_map_posterior hL r x, + integral_map (sourceEndpointPosterior_hasGaussianLaw hL r x).aemeasurable (by fun_prop)] + exact sourceEndpointPosterior_integral hL r x + +/-- The original Pi-space kernel covariance is the derived actual Schur covariance. -/ +theorem sourceEndpointKernel_covariance {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (x : Fin 6 → ℝ) (i j : Fin 6) : + covariance (fun v => v i) (fun v => v j) (sourceEndpointKernel L r x) = + sourceEndpointSchurAt L r i j := by + rw [sourceEndpointKernel_eq_map_posterior hL r x, + covariance_map_fun (by fun_prop) (by fun_prop) + (sourceEndpointPosterior_hasGaussianLaw hL r x).aemeasurable] + exact sourceEndpointPosterior_covariance hL r hG x i j + +''' +open('SourceEndpointParameters.lean','w').write(header+mbody+cbody+ibody+actual+'end I4SourceEndpointParameters\n') diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.failed1.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.failed1.lean.txt new file mode 100644 index 000000000..d84d39d0a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.failed1.lean.txt @@ -0,0 +1,68 @@ +import SourceEndpointRegression + +noncomputable section +open I4SmoothCarrier I4SourceEndpointRegression ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Matrix +open scoped BigOperators Matrix +set_option autoImplicit false + +namespace EndpointParametersCovarianceProbe + +theorem sourceEndpointResidual_covariance {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (i j : Fin 6) : + covariance (fun ω => sourceEndpointResidual L r ω i) + (fun ω => sourceEndpointResidual L r ω j) (smoothLaw 2 L) = + sourceEndpointCovarianceAt L r i j - + (sourceCrossCovarianceAt L r * (sourcePinCovarianceAt L r)⁻¹ * + (sourceCrossCovarianceAt L r)ᵀ) i j := by + let X := sourceConfluentPinsAt L r + let V := sourceAdjustedEndpoints L r + let B := sourceRegressionMapAt L r + let R := sourceEndpointResidual L r + let P := smoothLaw 2 L + have hXV := sourcePins_adjusted_hasGaussianLaw hL r + have : IsProbabilityMeasure P := hXV.isProbabilityMeasure + have hXi (l : Fin 6) : MemLp (fun ω => X ω l) 2 P := (hXV.fst.eval l).memLp_two + have hVi (l : Fin 6) : MemLp (fun ω => V ω l) 2 P := (hXV.snd.eval l).memLp_two + have hBi (l : Fin 6) : MemLp (fun ω => B (X ω) l) 2 P := + ((hXV.fst.map_fun B).eval l).memLp_two + have hRi (l : Fin 6) : MemLp (fun ω => R ω l) 2 P := by + exact (hVi l).sub (hBi l) + have hXR (l t : Fin 6) : + covariance (fun ω => X ω l) (fun ω => R ω t) P = 0 := by + simpa only [sub_zero] using + regression_residual_zero_covariance hXV B 0 0 + (sourceRegression_normal_equations L r hG) l t + have hBR (l t : Fin 6) : + covariance (fun ω => B (X ω) l) (fun ω => R ω t) P = 0 := by + have heq : (fun ω => B (X ω) l) = + (fun ω => ∑ a, B (Pi.single a 1) l * X ω a) := by + funext ω + exact regression_map_coordinate B (X ω) l + rw [heq, covariance_fun_sum_left (fun a => (hXi a).const_mul _) (hRi t)] + simp_rw [covariance_const_mul_left, hXR, mul_zero] + exact Finset.sum_const_zero + have hBV (l t : Fin 6) : + covariance (fun ω => B (X ω) l) (fun ω => V ω t) P = + (sourceRegressionMatrixAt L r * (sourceCrossCovarianceAt L r)ᵀ) l t := by + have heq : (fun ω => B (X ω) l) = + (fun ω => ∑ a, B (Pi.single a 1) l * X ω a) := by + funext ω + exact regression_map_coordinate B (X ω) l + rw [heq, covariance_fun_sum_left (fun a => (hXi a).const_mul _) (hVi t)] + simp_rw [covariance_const_mul_left] + change (∑ a, sourceRegressionMapAt L r (Pi.single a 1) l * + covariance (fun ω => X ω a) (fun ω => V ω t) P) = _ + simp only [sourceRegressionMapAt_single, Matrix.mul_apply, Matrix.transpose_apply] + apply Finset.sum_congr rfl + intro a _ + congr 1 + exact covariance_comm _ _ + change covariance (fun ω => R ω i) (fun ω => V ω j - B (X ω) j) P = _ + rw [covariance_fun_sub_right (hRi i) (hVi j) (hBi j), + covariance_comm (fun ω => R ω i) (fun ω => B (X ω) j), hBR j i, sub_zero] + change covariance (fun ω => V ω i - B (X ω) i) (fun ω => V ω j) P = _ + rw [covariance_fun_sub_left (hVi i) (hBi i) (hVi j), hBV i j] + rfl + +end EndpointParametersCovarianceProbe diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.failed1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.failed1.log new file mode 100644 index 000000000..16d139393 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.failed1.log @@ -0,0 +1,9 @@ +/workspace/scratch/regression-kernel/endpoint-parameters/covariance/ResidualCovariance.lean:33:4: error: Type mismatch: After simplification, term + regression_residual_zero_covariance hXV B 0 0 (sourceRegression_normal_equations L r hG) l t + has type + @Eq ℝ + cov[fun ω => sourceConfluentPinsAt L r ω l, fun ω => + (sourceAdjustedEndpoints L r ω - B (sourceConfluentPinsAt L r ω)) t; smoothLaw 2 L] + 0 +but is expected to have type + @Eq ℝ cov[fun ω => X ω l, fun ω => R ω t; P] 0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.lean.txt new file mode 100644 index 000000000..c221e90e2 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.lean.txt @@ -0,0 +1,68 @@ +import SourceEndpointRegression + +noncomputable section +open I4SmoothCarrier I4SourceEndpointRegression ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Matrix +open scoped BigOperators Matrix +set_option autoImplicit false + +namespace EndpointParametersCovarianceProbe + +theorem sourceEndpointResidual_covariance {L : ℝ} (hL : L ≠ 0) (r : ℝ) + (hG : IsUnit (sourcePinCovarianceAt L r)) (i j : Fin 6) : + covariance (fun ω => sourceEndpointResidual L r ω i) + (fun ω => sourceEndpointResidual L r ω j) (smoothLaw 2 L) = + sourceEndpointCovarianceAt L r i j - + (sourceCrossCovarianceAt L r * (sourcePinCovarianceAt L r)⁻¹ * + (sourceCrossCovarianceAt L r)ᵀ) i j := by + let X := sourceConfluentPinsAt L r + let V := sourceAdjustedEndpoints L r + let B := sourceRegressionMapAt L r + let R := sourceEndpointResidual L r + let P := smoothLaw 2 L + have hXV := sourcePins_adjusted_hasGaussianLaw hL r + have : IsProbabilityMeasure P := hXV.isProbabilityMeasure + have hXi (l : Fin 6) : MemLp (fun ω => X ω l) 2 P := (hXV.fst.eval l).memLp_two + have hVi (l : Fin 6) : MemLp (fun ω => V ω l) 2 P := (hXV.snd.eval l).memLp_two + have hBi (l : Fin 6) : MemLp (fun ω => B (X ω) l) 2 P := + ((hXV.fst.map_fun B).eval l).memLp_two + have hRi (l : Fin 6) : MemLp (fun ω => R ω l) 2 P := by + exact (hVi l).sub (hBi l) + have hXR (l t : Fin 6) : + covariance (fun ω => X ω l) (fun ω => R ω t) P = 0 := by + simpa only [sub_zero, X, R, P, sourceEndpointResidual] using + regression_residual_zero_covariance hXV B 0 0 + (sourceRegression_normal_equations L r hG) l t + have hBR (l t : Fin 6) : + covariance (fun ω => B (X ω) l) (fun ω => R ω t) P = 0 := by + have heq : (fun ω => B (X ω) l) = + (fun ω => ∑ a, B (Pi.single a 1) l * X ω a) := by + funext ω + exact regression_map_coordinate B (X ω) l + rw [heq, covariance_fun_sum_left (fun a => (hXi a).const_mul _) (hRi t)] + simp_rw [covariance_const_mul_left, hXR, mul_zero] + exact Finset.sum_const_zero + have hBV (l t : Fin 6) : + covariance (fun ω => B (X ω) l) (fun ω => V ω t) P = + (sourceRegressionMatrixAt L r * (sourceCrossCovarianceAt L r)ᵀ) l t := by + have heq : (fun ω => B (X ω) l) = + (fun ω => ∑ a, B (Pi.single a 1) l * X ω a) := by + funext ω + exact regression_map_coordinate B (X ω) l + rw [heq, covariance_fun_sum_left (fun a => (hXi a).const_mul _) (hVi t)] + simp_rw [covariance_const_mul_left] + change (∑ a, sourceRegressionMapAt L r (Pi.single a 1) l * + covariance (fun ω => X ω a) (fun ω => V ω t) P) = _ + simp only [sourceRegressionMapAt_single, Matrix.mul_apply, Matrix.transpose_apply] + apply Finset.sum_congr rfl + intro a _ + congr 1 + exact covariance_comm _ _ + change covariance (fun ω => R ω i) (fun ω => V ω j - B (X ω) j) P = _ + rw [covariance_fun_sub_right (hRi i) (hVi j) (hBi j), + covariance_comm (fun ω => R ω i) (fun ω => B (X ω) j), hBR j i, sub_zero] + change covariance (fun ω => V ω i - B (X ω) i) (fun ω => V ω j) P = _ + rw [covariance_fun_sub_left (hVi i) (hBi i) (hVi j), hBV i j] + rfl + +end EndpointParametersCovarianceProbe diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/covariance/ResidualCovariance.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means-01.log new file mode 100644 index 000000000..366da94de --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means-01.log @@ -0,0 +1,83 @@ +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:16:13: error: don't know how to synthesize implicit argument `μ` + @integral_map (SmoothSample d L) ℝ Real.normedAddCommGroup RCLike.toInnerProductSpaceReal.toNormedSpace + Subtype.instMeasurableSpace ?m.38 (SourceMode d → ℝ) MeasurableSpace.pi (sampleInclusion d L) + (Measurable.aemeasurable (sampleInclusion_measurable d L)) (scalarSeries (derivativeCoefficient d L k x u)) + (Measurable.aestronglyMeasurable (scalarSeries_measurable (derivativeCoefficient d L k x u))) +context: +d : ℕ +L : ℝ +hL : L ≠ 0 +k : ℕ +x : Fin d → ℝ +u : Fin k → Fin d → ℝ +⊢ Measure (SmoothSample d L) +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:17:4: error: don't know how to synthesize implicit argument `μ` + @Measurable.aestronglyMeasurable (SourceMode d → ℝ) ℝ PseudoMetricSpace.toUniformSpace.toTopologicalSpace + MeasurableSpace.pi MeasurableSpace.pi (Measure.map (sampleInclusion d L) ?m.38) + (scalarSeries (derivativeCoefficient d L k x u)) Real.measurableSpace PseudoEMetricSpace.pseudoMetrizableSpace + instSecondCountableTopologyReal BorelSpace.opensMeasurable + (scalarSeries_measurable (derivativeCoefficient d L k x u)) +context: +d : ℕ +L : ℝ +hL : L ≠ 0 +k : ℕ +x : Fin d → ℝ +u : Fin k → Fin d → ℝ +⊢ Measure (SourceMode d → ℝ) +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:16:26: error: don't know how to synthesize implicit argument `μ` + @Measurable.aemeasurable (SmoothSample d L) (SourceMode d → ℝ) Subtype.instMeasurableSpace MeasurableSpace.pi + (sampleInclusion d L) ?m.38 (sampleInclusion_measurable d L) +context: +d : ℕ +L : ℝ +hL : L ≠ 0 +k : ℕ +x : Fin d → ℝ +u : Fin k → Fin d → ℝ +⊢ Measure (SmoothSample d L) +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:16:7: error: failed to infer `have` declaration type +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:14:82: error: unsolved goals +d : ℕ +L : ℝ +hL : L ≠ 0 +k : ℕ +x : Fin d → ℝ +u : Fin k → Fin d → ℝ +⊢ ∫ (ω : SmoothSample d L), scalarSeries (derivativeCoefficient d L k x u) (sampleInclusion d L ω) ∂smoothLaw d L = 0 +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:32:6: error: Type mismatch: After simplification, term + smoothSourceField_derivative_integral_zero 2 hL 1 ?m.1408 ![directionY] + has type + ∫ (ω : SmoothSample 2 L), (fderiv ℝ (smoothSourceField 2 L ω) ?m.1408) (![directionY] 0) ∂smoothLaw 2 L = 0 +but is expected to have type + ∫ (ω : SmoothSample 2 L), (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionX ∂smoothLaw 2 L = 0 +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:32:6: error: Type mismatch: After simplification, term + smoothSourceField_derivative_integral_zero 2 hL 1 ?m.1408 ![directionY] + has type + ∫ (ω : SmoothSample 2 L), (fderiv ℝ (smoothSourceField 2 L ω) ?m.1408) (![directionY] 0) ∂smoothLaw 2 L = 0 +but is expected to have type + ∫ (ω : SmoothSample 2 L), (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionX ∂smoothLaw 2 L = 0 +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:32:6: error: Type mismatch: After simplification, term + smoothSourceField_derivative_integral_zero 2 hL 1 (endpointM r) ![directionY] + has type + @Eq ℝ (∫ (ω : SmoothSample 2 L), (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) (![directionY] 0) ∂smoothLaw 2 L) + 0 +but is expected to have type + @Eq ℝ (∫ (ω : SmoothSample 2 L), (fderiv ℝ (smoothSourceField 2 L ω) (endpointM r)) directionY ∂smoothLaw 2 L) 0 +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:32:6: error: Type mismatch: After simplification, term + smoothSourceField_derivative_integral_zero 2 hL 1 (endpointS r) ![directionY] + has type + @Eq ℝ (∫ (ω : SmoothSample 2 L), (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) (![directionY] 0) ∂smoothLaw 2 L) + 0 +but is expected to have type + @Eq ℝ (∫ (ω : SmoothSample 2 L), (fderiv ℝ (smoothSourceField 2 L ω) (endpointS r)) directionY ∂smoothLaw 2 L) 0 +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:38:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ?m.45 (∫ (x : SmoothSample 2 L), rawPinsWithHessians L r x ∂smoothLaw 2 L) +in the target expression + (∫ (ω : SmoothSample 2 L), rawPinsWithHessians L r ω ∂smoothLaw 2 L) i = 0 i + +L : ℝ +hL : L ≠ 0 +r : ℝ +i : Fin 12 +⊢ (∫ (ω : SmoothSample 2 L), rawPinsWithHessians L r ω ∂smoothLaw 2 L) i = 0 i diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means-02.log new file mode 100644 index 000000000..5127d218f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means-02.log @@ -0,0 +1,13 @@ +/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean:41:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + (ContinuousLinearMap.proj i) (∫ (x : SmoothSample 2 L), rawPinsWithHessians L r x ∂smoothLaw 2 L) +in the target expression + (∫ (ω : SmoothSample 2 L), rawPinsWithHessians L r ω ∂smoothLaw 2 L) i = 0 + +L : ℝ +hL : L ≠ 0 +r : ℝ +i : Fin 12 +hh : + ∫ (x : SmoothSample 2 L), (ContinuousLinearMap.proj i) (rawPinsWithHessians L r x) ∂smoothLaw 2 L = + (ContinuousLinearMap.proj i) (∫ (x : SmoothSample 2 L), rawPinsWithHessians L r x ∂smoothLaw 2 L) +⊢ (∫ (ω : SmoothSample 2 L), rawPinsWithHessians L r ω ∂smoothLaw 2 L) i = 0 diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means.lean.txt new file mode 100644 index 000000000..2fcf4eade --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointParameters/means/Means.lean.txt @@ -0,0 +1,75 @@ +import SourceEndpointRegression + +noncomputable section +open I4Foundation I4Source I4SourceGaussian I4SmoothCarrier I4SourceDerivativeGaussian +open I4SourceConfluentPins I4SourceEndpointRegression +open MeasureTheory ProbabilityTheory +open scoped BigOperators +set_option autoImplicit false + +namespace EndpointMeans + +theorem smoothSourceField_derivative_integral_zero (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (k : ℕ) (x : Fin d → ℝ) (u : Fin k → (Fin d → ℝ)) : + (∫ ω, iteratedFDeriv ℝ k (smoothSourceField d L ω) x u ∂smoothLaw d L) = 0 := by + simp_rw [smoothSourceField_derivative_series d hL] + have hm := integral_map (μ := smoothLaw d L) (sampleInclusion_measurable d L).aemeasurable + (scalarSeries_measurable (derivativeCoefficient d L k x u)).aestronglyMeasurable + rw [sampleInclusion_map d hL] at hm + rw [← hm] + exact scalarSeries_integral_zero _ (derivativeCoefficient_summable d hL k x u) + +theorem rawPinsWithHessians_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) (i : Fin 12) : + (∫ ω, rawPinsWithHessians L r ω i ∂smoothLaw 2 L) = 0 := by + fin_cases i + all_goals dsimp [rawPinsWithHessians, rawPins] + all_goals first + | exact smoothSourceField_derivative_integral_zero 2 hL 2 _ _ + | simpa only [iteratedFDeriv_zero_apply] using + smoothSourceField_derivative_integral_zero 2 hL 0 _ ![] + | simpa [iteratedFDeriv_one_apply] using + smoothSourceField_derivative_integral_zero 2 hL 1 _ ![directionX] + | simpa [iteratedFDeriv_one_apply] using + smoothSourceField_derivative_integral_zero 2 hL 1 _ ![directionY] + +theorem rawPinsWithHessians_integral_vector_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, rawPinsWithHessians L r ω ∂smoothLaw 2 L) = 0 := by + ext i + have hh := (ContinuousLinearMap.proj i : (Fin 12 → ℝ) →L[ℝ] ℝ).integral_comp_comm + (rawPinsWithHessians_hasGaussianLaw hL r).integrable + change (∫ ω, rawPinsWithHessians L r ω i ∂smoothLaw 2 L) = + (∫ ω, rawPinsWithHessians L r ω ∂smoothLaw 2 L) i at hh + change (∫ ω, rawPinsWithHessians L r ω ∂smoothLaw 2 L) i = 0 + rw [← hh] + exact rawPinsWithHessians_integral_zero hL r i + +theorem sourcePins_adjusted_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, (sourceConfluentPinsAt L r ω, sourceAdjustedEndpoints L r ω) ∂smoothLaw 2 L) = 0 := by + have hh := (endpointObservationMap r).integral_comp_comm + (rawPinsWithHessians_hasGaussianLaw hL r).integrable + simp_rw [endpointObservationMap_actual] at hh + rw [rawPinsWithHessians_integral_vector_zero hL r, map_zero] at hh + exact hh + +theorem sourceConfluentPinsAt_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, sourceConfluentPinsAt L r ω ∂smoothLaw 2 L) = 0 := by + have hh := congrArg Prod.fst (sourcePins_adjusted_integral_zero hL r) + rw [fst_integral (sourcePins_adjusted_hasGaussianLaw hL r).integrable] at hh + exact hh + +theorem sourceAdjustedEndpoints_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, sourceAdjustedEndpoints L r ω ∂smoothLaw 2 L) = 0 := by + have hh := congrArg Prod.snd (sourcePins_adjusted_integral_zero hL r) + rw [snd_integral (sourcePins_adjusted_hasGaussianLaw hL r).integrable] at hh + exact hh + +theorem sourceEndpointResidual_integral_zero {L : ℝ} (hL : L ≠ 0) (r : ℝ) : + (∫ ω, sourceEndpointResidual L r ω ∂smoothLaw 2 L) = 0 := by + have hG := sourcePins_adjusted_hasGaussianLaw hL r + unfold sourceEndpointResidual + rw [integral_sub hG.snd.integrable (hG.fst.map_fun (sourceRegressionMapAt L r)).integrable, + (sourceRegressionMapAt L r).integral_comp_comm hG.fst.integrable, + sourceAdjustedEndpoints_integral_zero hL r, sourceConfluentPinsAt_integral_zero hL r, + map_zero, sub_zero] + +end EndpointMeans diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/AxiomAudit-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/AxiomAudit-01.log new file mode 100644 index 000000000..cdb2768d5 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/AxiomAudit-01.log @@ -0,0 +1,23 @@ +'I4SourceEndpointRegression.sourceConfluentPinsAt_eq' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointRegression.endpointObservationMap_actual' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointRegression.sourcePins_adjusted_hasGaussianLaw' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointRegression.sourceRegressionMapAt_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointRegression.sourceRegressionMapAt_single' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointRegression.sourceRegression_normal_equations' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointRegression.sourceEndpointKernel_markov' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointRegression.sourceEndpointKernel_measurable' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointRegression.sourceEndpointKernel_apply' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointRegression.sourceEndpointKernel_hasCondDistrib' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointRegression.sourceEndpointKernel_spec' depends on axioms: [propext, Classical.choice, Quot.sound] +'I4SourceEndpointRegression.sourceAdjustedEndpoints_on_pin_fiber' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourceEndpointRegression.sourceAdjustedEndpoints_on_prescribed' depends on axioms: [propext, + Classical.choice, + Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/AxiomAudit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/AxiomAudit.lean.txt new file mode 100644 index 000000000..eb7298234 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/AxiomAudit.lean.txt @@ -0,0 +1,14 @@ +import SourceEndpointRegression +#print axioms I4SourceEndpointRegression.sourceConfluentPinsAt_eq +#print axioms I4SourceEndpointRegression.endpointObservationMap_actual +#print axioms I4SourceEndpointRegression.sourcePins_adjusted_hasGaussianLaw +#print axioms I4SourceEndpointRegression.sourceRegressionMapAt_apply +#print axioms I4SourceEndpointRegression.sourceRegressionMapAt_single +#print axioms I4SourceEndpointRegression.sourceRegression_normal_equations +#print axioms I4SourceEndpointRegression.sourceEndpointKernel_markov +#print axioms I4SourceEndpointRegression.sourceEndpointKernel_measurable +#print axioms I4SourceEndpointRegression.sourceEndpointKernel_apply +#print axioms I4SourceEndpointRegression.sourceEndpointKernel_hasCondDistrib +#print axioms I4SourceEndpointRegression.sourceEndpointKernel_spec +#print axioms I4SourceEndpointRegression.sourceAdjustedEndpoints_on_pin_fiber +#print axioms I4SourceEndpointRegression.sourceAdjustedEndpoints_on_prescribed diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/EVIDENCE.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/EVIDENCE.json new file mode 100644 index 000000000..c0cd44670 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/EVIDENCE.json @@ -0,0 +1,143 @@ +{ + "source": "SourceEndpointRegression.lean", + "source_sha256": "67e68553e004adc3b1b6beb987b3e7ad477b4541d96fec22ee481facd2038fce", + "source_contract_commit": "badd443", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "author": "OpenAI/Codex /root/regression_kernel with OpenAI/Codex /root/regression_kernel/gaussian_api matrix normal-equations API support", + "scientific_effect": "NONE; author-side evidence; current invertibility of actual pin covariance remains premise until covariance-limit join; different-family review separate", + "build": { + "command": "/home/agent/.elan/bin/lake env lean -o .lake/build/lib/lean/SourceEndpointRegression.olean SourceEndpointRegression.lean", + "exit": 0, + "log": "SourceEndpointRegression-03.log" + }, + "kernel": { + "command": "/home/agent/.elan/bin/lake env leanchecker SourceEndpointRegression", + "exit": 0, + "log": "KernelReplay-01.log" + }, + "axiom_audit": { + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-regression/AxiomAudit.lean", + "exit": 0, + "targets": { + "I4SourceEndpointRegression.sourceConfluentPinsAt_eq": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.endpointObservationMap_actual": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourcePins_adjusted_hasGaussianLaw": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceRegressionMapAt_apply": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceRegressionMapAt_single": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceRegression_normal_equations": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceEndpointKernel_markov": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceEndpointKernel_measurable": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceEndpointKernel_apply": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceEndpointKernel_hasCondDistrib": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceEndpointKernel_spec": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceAdjustedEndpoints_on_pin_fiber": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourceEndpointRegression.sourceAdjustedEndpoints_on_prescribed": [ + "Classical.choice", + "Quot.sound", + "propext" + ] + }, + "log": "AxiomAudit-01.log" + }, + "controls": [ + { + "file": "PositiveControls.lean", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-regression/PositiveControls.lean", + "capture_command": "set -o pipefail\n/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-regression/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/endpoint-regression/PositiveControls-01.log", + "exit": 0, + "log": "PositiveControls-01.log", + "reason": "actual joint Gaussian negative period; actual conditional law under current IsUnit; all-pin formula at prescribed d; actual pin-fiber identity" + }, + { + "file": "RejectedOmitPinAdjustment.lean", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-regression/RejectedOmitPinAdjustment.lean", + "capture_command": "set -o pipefail\n/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-regression/RejectedOmitPinAdjustment.lean 2>&1 | tee /workspace/scratch/regression-kernel/endpoint-regression/RejectedOmitPinAdjustment-01.log", + "exit": 1, + "log": "RejectedOmitPinAdjustment-01.log", + "reason": "concrete false raw12 witness: subtracting X3 matters" + }, + { + "file": "RejectedOmitCubicHeight.lean", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-regression/RejectedOmitCubicHeight.lean", + "capture_command": "set -o pipefail\n/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/endpoint-regression/RejectedOmitCubicHeight.lean 2>&1 | tee /workspace/scratch/regression-kernel/endpoint-regression/RejectedOmitCubicHeight-01.log", + "exit": 1, + "log": "RejectedOmitCubicHeight-01.log", + "reason": "concrete false prescribed height without r^3/12 correction" + } + ], + "logs": { + "SourceEndpointRegression-03.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "PositiveControls-01.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "RejectedOmitPinAdjustment-01.log": { + "bytes": 126, + "sha256": "6eb99e68203efc622ddb65169d6c5049c8c170ebfc6b8ac552c1f78362311d7e" + }, + "RejectedOmitCubicHeight-01.log": { + "bytes": 124, + "sha256": "0c54af1acff8e2137639a0bd4bb2cfe8819abdffa0c6c0c8dababdaa21df09f8" + }, + "AxiomAudit-01.log": { + "bytes": 1559, + "sha256": "38483bee68f06d58e2725d4105a15101f158630030f8bff60696db492ee74142" + }, + "KernelReplay-01.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + } + }, + "reuse": "Completed source and vendored GaussianRegressionKernel proofs/objects reused; this verification concerns only the new endpoint regression module and its controls." +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/KernelReplay-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/KernelReplay-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/PositiveControls-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/PositiveControls-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/PositiveControls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/PositiveControls.lean.txt new file mode 100644 index 000000000..4e4a56710 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/PositiveControls.lean.txt @@ -0,0 +1,21 @@ +import SourceEndpointRegression +open I4SourceEndpointRegression I4SmoothCarrier ProbabilityTheory MeasureTheory + +example : HasGaussianLaw + (fun ω => (sourceConfluentPinsAt (-1) 1 ω, sourceAdjustedEndpoints (-1) 1 ω)) + (smoothLaw 2 (-1)) := sourcePins_adjusted_hasGaussianLaw (by norm_num) 1 + +example (hG : IsUnit (sourcePinCovarianceAt 1 1)) : + HasCondDistrib (sourceAdjustedEndpoints 1 1) (sourceConfluentPinsAt 1 1) + (sourceEndpointKernel 1 1) (smoothLaw 2 1) := + sourceEndpointKernel_hasCondDistrib (by norm_num) 1 hG + +example (b : ℝ) : sourceEndpointKernel 1 1 (sourcePrescribedPins 1 b) = + ((smoothLaw 2 1).map (sourceEndpointResidual 1 1)).map + (fun v => v + sourceRegressionMapAt 1 1 (sourcePrescribedPins 1 b)) := + sourceEndpointKernel_apply (by norm_num) 1 _ + +example (ω : SmoothSample 2 1) (b : ℝ) + (hp : sourceConfluentPinsAt 1 1 ω = sourcePrescribedPins 1 b) : + sourceAdjustedEndpoints 1 1 ω = sourceNormalizedEndpoints 1 1 ω := + sourceAdjustedEndpoints_on_prescribed 1 1 b ω hp diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitCubicHeight-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitCubicHeight-01.log new file mode 100644 index 000000000..2c980116f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitCubicHeight-01.log @@ -0,0 +1,2 @@ +/workspace/scratch/regression-kernel/endpoint-regression/RejectedOmitCubicHeight.lean:3:55: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitCubicHeight.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitCubicHeight.lean.txt new file mode 100644 index 000000000..4c3074bae --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitCubicHeight.lean.txt @@ -0,0 +1,4 @@ +import SourceEndpointRegression +open I4SourceEndpointRegression +example : sourcePrescribedPins 1 0 = ![0,0,0,0,0,2] := by + norm_num [sourcePrescribedPins] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitPinAdjustment-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitPinAdjustment-01.log new file mode 100644 index 000000000..157e464db --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitPinAdjustment-01.log @@ -0,0 +1,2 @@ +/workspace/scratch/regression-kernel/endpoint-regression/RejectedOmitPinAdjustment.lean:3:75: error: unsolved goals +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitPinAdjustment.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitPinAdjustment.lean.txt new file mode 100644 index 000000000..a6b67f635 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/RejectedOmitPinAdjustment.lean.txt @@ -0,0 +1,4 @@ +import SourceEndpointRegression +open I4SourceEndpointRegression I4SourceConfluentPins +example : (endpointObservationMap 1 ![0,0,0,1,0,0,0,0,0,0,0,0]).2 0 = 0 := by + norm_num [endpointObservationMap, confluentTransform] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/SourceEndpointRegression-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/SourceEndpointRegression-01.log new file mode 100644 index 000000000..d8d083377 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/SourceEndpointRegression-01.log @@ -0,0 +1,7 @@ +SourceEndpointRegression.lean:74:4: error: `fun_prop` was unable to prove `Continuous fun v => + let x := confluentTransform r fun i => v (Fin.castAdd 6 i); + (x, ![(v 6 - x 3) / r, (v 9 - x 3) / r, (v 7 - x 4) / r, (v 10 - x 4) / r, v 8, v 11])` + +Issues: + No theorems found for `I4SourceConfluentPins.confluentTransform` in order to prove `Continuous fun x1 => confluentTransform r x1` + No theorems found for `I4SourceConfluentPins.confluentTransform` in order to prove `Continuous fun x => confluentTransform r fun i => x (Fin.castAdd 6 i)` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/SourceEndpointRegression-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/SourceEndpointRegression-02.log new file mode 100644 index 000000000..03788fe06 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/SourceEndpointRegression-02.log @@ -0,0 +1,16 @@ +SourceEndpointRegression.lean:169:2: error: Type mismatch: After simplification, term + hcond + has type + @HasCondDistrib (SmoothSample 2 L) (Fin 6 → ℝ) (Fin 6 → ℝ) Subtype.instMeasurableSpace MeasurableSpace.pi + MeasurableSpace.pi (sourceAdjustedEndpoints L r) (sourceConfluentPinsAt L r) + (regressionKernel + (Measure.map + (fun ω => sourceAdjustedEndpoints L r ω - sourceRegressionMatrixAt L r *ᵥ sourceConfluentPinsAt L r ω) + (smoothLaw 2 L)) + (sourceRegressionMapAt L r) 0 0) + (smoothLaw 2 L) +but is expected to have type + @HasCondDistrib (SmoothSample 2 L) (Fin 6 → ℝ) (Fin 6 → ℝ) Subtype.instMeasurableSpace MeasurableSpace.pi + MeasurableSpace.pi (sourceAdjustedEndpoints L r) (sourceConfluentPinsAt L r) + (regressionKernel (Measure.map (sourceEndpointResidual L r) (smoothLaw 2 L)) (sourceRegressionMapAt L r) 0 0) + (smoothLaw 2 L) diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/SourceEndpointRegression-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/SourceEndpointRegression-03.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.failed1.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.failed1.lean.txt new file mode 100644 index 000000000..0bab486e8 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.failed1.lean.txt @@ -0,0 +1,56 @@ +import Mathlib.Probability.Moments.Covariance +import Mathlib.LinearAlgebra.Matrix.NonsingularInverse +import Mathlib.LinearAlgebra.Matrix.ToLin +import Mathlib.Topology.Algebra.Module.FiniteDimension + +open MeasureTheory ProbabilityTheory Matrix +open scoped ProbabilityTheory BigOperators Matrix +set_option autoImplicit false + +namespace EndpointRegressionAPI + +noncomputable def matrixCLM {n k : ℕ} (M : Matrix (Fin k) (Fin n) ℝ) : + (Fin n → ℝ) →L[ℝ] (Fin k → ℝ) := + (Matrix.toLin' M).toContinuousLinearMap + +lemma matrixCLM_apply {n k : ℕ} (M : Matrix (Fin k) (Fin n) ℝ) (x : Fin n → ℝ) : + matrixCLM M x = M *ᵥ x := rfl + +lemma matrixCLM_single {n k : ℕ} (M : Matrix (Fin k) (Fin n) ℝ) (l : Fin n) + (j : Fin k) : matrixCLM M (Pi.single l 1) j = M j l := by + simp [matrixCLM, Matrix.mulVec_single_one] + +theorem covariance_normal_equations {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} (X : Ω → Fin n → ℝ) (V : Ω → Fin k → ℝ) + (hG : IsUnit (Matrix.of (fun i j => + covariance (fun ω => X ω i) (fun ω => X ω j) P) : Matrix (Fin n) (Fin n) ℝ)) : + let G : Matrix (Fin n) (Fin n) ℝ := fun i j => + covariance (fun ω => X ω i) (fun ω => X ω j) P + let C : Matrix (Fin k) (Fin n) ℝ := fun i j => + covariance (fun ω => V ω i) (fun ω => X ω j) P + let B := matrixCLM (C * G⁻¹) + ∀ i j, covariance (fun ω => X ω i) (fun ω => V ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P := by + let G : Matrix (Fin n) (Fin n) ℝ := fun i j => + covariance (fun ω => X ω i) (fun ω => X ω j) P + let C : Matrix (Fin k) (Fin n) ℝ := fun i j => + covariance (fun ω => V ω i) (fun ω => X ω j) P + change ∀ i j, covariance (fun ω => X ω i) (fun ω => V ω j) P = + ∑ l, matrixCLM (C * G⁻¹) (Pi.single l 1) j * + covariance (fun ω => X ω i) (fun ω => X ω l) P + have hG' : IsUnit G := hG + let _ : Invertible G := hG'.invertible + have hprod : C * G⁻¹ * G = C := Matrix.inv_mul_cancel_right_of_invertible G C + intro i j + calc + covariance (fun ω => X ω i) (fun ω => V ω j) P = C j i := covariance_comm + _ = ∑ l, (C * G⁻¹) j l * G l i := (congrArg (fun M => M j i) hprod).symm + _ = ∑ l, matrixCLM (C * G⁻¹) (Pi.single l 1) j * + covariance (fun ω => X ω i) (fun ω => X ω l) P := by + apply Finset.sum_congr rfl + intro l _ + rw [matrixCLM_single] + congr 1 + exact covariance_comm + +end EndpointRegressionAPI diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.failed1.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.failed1.log new file mode 100644 index 000000000..820ecf6de --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.failed1.log @@ -0,0 +1,19 @@ +/workspace/scratch/regression-kernel/endpoint-regression/api/NormalEquations.lean:21:19: warning: This simp argument is unused: + Matrix.mulVec_single_one + +Hint: Omit it from the simp argument list. + [apply] simp [matrixCLM] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` +/workspace/scratch/regression-kernel/endpoint-regression/api/NormalEquations.lean:46:62: error: Type mismatch + covariance_comm +has type + ∀ (X Y : ?m.158 → ℝ) {μ : Measure ?m.158}, cov[X, Y; μ] = cov[Y, X; μ] +but is expected to have type + cov[fun ω => X ω i, fun ω => V ω j; P] = C j i +/workspace/scratch/regression-kernel/endpoint-regression/api/NormalEquations.lean:54:6: error: Type mismatch + covariance_comm +has type + ∀ (X Y : ?m.260 → ℝ) {μ : Measure ?m.260}, cov[X, Y; μ] = cov[Y, X; μ] +but is expected to have type + G l i = cov[fun ω => X ω i, fun ω => X ω l; P] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.lean.txt new file mode 100644 index 000000000..2a6b6e547 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.lean.txt @@ -0,0 +1,56 @@ +import Mathlib.Probability.Moments.Covariance +import Mathlib.LinearAlgebra.Matrix.NonsingularInverse +import Mathlib.LinearAlgebra.Matrix.ToLin +import Mathlib.Topology.Algebra.Module.FiniteDimension + +open MeasureTheory ProbabilityTheory Matrix +open scoped ProbabilityTheory BigOperators Matrix +set_option autoImplicit false + +namespace EndpointRegressionAPI + +noncomputable def matrixCLM {n k : ℕ} (M : Matrix (Fin k) (Fin n) ℝ) : + (Fin n → ℝ) →L[ℝ] (Fin k → ℝ) := + (Matrix.toLin' M).toContinuousLinearMap + +lemma matrixCLM_apply {n k : ℕ} (M : Matrix (Fin k) (Fin n) ℝ) (x : Fin n → ℝ) : + matrixCLM M x = M *ᵥ x := rfl + +lemma matrixCLM_single {n k : ℕ} (M : Matrix (Fin k) (Fin n) ℝ) (l : Fin n) + (j : Fin k) : matrixCLM M (Pi.single l 1) j = M j l := by + simp [matrixCLM] + +theorem covariance_normal_equations {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} (X : Ω → Fin n → ℝ) (V : Ω → Fin k → ℝ) + (hG : IsUnit (Matrix.of (fun i j => + covariance (fun ω => X ω i) (fun ω => X ω j) P) : Matrix (Fin n) (Fin n) ℝ)) : + let G : Matrix (Fin n) (Fin n) ℝ := fun i j => + covariance (fun ω => X ω i) (fun ω => X ω j) P + let C : Matrix (Fin k) (Fin n) ℝ := fun i j => + covariance (fun ω => V ω i) (fun ω => X ω j) P + let B := matrixCLM (C * G⁻¹) + ∀ i j, covariance (fun ω => X ω i) (fun ω => V ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P := by + let G : Matrix (Fin n) (Fin n) ℝ := fun i j => + covariance (fun ω => X ω i) (fun ω => X ω j) P + let C : Matrix (Fin k) (Fin n) ℝ := fun i j => + covariance (fun ω => V ω i) (fun ω => X ω j) P + change ∀ i j, covariance (fun ω => X ω i) (fun ω => V ω j) P = + ∑ l, matrixCLM (C * G⁻¹) (Pi.single l 1) j * + covariance (fun ω => X ω i) (fun ω => X ω l) P + have hG' : IsUnit G := hG + let _ : Invertible G := hG'.invertible + have hprod : C * G⁻¹ * G = C := Matrix.inv_mul_cancel_right_of_invertible G C + intro i j + calc + covariance (fun ω => X ω i) (fun ω => V ω j) P = C j i := covariance_comm _ _ + _ = ∑ l, (C * G⁻¹) j l * G l i := (congrArg (fun M => M j i) hprod).symm + _ = ∑ l, matrixCLM (C * G⁻¹) (Pi.single l 1) j * + covariance (fun ω => X ω i) (fun ω => X ω l) P := by + apply Finset.sum_congr rfl + intro l _ + rw [matrixCLM_single] + congr 1 + exact covariance_comm _ _ + +end EndpointRegressionAPI diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourceEndpointRegression/api/NormalEquations.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/AxiomAudit-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/AxiomAudit-01.log new file mode 100644 index 000000000..477b26fee --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/AxiomAudit-01.log @@ -0,0 +1,27 @@ +'I4SourcePhysicalEndpointKernel.sourceNormalizedEndpoints_eq_adjusted' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_markov' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_measurable' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_apply' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_hasCondDistrib' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourcePhysicalEndpointKernel.sourcePhysicalCorrection_prescribed' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_at_prescribed' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_integral' depends on axioms: [propext, + Classical.choice, + Quot.sound] +'I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_spec' depends on axioms: [propext, + Classical.choice, + Quot.sound] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/AxiomAudit.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/AxiomAudit.lean.txt new file mode 100644 index 000000000..55601b21c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/AxiomAudit.lean.txt @@ -0,0 +1,10 @@ +import SourcePhysicalEndpointKernel +#print axioms I4SourcePhysicalEndpointKernel.sourceNormalizedEndpoints_eq_adjusted +#print axioms I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_markov +#print axioms I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_measurable +#print axioms I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_apply +#print axioms I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_hasCondDistrib +#print axioms I4SourcePhysicalEndpointKernel.sourcePhysicalCorrection_prescribed +#print axioms I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_at_prescribed +#print axioms I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_integral +#print axioms I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_spec diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/EVIDENCE.json b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/EVIDENCE.json new file mode 100644 index 000000000..ee2abbe1b --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/EVIDENCE.json @@ -0,0 +1,124 @@ +{ + "source": "SourcePhysicalEndpointKernel.lean", + "source_sha256": "689ced2a760b38333b0224f19af52812d0057527d5a790ba50116cb5f07b7e3a", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "source_contract_commit": "1819e38", + "author": "OpenAI/Codex /root/regression_kernel", + "scientific_effect": "NONE; author-side verification only; whole-field Palm and event joins remain open; different-family review separate", + "build": { + "command": "/home/agent/.elan/bin/lake env lean -o .lake/build/lib/lean/SourcePhysicalEndpointKernel.olean SourcePhysicalEndpointKernel.lean", + "exit": 0, + "log": "SourcePhysicalEndpointKernel-04.log" + }, + "kernel": { + "command": "/home/agent/.elan/bin/lake env leanchecker SourcePhysicalEndpointKernel", + "exit": 0, + "log": "KernelReplay-01.log" + }, + "axioms": { + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/physical-endpoint/AxiomAudit.lean", + "exit": 0, + "targets": { + "I4SourcePhysicalEndpointKernel.sourceNormalizedEndpoints_eq_adjusted": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_markov": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_measurable": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_apply": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_hasCondDistrib": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourcePhysicalEndpointKernel.sourcePhysicalCorrection_prescribed": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_at_prescribed": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_integral": [ + "Classical.choice", + "Quot.sound", + "propext" + ], + "I4SourcePhysicalEndpointKernel.sourcePhysicalEndpointKernel_spec": [ + "Classical.choice", + "Quot.sound", + "propext" + ] + }, + "log": "AxiomAudit-01.log" + }, + "controls": [ + { + "file": "PositiveControls.lean", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/physical-endpoint/PositiveControls.lean", + "capture_command": "set -o pipefail\n/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/physical-endpoint/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/physical-endpoint/PositiveControls-01.log", + "exit": 0, + "log": "PositiveControls-01.log", + "reason": "arbitrary-height prescribed literal measure equality; actual physical HasCondDistrib at negative period; nonprescribed D nonzero; actual physical kernel differs from adjusted kernel at e3 by unequal means" + }, + { + "file": "RejectedAllPinCorrectionZero.lean", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/physical-endpoint/RejectedAllPinCorrectionZero.lean", + "capture_command": "set -o pipefail\n/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/physical-endpoint/RejectedAllPinCorrectionZero.lean 2>&1 | tee /workspace/scratch/regression-kernel/physical-endpoint/RejectedAllPinCorrectionZero-01.log", + "exit": 1, + "log": "RejectedAllPinCorrectionZero-01.log", + "reason": "D_1(e3) has first2 coordinates1, not0" + }, + { + "file": "RejectedOmitPhysicalMeanCorrection.lean", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/physical-endpoint/RejectedOmitPhysicalMeanCorrection.lean", + "capture_command": "set -o pipefail\n/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/physical-endpoint/RejectedOmitPhysicalMeanCorrection.lean 2>&1 | tee /workspace/scratch/regression-kernel/physical-endpoint/RejectedOmitPhysicalMeanCorrection-01.log", + "exit": 1, + "log": "RejectedOmitPhysicalMeanCorrection-01.log", + "reason": "physical kernel mean at e3 differs by the nonzero D correction" + } + ], + "control_warnings": "PositiveControls and RejectedOmitPhysicalMeanCorrection report one harmless unused Pi.single_apply simp-argument warning; mathematical outcomes recorded exactly, no repeat for warning alone.", + "logs": { + "SourcePhysicalEndpointKernel-04.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + "PositiveControls-01.log": { + "bytes": 314, + "sha256": "58d90f2859b3d419465abcb614b267185bc5955917255feac90433312a6da1e6" + }, + "RejectedAllPinCorrectionZero-01.log": { + "bytes": 160, + "sha256": "bc3eeb3a71cb384d141f3be4ea9fc977d6915ffaa14c5713ea4f0528de941719" + }, + "RejectedOmitPhysicalMeanCorrection-01.log": { + "bytes": 491, + "sha256": "0f1c3137368ade94ad6d986fa47a713648174a6ec80f81deeb9f2072215d8f89" + }, + "AxiomAudit-01.log": { + "bytes": 1190, + "sha256": "af2209a06a8b3b470bbf89b2ac97462f02da639ba573905fa1f627c361f6be17" + }, + "KernelReplay-01.log": { + "bytes": 0, + "sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + } + }, + "reuse": "Completed Gaussian/kernel/parameter source artifacts reused unchanged; only this new physical kernel module and its new controls checked." +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/KernelReplay-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/KernelReplay-01.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/PositiveControls-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/PositiveControls-01.log new file mode 100644 index 000000000..954044176 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/PositiveControls-01.log @@ -0,0 +1,7 @@ +/workspace/scratch/regression-kernel/physical-endpoint/PositiveControls.lean:25:34: warning: This simp argument is unused: + Pi.single_apply + +Hint: Omit it from the simp argument list. + [apply] simp [sourcePhysicalCorrection] at hh + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/PositiveControls.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/PositiveControls.lean.txt new file mode 100644 index 000000000..bb4ddae89 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/PositiveControls.lean.txt @@ -0,0 +1,25 @@ +import SourcePhysicalEndpointKernel +open I4SourceEndpointRegression I4SourceEndpointParameters I4SourcePhysicalEndpointKernel I4SmoothCarrier +open MeasureTheory ProbabilityTheory Matrix + +example (r b : ℝ) : sourcePhysicalEndpointKernel 1 r (sourcePrescribedPins r b) = + sourceEndpointKernel 1 r (sourcePrescribedPins r b) := + sourcePhysicalEndpointKernel_at_prescribed (by norm_num) r b + +example (r : ℝ) (hG : IsUnit (sourcePinCovarianceAt (-1) r)) : + HasCondDistrib (sourceNormalizedEndpoints (-1) r) (sourceConfluentPinsAt (-1) r) + (sourcePhysicalEndpointKernel (-1) r) (smoothLaw 2 (-1)) := + sourcePhysicalEndpointKernel_hasCondDistrib (by norm_num) r hG + +example : sourcePhysicalCorrection 1 (Pi.single (3 : Fin 6) (1 : ℝ)) ≠ 0 := by + intro h + have hh := congrFun h (0 : Fin 6) + norm_num [sourcePhysicalCorrection, Pi.single_apply] at hh + +example : sourcePhysicalEndpointKernel 1 1 (Pi.single (3 : Fin 6) (1 : ℝ)) ≠ + sourceEndpointKernel 1 1 (Pi.single (3 : Fin 6) (1 : ℝ)) := by + intro h + have hm := congrArg (fun μ : Measure (Fin 6 → ℝ) => ∫ v, v ∂μ) h + rw [sourcePhysicalEndpointKernel_integral (by norm_num), sourceEndpointKernel_integral (by norm_num)] at hm + have hh := congrFun hm (0 : Fin 6) + simp [sourcePhysicalCorrection, Pi.single_apply] at hh diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedAllPinCorrectionZero-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedAllPinCorrectionZero-01.log new file mode 100644 index 000000000..7f83315a9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedAllPinCorrectionZero-01.log @@ -0,0 +1,6 @@ +/workspace/scratch/regression-kernel/physical-endpoint/RejectedAllPinCorrectionZero.lean:3:76: error: unsolved goals +case «0» +⊢ False + +case «1» +⊢ False diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedAllPinCorrectionZero.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedAllPinCorrectionZero.lean.txt new file mode 100644 index 000000000..93c2149fb --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedAllPinCorrectionZero.lean.txt @@ -0,0 +1,5 @@ +import SourcePhysicalEndpointKernel +open I4SourcePhysicalEndpointKernel +example : sourcePhysicalCorrection 1 (Pi.single (3 : Fin 6) (1 : ℝ)) = 0 := by + ext i + fin_cases i <;> norm_num [sourcePhysicalCorrection, Pi.single_apply] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedOmitPhysicalMeanCorrection-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedOmitPhysicalMeanCorrection-01.log new file mode 100644 index 000000000..7cdf31e46 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedOmitPhysicalMeanCorrection-01.log @@ -0,0 +1,13 @@ +/workspace/scratch/regression-kernel/physical-endpoint/RejectedOmitPhysicalMeanCorrection.lean:4:65: error: unsolved goals +case «0» +⊢ False + +case «1» +⊢ False +/workspace/scratch/regression-kernel/physical-endpoint/RejectedOmitPhysicalMeanCorrection.lean:7:50: warning: This simp argument is unused: + Pi.single_apply + +Hint: Omit it from the simp argument list. + [apply] simp [sourcePhysicalCorrection] + +Note: This linter can be disabled with `set_option linter.unusedSimpArgs false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedOmitPhysicalMeanCorrection.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedOmitPhysicalMeanCorrection.lean.txt new file mode 100644 index 000000000..da2f48cb9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/RejectedOmitPhysicalMeanCorrection.lean.txt @@ -0,0 +1,7 @@ +import SourcePhysicalEndpointKernel +open I4SourceEndpointRegression I4SourcePhysicalEndpointKernel MeasureTheory +example : (∫ v, v ∂sourcePhysicalEndpointKernel 1 1 (Pi.single (3 : Fin 6) (1 : ℝ))) = + sourceRegressionMapAt 1 1 (Pi.single (3 : Fin 6) (1 : ℝ)) := by + rw [sourcePhysicalEndpointKernel_integral (by norm_num)] + ext i + fin_cases i <;> simp [sourcePhysicalCorrection, Pi.single_apply] diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-01.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-01.log new file mode 100644 index 000000000..4f0b1030c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-01.log @@ -0,0 +1,6 @@ +SourcePhysicalEndpointKernel.lean:62:4: warning: `ContinuousLinearMap.add_apply` has been deprecated: Use `_root_.add_apply` instead + +Note: The updated constant is in a different namespace. Dot notation may need to be changed (e.g., from `x.add_apply` to `_root_.add_apply x`). +SourcePhysicalEndpointKernel.lean:87:4: warning: `ContinuousLinearMap.add_apply` has been deprecated: Use `_root_.add_apply` instead + +Note: The updated constant is in a different namespace. Dot notation may need to be changed (e.g., from `x.add_apply` to `_root_.add_apply x`). diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-02.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-02.log new file mode 100644 index 000000000..59fa836ca --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-02.log @@ -0,0 +1 @@ +SourcePhysicalEndpointKernel.lean:112:2: error: `simp` made no progress diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-03.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-03.log new file mode 100644 index 000000000..d2c7d514f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-03.log @@ -0,0 +1,16 @@ +SourcePhysicalEndpointKernel.lean:112:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern + ∫ (a : SmoothSample 2 L), (sourceEndpointResidual L r + fun x => ?m.163) a + ?m.170 ∂smoothLaw 2 L +in the target expression + ∫ (x_1 : SmoothSample 2 L), + sourceEndpointResidual L r x_1 + (sourceRegressionMapAt L r) x + (sourcePhysicalCorrection r) x ∂smoothLaw 2 L = + (sourceRegressionMapAt L r) x + (sourcePhysicalCorrection r) x + +L : ℝ +hL : L ≠ 0 +r : ℝ +x : Fin 6 → ℝ +this : IsProbabilityMeasure (smoothLaw 2 L) +hR : HasGaussianLaw (sourceEndpointResidual L r) (smoothLaw 2 L) +⊢ ∫ (x_1 : SmoothSample 2 L), + sourceEndpointResidual L r x_1 + (sourceRegressionMapAt L r) x + (sourcePhysicalCorrection r) x ∂smoothLaw 2 L = + (sourceRegressionMapAt L r) x + (sourcePhysicalCorrection r) x diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-04.log b/frontiers/source_derivative_gaussian_20261025/evidence/SourcePhysicalEndpointKernel/SourcePhysicalEndpointKernel-04.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.failed1.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.failed1.lean.txt new file mode 100644 index 000000000..e4b3ea5d9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.failed1.lean.txt @@ -0,0 +1,30 @@ +import Mathlib.LinearAlgebra.Matrix.PosDef + +open Matrix +open scoped Matrix +set_option autoImplicit false + +namespace ContactSchurProbe + +theorem block_posDef_schur_pos (A : Matrix (Fin 6) (Fin 6) ℝ) + (B : Matrix (Fin 6) Unit ℝ) (D : Matrix Unit Unit ℝ) + (hG : (Matrix.fromBlocks A B Bᴴ D).PosDef) : + A.PosDef ∧ 0 < (D - Bᴴ * A⁻¹ * B) () () := by + have hA : A.PosDef := by + simpa [Matrix.submatrix] using hG.submatrix (e := Sum.inl) Sum.inl_injective + letI : Invertible A := hA.isUnit.invertible + let y : Unit → ℝ := fun _ => 1 + let x : Fin 6 → ℝ := -((A⁻¹ * B) *ᵥ y) + have hxy : Sum.elim x y ≠ 0 := by + intro h + have hh := congrFun h (Sum.inr ()) + simpa [y] using hh + have hp := hG.dotProduct_mulVec_pos hxy + rw [Matrix.dotProduct_mulVec, + Matrix.schur_complement_eq₁₁ B D x y hA.isHermitian] at hp + dsimp [x] at hp + rw [neg_add_cancel, dotProduct_zero, zero_add] at hp + refine ⟨hA, ?_⟩ + simpa [y, Matrix.vecMul, dotProduct] using hp + +end ContactSchurProbe diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.failed1.log b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.failed1.log new file mode 100644 index 000000000..a04fe3f32 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.failed1.log @@ -0,0 +1,17 @@ +/workspace/scratch/regression-kernel/contact-schur/SchurProbe.lean:14:4: error: Type mismatch: After simplification, term + PosDef.submatrix hG Sum.inl_injective + has type + @PosDef (Fin 6) ℝ Real.instRing Real.partialOrder instStarRingReal (of fun i j => A i j) +but is expected to have type + @PosDef (Fin 6) ℝ Real.instRing Real.partialOrder instStarRingReal A +/workspace/scratch/regression-kernel/contact-schur/SchurProbe.lean:15:2: warning: Try this: + letI̵ + +The goal is a proposition, so `let` is preferred over `letI`. +The difference between `let` and `letI` is that `letI` inlines the value. +But this is not relevant for proofs because of proof irrelevance. + +Note: This linter can be disabled with `set_option linter.style.haveILetI false` +/workspace/scratch/regression-kernel/contact-schur/SchurProbe.lean:21:4: warning: Try `simp at hh` instead of `simpa using hh` + +Note: This linter can be disabled with `set_option linter.unnecessarySimpa false` diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.lean.txt new file mode 100644 index 000000000..b14d8e445 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.lean.txt @@ -0,0 +1,32 @@ +import Mathlib.LinearAlgebra.Matrix.PosDef + +open Matrix +open scoped Matrix +set_option autoImplicit false + +namespace ContactSchurProbe + +theorem block_posDef_schur_pos (A : Matrix (Fin 6) (Fin 6) ℝ) + (B : Matrix (Fin 6) Unit ℝ) (D : Matrix Unit Unit ℝ) + (hG : (Matrix.fromBlocks A B Bᴴ D).PosDef) : + A.PosDef ∧ 0 < (D - Bᴴ * A⁻¹ * B) () () := by + have hA : A.PosDef := by + convert hG.submatrix (e := Sum.inl) Sum.inl_injective using 1 + ext i j + rfl + let _ : Invertible A := hA.isUnit.invertible + let y : Unit → ℝ := fun _ => 1 + let x : Fin 6 → ℝ := -((A⁻¹ * B) *ᵥ y) + have hxy : Sum.elim x y ≠ 0 := by + intro h + have hh := congrFun h (Sum.inr ()) + simp [y] at hh + have hp := hG.dotProduct_mulVec_pos hxy + rw [Matrix.dotProduct_mulVec, + Matrix.schur_complement_eq₁₁ B D x y hA.isHermitian] at hp + dsimp [x] at hp + rw [neg_add_cancel, dotProduct_zero, zero_add] at hp + refine ⟨hA, ?_⟩ + simpa [y, Matrix.vecMul, dotProduct] using hp + +end ContactSchurProbe diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.log b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/SchurProbe.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.failed1.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.failed1.lean.txt new file mode 100644 index 000000000..89ef2f1cd --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.failed1.lean.txt @@ -0,0 +1,54 @@ +import Mathlib.Probability.Moments.Variance +import Mathlib.LinearAlgebra.Matrix.PosDef +import Mathlib.Tactic.Ring + +open MeasureTheory ProbabilityTheory Matrix +open scoped ProbabilityTheory BigOperators Matrix +set_option autoImplicit false + +namespace ResidualVarianceProbe + +theorem residual_variance_of_isUnit_covariance {Ω ι : Type*} [MeasurableSpace Ω] + [Fintype ι] [DecidableEq ι] {P : Measure Ω} [IsFiniteMeasure P] + (J : Ω → ι → ℝ) (Q : Ω → ℝ) + (hJ : ∀ i, MemLp (fun ω => J ω i) 2 P) (hQ : MemLp Q 2 P) + (hG : IsUnit (fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P : + Matrix ι ι ℝ)) : + let G : Matrix ι ι ℝ := fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P + let b : ι → ℝ := fun i => covariance (fun ω => J ω i) Q P + variance (fun ω => Q ω - (G⁻¹ *ᵥ b) ⬝ᵥ J ω) P = + variance Q P - b ⬝ᵥ (G⁻¹ *ᵥ b) := by + classical + let G : Matrix ι ι ℝ := fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P + let b : ι → ℝ := fun i => covariance (fun ω => J ω i) Q P + let a : ι → ℝ := G⁻¹ *ᵥ b + let U : Ω → ℝ := fun ω => a ⬝ᵥ J ω + change variance (fun ω => Q ω - U ω) P = variance Q P - b ⬝ᵥ a + have hG' : IsUnit G := hG + let _ : Invertible G := hG'.invertible + have hnormal : G *ᵥ a = b := by + dsimp [a] + rw [Matrix.mulVec_mulVec, Matrix.mul_inv_of_invertible, Matrix.one_mulVec] + have hterms (i : ι) : MemLp (fun ω => a i * J ω i) 2 P := + (hJ i).const_mul _ + have hU : MemLp U 2 P := memLp_finsetSum Finset.univ (fun i _ => hterms i) + have hJU (i : ι) : covariance (fun ω => J ω i) U P = b i := by + change covariance (fun ω => J ω i) (fun ω => ∑ j, a j * J ω j) P = b i + rw [covariance_fun_sum_right hterms (hJ i)] + simp_rw [covariance_const_mul_right] + simpa [G, Matrix.mulVec, dotProduct, mul_comm] using congrFun hnormal i + have hQU : covariance Q U P = a ⬝ᵥ b := by + change covariance Q (fun ω => ∑ i, a i * J ω i) P = a ⬝ᵥ b + rw [covariance_fun_sum_right hterms hQ] + simp_rw [covariance_const_mul_right] + simp only [b, dotProduct, covariance_comm Q] + have hvarU : variance U P = a ⬝ᵥ b := by + rw [← covariance_self hU.aemeasurable] + change covariance (fun ω => ∑ i, a i * J ω i) U P = a ⬝ᵥ b + rw [covariance_fun_sum_left hterms hU] + simp_rw [covariance_const_mul_left, hJU] + rfl + rw [variance_fun_sub hQ hU, hQU, hvarU, dotProduct_comm b a] + ring + +end ResidualVarianceProbe diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.failed1.log b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.failed1.log new file mode 100644 index 000000000..8030f5184 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.failed1.log @@ -0,0 +1,6 @@ +/workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.lean:27:25: error: Type mismatch + hG +has type + @IsUnit (ι → ι → ℝ) Pi.monoid fun i j => cov[fun ω => J ω i, fun ω => J ω j; P] +but is expected to have type + @IsUnit (Matrix ι ι ℝ) semiring.toMonoid G diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.lean.txt b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.lean.txt new file mode 100644 index 000000000..b87a7dbc7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.lean.txt @@ -0,0 +1,54 @@ +import Mathlib.Probability.Moments.Variance +import Mathlib.LinearAlgebra.Matrix.PosDef +import Mathlib.Tactic.Ring + +open MeasureTheory ProbabilityTheory Matrix +open scoped ProbabilityTheory BigOperators Matrix +set_option autoImplicit false + +namespace ResidualVarianceProbe + +theorem residual_variance_of_isUnit_covariance {Ω ι : Type*} [MeasurableSpace Ω] + [Fintype ι] [DecidableEq ι] {P : Measure Ω} [IsFiniteMeasure P] + (J : Ω → ι → ℝ) (Q : Ω → ℝ) + (hJ : ∀ i, MemLp (fun ω => J ω i) 2 P) (hQ : MemLp Q 2 P) + (hG : IsUnit (Matrix.of (fun i j => + covariance (fun ω => J ω i) (fun ω => J ω j) P) : Matrix ι ι ℝ)) : + let G : Matrix ι ι ℝ := fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P + let b : ι → ℝ := fun i => covariance (fun ω => J ω i) Q P + variance (fun ω => Q ω - (G⁻¹ *ᵥ b) ⬝ᵥ J ω) P = + variance Q P - b ⬝ᵥ (G⁻¹ *ᵥ b) := by + classical + let G : Matrix ι ι ℝ := fun i j => covariance (fun ω => J ω i) (fun ω => J ω j) P + let b : ι → ℝ := fun i => covariance (fun ω => J ω i) Q P + let a : ι → ℝ := G⁻¹ *ᵥ b + let U : Ω → ℝ := fun ω => a ⬝ᵥ J ω + change variance (fun ω => Q ω - U ω) P = variance Q P - b ⬝ᵥ a + have hG' : IsUnit G := hG + let _ : Invertible G := hG'.invertible + have hnormal : G *ᵥ a = b := by + dsimp [a] + rw [Matrix.mulVec_mulVec, Matrix.mul_inv_of_invertible, Matrix.one_mulVec] + have hterms (i : ι) : MemLp (fun ω => a i * J ω i) 2 P := + (hJ i).const_mul _ + have hU : MemLp U 2 P := memLp_finsetSum Finset.univ (fun i _ => hterms i) + have hJU (i : ι) : covariance (fun ω => J ω i) U P = b i := by + change covariance (fun ω => J ω i) (fun ω => ∑ j, a j * J ω j) P = b i + rw [covariance_fun_sum_right hterms (hJ i)] + simp_rw [covariance_const_mul_right] + simpa [G, Matrix.mulVec, dotProduct, mul_comm] using congrFun hnormal i + have hQU : covariance Q U P = a ⬝ᵥ b := by + change covariance Q (fun ω => ∑ i, a i * J ω i) P = a ⬝ᵥ b + rw [covariance_fun_sum_right hterms hQ] + simp_rw [covariance_const_mul_right] + simp only [b, dotProduct, covariance_comm Q] + have hvarU : variance U P = a ⬝ᵥ b := by + rw [← covariance_self hU.aemeasurable] + change covariance (fun ω => ∑ i, a i * J ω i) U P = a ⬝ᵥ b + rw [covariance_fun_sum_left hterms hU] + simp_rw [covariance_const_mul_left, hJU] + rfl + rw [variance_fun_sub hQ hU, hQU, hvarU, dotProduct_comm b a] + ring + +end ResidualVarianceProbe diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.log b/frontiers/source_derivative_gaussian_20261025/evidence/author_helper_gaussian_api/actual-residual/ResidualVariance.log new file mode 100644 index 000000000..e69de29bb diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/AUTHOR_HISTORY_SUPPLEMENT.json b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/AUTHOR_HISTORY_SUPPLEMENT.json new file mode 100644 index 000000000..90befdcb9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/AUTHOR_HISTORY_SUPPLEMENT.json @@ -0,0 +1,110 @@ +{ + "record_type": "author tool-history reports; not a machine-created execution receipt", + "scientific_effect": "NONE", + "reports": [ + { + "author": "OpenAI/Codex /root/uniform_jet_moment_sources", + "scope": [ + "SourceDerivativeGaussian", + "SourceContactCovariance", + "SourceDerivativeContinuity" + ], + "verbatim_report": "Author-history confirmation: all three exact SHA values you list are my final source bytes, unchanged by me since their recorded final compile/controls/axiom/kernel executions. No existing JSON receipt/sourcehash record file was created by me; SHA values were read by sha256sum after source freeze and reported in my source-bound handoffs to root. Treat that as author-history confirmation, not a machine-created execution receipt. Existing command/exit/log triples are the only machine artifacts I produced; please preserve those honest limits and do not rerun.", + "confirmed_source_hashes": { + "SourceDerivativeGaussian.lean": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "SourceContactCovariance.lean": "4189cbaf4c18c55f21e3af2c96b0b2842d83161f44596eb691c507dcc1a439c8", + "SourceDerivativeContinuity.lean": "552511e0aee227133b7d067faeceaab9c43c0094b07d68b8eddab0ab2419775e" + } + }, + { + "author": "OpenAI/Codex /root/regression_kernel", + "scope": [ + "ContactJetCovariance", + "SourceContactSchur" + ], + "control_cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "capture_shell_prelude": "set -o pipefail", + "controls": [ + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log", + "observed_exit": 0 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency-01.log", + "observed_exit": 1 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls-01.log", + "observed_exit": 0 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex-01.log", + "observed_exit": 1 + } + ], + "report_transcription": "Actual control commands, no rerun: cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025; each ran bash set -o pipefail then /home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log observed exit0; corresponding contact-gram RejectedZeroFrequency.lean / RejectedZeroAmplitude.lean with matching -01.log each observed exit1. Schur controls same command pattern under scratch/regression-kernel/source-contact-schur/: PositiveControls.lean to PositiveControls-01.log observedexit0; RejectedZeroPeriod.lean and RejectedWrongQIndex.lean to matching-01.log each observedexit1. These are actual tool-observed command/exits from my current transcript. Helper evidence: /workspace/scratch/regression-kernel/contact-schur/SchurProbe.{lean,log} final exit0 and failed1 retained; /workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.{lean,log} final exit0 and ResidualVariance.failed1.* retained. Helpers coauthored OpenAI/Codex gaussian_api. Earlier failedproduction ContactJetCovariance-01 was assembly-script ValueError/no sourcefile, exit1; ContactJetCovariance-02 success. SourceContactSchur-01/-02/-03 failed author attempts exit1; -04 success.", + "transcription_note": "Faithful plain-text transcription of author report; punctuation/arrow notation normalized. Not a raw machine log." + }, + { + "author": "OpenAI/Codex /root/normalizer_palm_research", + "scope": [ + "GaussianRegressionLimits" + ], + "record_type": "actual author tool-history report; not a machine-created execution receipt", + "verbatim_report": "Author tool-history report (not invented machine receipt): cwd /workspace/Math-/frontiers/source_derivative_gaussian_20261025. Final compile `lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean` exit 0, captured /workspace/scratch/regression-limits/regression-limits-03.log empty. Kernel `lake env leanchecker GaussianRegressionLimits` exit 0, regression-limits-leanchecker.log empty. Final controls `lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean` exit 0, regression-limits-controls-02.log empty. Imported-module #print axioms outputs for regression_parameter_limits and regression_uniform_bounds are in controls-01.log (that command exit 1 for unrelated untyped control index); both only propext, Classical.choice, Quot.sound. No production changes occurred after final successful compile/kernel. Current source SHA256 read-only measured 26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be. All three success logs empty SHA e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855; controls01 SHA c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be. Exact source compiled was this source, author-history association; no rerun.", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "confirmed_source_sha256": "26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be", + "source_unchanged_after_final_compile_kernel": true, + "runs": [ + { + "role": "final_compile", + "command": "lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-03.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "named_kernel", + "command": "lake env leanchecker GaussianRegressionLimits", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-leanchecker.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "corrected_controls", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-02.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "initial_controls_container_with_valid_printed_axioms", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 1, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-01.log", + "raw_log_sha256": "c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be" + } + ] + } + ], + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "reuse": "No checks repeated. These metadata limits do not invalidate completed applicable mathematical evidence or authorize a repeated proof campaign." +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/DRAFT_README.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/DRAFT_README.md new file mode 100644 index 000000000..b97235727 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/DRAFT_README.md @@ -0,0 +1,27 @@ +# Actual Fourier-source derivatives and contact conditioning prerequisites + +This additive Lean package proves joint Gaussianity and covariance formulas for actual derivatives of the existing smooth Fourier source. It also proves positive definiteness of the literal seven-coordinate contact covariance, invertibility of its six-pin block, and strictly positive curvature residual variance. The original research goal remains active; the remaining conditioning, Palm, event and parent joins are part of that goal. + +The source is `I4SmoothCarrier.smoothSourceField` under its unchanged `smoothLaw`. `INPUT_BINDING.json` pins twelve byte-identical smooth-source inputs to Math- commit `e75cb81e8961d178f43aa9967dddd11cd579dc8a`. It separately pins the reused canonical Gaussian regression module and its execution receipt. The historical sources and reviews remain unchanged. Lean is pinned to v4.34.1 and Mathlib to `d13f23b723b8a846827a245b89c10fc7d3f11612`. + +The nine completed modules provide: + +| Module | Result and principal hypotheses | +|---|---| +| `SourceDerivativeGaussian` | For every natural dimension, nonzero period, and arbitrary finite collection of derivative orders, points and direction lists, the actual observations are jointly Gaussian. Every smooth sample satisfies the exact differentiated Fourier-series identity. Covariances equal convergent products of the actual differentiated coefficients. Empty families, order zero and repeated observations are retained. | +| `ContactJetCovariance` | A seven-mode polynomial witness forces positivity of a row-summable infinite coefficient Gram matrix. The coefficient identities and nonzero mode amplitudes are explicit hypotheses of this generic certificate. | +| `SourceContactCovariance` | Instantiates that certificate on the literal origin observations `(f,fx,fy,fxx,fxy,fxxx,fyy)` in dimension two. Canonical half-lattice membership, exact selected derivative coefficients and positive actual Fourier amplitudes are proved. For `L != 0`, the full covariance is positive definite. | +| `SourceConfluentPins` | The actual six endpoint pins admit an explicit continuous linear equivalence for `r != 0`. The literal target `(b,b-r^3/6,0,0,0,0)` maps to `(b-r^3/12,0,0,0,0,2)`. The raw pins, confluent pins and joint twelve-vector of raw pins plus endpoint Hessians have their Gaussianity derived from actual derivatives. | +| `SourceContactSchur` | For `L != 0`, the contact pin covariance of `J=(f,fx,fy,fxx,fxy,fxxx)` is positive definite and invertible. Its exact `6+1` decomposition with `Q=fyy` yields a positive Schur complement, identified with the actual least-squares curvature residual variance. | +| `SourceDerivativeContinuity` | Derivative covariance is continuous in both evaluation points, using point-independent summable coefficient majorants. Every actual directional derivative has zero unconditional mean. This does not substitute zero for conditional regression means. | +| `ConfluentIntegralIdentities` | For real scalar curves `F` of class C3 and `H` of class C2, exact average and Peano-kernel formulas remove the pin and adjusted-endpoint divided differences. The identities preserve signed nonzero `r`. | +| `GaussianRegressionLimits` | From finite block limits for `G,C,B,d` and invertibility of `G0`, derives eventual current `G` invertibility, convergence of `C G^-1 d` and `B-C G^-1 C^T`, and uniform small-radius mean/diagonal bounds. Empty dimensions and singular target covariance are retained; concrete source block limits remain an application obligation. | +| `SourceEndpointRegression` | For the actual source, derives joint Gaussianity of confluent pins and adjusted endpoints, proves the actual matrix normal equations, and constructs the explicit measurable Markov residual kernel at every pin value. `HasCondDistrib` uses the explicit current actual-pin covariance invertibility premise. On the prescribed fiber, adjusted endpoints equal the physical normalized six-vector. | + +The preimplementation contracts are the `SOURCE_*.md` files. They distinguish completed statements from planned successors. The active `SourceConfluentLimits` and `SourceEndpointParameters` successors are not included in this completed inventory. + +The remaining original-goal work includes proving and joining actual small-radius confluent covariance limits, eventual current pin invertibility, actual regression parameter limits and uniform bounds, joining the canonical endpoint kernel to its Gaussian parameters and common coupling, the physical weighted normalizer, whole-field/geometric Palm identification, event-tail transfer and the parent theorems. Continuity of unadjusted derivative covariances alone does not justify singular `1/r` quotient limits. The midpoint quartic nine-vector is not substituted for the physical endpoint six-vector. + +All implementation and technical assistance are OpenAI author-side work, including `/root`, `/root/uniform_jet_moment_sources`, `/root/regression_kernel`, `/root/dm2_landed_composition`, `/root/normalizer_palm_research`, and the `gaussian_api` and `peano_strategy` helpers. Different-family scientific alignment review remains pending. Existing module compilation, named kernel replay, axiom audits and mathematical controls are recorded in the evidence catalog and reused. This packaging adds no combined hosted run or independent scientific acceptance. Scientific effect: NONE. No parent or scientific-status acceptance is claimed. + +This source package has 66 public theorem/lemma targets across nine modules. Its inventory is distinct from the sibling determinant-moment packet, which independently also has 66 targets; the matching counts do not transfer receipts or reviews between the two packages. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/DRAFT_SCOPE.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/DRAFT_SCOPE.md new file mode 100644 index 000000000..bcf764c76 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/DRAFT_SCOPE.md @@ -0,0 +1,27 @@ +# Exact nine-module research scope + +This document describes the nine completed modules named below. Later modules need their own exact statements and execution evidence before being added. The full original research goal remains active. + +`SourceDerivativeGaussian` works with the literal `SmoothSample d L`, `smoothLaw d L` and `smoothSourceField d L`. Its model premise is `L != 0`; dimension, family size, orders, points and direction lists are arbitrary. Differentiated deterministic coefficient summability and the every-sample derivative-series identity are conclusions. Joint Gaussianity is derived from the preserved infinite coefficient law. The covariance conclusion is the infinite differentiated-coefficient pairing; it is not yet an identification with derivatives of a theta-kernel expression. + +`ContactJetCovariance` is a generic coefficient certificate. Row summability, seven specified witness columns, nonzero column amplitudes and nonzero frequency scale imply positive definiteness of the full infinite Gram matrix. The proof uses the sum-of-squares quadratic form and the explicit witness's separating columns. It does not replace the infinite source by those seven modes. + +`SourceContactCovariance` supplies those hypotheses for the actual two-dimensional source. Its origin coordinate order is `(f,fx,fy,fxx,fxy,fxxx,fyy)`. The selected modes are DC, cos(1,0), cos(0,1), cos(1,1), sin(1,0), sin(2,0), sin(0,1), with their actual source amplitudes and canonical half-lattice membership. From `L != 0` it derives positive definiteness of the literal contact covariance; no covariance positivity is assumed. + +`SourceContactSchur` defines `J=(f,fx,fy,fxx,fxy,fxxx)`, `Q=fyy`, their covariance blocks `G,b,d`, and `d-b dotProduct (G inverse mulVec b)`. An exact coordinate equivalence proves the `6+1` block identity. From `L != 0` it derives contact `G` positivity/invertibility, Schur positivity and equality with the variance of the actual curvature residual `Q-(G inverse mulVec b) dotProduct J`. These are contact claims. They do not alone prove current invertibility at all sufficiently small nonzero separations or select a conditional distribution at a prescribed null pin. + +`SourceConfluentPins` fixes endpoints `(-r/2,0),(r/2,0)` and raw order `(fM,fS,fxM,fxS,fyM,fyS)`. Its explicit transform is invertible for `r != 0`; the Gaussian laws are consequences of actual derivative observations. Arbitrary real height `b` is retained in the exact target `(b-r^3/12,0,0,0,0,2)`. The twelve-observation Gaussian statement includes endpoint Hessian order `(fxxM,fxyM,fyyM,fxxS,fxyS,fyyS)` jointly with the raw pins. It does not assume independence or a physical-endpoint identity with a midpoint quartic jet. + +`SourceDerivativeContinuity` proves continuity of every fixed pair of directional-derivative covariances in its evaluation points using global summable bounds. Zero means refer to the unchanged unconditional source law. Conditional torus means are not replaced by zero or a planar reference value. + +`ConfluentIntegralIdentities` is a deterministic scalar result for `F : Real -> Real` of class C3, `H : Real -> Real` of class C2 and signed `r != 0`. It proves the exact integral formulas for the second and transverse divided differences, cubic Peano pin, and left/right adjusted endpoints. The transverse curve `H` denotes the y-derivative curve, so its pin uses `H` itself. Neither an integral representation nor a quotient limit is assumed. Actual-source instantiation and justified infinite-sum limiting arguments remain necessary for stochastic block limits. + +`GaussianRegressionLimits` is a deterministic matrix theorem. Given convergence of finite pin, cross, target and prescribed-value blocks `G,C,B,d` to `G0,C0,B0,d0`, and only `IsUnit G0`, it derives eventual current invertibility, inverse convergence, regression mean/covariance convergence and one interval of simultaneous coordinate bounds. The formulas describe a centered prior with arbitrary prescribed target `d`; conditional means need not vanish. It assumes neither target covariance invertibility nor current pin invertibility. Actual source block convergence is not supplied by this generic theorem. + +`SourceEndpointRegression` uses the literal actual source observations. In LM006 order, its adjusted endpoint vector is `((xxM-X3)/r,(xxS-X3)/r,(xyM-X4)/r,(xyS-X4)/r,yyM,yyS)`. Joint Gaussianity comes from the actual twelve-derivative vector and a proved continuous linear image. For current actual pin covariance `G`, the explicit premise `IsUnit G` supplies matrix inverse normal equations; residual Gaussianity and independence are derived through the reused canonical regression theorem. The resulting Markov measurable kernel has its literal residual pushforward at every pin value, including the prescribed null pin, and satisfies `HasCondDistrib`. The physical endpoint identity is proved deterministically on the `X3=X4=0` fiber. This does not assert eventual current invertibility or a canonical whole-field conditioning theorem. + +Still required within the original goal: actual confluent pin/adjusted-endpoint covariance limits; their join to contact positivity and finite-radius regression; joining the explicit endpoint kernel to its Gaussian parameters and common coupling; uniform conditional parameters and the positive physical normalizer; whole-field and geometric Palm/event identification; event-tail and parent conclusions. No theorem adds one of those desired conclusions as a premise and calls that conclusion proved. The active `SourceConfluentLimits` and `SourceEndpointParameters` successors are not included in this completed inventory. + +Applicable completed verification on unchanged sources is reusable. Completed module compilations, named kernel replays, transitive axiom audits and meaningful controls are recorded separately from hosted execution and different-family alignment. This packaging does not rerun or reclassify those checks. All present authors/helpers are OpenAI; `gaussian_api` and `peano_strategy` assistance is author-side. Different-family scientific review remains pending. Scientific effect NONE; no status-register changes or parent acceptance follow from this packet. + +The 66 public theorem/lemma names inventoried here belong to this nine-module source package. The sibling determinant-moment packet independently has a different 66-target inventory. Neither mathematical evidence nor scientific review is transferred merely because those counts agree. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/EVIDENCE_CATALOG.json b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/EVIDENCE_CATALOG.json new file mode 100644 index 000000000..2ce2baee4 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/EVIDENCE_CATALOG.json @@ -0,0 +1,4454 @@ +{ + "schema_version": 1, + "role": "read-only projection of existing mathematical evidence; no new execution", + "scientific_effect": "NONE", + "goal_status": "FULL_ORIGINAL_RESEARCH_GOAL_ACTIVE", + "source_root": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "scope": [ + "SourceDerivativeGaussian", + "ContactJetCovariance", + "SourceContactCovariance", + "SourceConfluentPins", + "SourceContactSchur", + "SourceDerivativeContinuity", + "ConfluentIntegralIdentities", + "SourceEndpointRegression", + "GaussianRegressionLimits" + ], + "not_in_this_completed_inventory": [ + "SourceConfluentLimits", + "SourceEndpointParameters" + ], + "source_inputs_binding": { + "path": "INPUT_BINDING.json", + "sha256": "fd8aed39de7ba48db4813887b85c18d6b98356c1a8ea82c87a1d06b49bc29dff", + "input_count": 13, + "record": { + "repository": "d6g8k5htny-coder/Math-", + "commit": "e75cb81e8961d178f43aa9967dddd11cd579dc8a", + "files": { + "inputs/GaussianSeriesFoundation.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/GaussianSeriesFoundation.lean", + "sha256": "feea566d5aeba58459bd4a1f20f070bdcd80ee30712817324cdd594e3e232c95" + }, + "inputs/SpectralWeights.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SpectralWeights.lean", + "sha256": "92af14a2d83304de3f8576ee10b3d376587e93c4464f8c715b8d2e32360f44b0" + }, + "inputs/SourceFourierSeries.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceFourierSeries.lean", + "sha256": "5f145e8e8693bfe4d1554092c8258dbda4f17f8e078c503029e4e8ff2c018014" + }, + "inputs/SourceC4.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceC4.lean", + "sha256": "b15e53b825c2e2d6af2af18182da227d8794dddcc3e1b29611a913f89a889397" + }, + "inputs/SourceGaussianLaw.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceGaussianLaw.lean", + "sha256": "a60ab6099d3cef0eeecaefa9e35d95ec14ec54ffd6c89e42dfcf6170907166e8" + }, + "inputs/SourceLattice.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceLattice.lean", + "sha256": "c644305a426bc876f609c055ce999b8008f028c67a1091639d919f4122981b7b" + }, + "inputs/SourceKernel.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceKernel.lean", + "sha256": "5fa3abc8fb0db0c14c67b4ee840e659cdf80f7d22c9fd4cb5e6c1c7b204fbd21" + }, + "inputs/SourceCovariance.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceCovariance.lean", + "sha256": "30560009582659a100637b5510c657cf1fa762f68a37bae8fab42c1eeee6aee5" + }, + "inputs/SourceTorus.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceTorus.lean", + "sha256": "02857b44552ab0e51a756b3b095c4ba43f8168ac332b069acb5bc8146d337eef" + }, + "inputs/SourceTorusC4.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceTorusC4.lean", + "sha256": "b67b13030389dea4521e8e2afcac30ea44f84abd65c374bf208067b288fc64a1" + }, + "inputs/SourceSmooth.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceSmooth.lean", + "sha256": "12ad7263267a970bd5d5c95d7c7aa62d9c971f7beebf28f1aa7ed16d3a112f1b" + }, + "inputs/SourceSmoothCarrier.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/SourceSmoothCarrier.lean", + "sha256": "5defdf223e85e7ccd23b5a903a1a306fe7f723727eb6aabf62eef523ef8639ed" + }, + 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Different-family scientific alignment review remains pending.", + "verification_reuse": "All saved compilation, named kernel, axiom and controls evidence reused. No Lean, kernel replay, tests, or old verification executed during this projection.", + "packaging_note": "Original raw logs, commands, metadata, failures and control/helper source bytes preserved. Lean source artifacts have .lean.txt projection suffix; actual original paths remain in records. This index is not a hosted receipt or accepted scientific alignment record.", + "author_history_supplement": { + "path": "AUTHOR_HISTORY_SUPPLEMENT.json", + "sha256": "e3c4a4fbf56f5938f65138d8c61c632f921ee84cc6e9f1b6ebd7af70f8da8052", + "record": { + "record_type": "author tool-history reports; not a machine-created execution receipt", + "scientific_effect": "NONE", + "reports": [ + { + "author": "OpenAI/Codex /root/uniform_jet_moment_sources", + "scope": [ + "SourceDerivativeGaussian", + "SourceContactCovariance", + "SourceDerivativeContinuity" + ], + "verbatim_report": "Author-history confirmation: all three exact SHA values you list are my final source bytes, unchanged by me since their recorded final compile/controls/axiom/kernel executions. No existing JSON receipt/sourcehash record file was created by me; SHA values were read by sha256sum after source freeze and reported in my source-bound handoffs to root. Treat that as author-history confirmation, not a machine-created execution receipt. Existing command/exit/log triples are the only machine artifacts I produced; please preserve those honest limits and do not rerun.", + "confirmed_source_hashes": { + "SourceDerivativeGaussian.lean": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "SourceContactCovariance.lean": "4189cbaf4c18c55f21e3af2c96b0b2842d83161f44596eb691c507dcc1a439c8", + "SourceDerivativeContinuity.lean": "552511e0aee227133b7d067faeceaab9c43c0094b07d68b8eddab0ab2419775e" + } + }, + { + "author": "OpenAI/Codex /root/regression_kernel", + "scope": [ + "ContactJetCovariance", + "SourceContactSchur" + ], + "control_cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "capture_shell_prelude": "set -o pipefail", + "controls": [ + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log", + "observed_exit": 0 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency-01.log", + "observed_exit": 1 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls-01.log", + "observed_exit": 0 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex-01.log", + "observed_exit": 1 + } + ], + "report_transcription": "Actual control commands, no rerun: cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025; each ran bash set -o pipefail then /home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log observed exit0; corresponding contact-gram RejectedZeroFrequency.lean / RejectedZeroAmplitude.lean with matching -01.log each observed exit1. Schur controls same command pattern under scratch/regression-kernel/source-contact-schur/: PositiveControls.lean to PositiveControls-01.log observedexit0; RejectedZeroPeriod.lean and RejectedWrongQIndex.lean to matching-01.log each observedexit1. These are actual tool-observed command/exits from my current transcript. Helper evidence: /workspace/scratch/regression-kernel/contact-schur/SchurProbe.{lean,log} final exit0 and failed1 retained; /workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.{lean,log} final exit0 and ResidualVariance.failed1.* retained. Helpers coauthored OpenAI/Codex gaussian_api. Earlier failedproduction ContactJetCovariance-01 was assembly-script ValueError/no sourcefile, exit1; ContactJetCovariance-02 success. SourceContactSchur-01/-02/-03 failed author attempts exit1; -04 success.", + "transcription_note": "Faithful plain-text transcription of author report; punctuation/arrow notation normalized. Not a raw machine log." + }, + { + "author": "OpenAI/Codex /root/normalizer_palm_research", + "scope": [ + "GaussianRegressionLimits" + ], + "record_type": "actual author tool-history report; not a machine-created execution receipt", + "verbatim_report": "Author tool-history report (not invented machine receipt): cwd /workspace/Math-/frontiers/source_derivative_gaussian_20261025. Final compile `lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean` exit 0, captured /workspace/scratch/regression-limits/regression-limits-03.log empty. Kernel `lake env leanchecker GaussianRegressionLimits` exit 0, regression-limits-leanchecker.log empty. Final controls `lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean` exit 0, regression-limits-controls-02.log empty. Imported-module #print axioms outputs for regression_parameter_limits and regression_uniform_bounds are in controls-01.log (that command exit 1 for unrelated untyped control index); both only propext, Classical.choice, Quot.sound. No production changes occurred after final successful compile/kernel. Current source SHA256 read-only measured 26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be. All three success logs empty SHA e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855; controls01 SHA c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be. Exact source compiled was this source, author-history association; no rerun.", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "confirmed_source_sha256": "26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be", + "source_unchanged_after_final_compile_kernel": true, + "runs": [ + { + "role": "final_compile", + "command": "lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-03.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "named_kernel", + "command": "lake env leanchecker GaussianRegressionLimits", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-leanchecker.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "corrected_controls", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-02.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "initial_controls_container_with_valid_printed_axioms", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 1, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-01.log", + "raw_log_sha256": "c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be" + } + ] + } + ], + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "reuse": "No checks repeated. These metadata limits do not invalidate completed applicable mathematical evidence or authorize a repeated proof campaign." + } + }, + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "append_scope": "Original seven module entries/evidence preserved. Only SourceEndpointRegression and GaussianRegressionLimits appended; original seven-module catalog/drafts retained under historical-seven-module/.", + "target_inventory_identity": { + "package": "frontiers/source_derivative_gaussian_20261025", + "public_theorem_lemma_count": 66, + "distinct_from": "frontiers/determinant_moment_20261025 has its own different 66-target inventory; equal counts do not transfer receipts/reviews" + } +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/EVIDENCE_SUMMARY.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/EVIDENCE_SUMMARY.md new file mode 100644 index 000000000..f8aaaa448 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-nine-module/EVIDENCE_SUMMARY.md @@ -0,0 +1,53 @@ +# Read-only nine-module evidence projection + +The catalog inventories **66 public theorem/lemma targets** in nine completed +modules. Saved transitive axiom reports cover every such target. The +`SourceContactSchur` audit also includes the actual inverse definition, giving +67 distinct reported public declarations. Only `propext`, `Classical.choice` +and `Quot.sound` appear in these reports. Named module kernel replay and final +module compilation records each report exit 0. No checks were run during this +projection; completed applicable evidence is reused. + +| Module | Public theorem/lemma targets | Principal saved evidence | +|---|---:|---| +| SourceDerivativeGaussian | 5 | `05-covariance-compile`, `06-controls`, `07-axiom-audit`, `08-kernel-replay` | +| ContactJetCovariance | 8 | `EVIDENCE.json`, final `ContactJetCovariance-02`, positive controls, zero-frequency/zero-amplitude rejections | +| SourceContactCovariance | 7 | `05-column-identities-compile`, `06-controls`, `07-axiom-audit`, `08-kernel-replay` | +| SourceConfluentPins | 8 | `qualification_receipt.json`, earlier final compile, signed-radius/target controls and zero-radius rejections | +| SourceContactSchur | 10 | `EVIDENCE.json`, final `SourceContactSchur-04`, positive controls and zero-period/wrong-Q-index rejections | +| SourceDerivativeContinuity | 6 | `02-evaluation-and-law-compile`, corrected `05-controls-explicit-path`, `04-axiom-audit`, `06-kernel-replay` | +| ConfluentIntegralIdentities | 7 | `qualification_receipt.json`, final compile, positive controls and cubic/transverse false controls | +| SourceEndpointRegression | 13 | `EVIDENCE.json`, final `SourceEndpointRegression-03`, named kernel, thirteen exported axiom reports, positive controls and two adjustment/height rejections | +| GaussianRegressionLimits | 2 | final compile `regression-limits-03`, named kernel, corrected `controls-02`; two valid imported-module axiom reports retained in failed control container `controls-01` | + +`EVIDENCE_CATALOG.json` retains exact source SHA256 values, public declarations +with source lines, recorded commands/exits, receipt contents, raw-log hashes and +an index of **229 original evidence/helper files** copied beneath `evidence/`. +Earlier failures remain included. In particular, the integral module's final +compilation has one harmless `unnecessarySeqFocus` warning; it is not represented +as a warning-free run. Projection filenames for Lean controls/helpers end in +`.lean.txt`; original paths and bytes remain bound in the catalog. + +The actual `gaussian_api` optional Schur/residual/matrix-normal-equations proof helper and +`peano_strategy` Peano proof helper are included in the author lineage and raw +helper evidence. README/SCOPE drafting assistance from `density_api` is also +identified as OpenAI author-side work. No organizational or scientific +independence is inferred from those helpers. + +`AUTHOR_HISTORY_SUPPLEMENT.json` discloses two metadata limits: the three +`uniform_jet_moment_sources` groups have command/exit/raw-log triples but no +historical JSON source-hash receipt; their unchanged final source association +was confirmed by the author. The two `regression_kernel` groups' precise control +capture commands come from the author's actual tool history, alongside the +existing raw logs and exit-bearing receipts. Those reports are not relabeled as +machine-generated receipts. The new GaussianRegressionLimits source association and exact commands/exits were also confirmed by its author’s actual tool history. Its controls-01 process exited 1 for an unrelated control elaboration error while printing two valid imported-module axiom reports. Corrected controls-02 separately exited 0; no exit-0 axiom-only command is invented. No referenced evidence files or saved public target axiom reports were missing in the nine-module inventory. + +`INPUT_BINDING.json` describes thirteen reused inputs: twelve unchanged source +files and the separately verified Gaussian regression module. The catalog binds +that file without changing or re-executing the inputs. Historical origin +manifests/reviews remain provenance and retain their own scope. + +The original research goal remains active. Different-family scientific alignment +review is pending. This projection does not award parent acceptance, a hosted execution result or a scientific-status change. `SourceConfluentLimits` and `SourceEndpointParameters` remain outside this qualified inventory; root may append their separate actual qualification. The unchanged original seven-module catalog/drafts remain under `historical-seven-module/`. + +The nine-module source-package inventory here is distinct from the sibling determinant-moment packet. Each independently has 66 public theorem/lemma targets; this numerical coincidence transfers no execution receipt, target coverage or review between packages. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/AUTHOR_HISTORY_SUPPLEMENT.json b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/AUTHOR_HISTORY_SUPPLEMENT.json new file mode 100644 index 000000000..28e1eb274 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/AUTHOR_HISTORY_SUPPLEMENT.json @@ -0,0 +1,68 @@ +{ + "record_type": "author tool-history reports; not a machine-created execution receipt", + "scientific_effect": "NONE", + "reports": [ + { + "author": "OpenAI/Codex /root/uniform_jet_moment_sources", + "scope": [ + "SourceDerivativeGaussian", + "SourceContactCovariance", + "SourceDerivativeContinuity" + ], + "verbatim_report": "Author-history confirmation: all three exact SHA values you list are my final source bytes, unchanged by me since their recorded final compile/controls/axiom/kernel executions. No existing JSON receipt/sourcehash record file was created by me; SHA values were read by sha256sum after source freeze and reported in my source-bound handoffs to root. Treat that as author-history confirmation, not a machine-created execution receipt. Existing command/exit/log triples are the only machine artifacts I produced; please preserve those honest limits and do not rerun.", + "confirmed_source_hashes": { + "SourceDerivativeGaussian.lean": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "SourceContactCovariance.lean": "4189cbaf4c18c55f21e3af2c96b0b2842d83161f44596eb691c507dcc1a439c8", + "SourceDerivativeContinuity.lean": "552511e0aee227133b7d067faeceaab9c43c0094b07d68b8eddab0ab2419775e" + } + }, + { + "author": "OpenAI/Codex /root/regression_kernel", + "scope": [ + "ContactJetCovariance", + "SourceContactSchur" + ], + "control_cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "capture_shell_prelude": "set -o pipefail", + "controls": [ + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log", + "observed_exit": 0 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency-01.log", + "observed_exit": 1 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls-01.log", + "observed_exit": 0 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex-01.log", + "observed_exit": 1 + } + ], + "report_transcription": "Actual control commands, no rerun: cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025; each ran bash set -o pipefail then /home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log observed exit0; corresponding contact-gram RejectedZeroFrequency.lean / RejectedZeroAmplitude.lean with matching -01.log each observed exit1. Schur controls same command pattern under scratch/regression-kernel/source-contact-schur/: PositiveControls.lean to PositiveControls-01.log observedexit0; RejectedZeroPeriod.lean and RejectedWrongQIndex.lean to matching-01.log each observedexit1. These are actual tool-observed command/exits from my current transcript. Helper evidence: /workspace/scratch/regression-kernel/contact-schur/SchurProbe.{lean,log} final exit0 and failed1 retained; /workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.{lean,log} final exit0 and ResidualVariance.failed1.* retained. Helpers coauthored OpenAI/Codex gaussian_api. Earlier failedproduction ContactJetCovariance-01 was assembly-script ValueError/no sourcefile, exit1; ContactJetCovariance-02 success. SourceContactSchur-01/-02/-03 failed author attempts exit1; -04 success.", + "transcription_note": "Faithful plain-text transcription of author report; punctuation/arrow notation normalized. Not a raw machine log." + } + ], + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses." + ], + "reuse": "No checks repeated. These metadata limits do not invalidate completed applicable mathematical evidence or authorize a repeated proof campaign." +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/DRAFT_README.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/DRAFT_README.md new file mode 100644 index 000000000..875a2b0d9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/DRAFT_README.md @@ -0,0 +1,23 @@ +# Actual Fourier-source derivatives and contact conditioning prerequisites + +This additive Lean package proves joint Gaussianity and covariance formulas for actual derivatives of the existing smooth Fourier source. It also proves positive definiteness of the literal seven-coordinate contact covariance, invertibility of its six-pin block, and strictly positive curvature residual variance. The original research goal remains active; the remaining conditioning, Palm, event and parent joins are part of that goal. + +The source is `I4SmoothCarrier.smoothSourceField` under its unchanged `smoothLaw`. `INPUT_BINDING.json` pins twelve byte-identical smooth-source inputs to Math- commit `e75cb81e8961d178f43aa9967dddd11cd579dc8a`. It separately pins the reused canonical Gaussian regression module and its execution receipt. The historical sources and reviews remain unchanged. Lean is pinned to v4.34.1 and Mathlib to `d13f23b723b8a846827a245b89c10fc7d3f11612`. + +The seven completed modules provide: + +| Module | Result and principal hypotheses | +|---|---| +| `SourceDerivativeGaussian` | For every natural dimension, nonzero period, and arbitrary finite collection of derivative orders, points and direction lists, the actual observations are jointly Gaussian. Every smooth sample satisfies the exact differentiated Fourier-series identity. Covariances equal convergent products of the actual differentiated coefficients. Empty families, order zero and repeated observations are retained. | +| `ContactJetCovariance` | A seven-mode polynomial witness forces positivity of a row-summable infinite coefficient Gram matrix. The coefficient identities and nonzero mode amplitudes are explicit hypotheses of this generic certificate. | +| `SourceContactCovariance` | Instantiates that certificate on the literal origin observations `(f,fx,fy,fxx,fxy,fxxx,fyy)` in dimension two. Canonical half-lattice membership, exact selected derivative coefficients and positive actual Fourier amplitudes are proved. For `L != 0`, the full covariance is positive definite. | +| `SourceConfluentPins` | The actual six endpoint pins admit an explicit continuous linear equivalence for `r != 0`. The literal target `(b,b-r^3/6,0,0,0,0)` maps to `(b-r^3/12,0,0,0,0,2)`. The raw pins, confluent pins and joint twelve-vector of raw pins plus endpoint Hessians have their Gaussianity derived from actual derivatives. | +| `SourceContactSchur` | For `L != 0`, the contact pin covariance of `J=(f,fx,fy,fxx,fxy,fxxx)` is positive definite and invertible. Its exact `6+1` decomposition with `Q=fyy` yields a positive Schur complement, identified with the actual least-squares curvature residual variance. | +| `SourceDerivativeContinuity` | Derivative covariance is continuous in both evaluation points, using point-independent summable coefficient majorants. Every actual directional derivative has zero unconditional mean. This does not substitute zero for conditional regression means. | +| `ConfluentIntegralIdentities` | For real scalar curves `F` of class C3 and `H` of class C2, exact average and Peano-kernel formulas remove the pin and adjusted-endpoint divided differences. The identities preserve signed nonzero `r`. | + +The preimplementation contracts are the `SOURCE_*.md` files. They distinguish completed statements from planned successors. In particular, the seven-module description above does not assert completion of `GaussianRegressionLimits`, `SourceEndpointRegression` or `SourceConfluentLimits`. + +The remaining original-goal work includes proving and joining actual small-radius confluent covariance limits, eventual current pin invertibility, actual regression parameter limits and uniform bounds, the canonical conditional endpoint law and its common coupling, the physical weighted normalizer, whole-field/geometric Palm identification, event-tail transfer and the parent theorems. Continuity of unadjusted derivative covariances alone does not justify singular `1/r` quotient limits. The midpoint quartic nine-vector is not substituted for the physical endpoint six-vector. + +All implementation and technical assistance are OpenAI author-side work, including `/root`, `/root/uniform_jet_moment_sources`, `/root/regression_kernel`, `/root/dm2_landed_composition` and the `gaussian_api` and `peano_strategy` helpers. Different-family scientific alignment review remains pending. Existing module compilation, named kernel replay, axiom audits and mathematical controls are recorded in the evidence catalog and reused. This packaging adds no combined hosted run or independent scientific acceptance. Scientific effect: NONE. No parent or scientific-status acceptance is claimed. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/DRAFT_SCOPE.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/DRAFT_SCOPE.md new file mode 100644 index 000000000..7fb658bb7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/DRAFT_SCOPE.md @@ -0,0 +1,21 @@ +# Exact seven-module research scope + +This document describes the seven completed modules named below. Later modules need their own exact statements and execution evidence before being added. The full original research goal remains active. + +`SourceDerivativeGaussian` works with the literal `SmoothSample d L`, `smoothLaw d L` and `smoothSourceField d L`. Its model premise is `L != 0`; dimension, family size, orders, points and direction lists are arbitrary. Differentiated deterministic coefficient summability and the every-sample derivative-series identity are conclusions. Joint Gaussianity is derived from the preserved infinite coefficient law. The covariance conclusion is the infinite differentiated-coefficient pairing; it is not yet an identification with derivatives of a theta-kernel expression. + +`ContactJetCovariance` is a generic coefficient certificate. Row summability, seven specified witness columns, nonzero column amplitudes and nonzero frequency scale imply positive definiteness of the full infinite Gram matrix. The proof uses the sum-of-squares quadratic form and the explicit witness's separating columns. It does not replace the infinite source by those seven modes. + +`SourceContactCovariance` supplies those hypotheses for the actual two-dimensional source. Its origin coordinate order is `(f,fx,fy,fxx,fxy,fxxx,fyy)`. The selected modes are DC, cos(1,0), cos(0,1), cos(1,1), sin(1,0), sin(2,0), sin(0,1), with their actual source amplitudes and canonical half-lattice membership. From `L != 0` it derives positive definiteness of the literal contact covariance; no covariance positivity is assumed. + +`SourceContactSchur` defines `J=(f,fx,fy,fxx,fxy,fxxx)`, `Q=fyy`, their covariance blocks `G,b,d`, and `d-b dotProduct (G inverse mulVec b)`. An exact coordinate equivalence proves the `6+1` block identity. From `L != 0` it derives contact `G` positivity/invertibility, Schur positivity and equality with the variance of the actual curvature residual `Q-(G inverse mulVec b) dotProduct J`. These are contact claims. They do not alone prove current invertibility at all sufficiently small nonzero separations or select a conditional distribution at a prescribed null pin. + +`SourceConfluentPins` fixes endpoints `(-r/2,0),(r/2,0)` and raw order `(fM,fS,fxM,fxS,fyM,fyS)`. Its explicit transform is invertible for `r != 0`; the Gaussian laws are consequences of actual derivative observations. Arbitrary real height `b` is retained in the exact target `(b-r^3/12,0,0,0,0,2)`. The twelve-observation Gaussian statement includes endpoint Hessian order `(fxxM,fxyM,fyyM,fxxS,fxyS,fyyS)` jointly with the raw pins. It does not assume independence or a physical-endpoint identity with a midpoint quartic jet. + +`SourceDerivativeContinuity` proves continuity of every fixed pair of directional-derivative covariances in its evaluation points using global summable bounds. Zero means refer to the unchanged unconditional source law. Conditional torus means are not replaced by zero or a planar reference value. + +`ConfluentIntegralIdentities` is a deterministic scalar result for `F : Real -> Real` of class C3, `H : Real -> Real` of class C2 and signed `r != 0`. It proves the exact integral formulas for the second and transverse divided differences, cubic Peano pin, and left/right adjusted endpoints. The transverse curve `H` denotes the y-derivative curve, so its pin uses `H` itself. Neither an integral representation nor a quotient limit is assumed. Actual-source instantiation and justified infinite-sum limiting arguments remain necessary for stochastic block limits. + +Still required within the original goal: actual confluent pin/adjusted-endpoint covariance limits; their join to contact positivity and finite-radius regression; the literal canonical endpoint law and common coupling; uniform conditional parameters and the positive physical normalizer; whole-field and geometric Palm/event identification; event-tail and parent conclusions. No theorem adds one of those desired conclusions as a premise and calls that conclusion proved. The in-progress `GaussianRegressionLimits`, `SourceEndpointRegression` and `SourceConfluentLimits` are not included in this completed inventory. + +Applicable completed verification on unchanged sources is reusable. Completed module compilations, named kernel replays, transitive axiom audits and meaningful controls are recorded separately from hosted execution and different-family alignment. This packaging does not rerun or reclassify those checks. All present authors/helpers are OpenAI; `gaussian_api` and `peano_strategy` assistance is author-side. Different-family scientific review remains pending. 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Different-family scientific alignment review remains pending.", + "verification_reuse": "All saved compilation, named kernel, axiom and controls evidence reused. No Lean, kernel replay, tests, or old verification executed during this projection.", + "packaging_note": "Original raw logs, commands, metadata, failures and control/helper source bytes preserved. Lean source artifacts have .lean.txt projection suffix; actual original paths remain in records. This index is not a hosted receipt or accepted scientific alignment record.", + "author_history_supplement": { + "path": "AUTHOR_HISTORY_SUPPLEMENT.json", + "sha256": "3b8f3b7505c9646198c7427f5fc02e1ccd036ec38ec0d95181159d9c1ebc720b", + "record": { + "record_type": "author tool-history reports; not a machine-created execution receipt", + "scientific_effect": "NONE", + "reports": [ + { + "author": "OpenAI/Codex /root/uniform_jet_moment_sources", + "scope": [ + "SourceDerivativeGaussian", + "SourceContactCovariance", + "SourceDerivativeContinuity" + ], + "verbatim_report": "Author-history confirmation: all three exact SHA values you list are my final source bytes, unchanged by me since their recorded final compile/controls/axiom/kernel executions. No existing JSON receipt/sourcehash record file was created by me; SHA values were read by sha256sum after source freeze and reported in my source-bound handoffs to root. Treat that as author-history confirmation, not a machine-created execution receipt. Existing command/exit/log triples are the only machine artifacts I produced; please preserve those honest limits and do not rerun.", + "confirmed_source_hashes": { + "SourceDerivativeGaussian.lean": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "SourceContactCovariance.lean": "4189cbaf4c18c55f21e3af2c96b0b2842d83161f44596eb691c507dcc1a439c8", + "SourceDerivativeContinuity.lean": "552511e0aee227133b7d067faeceaab9c43c0094b07d68b8eddab0ab2419775e" + } + }, + { + "author": "OpenAI/Codex /root/regression_kernel", + "scope": [ + "ContactJetCovariance", + "SourceContactSchur" + ], + "control_cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "capture_shell_prelude": "set -o pipefail", + "controls": [ + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log", + "observed_exit": 0 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency-01.log", + "observed_exit": 1 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls-01.log", + "observed_exit": 0 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex-01.log", + "observed_exit": 1 + } + ], + "report_transcription": "Actual control commands, no rerun: cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025; each ran bash set -o pipefail then /home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log observed exit0; corresponding contact-gram RejectedZeroFrequency.lean / RejectedZeroAmplitude.lean with matching -01.log each observed exit1. Schur controls same command pattern under scratch/regression-kernel/source-contact-schur/: PositiveControls.lean to PositiveControls-01.log observedexit0; RejectedZeroPeriod.lean and RejectedWrongQIndex.lean to matching-01.log each observedexit1. These are actual tool-observed command/exits from my current transcript. Helper evidence: /workspace/scratch/regression-kernel/contact-schur/SchurProbe.{lean,log} final exit0 and failed1 retained; /workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.{lean,log} final exit0 and ResidualVariance.failed1.* retained. Helpers coauthored OpenAI/Codex gaussian_api. Earlier failedproduction ContactJetCovariance-01 was assembly-script ValueError/no sourcefile, exit1; ContactJetCovariance-02 success. SourceContactSchur-01/-02/-03 failed author attempts exit1; -04 success.", + "transcription_note": "Faithful plain-text transcription of author report; punctuation/arrow notation normalized. Not a raw machine log." + } + ], + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses." + ], + "reuse": "No checks repeated. These metadata limits do not invalidate completed applicable mathematical evidence or authorize a repeated proof campaign." + } + }, + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses." + ] +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/EVIDENCE_SUMMARY.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/EVIDENCE_SUMMARY.md new file mode 100644 index 000000000..64334dc00 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-seven-module/EVIDENCE_SUMMARY.md @@ -0,0 +1,54 @@ +# Read-only seven-module evidence projection + +The catalog inventories **51 public theorem/lemma targets** in seven completed +modules. Saved transitive axiom reports cover every such target. The +`SourceContactSchur` audit also includes the actual inverse definition, giving +52 distinct reported public declarations. Only `propext`, `Classical.choice` +and `Quot.sound` appear in these reports. Named module kernel replay and final +module compilation records each report exit 0. No checks were run during this +projection; completed applicable evidence is reused. + +| Module | Public theorem/lemma targets | Principal saved evidence | +|---|---:|---| +| SourceDerivativeGaussian | 5 | `05-covariance-compile`, `06-controls`, `07-axiom-audit`, `08-kernel-replay` | +| ContactJetCovariance | 8 | `EVIDENCE.json`, final `ContactJetCovariance-02`, positive controls, zero-frequency/zero-amplitude rejections | +| SourceContactCovariance | 7 | `05-column-identities-compile`, `06-controls`, `07-axiom-audit`, `08-kernel-replay` | +| SourceConfluentPins | 8 | `qualification_receipt.json`, earlier final compile, signed-radius/target controls and zero-radius rejections | +| SourceContactSchur | 10 | `EVIDENCE.json`, final `SourceContactSchur-04`, positive controls and zero-period/wrong-Q-index rejections | +| SourceDerivativeContinuity | 6 | `02-evaluation-and-law-compile`, corrected `05-controls-explicit-path`, `04-axiom-audit`, `06-kernel-replay` | +| ConfluentIntegralIdentities | 7 | `qualification_receipt.json`, final compile, positive controls and cubic/transverse false controls | + +`EVIDENCE_CATALOG.json` retains exact source SHA256 values, public declarations +with source lines, recorded commands/exits, receipt contents, raw-log hashes and +an index of **191 original evidence/helper files** copied beneath `evidence/`. +Earlier failures remain included. In particular, the integral module's final +compilation has one harmless `unnecessarySeqFocus` warning; it is not represented +as a warning-free run. Projection filenames for Lean controls/helpers end in +`.lean.txt`; original paths and bytes remain bound in the catalog. + +The actual `gaussian_api` optional Schur/residual proof helper and +`peano_strategy` Peano proof helper are included in the author lineage and raw +helper evidence. README/SCOPE drafting assistance from `density_api` is also +identified as OpenAI author-side work. No organizational or scientific +independence is inferred from those helpers. + +`AUTHOR_HISTORY_SUPPLEMENT.json` discloses two metadata limits: the three +`uniform_jet_moment_sources` groups have command/exit/raw-log triples but no +historical JSON source-hash receipt; their unchanged final source association +was confirmed by the author. The two `regression_kernel` groups' precise control +capture commands come from the author's actual tool history, alongside the +existing raw logs and exit-bearing receipts. Those reports are not relabeled as +machine-generated receipts. No referenced evidence files or saved public target +axiom reports were missing in the seven-module inventory. + +`INPUT_BINDING.json` describes thirteen reused inputs: twelve unchanged source +files and the separately verified Gaussian regression module. The catalog binds +that file without changing or re-executing the inputs. Historical origin +manifests/reviews remain provenance and retain their own scope. + +The original research goal remains active. Different-family scientific alignment +review is pending. This projection does not award parent acceptance, a hosted +execution result or a scientific-status change. `GaussianRegressionLimits`, +`SourceEndpointRegression` and `SourceConfluentLimits` are deliberately excluded +from this seven-module completed inventory; root may append their actual +qualification when available. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/AUTHOR_HISTORY_SUPPLEMENT.json b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/AUTHOR_HISTORY_SUPPLEMENT.json new file mode 100644 index 000000000..90befdcb9 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/AUTHOR_HISTORY_SUPPLEMENT.json @@ -0,0 +1,110 @@ +{ + "record_type": "author tool-history reports; not a machine-created execution receipt", + "scientific_effect": "NONE", + "reports": [ + { + "author": "OpenAI/Codex /root/uniform_jet_moment_sources", + "scope": [ + "SourceDerivativeGaussian", + "SourceContactCovariance", + "SourceDerivativeContinuity" + ], + "verbatim_report": "Author-history confirmation: all three exact SHA values you list are my final source bytes, unchanged by me since their recorded final compile/controls/axiom/kernel executions. No existing JSON receipt/sourcehash record file was created by me; SHA values were read by sha256sum after source freeze and reported in my source-bound handoffs to root. Treat that as author-history confirmation, not a machine-created execution receipt. Existing command/exit/log triples are the only machine artifacts I produced; please preserve those honest limits and do not rerun.", + "confirmed_source_hashes": { + "SourceDerivativeGaussian.lean": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "SourceContactCovariance.lean": "4189cbaf4c18c55f21e3af2c96b0b2842d83161f44596eb691c507dcc1a439c8", + "SourceDerivativeContinuity.lean": "552511e0aee227133b7d067faeceaab9c43c0094b07d68b8eddab0ab2419775e" + } + }, + { + "author": "OpenAI/Codex /root/regression_kernel", + "scope": [ + "ContactJetCovariance", + "SourceContactSchur" + ], + "control_cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "capture_shell_prelude": "set -o pipefail", + "controls": [ + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log", + "observed_exit": 0 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency-01.log", + "observed_exit": 1 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls-01.log", + "observed_exit": 0 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex-01.log", + "observed_exit": 1 + } + ], + "report_transcription": "Actual control commands, no rerun: cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025; each ran bash set -o pipefail then /home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log observed exit0; corresponding contact-gram RejectedZeroFrequency.lean / RejectedZeroAmplitude.lean with matching -01.log each observed exit1. Schur controls same command pattern under scratch/regression-kernel/source-contact-schur/: PositiveControls.lean to PositiveControls-01.log observedexit0; RejectedZeroPeriod.lean and RejectedWrongQIndex.lean to matching-01.log each observedexit1. These are actual tool-observed command/exits from my current transcript. Helper evidence: /workspace/scratch/regression-kernel/contact-schur/SchurProbe.{lean,log} final exit0 and failed1 retained; /workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.{lean,log} final exit0 and ResidualVariance.failed1.* retained. Helpers coauthored OpenAI/Codex gaussian_api. Earlier failedproduction ContactJetCovariance-01 was assembly-script ValueError/no sourcefile, exit1; ContactJetCovariance-02 success. SourceContactSchur-01/-02/-03 failed author attempts exit1; -04 success.", + "transcription_note": "Faithful plain-text transcription of author report; punctuation/arrow notation normalized. Not a raw machine log." + }, + { + "author": "OpenAI/Codex /root/normalizer_palm_research", + "scope": [ + "GaussianRegressionLimits" + ], + "record_type": "actual author tool-history report; not a machine-created execution receipt", + "verbatim_report": "Author tool-history report (not invented machine receipt): cwd /workspace/Math-/frontiers/source_derivative_gaussian_20261025. Final compile `lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean` exit 0, captured /workspace/scratch/regression-limits/regression-limits-03.log empty. Kernel `lake env leanchecker GaussianRegressionLimits` exit 0, regression-limits-leanchecker.log empty. Final controls `lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean` exit 0, regression-limits-controls-02.log empty. Imported-module #print axioms outputs for regression_parameter_limits and regression_uniform_bounds are in controls-01.log (that command exit 1 for unrelated untyped control index); both only propext, Classical.choice, Quot.sound. No production changes occurred after final successful compile/kernel. Current source SHA256 read-only measured 26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be. All three success logs empty SHA e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855; controls01 SHA c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be. Exact source compiled was this source, author-history association; no rerun.", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "confirmed_source_sha256": "26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be", + "source_unchanged_after_final_compile_kernel": true, + "runs": [ + { + "role": "final_compile", + "command": "lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-03.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "named_kernel", + "command": "lake env leanchecker GaussianRegressionLimits", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-leanchecker.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "corrected_controls", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-02.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "initial_controls_container_with_valid_printed_axioms", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 1, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-01.log", + "raw_log_sha256": "c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be" + } + ] + } + ], + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "reuse": "No checks repeated. These metadata limits do not invalidate completed applicable mathematical evidence or authorize a repeated proof campaign." +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/DRAFT_README.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/DRAFT_README.md new file mode 100644 index 000000000..788ec2a3a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/DRAFT_README.md @@ -0,0 +1,28 @@ +# Actual Fourier-source derivatives and contact conditioning prerequisites + +This additive Lean package proves joint Gaussianity and covariance formulas for actual derivatives of the existing smooth Fourier source. It also proves positive definiteness of the literal seven-coordinate contact covariance, invertibility of its six-pin block, strictly positive curvature residual variance, and the canonical actual endpoint Gaussian law with its derived mean and Schur covariance under explicit current pin invertibility. The original research goal remains active; the remaining conditioning, Palm, event and parent joins are part of that goal. + +The source is `I4SmoothCarrier.smoothSourceField` under its unchanged `smoothLaw`. `INPUT_BINDING.json` pins twelve byte-identical smooth-source inputs to Math- commit `e75cb81e8961d178f43aa9967dddd11cd579dc8a`. It separately pins eleven exact input modules from the sibling determinant-moment packet at `16f81b973a02783f8be83081aa7f198724cf5767`, under its existing execution receipt `95e502472e3d748c8b5dca3328a8644a9cb7cfc8fe40d5f9cce95a32814e5ec6`. The current binding has 23 reused inputs in total; their completed applicable verification is reused. The historical sources and reviews remain unchanged. Lean is pinned to v4.34.1 and Mathlib to `d13f23b723b8a846827a245b89c10fc7d3f11612`. + +The ten completed modules provide: + +| Module | Result and principal hypotheses | +|---|---| +| `SourceDerivativeGaussian` | For every natural dimension, nonzero period, and arbitrary finite collection of derivative orders, points and direction lists, the actual observations are jointly Gaussian. Every smooth sample satisfies the exact differentiated Fourier-series identity. Covariances equal convergent products of the actual differentiated coefficients. Empty families, order zero and repeated observations are retained. | +| `ContactJetCovariance` | A seven-mode polynomial witness forces positivity of a row-summable infinite coefficient Gram matrix. The coefficient identities and nonzero mode amplitudes are explicit hypotheses of this generic certificate. | +| `SourceContactCovariance` | Instantiates that certificate on the literal origin observations `(f,fx,fy,fxx,fxy,fxxx,fyy)` in dimension two. Canonical half-lattice membership, exact selected derivative coefficients and positive actual Fourier amplitudes are proved. For `L != 0`, the full covariance is positive definite. | +| `SourceConfluentPins` | The actual six endpoint pins admit an explicit continuous linear equivalence for `r != 0`. The literal target `(b,b-r^3/6,0,0,0,0)` maps to `(b-r^3/12,0,0,0,0,2)`. The raw pins, confluent pins and joint twelve-vector of raw pins plus endpoint Hessians have their Gaussianity derived from actual derivatives. | +| `SourceContactSchur` | For `L != 0`, the contact pin covariance of `J=(f,fx,fy,fxx,fxy,fxxx)` is positive definite and invertible. Its exact `6+1` decomposition with `Q=fyy` yields a positive Schur complement, identified with the actual least-squares curvature residual variance. | +| `SourceDerivativeContinuity` | Derivative covariance is continuous in both evaluation points, using point-independent summable coefficient majorants. Every actual directional derivative has zero unconditional mean. This does not substitute zero for conditional regression means. | +| `ConfluentIntegralIdentities` | For real scalar curves `F` of class C3 and `H` of class C2, exact average and Peano-kernel formulas remove the pin and adjusted-endpoint divided differences. The identities preserve signed nonzero `r`. | +| `GaussianRegressionLimits` | From finite block limits for `G,C,B,d` and invertibility of `G0`, derives eventual current `G` invertibility, convergence of `C G^-1 d` and `B-C G^-1 C^T`, and uniform small-radius mean/diagonal bounds. Empty dimensions and singular target covariance are retained; concrete source block limits remain an application obligation. | +| `SourceEndpointRegression` | For the actual source, derives joint Gaussianity of confluent pins and adjusted endpoints, proves the actual matrix normal equations, and constructs the explicit measurable Markov residual kernel at every pin value. `HasCondDistrib` uses the explicit current actual-pin covariance invertibility premise. On the prescribed fiber, adjusted endpoints equal the physical normalized six-vector. | +| `SourceEndpointParameters` | Derives actual zero unconditional means, Gaussian residual/posterior laws, the exact regression mean and Schur covariance, and PSD of that covariance. Under current actual pin invertibility, the canonical all-pin kernel becomes the literal `multivariateGaussian` after the specified Euclidean coordinate transport, including singular laws and noncentered prescribed-pin means. | + +The preimplementation contracts are the `SOURCE_*.md` files. They distinguish completed statements from planned successors. `SourceConfluentLimits`, `SourceEndpointEstimates` and `SourceContactEndpointParameters` remain outside this qualified ten-module inventory; their separate work does not receive a qualification claim from this catalog. + +The remaining original-goal work includes proving and joining actual small-radius confluent covariance limits, eventual current pin invertibility, actual regression parameter limits and uniform bounds, joining the actual endpoint Gaussian parameters to their contact limits and common coupling, the physical weighted normalizer, whole-field/geometric Palm identification, event-tail transfer and the parent theorems. Continuity of unadjusted derivative covariances alone does not justify singular `1/r` quotient limits. The midpoint quartic nine-vector is not substituted for the physical endpoint six-vector. + +All implementation and technical assistance are OpenAI author-side work, including `/root`, `/root/uniform_jet_moment_sources`, `/root/regression_kernel`, `/root/dm2_landed_composition`, `/root/normalizer_palm_research`, and the `gaussian_api`, `endpoint_means` and `peano_strategy` helpers. Different-family scientific alignment review remains pending. Existing module compilation, named kernel replay, axiom audits and mathematical controls are recorded in the evidence catalog and reused. This packaging adds no combined hosted run or independent scientific acceptance. Scientific effect: NONE. No parent or scientific-status acceptance is claimed. + +This actual-source package has 85 public theorem/lemma targets across ten modules. Its inventory is distinct from the sibling determinant-moment packet’s 66 targets. Earlier sibling scientific reviews retain their original scope and do not cover this actual-source inventory. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/DRAFT_SCOPE.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/DRAFT_SCOPE.md new file mode 100644 index 000000000..5c4dec9f0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/DRAFT_SCOPE.md @@ -0,0 +1,31 @@ +# Exact ten-module research scope + +This document describes the ten completed modules named below. Later modules need their own exact statements and execution evidence before being added. The full original research goal remains active. + +`SourceDerivativeGaussian` works with the literal `SmoothSample d L`, `smoothLaw d L` and `smoothSourceField d L`. Its model premise is `L != 0`; dimension, family size, orders, points and direction lists are arbitrary. Differentiated deterministic coefficient summability and the every-sample derivative-series identity are conclusions. Joint Gaussianity is derived from the preserved infinite coefficient law. The covariance conclusion is the infinite differentiated-coefficient pairing; it is not yet an identification with derivatives of a theta-kernel expression. + +`ContactJetCovariance` is a generic coefficient certificate. Row summability, seven specified witness columns, nonzero column amplitudes and nonzero frequency scale imply positive definiteness of the full infinite Gram matrix. The proof uses the sum-of-squares quadratic form and the explicit witness's separating columns. It does not replace the infinite source by those seven modes. + +`SourceContactCovariance` supplies those hypotheses for the actual two-dimensional source. Its origin coordinate order is `(f,fx,fy,fxx,fxy,fxxx,fyy)`. The selected modes are DC, cos(1,0), cos(0,1), cos(1,1), sin(1,0), sin(2,0), sin(0,1), with their actual source amplitudes and canonical half-lattice membership. From `L != 0` it derives positive definiteness of the literal contact covariance; no covariance positivity is assumed. + +`SourceContactSchur` defines `J=(f,fx,fy,fxx,fxy,fxxx)`, `Q=fyy`, their covariance blocks `G,b,d`, and `d-b dotProduct (G inverse mulVec b)`. An exact coordinate equivalence proves the `6+1` block identity. From `L != 0` it derives contact `G` positivity/invertibility, Schur positivity and equality with the variance of the actual curvature residual `Q-(G inverse mulVec b) dotProduct J`. These are contact claims. They do not alone prove current invertibility at all sufficiently small nonzero separations or select a conditional distribution at a prescribed null pin. + +`SourceConfluentPins` fixes endpoints `(-r/2,0),(r/2,0)` and raw order `(fM,fS,fxM,fxS,fyM,fyS)`. Its explicit transform is invertible for `r != 0`; the Gaussian laws are consequences of actual derivative observations. Arbitrary real height `b` is retained in the exact target `(b-r^3/12,0,0,0,0,2)`. The twelve-observation Gaussian statement includes endpoint Hessian order `(fxxM,fxyM,fyyM,fxxS,fxyS,fyyS)` jointly with the raw pins. It does not assume independence or a physical-endpoint identity with a midpoint quartic jet. + +`SourceDerivativeContinuity` proves continuity of every fixed pair of directional-derivative covariances in its evaluation points using global summable bounds. Zero means refer to the unchanged unconditional source law. Conditional torus means are not replaced by zero or a planar reference value. + +`ConfluentIntegralIdentities` is a deterministic scalar result for `F : Real -> Real` of class C3, `H : Real -> Real` of class C2 and signed `r != 0`. It proves the exact integral formulas for the second and transverse divided differences, cubic Peano pin, and left/right adjusted endpoints. The transverse curve `H` denotes the y-derivative curve, so its pin uses `H` itself. Neither an integral representation nor a quotient limit is assumed. Actual-source instantiation and justified infinite-sum limiting arguments remain necessary for stochastic block limits. + +`GaussianRegressionLimits` is a deterministic matrix theorem. Given convergence of finite pin, cross, target and prescribed-value blocks `G,C,B,d` to `G0,C0,B0,d0`, and only `IsUnit G0`, it derives eventual current invertibility, inverse convergence, regression mean/covariance convergence and one interval of simultaneous coordinate bounds. The formulas describe a centered prior with arbitrary prescribed target `d`; conditional means need not vanish. It assumes neither target covariance invertibility nor current pin invertibility. Actual source block convergence is not supplied by this generic theorem. + +`SourceEndpointRegression` uses the literal actual source observations. In LM006 order, its adjusted endpoint vector is `((xxM-X3)/r,(xxS-X3)/r,(xyM-X4)/r,(xyS-X4)/r,yyM,yyS)`. Joint Gaussianity comes from the actual twelve-derivative vector and a proved continuous linear image. For current actual pin covariance `G`, the explicit premise `IsUnit G` supplies matrix inverse normal equations; residual Gaussianity and independence are derived through the reused canonical regression theorem. The resulting Markov measurable kernel has its literal residual pushforward at every pin value, including the prescribed null pin, and satisfies `HasCondDistrib`. The physical endpoint identity is proved deterministically on the `X3=X4=0` fiber. This does not assert eventual current invertibility or a canonical whole-field conditioning theorem. + +`SourceEndpointParameters` derives actual unconditional zero means rather than assuming them, then proves residual and all-pin posterior Gaussianity. Under the explicit current actual pin covariance invertibility premise, covariance bilinearity and the normal equations identify the residual covariance with `B-C G^-1 C^T`. PSD follows from that actual covariance; no residual nondegeneracy is assumed. The literal canonical kernel has mean `C G^-1 x` at every pin value and, after the specified `toLp 2` Euclidean coordinate transport, equals the pinned `multivariateGaussian` of that mean and Schur covariance. Singular covariance, arbitrary prescribed pin and nonzero posterior mean are preserved. The included generic finite-Gaussian parameter lemma also retains dimension zero. Current invertibility and actual small-radius block limits still require their separate source joins. + +Still required within the original goal: actual confluent pin/adjusted-endpoint covariance limits; their join to contact positivity and finite-radius regression; joining the derived actual endpoint Gaussian parameters to their contact limits and common coupling; uniform conditional parameters and the positive physical normalizer; whole-field and geometric Palm/event identification; event-tail and parent conclusions. No theorem adds one of those desired conclusions as a premise and calls that conclusion proved. `SourceConfluentLimits`, `SourceEndpointEstimates` and `SourceContactEndpointParameters` are outside this ten-module qualified inventory and require their separate evidence/scope before inclusion. + +Applicable completed verification on unchanged sources is reusable. Completed module compilations, named kernel replays, transitive axiom audits and meaningful controls are recorded separately from hosted execution and different-family alignment. This packaging does not rerun or reclassify those checks. All present authors/helpers are OpenAI; `gaussian_api`, `endpoint_means` and `peano_strategy` assistance is author-side. Different-family scientific review remains pending. Scientific effect NONE; no status-register changes or parent acceptance follow from this packet. + +The 85 public theorem/lemma names inventoried here belong to this ten-module actual-source package. The sibling determinant-moment packet has a separate 66-target inventory and its reviews retain their own exact scope. Neither execution receipts nor scientific review transfer between those inventories. + +The current input binding covers 23 reused sources: twelve landed smooth-source files plus eleven exact sibling-packet modules at commit `16f81b973a02783f8be83081aa7f198724cf5767`, under execution receipt `95e502472e3d748c8b5dca3328a8644a9cb7cfc8fe40d5f9cce95a32814e5ec6`. This metadata expansion does not repeat or broaden the completed verification or its scientific review. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/EVIDENCE_CATALOG.json b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/EVIDENCE_CATALOG.json new file mode 100644 index 000000000..0a55ed864 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/EVIDENCE_CATALOG.json @@ -0,0 +1,5183 @@ +{ + "schema_version": 1, + "role": "read-only projection of existing mathematical evidence; no new execution", + "scientific_effect": "NONE", + "goal_status": "FULL_ORIGINAL_RESEARCH_GOAL_ACTIVE", + "source_root": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "scope": [ + "SourceDerivativeGaussian", + "ContactJetCovariance", + "SourceContactCovariance", + "SourceConfluentPins", + "SourceContactSchur", + 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No new proof checks performed." + }, + "modules": [ + { + "module": "SourceDerivativeGaussian", + "namespace": "I4SourceDerivativeGaussian", + "source_path": "SourceDerivativeGaussian.lean", + "source_bytes": 9874, + "source_sha256": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "imports": [ + "SourceSmoothCarrier", + "Mathlib.Analysis.Calculus.SmoothSeries", + "Mathlib.Topology.Algebra.Module.Multilinear.Topology" + ], + "public_targets": [ + "I4SourceDerivativeGaussian.derivativeCoefficient_summable", + "I4SourceDerivativeGaussian.smoothSourceField_derivative_series", + "I4SourceDerivativeGaussian.scalarSeries_finiteVector_hasGaussianLaw", + "I4SourceDerivativeGaussian.smoothSourceField_derivativeVector_hasGaussianLaw", + "I4SourceDerivativeGaussian.smoothSourceField_derivative_covariance" + ], + "public_declarations": [ + { + "kind": "def", + "name": "I4SourceDerivativeGaussian.derivativeCoefficient", + "line": 53 + }, + { + "kind": "theorem", + "name": 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+ }, + { + "original_path": "/workspace/scratch/regression-kernel/endpoint-parameters/means/Means.lean", + "projection_path": "evidence/SourceEndpointParameters/means/Means.lean.txt", + "bytes": 3764, + "sha256": "fe957a1e6271af99b45dca727f81bf418551a13473d5e0776f8b032ca09f62cc" + } + ], + "known_missing_records": [], + "author_lineage": [ + "OpenAI/Codex /root: source contracts, integration and packaging", + "OpenAI/Codex /root/uniform_jet_moment_sources: actual derivative Gaussianity, actual source contact covariance, derivative continuity", + "OpenAI/Codex /root/regression_kernel: contact Gram and source contact Schur", + "OpenAI/Codex /root/regression_kernel/gaussian_api: optional Schur and generic L2 residual-variance proof support", + "OpenAI/Codex /root/dm2_landed_composition: confluent pin equivalence and integral identities", + "OpenAI/Codex /root/dm2_landed_composition/peano_strategy: actual generic Peano scratch proof helper", + "OpenAI/Codex /root/palm_measure_transfer: read-only evidence projection", + "OpenAI/Codex /root/palm_measure_transfer/density_api: read-only README/SCOPE draft assistance", + "OpenAI/Codex /root/normalizer_palm_research: Gaussian regression parameter limits and uniform bounds", + "OpenAI/Codex /root/regression_kernel/gaussian_api: actual SourceEndpointRegression matrix normal-equations API support", + "OpenAI/Codex /root/regression_kernel/endpoint_means: actual endpoint unconditional zero-mean proofs, retained means/Means.lean and Means-03.log", + "OpenAI/Codex /root/regression_kernel/gaussian_api: actual SourceEndpointParameters residual covariance proof support, retained covariance/ResidualCovariance.lean/.log" + ], + "review": "All above work is OpenAI author-side. Different-family scientific alignment review remains pending.", + "verification_reuse": "All saved compilation, named kernel, axiom and controls evidence reused. No Lean, kernel replay, tests, or old verification executed during this projection.", + "packaging_note": "Original raw logs, commands, metadata, failures and control/helper source bytes preserved. Lean source artifacts have .lean.txt projection suffix; actual original paths remain in records. This index is not a hosted receipt or accepted scientific alignment record.", + "author_history_supplement": { + "path": "AUTHOR_HISTORY_SUPPLEMENT.json", + "sha256": "e3c4a4fbf56f5938f65138d8c61c632f921ee84cc6e9f1b6ebd7af70f8da8052", + "record": { + "record_type": "author tool-history reports; not a machine-created execution receipt", + "scientific_effect": "NONE", + "reports": [ + { + "author": "OpenAI/Codex /root/uniform_jet_moment_sources", + "scope": [ + "SourceDerivativeGaussian", + "SourceContactCovariance", + "SourceDerivativeContinuity" + ], + "verbatim_report": "Author-history confirmation: all three exact SHA values you list are my final source bytes, unchanged by me since their recorded final compile/controls/axiom/kernel executions. No existing JSON receipt/sourcehash record file was created by me; SHA values were read by sha256sum after source freeze and reported in my source-bound handoffs to root. Treat that as author-history confirmation, not a machine-created execution receipt. Existing command/exit/log triples are the only machine artifacts I produced; please preserve those honest limits and do not rerun.", + "confirmed_source_hashes": { + "SourceDerivativeGaussian.lean": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "SourceContactCovariance.lean": "4189cbaf4c18c55f21e3af2c96b0b2842d83161f44596eb691c507dcc1a439c8", + "SourceDerivativeContinuity.lean": "552511e0aee227133b7d067faeceaab9c43c0094b07d68b8eddab0ab2419775e" + } + }, + { + "author": "OpenAI/Codex /root/regression_kernel", + "scope": [ + "ContactJetCovariance", + "SourceContactSchur" + ], + "control_cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "capture_shell_prelude": "set -o pipefail", + "controls": [ + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log", + "observed_exit": 0 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency-01.log", + "observed_exit": 1 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls-01.log", + "observed_exit": 0 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex-01.log", + "observed_exit": 1 + } + ], + "report_transcription": "Actual control commands, no rerun: cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025; each ran bash set -o pipefail then /home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log observed exit0; corresponding contact-gram RejectedZeroFrequency.lean / RejectedZeroAmplitude.lean with matching -01.log each observed exit1. Schur controls same command pattern under scratch/regression-kernel/source-contact-schur/: PositiveControls.lean to PositiveControls-01.log observedexit0; RejectedZeroPeriod.lean and RejectedWrongQIndex.lean to matching-01.log each observedexit1. These are actual tool-observed command/exits from my current transcript. Helper evidence: /workspace/scratch/regression-kernel/contact-schur/SchurProbe.{lean,log} final exit0 and failed1 retained; /workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.{lean,log} final exit0 and ResidualVariance.failed1.* retained. Helpers coauthored OpenAI/Codex gaussian_api. Earlier failedproduction ContactJetCovariance-01 was assembly-script ValueError/no sourcefile, exit1; ContactJetCovariance-02 success. SourceContactSchur-01/-02/-03 failed author attempts exit1; -04 success.", + "transcription_note": "Faithful plain-text transcription of author report; punctuation/arrow notation normalized. Not a raw machine log." + }, + { + "author": "OpenAI/Codex /root/normalizer_palm_research", + "scope": [ + "GaussianRegressionLimits" + ], + "record_type": "actual author tool-history report; not a machine-created execution receipt", + "verbatim_report": "Author tool-history report (not invented machine receipt): cwd /workspace/Math-/frontiers/source_derivative_gaussian_20261025. Final compile `lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean` exit 0, captured /workspace/scratch/regression-limits/regression-limits-03.log empty. Kernel `lake env leanchecker GaussianRegressionLimits` exit 0, regression-limits-leanchecker.log empty. Final controls `lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean` exit 0, regression-limits-controls-02.log empty. Imported-module #print axioms outputs for regression_parameter_limits and regression_uniform_bounds are in controls-01.log (that command exit 1 for unrelated untyped control index); both only propext, Classical.choice, Quot.sound. No production changes occurred after final successful compile/kernel. Current source SHA256 read-only measured 26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be. All three success logs empty SHA e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855; controls01 SHA c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be. Exact source compiled was this source, author-history association; no rerun.", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "confirmed_source_sha256": "26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be", + "source_unchanged_after_final_compile_kernel": true, + "runs": [ + { + "role": "final_compile", + "command": "lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-03.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "named_kernel", + "command": "lake env leanchecker GaussianRegressionLimits", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-leanchecker.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "corrected_controls", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-02.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "initial_controls_container_with_valid_printed_axioms", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 1, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-01.log", + "raw_log_sha256": "c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be" + } + ] + } + ], + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "reuse": "No checks repeated. These metadata limits do not invalidate completed applicable mathematical evidence or authorize a repeated proof campaign." + } + }, + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "append_scope": "Original seven- and nine-module catalog snapshots preserved under historical-seven-module/ and historical-nine-module/. Only SourceEndpointParameters appended at this step; INPUT_BINDING metadata updated to current23 reused sources.", + "target_inventory_identity": { + "package": "frontiers/source_derivative_gaussian_20261025", + "public_theorem_lemma_count": 85, + "distinct_from": "frontiers/determinant_moment_20261025 has a separate 66-target inventory; source/receipt/review scopes must not be transferred between packages" + }, + "historical_catalogs": [ + { + "path": "historical-seven-module/EVIDENCE_CATALOG.json", + "sha256": "f4977b7a28a714112df72a9249dbdb0c81f80539e5f19f43a848c59e9393f963" + }, + { + "path": "historical-nine-module/EVIDENCE_CATALOG.json", + "sha256": "74509aaf3eb1e17bf8fd3b10880aa2eedc9e14b257dc0cb0b693cd76a4770cbb" + } + ] +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/EVIDENCE_SUMMARY.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/EVIDENCE_SUMMARY.md new file mode 100644 index 000000000..0b782eb19 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/EVIDENCE_SUMMARY.md @@ -0,0 +1,53 @@ +# Read-only ten-module evidence projection + +The catalog inventories **85 public theorem/lemma targets** in ten completed +modules. Saved transitive axiom reports cover every such target. The +`SourceContactSchur` audit also includes the actual inverse definition, giving +86 distinct reported public declarations. Only `propext`, `Classical.choice` +and `Quot.sound` appear in these reports. Named module kernel replay and final +module compilation records each report exit 0. No checks were run during this +projection; completed applicable evidence is reused. + +| Module | Public theorem/lemma targets | Principal saved evidence | +|---|---:|---| +| SourceDerivativeGaussian | 5 | `05-covariance-compile`, `06-controls`, `07-axiom-audit`, `08-kernel-replay` | +| ContactJetCovariance | 8 | `EVIDENCE.json`, final `ContactJetCovariance-02`, positive controls, zero-frequency/zero-amplitude rejections | +| SourceContactCovariance | 7 | `05-column-identities-compile`, `06-controls`, `07-axiom-audit`, `08-kernel-replay` | +| SourceConfluentPins | 8 | `qualification_receipt.json`, earlier final compile, signed-radius/target controls and zero-radius rejections | +| SourceContactSchur | 10 | `EVIDENCE.json`, final `SourceContactSchur-04`, positive controls and zero-period/wrong-Q-index rejections | +| SourceDerivativeContinuity | 6 | `02-evaluation-and-law-compile`, corrected `05-controls-explicit-path`, `04-axiom-audit`, `06-kernel-replay` | +| ConfluentIntegralIdentities | 7 | `qualification_receipt.json`, final compile, positive controls and cubic/transverse false controls | +| SourceEndpointRegression | 13 | `EVIDENCE.json`, final `SourceEndpointRegression-03`, named kernel, thirteen exported axiom reports, positive controls and two adjustment/height rejections | +| GaussianRegressionLimits | 2 | final compile `regression-limits-03`, named kernel, corrected `controls-02`; two valid imported-module axiom reports retained in failed control container `controls-01` | +| SourceEndpointParameters | 19 | `EVIDENCE.json`, final `SourceEndpointParameters-02`, named kernel, nineteen exported axiom reports, positive singular/noncentered/empty controls and two false-parameter rejections | + +`EVIDENCE_CATALOG.json` retains exact source SHA256 values, public declarations +with source lines, recorded commands/exits, receipt contents, raw-log hashes and +an index of **255 original evidence/helper files** copied beneath `evidence/`. +Earlier failures remain included. In particular, the integral module's final +compilation has one harmless `unnecessarySeqFocus` warning; it is not represented +as a warning-free run. Projection filenames for Lean controls/helpers end in +`.lean.txt`; original paths and bytes remain bound in the catalog. + +The actual `gaussian_api` optional Schur/residual/matrix-normal-equations proof helper and +`peano_strategy` Peano proof helper are included in the author lineage and raw +helper evidence. README/SCOPE drafting assistance from `density_api` is also +identified as OpenAI author-side work. The actual `endpoint_means` zero-mean proof helper and the `gaussian_api` residual-covariance helper are included with their original `means/` and `covariance/` evidence. No organizational or scientific +independence is inferred from those helpers. + +`AUTHOR_HISTORY_SUPPLEMENT.json` discloses two metadata limits: the three +`uniform_jet_moment_sources` groups have command/exit/raw-log triples but no +historical JSON source-hash receipt; their unchanged final source association +was confirmed by the author. The two `regression_kernel` groups' precise control +capture commands come from the author's actual tool history, alongside the +existing raw logs and exit-bearing receipts. Those reports are not relabeled as +machine-generated receipts. The new GaussianRegressionLimits source association and exact commands/exits were also confirmed by its author’s actual tool history. Its controls-01 process exited 1 for an unrelated control elaboration error while printing two valid imported-module axiom reports. Corrected controls-02 separately exited 0; no exit-0 axiom-only command is invented. No referenced evidence files or saved public target axiom reports were missing in the ten-module inventory. + +`INPUT_BINDING.json` now describes 23 reused inputs: twelve unchanged landed smooth-source files plus eleven exact sibling-packet modules at commit `16f81b973a02783f8be83081aa7f198724cf5767`, under existing execution receipt `95e502472e3d748c8b5dca3328a8644a9cb7cfc8fe40d5f9cce95a32814e5ec6`. Its current SHA256 is `42292a58cf84e118b34f4b8b952fa3b650126dc3d167cee85d8048b9bc6d4b13`. The catalog binds +that file without changing or re-executing the inputs. Historical origin +manifests/reviews remain provenance and retain their own scope. + +The original research goal remains active. Different-family scientific alignment +review is pending. This projection does not award parent acceptance, a hosted execution result or a scientific-status change. `SourceConfluentLimits`, `SourceEndpointEstimates` and `SourceContactEndpointParameters` remain outside this qualified inventory; root may append their separate actual qualification. The unchanged seven- and nine-module catalog/draft snapshots remain under `historical-seven-module/` and `historical-nine-module/`. The nine-module catalog retains SHA256 `74509aaf3eb1e17bf8fd3b10880aa2eedc9e14b257dc0cb0b693cd76a4770cbb`. + +The ten-module actual-source inventory here has 85 public theorem/lemma targets and is distinct from the sibling determinant-moment packet’s 66 targets. No sibling receipt, target coverage or scientific review is transferred to this actual-source inventory. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/EVIDENCE_CATALOG.json b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/EVIDENCE_CATALOG.json new file mode 100644 index 000000000..0a55ed864 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/EVIDENCE_CATALOG.json @@ -0,0 +1,5183 @@ +{ + "schema_version": 1, + "role": "read-only projection of existing mathematical evidence; no new execution", + "scientific_effect": "NONE", + "goal_status": "FULL_ORIGINAL_RESEARCH_GOAL_ACTIVE", + "source_root": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "scope": [ + "SourceDerivativeGaussian", + "ContactJetCovariance", + "SourceContactCovariance", + "SourceConfluentPins", + "SourceContactSchur", + "SourceDerivativeContinuity", + "ConfluentIntegralIdentities", + "SourceEndpointRegression", + "GaussianRegressionLimits", + "SourceEndpointParameters" + ], + "not_in_this_completed_inventory": [ + "SourceConfluentLimits", + "SourceEndpointEstimates", + "SourceContactEndpointParameters" + ], + "source_inputs_binding": { + "path": "INPUT_BINDING.json", + "sha256": "42292a58cf84e118b34f4b8b952fa3b650126dc3d167cee85d8048b9bc6d4b13", + "input_count": 23, + "record": { + "repository": "d6g8k5htny-coder/Math-", + "commit": "e75cb81e8961d178f43aa9967dddd11cd579dc8a", + "files": { + "inputs/GaussianSeriesFoundation.lean": { + "origin_path": "frontiers/source_smooth_realization_20261011/GaussianSeriesFoundation.lean", + "sha256": "feea566d5aeba58459bd4a1f20f070bdcd80ee30712817324cdd594e3e232c95" + }, + "inputs/SpectralWeights.lean": { + "origin_path": 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read-only evidence projection", + "OpenAI/Codex /root/palm_measure_transfer/density_api: read-only README/SCOPE draft assistance", + "OpenAI/Codex /root/normalizer_palm_research: Gaussian regression parameter limits and uniform bounds", + "OpenAI/Codex /root/regression_kernel/gaussian_api: actual SourceEndpointRegression matrix normal-equations API support", + "OpenAI/Codex /root/regression_kernel/endpoint_means: actual endpoint unconditional zero-mean proofs, retained means/Means.lean and Means-03.log", + "OpenAI/Codex /root/regression_kernel/gaussian_api: actual SourceEndpointParameters residual covariance proof support, retained covariance/ResidualCovariance.lean/.log" + ], + "review": "All above work is OpenAI author-side. Different-family scientific alignment review remains pending.", + "verification_reuse": "All saved compilation, named kernel, axiom and controls evidence reused. No Lean, kernel replay, tests, or old verification executed during this projection.", + "packaging_note": "Original raw logs, commands, metadata, failures and control/helper source bytes preserved. Lean source artifacts have .lean.txt projection suffix; actual original paths remain in records. This index is not a hosted receipt or accepted scientific alignment record.", + "author_history_supplement": { + "path": "AUTHOR_HISTORY_SUPPLEMENT.json", + "sha256": "e3c4a4fbf56f5938f65138d8c61c632f921ee84cc6e9f1b6ebd7af70f8da8052", + "record": { + "record_type": "author tool-history reports; not a machine-created execution receipt", + "scientific_effect": "NONE", + "reports": [ + { + "author": "OpenAI/Codex /root/uniform_jet_moment_sources", + "scope": [ + "SourceDerivativeGaussian", + "SourceContactCovariance", + "SourceDerivativeContinuity" + ], + "verbatim_report": "Author-history confirmation: all three exact SHA values you list are my final source bytes, unchanged by me since their recorded final compile/controls/axiom/kernel executions. No existing JSON receipt/sourcehash record file was created by me; SHA values were read by sha256sum after source freeze and reported in my source-bound handoffs to root. Treat that as author-history confirmation, not a machine-created execution receipt. Existing command/exit/log triples are the only machine artifacts I produced; please preserve those honest limits and do not rerun.", + "confirmed_source_hashes": { + "SourceDerivativeGaussian.lean": "c9754debcbf2aab7ff8d9604f2c13726723e8a8c1921ae68f83349205e43885d", + "SourceContactCovariance.lean": "4189cbaf4c18c55f21e3af2c96b0b2842d83161f44596eb691c507dcc1a439c8", + "SourceDerivativeContinuity.lean": "552511e0aee227133b7d067faeceaab9c43c0094b07d68b8eddab0ab2419775e" + } + }, + { + "author": "OpenAI/Codex /root/regression_kernel", + "scope": [ + "ContactJetCovariance", + "SourceContactSchur" + ], + "control_cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "capture_shell_prelude": "set -o pipefail", + "controls": [ + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log", + "observed_exit": 0 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroFrequency-01.log", + "observed_exit": 1 + }, + { + "module": "ContactJetCovariance", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/RejectedZeroAmplitude-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/PositiveControls-01.log", + "observed_exit": 0 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedZeroPeriod-01.log", + "observed_exit": 1 + }, + { + "module": "SourceContactSchur", + "command": "/home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex.lean 2>&1 | tee /workspace/scratch/regression-kernel/source-contact-schur/RejectedWrongQIndex-01.log", + "observed_exit": 1 + } + ], + "report_transcription": "Actual control commands, no rerun: cwd=/workspace/Math-/frontiers/source_derivative_gaussian_20261025; each ran bash set -o pipefail then /home/agent/.elan/bin/lake env lean /workspace/scratch/regression-kernel/contact-gram/PositiveControls.lean 2>&1 | tee /workspace/scratch/regression-kernel/contact-gram/PositiveControls-03.log observed exit0; corresponding contact-gram RejectedZeroFrequency.lean / RejectedZeroAmplitude.lean with matching -01.log each observed exit1. Schur controls same command pattern under scratch/regression-kernel/source-contact-schur/: PositiveControls.lean to PositiveControls-01.log observedexit0; RejectedZeroPeriod.lean and RejectedWrongQIndex.lean to matching-01.log each observedexit1. These are actual tool-observed command/exits from my current transcript. Helper evidence: /workspace/scratch/regression-kernel/contact-schur/SchurProbe.{lean,log} final exit0 and failed1 retained; /workspace/scratch/regression-kernel/contact-schur/actual-residual/ResidualVariance.{lean,log} final exit0 and ResidualVariance.failed1.* retained. Helpers coauthored OpenAI/Codex gaussian_api. Earlier failedproduction ContactJetCovariance-01 was assembly-script ValueError/no sourcefile, exit1; ContactJetCovariance-02 success. SourceContactSchur-01/-02/-03 failed author attempts exit1; -04 success.", + "transcription_note": "Faithful plain-text transcription of author report; punctuation/arrow notation normalized. Not a raw machine log." + }, + { + "author": "OpenAI/Codex /root/normalizer_palm_research", + "scope": [ + "GaussianRegressionLimits" + ], + "record_type": "actual author tool-history report; not a machine-created execution receipt", + "verbatim_report": "Author tool-history report (not invented machine receipt): cwd /workspace/Math-/frontiers/source_derivative_gaussian_20261025. Final compile `lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean` exit 0, captured /workspace/scratch/regression-limits/regression-limits-03.log empty. Kernel `lake env leanchecker GaussianRegressionLimits` exit 0, regression-limits-leanchecker.log empty. Final controls `lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean` exit 0, regression-limits-controls-02.log empty. Imported-module #print axioms outputs for regression_parameter_limits and regression_uniform_bounds are in controls-01.log (that command exit 1 for unrelated untyped control index); both only propext, Classical.choice, Quot.sound. No production changes occurred after final successful compile/kernel. Current source SHA256 read-only measured 26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be. All three success logs empty SHA e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855; controls01 SHA c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be. Exact source compiled was this source, author-history association; no rerun.", + "cwd": "/workspace/Math-/frontiers/source_derivative_gaussian_20261025", + "confirmed_source_sha256": "26c403ea1104fbce89f5c2833f603edd4d2703669d8a94a77b2c9a2a06b572be", + "source_unchanged_after_final_compile_kernel": true, + "runs": [ + { + "role": "final_compile", + "command": "lake env lean -o .lake/build/lib/lean/GaussianRegressionLimits.olean GaussianRegressionLimits.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-03.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "named_kernel", + "command": "lake env leanchecker GaussianRegressionLimits", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-leanchecker.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "corrected_controls", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 0, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-02.log", + "raw_log_sha256": "e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855" + }, + { + "role": "initial_controls_container_with_valid_printed_axioms", + "command": "lake env lean /workspace/scratch/regression-limits/regression-limits-controls.lean", + "observed_exit": 1, + "raw_log_path": "/workspace/scratch/regression-limits/regression-limits-controls-01.log", + "raw_log_sha256": "c705e3d645105af0a96a8b0463d6bdf18d5cd6d1d911d9cd617cc031302ba8be" + } + ] + } + ], + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "reuse": "No checks repeated. These metadata limits do not invalidate completed applicable mathematical evidence or authorize a repeated proof campaign." + } + }, + "metadata_limits": [ + "The three uniform_jet_moment_sources groups preserve exact command/exit/raw-log triples. Their source-hash association is an author-history confirmation plus current projection hashes, not a newly invented historical machine receipt.", + "The two regression_kernel control command strings above come from the author's actual tool-history report. Existing EVIDENCE.json files and raw logs separately retain their observed exit statuses.", + "GaussianRegressionLimits final-source command/exit association is an explicit author tool-history report, not a historical machine JSON receipt. Its initial controls container exited 1 for an unrelated control elaboration error while printing valid imported-module axiom reports; final corrected controls exited 0 separately." + ], + "append_scope": "Original seven- and nine-module catalog snapshots preserved under historical-seven-module/ and historical-nine-module/. Only SourceEndpointParameters appended at this step; INPUT_BINDING metadata updated to current23 reused sources.", + "target_inventory_identity": { + "package": "frontiers/source_derivative_gaussian_20261025", + "public_theorem_lemma_count": 85, + "distinct_from": "frontiers/determinant_moment_20261025 has a separate 66-target inventory; source/receipt/review scopes must not be transferred between packages" + }, + "historical_catalogs": [ + { + "path": "historical-seven-module/EVIDENCE_CATALOG.json", + "sha256": "f4977b7a28a714112df72a9249dbdb0c81f80539e5f19f43a848c59e9393f963" + }, + { + "path": "historical-nine-module/EVIDENCE_CATALOG.json", + "sha256": "74509aaf3eb1e17bf8fd3b10880aa2eedc9e14b257dc0cb0b693cd76a4770cbb" + } + ] +} diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/INPUT_BINDING.json 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b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/README.md @@ -0,0 +1,28 @@ +# Actual Fourier-source derivatives and contact conditioning prerequisites + +This additive Lean package proves joint Gaussianity and covariance formulas for actual derivatives of the existing smooth Fourier source. It also proves positive definiteness of the literal seven-coordinate contact covariance, invertibility of its six-pin block, strictly positive curvature residual variance, and the canonical actual endpoint Gaussian law with its derived mean and Schur covariance under explicit current pin invertibility. The original research goal remains active; the remaining conditioning, Palm, event and parent joins are part of that goal. + +The source is `I4SmoothCarrier.smoothSourceField` under its unchanged `smoothLaw`. `INPUT_BINDING.json` pins twelve byte-identical smooth-source inputs to Math- commit `e75cb81e8961d178f43aa9967dddd11cd579dc8a`. It separately pins eleven exact input modules from the sibling determinant-moment packet at `16f81b973a02783f8be83081aa7f198724cf5767`, under its existing execution receipt `95e502472e3d748c8b5dca3328a8644a9cb7cfc8fe40d5f9cce95a32814e5ec6`. The current binding has 23 reused inputs in total; their completed applicable verification is reused. The historical sources and reviews remain unchanged. Lean is pinned to v4.34.1 and Mathlib to `d13f23b723b8a846827a245b89c10fc7d3f11612`. + +The ten completed modules provide: + +| Module | Result and principal hypotheses | +|---|---| +| `SourceDerivativeGaussian` | For every natural dimension, nonzero period, and arbitrary finite collection of derivative orders, points and direction lists, the actual observations are jointly Gaussian. Every smooth sample satisfies the exact differentiated Fourier-series identity. Covariances equal convergent products of the actual differentiated coefficients. Empty families, order zero and repeated observations are retained. | +| `ContactJetCovariance` | A seven-mode polynomial witness forces positivity of a row-summable infinite coefficient Gram matrix. The coefficient identities and nonzero mode amplitudes are explicit hypotheses of this generic certificate. | +| `SourceContactCovariance` | Instantiates that certificate on the literal origin observations `(f,fx,fy,fxx,fxy,fxxx,fyy)` in dimension two. Canonical half-lattice membership, exact selected derivative coefficients and positive actual Fourier amplitudes are proved. For `L != 0`, the full covariance is positive definite. | +| `SourceConfluentPins` | The actual six endpoint pins admit an explicit continuous linear equivalence for `r != 0`. The literal target `(b,b-r^3/6,0,0,0,0)` maps to `(b-r^3/12,0,0,0,0,2)`. The raw pins, confluent pins and joint twelve-vector of raw pins plus endpoint Hessians have their Gaussianity derived from actual derivatives. | +| `SourceContactSchur` | For `L != 0`, the contact pin covariance of `J=(f,fx,fy,fxx,fxy,fxxx)` is positive definite and invertible. Its exact `6+1` decomposition with `Q=fyy` yields a positive Schur complement, identified with the actual least-squares curvature residual variance. | +| `SourceDerivativeContinuity` | Derivative covariance is continuous in both evaluation points, using point-independent summable coefficient majorants. Every actual directional derivative has zero unconditional mean. This does not substitute zero for conditional regression means. | +| `ConfluentIntegralIdentities` | For real scalar curves `F` of class C3 and `H` of class C2, exact average and Peano-kernel formulas remove the pin and adjusted-endpoint divided differences. The identities preserve signed nonzero `r`. | +| `GaussianRegressionLimits` | From finite block limits for `G,C,B,d` and invertibility of `G0`, derives eventual current `G` invertibility, convergence of `C G^-1 d` and `B-C G^-1 C^T`, and uniform small-radius mean/diagonal bounds. Empty dimensions and singular target covariance are retained; concrete source block limits remain an application obligation. | +| `SourceEndpointRegression` | For the actual source, derives joint Gaussianity of confluent pins and adjusted endpoints, proves the actual matrix normal equations, and constructs the explicit measurable Markov residual kernel at every pin value. `HasCondDistrib` uses the explicit current actual-pin covariance invertibility premise. On the prescribed fiber, adjusted endpoints equal the physical normalized six-vector. | +| `SourceEndpointParameters` | Derives actual zero unconditional means, Gaussian residual/posterior laws, the exact regression mean and Schur covariance, and PSD of that covariance. Under current actual pin invertibility, the canonical all-pin kernel becomes the literal `multivariateGaussian` after the specified Euclidean coordinate transport, including singular laws and noncentered prescribed-pin means. | + +The preimplementation contracts are the `SOURCE_*.md` files. They distinguish completed statements from planned successors. `SourceConfluentLimits`, `SourceEndpointEstimates` and `SourceContactEndpointParameters` remain outside this qualified ten-module inventory; their separate work does not receive a qualification claim from this catalog. + +The remaining original-goal work includes proving and joining actual small-radius confluent covariance limits, eventual current pin invertibility, actual regression parameter limits and uniform bounds, joining the actual endpoint Gaussian parameters to their contact limits and common coupling, the physical weighted normalizer, whole-field/geometric Palm identification, event-tail transfer and the parent theorems. Continuity of unadjusted derivative covariances alone does not justify singular `1/r` quotient limits. The midpoint quartic nine-vector is not substituted for the physical endpoint six-vector. + +All implementation and technical assistance are OpenAI author-side work, including `/root`, `/root/uniform_jet_moment_sources`, `/root/regression_kernel`, `/root/dm2_landed_composition`, `/root/normalizer_palm_research`, and the `gaussian_api`, `endpoint_means` and `peano_strategy` helpers. Different-family scientific alignment review remains pending. Existing module compilation, named kernel replay, axiom audits and mathematical controls are recorded in the evidence catalog and reused. This packaging adds no combined hosted run or independent scientific acceptance. Scientific effect: NONE. No parent or scientific-status acceptance is claimed. + +This actual-source package has 85 public theorem/lemma targets across ten modules. Its inventory is distinct from the sibling determinant-moment packet’s 66 targets. Earlier sibling scientific reviews retain their original scope and do not cover this actual-source inventory. diff --git a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/SCOPE.md b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/SCOPE.md new file mode 100644 index 000000000..5c4dec9f0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/SCOPE.md @@ -0,0 +1,31 @@ +# Exact ten-module research scope + +This document describes the ten completed modules named below. Later modules need their own exact statements and execution evidence before being added. The full original research goal remains active. + +`SourceDerivativeGaussian` works with the literal `SmoothSample d L`, `smoothLaw d L` and `smoothSourceField d L`. Its model premise is `L != 0`; dimension, family size, orders, points and direction lists are arbitrary. Differentiated deterministic coefficient summability and the every-sample derivative-series identity are conclusions. Joint Gaussianity is derived from the preserved infinite coefficient law. The covariance conclusion is the infinite differentiated-coefficient pairing; it is not yet an identification with derivatives of a theta-kernel expression. + +`ContactJetCovariance` is a generic coefficient certificate. Row summability, seven specified witness columns, nonzero column amplitudes and nonzero frequency scale imply positive definiteness of the full infinite Gram matrix. The proof uses the sum-of-squares quadratic form and the explicit witness's separating columns. It does not replace the infinite source by those seven modes. + +`SourceContactCovariance` supplies those hypotheses for the actual two-dimensional source. Its origin coordinate order is `(f,fx,fy,fxx,fxy,fxxx,fyy)`. The selected modes are DC, cos(1,0), cos(0,1), cos(1,1), sin(1,0), sin(2,0), sin(0,1), with their actual source amplitudes and canonical half-lattice membership. From `L != 0` it derives positive definiteness of the literal contact covariance; no covariance positivity is assumed. + +`SourceContactSchur` defines `J=(f,fx,fy,fxx,fxy,fxxx)`, `Q=fyy`, their covariance blocks `G,b,d`, and `d-b dotProduct (G inverse mulVec b)`. An exact coordinate equivalence proves the `6+1` block identity. From `L != 0` it derives contact `G` positivity/invertibility, Schur positivity and equality with the variance of the actual curvature residual `Q-(G inverse mulVec b) dotProduct J`. These are contact claims. They do not alone prove current invertibility at all sufficiently small nonzero separations or select a conditional distribution at a prescribed null pin. + +`SourceConfluentPins` fixes endpoints `(-r/2,0),(r/2,0)` and raw order `(fM,fS,fxM,fxS,fyM,fyS)`. Its explicit transform is invertible for `r != 0`; the Gaussian laws are consequences of actual derivative observations. Arbitrary real height `b` is retained in the exact target `(b-r^3/12,0,0,0,0,2)`. The twelve-observation Gaussian statement includes endpoint Hessian order `(fxxM,fxyM,fyyM,fxxS,fxyS,fyyS)` jointly with the raw pins. It does not assume independence or a physical-endpoint identity with a midpoint quartic jet. + +`SourceDerivativeContinuity` proves continuity of every fixed pair of directional-derivative covariances in its evaluation points using global summable bounds. Zero means refer to the unchanged unconditional source law. Conditional torus means are not replaced by zero or a planar reference value. + +`ConfluentIntegralIdentities` is a deterministic scalar result for `F : Real -> Real` of class C3, `H : Real -> Real` of class C2 and signed `r != 0`. It proves the exact integral formulas for the second and transverse divided differences, cubic Peano pin, and left/right adjusted endpoints. The transverse curve `H` denotes the y-derivative curve, so its pin uses `H` itself. Neither an integral representation nor a quotient limit is assumed. Actual-source instantiation and justified infinite-sum limiting arguments remain necessary for stochastic block limits. + +`GaussianRegressionLimits` is a deterministic matrix theorem. Given convergence of finite pin, cross, target and prescribed-value blocks `G,C,B,d` to `G0,C0,B0,d0`, and only `IsUnit G0`, it derives eventual current invertibility, inverse convergence, regression mean/covariance convergence and one interval of simultaneous coordinate bounds. The formulas describe a centered prior with arbitrary prescribed target `d`; conditional means need not vanish. It assumes neither target covariance invertibility nor current pin invertibility. Actual source block convergence is not supplied by this generic theorem. + +`SourceEndpointRegression` uses the literal actual source observations. In LM006 order, its adjusted endpoint vector is `((xxM-X3)/r,(xxS-X3)/r,(xyM-X4)/r,(xyS-X4)/r,yyM,yyS)`. Joint Gaussianity comes from the actual twelve-derivative vector and a proved continuous linear image. For current actual pin covariance `G`, the explicit premise `IsUnit G` supplies matrix inverse normal equations; residual Gaussianity and independence are derived through the reused canonical regression theorem. The resulting Markov measurable kernel has its literal residual pushforward at every pin value, including the prescribed null pin, and satisfies `HasCondDistrib`. The physical endpoint identity is proved deterministically on the `X3=X4=0` fiber. This does not assert eventual current invertibility or a canonical whole-field conditioning theorem. + +`SourceEndpointParameters` derives actual unconditional zero means rather than assuming them, then proves residual and all-pin posterior Gaussianity. Under the explicit current actual pin covariance invertibility premise, covariance bilinearity and the normal equations identify the residual covariance with `B-C G^-1 C^T`. PSD follows from that actual covariance; no residual nondegeneracy is assumed. The literal canonical kernel has mean `C G^-1 x` at every pin value and, after the specified `toLp 2` Euclidean coordinate transport, equals the pinned `multivariateGaussian` of that mean and Schur covariance. Singular covariance, arbitrary prescribed pin and nonzero posterior mean are preserved. The included generic finite-Gaussian parameter lemma also retains dimension zero. Current invertibility and actual small-radius block limits still require their separate source joins. + +Still required within the original goal: actual confluent pin/adjusted-endpoint covariance limits; their join to contact positivity and finite-radius regression; joining the derived actual endpoint Gaussian parameters to their contact limits and common coupling; uniform conditional parameters and the positive physical normalizer; whole-field and geometric Palm/event identification; event-tail and parent conclusions. No theorem adds one of those desired conclusions as a premise and calls that conclusion proved. `SourceConfluentLimits`, `SourceEndpointEstimates` and `SourceContactEndpointParameters` are outside this ten-module qualified inventory and require their separate evidence/scope before inclusion. + +Applicable completed verification on unchanged sources is reusable. Completed module compilations, named kernel replays, transitive axiom audits and meaningful controls are recorded separately from hosted execution and different-family alignment. This packaging does not rerun or reclassify those checks. All present authors/helpers are OpenAI; `gaussian_api`, `endpoint_means` and `peano_strategy` assistance is author-side. Different-family scientific review remains pending. Scientific effect NONE; no status-register changes or parent acceptance follow from this packet. + +The 85 public theorem/lemma names inventoried here belong to this ten-module actual-source package. The sibling determinant-moment packet has a separate 66-target inventory and its reviews retain their own exact scope. Neither execution receipts nor scientific review transfer between those inventories. + +The current input binding covers 23 reused sources: twelve landed smooth-source files plus eleven exact sibling-packet modules at commit `16f81b973a02783f8be83081aa7f198724cf5767`, under execution receipt `95e502472e3d748c8b5dca3328a8644a9cb7cfc8fe40d5f9cce95a32814e5ec6`. 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a/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/lakefile.toml b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/lakefile.toml new file mode 100644 index 000000000..78846db6e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/evidence/catalog-history/historical-ten-module/published-package-metadata/lakefile.toml @@ -0,0 +1,43 @@ +name = "source_derivative_gaussian" +version = "0.1.0" +defaultTargets = ["SourceDerivativeGaussian", "ContactJetCovariance", "SourceContactCovariance", "SourceConfluentPins", "SourceContactSchur", "SourceDerivativeContinuity", "ConfluentIntegralIdentities", "SourceEndpointRegression", "GaussianRegressionLimits", "SourceEndpointParameters"] + +[[require]] +name = "mathlib" +git = "https://github.com/leanprover-community/mathlib4.git" +rev = "d13f23b723b8a846827a245b89c10fc7d3f11612" + +[[lean_lib]] +name = "SourceDerivativeInputs" +srcDir = "inputs" +roots = ["GaussianSeriesFoundation", "SpectralWeights", "SourceFourierSeries", "SourceC4", "SourceGaussianLaw", "SourceLattice", "SourceKernel", "SourceCovariance", "SourceTorus", "SourceTorusC4", "SourceSmooth", "SourceSmoothCarrier", "ResearchFormalCoreR1.DeterminantMoment", "ResearchFormalCoreR1.GaussianScalarMoments", "ResearchFormalCoreR1.GaussianJetMoments", "ResearchFormalCoreR1.GaussianNormalizer", "ResearchFormalCoreR1.EndpointGaussianMoments", "ResearchFormalCoreR1.NormalizerFloor", "ResearchFormalCoreR1.EndpointNormalizerBridge", "ResearchFormalCoreR1.GaussianCoupling", "ResearchFormalCoreR1.GaussianNormalizerLimit", "ResearchFormalCoreR1.PalmReweighting", "ResearchFormalCoreR1.GaussianRegressionKernel"] + +[[lean_lib]] +name = "SourceDerivativeGaussian" + +[[lean_lib]] +name = "ContactJetCovariance" + +[[lean_lib]] +name = "SourceContactCovariance" + +[[lean_lib]] +name = "SourceConfluentPins" + +[[lean_lib]] +name = "SourceContactSchur" + +[[lean_lib]] +name = "SourceDerivativeContinuity" + +[[lean_lib]] +name = "ConfluentIntegralIdentities" + +[[lean_lib]] +name = "SourceEndpointRegression" + +[[lean_lib]] +name = "GaussianRegressionLimits" + +[[lean_lib]] +name = "SourceEndpointParameters" diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/GaussianSeriesFoundation.lean b/frontiers/source_derivative_gaussian_20261025/inputs/GaussianSeriesFoundation.lean new file mode 100644 index 000000000..93651da7f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/GaussianSeriesFoundation.lean @@ -0,0 +1,96 @@ +import Mathlib.Probability.Distributions.Gaussian.Real +import Mathlib.Probability.Independence.InfinitePi +import Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum +import Mathlib.MeasureTheory.Function.LpSeminorm.SMul + +/-! The actual product law and the almost-sure absolute-series foundation. +This does not assert a Fourier covariance, sample smoothness or finite-jet rank. -/ + +noncomputable section + +open MeasureTheory ProbabilityTheory +open scoped ENNReal + +namespace I4Foundation + +/-- The same actual probability measure used for every coefficient and series assertion. -/ +def coefficientLaw (ι : Type*) : Measure (ι → ℝ) := + Measure.infinitePi (fun _ : ι => gaussianReal 0 1) + +instance coefficientLaw_isProbabilityMeasure (ι : Type*) : + IsProbabilityMeasure (coefficientLaw ι) := by + unfold coefficientLaw + infer_instance + +theorem coordinate_measurePreserving {ι : Type*} (i : ι) : + MeasurePreserving (fun ω : ι → ℝ => ω i) (coefficientLaw ι) + (gaussianReal 0 1) := by + exact measurePreserving_eval_infinitePi (fun _ : ι => gaussianReal 0 1) i + +theorem coordinate_marginal {ι : Type*} (i : ι) : + (coefficientLaw ι).map (fun ω => ω i) = gaussianReal 0 1 := by + exact (coordinate_measurePreserving i).map_eq + +theorem coordinates_independent (ι : Type*) : + iIndepFun (fun i (ω : ι → ℝ) => ω i) (coefficientLaw ι) := by + exact iIndepFun_infinitePi (P := fun _ : ι => gaussianReal 0 1) + (X := fun _ : ι => id) (fun _ => measurable_id) + +theorem coordinate_memLp {ι : Type*} (i : ι) (p : ℝ≥0∞) (hp : p ≠ ∞) : + MemLp (fun ω : ι → ℝ => ω i) p (coefficientLaw ι) := by + exact (memLp_id_gaussianReal' p hp).comp_measurePreserving + (coordinate_measurePreserving i) + +theorem coordinate_integrable {ι : Type*} (i : ι) : + Integrable (fun ω : ι → ℝ => ω i) (coefficientLaw ι) := by + exact memLp_one_iff_integrable.mp (coordinate_memLp i 1 (by simp)) + +theorem coordinate_abs_integrable {ι : Type*} (i : ι) : + Integrable (fun ω : ι → ℝ => |ω i|) (coefficientLaw ι) := by + simpa only [Real.norm_eq_abs] using (coordinate_integrable i).norm + +theorem weighted_abs_summable_ae {ι : Type*} [Countable ι] + (a : ι → ℝ) (ha : ∀ i, 0 ≤ a i) (hs : Summable a) : + ∀ᵐ ω ∂coefficientLaw ι, Summable (fun i => a i * |ω i|) := by + let C := eLpNorm id 1 (gaussianReal 0 1) + have hC : C < ∞ := (memLp_id_gaussianReal' 1 (by simp)).eLpNorm_lt_top + have hcoord (i : ι) : + eLpNorm (fun ω : ι → ℝ => ω i) 1 (coefficientLaw ι) = C := by + exact eLpNorm_comp_measurePreserving + (memLp_id_gaussianReal' 1 (by simp)).aestronglyMeasurable + (coordinate_measurePreserving i) + have hnorm (i : ι) : + eLpNorm (fun ω : ι → ℝ => a i * ω i) 1 (coefficientLaw ι) = + ENNReal.ofReal (a i) * C := by + change eLpNorm (a i • (fun ω : ι → ℝ => ω i)) 1 (coefficientLaw ι) = _ + rw [eLpNorm_const_smul, Real.enorm_eq_ofReal (ha i), hcoord] + have hsum : (∑' i, eLpNorm (fun ω : ι → ℝ => a i * ω i) 1 + (coefficientLaw ι)) ≠ ∞ := by + simp_rw [hnorm] + rw [ENNReal.tsum_mul_right] + exact ENNReal.mul_ne_top hs.tsum_ofReal_ne_top hC.ne + have h := summable_norm_of_tsum_eLpNorm_ne_top (p := 1) (le_refl 1) hsum + filter_upwards [h] with ω hω + simpa only [Real.norm_eq_abs, abs_mul, abs_of_nonneg (ha _)] using hω + +/-- The actual scalar sum; its convergence is proved separately on the product law. -/ +def scalarSeries {ι : Type*} (a : ι → ℝ) (ω : ι → ℝ) : ℝ := + ∑' i, a i * ω i + +theorem scalarSeries_measurable {ι : Type*} [Countable ι] (a : ι → ℝ) : + Measurable (scalarSeries a) := by + unfold scalarSeries + apply Measurable.tsum + intro i + exact measurable_const.mul (measurable_pi_apply i) + +theorem scalarSeries_hasSum_ae {ι : Type*} [Countable ι] + (a : ι → ℝ) (hs : Summable a) : + ∀ᵐ ω ∂coefficientLaw ι, HasSum (fun i => a i * ω i) (scalarSeries a ω) := by + filter_upwards [weighted_abs_summable_ae (fun i => |a i|) + (fun i => abs_nonneg _) hs.abs] with ω hω + have hnorm : Summable (fun i => ‖a i * ω i‖) := by + simpa only [Real.norm_eq_abs, abs_mul] using hω + exact hnorm.of_norm.hasSum + +end I4Foundation diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/DeterminantMoment.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/DeterminantMoment.lean new file mode 100644 index 000000000..93e296ca1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/DeterminantMoment.lean @@ -0,0 +1,218 @@ +import Mathlib.MeasureTheory.Integral.Bochner.Basic +import Mathlib.LinearAlgebra.Matrix.Determinant.Basic +import Mathlib.Topology.Instances.Matrix +import Mathlib.MeasureTheory.Order.Group.Lattice +import Mathlib.Tactic.FieldSimp +import Mathlib.Tactic.Ring +import Mathlib.Tactic.NormNum +import Mathlib.Tactic.Positivity +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.FinCases +import Mathlib.Tactic.FunProp +import Mathlib.Tactic.Push + +namespace ResearchFormalCoreR1 + +set_option autoImplicit false + +open MeasureTheory +open scoped BigOperators + +abbrev DM2Jet := Fin 9 → ℝ + +def dm2Radius (j : DM2Jet) : ℝ := 1 + ∑ i, |j i| + +noncomputable def dm2HM (r : ℝ) (j : DM2Jet) : Matrix (Fin 2) (Fin 2) ℝ := + !![-r + r^2*j 4/12, -j 1*r/2 + r^2*j 5/12; + -j 1*r/2 + r^2*j 5/12, j 0-r*j 2/2+r^2*j 6/8] + +noncomputable def dm2HS (r : ℝ) (j : DM2Jet) : Matrix (Fin 2) (Fin 2) ℝ := + !![r + r^2*j 4/12, j 1*r/2 + r^2*j 5/12; + j 1*r/2 + r^2*j 5/12, j 0+r*j 2/2+r^2*j 6/8] + +/-- The source's negative-definite condition, stated as the quadratic form. -/ +def dm2Negative (H : Matrix (Fin 2) (Fin 2) ℝ) : Prop := + ∀ v : Fin 2 → ℝ, v ≠ 0 → ∑ i, ∑ k, v i * H i k * v k < 0 + +noncomputable def dm2Weight (r : ℝ) (j : DM2Jet) : ℝ := by + classical + exact if dm2Negative (dm2HM r j) ∧ (dm2HS r j).det < 0 then + |(dm2HM r j).det * (dm2HS r j).det| else 0 + +private lemma dm2_scale (r x : ℝ) (hr : 0 ≤ r) (hr1 : r ≤ 1) : + |r*x| ≤ |x| := by + rw [abs_mul, abs_of_nonneg hr] + nlinarith [abs_nonneg x] + +private lemma dm2_factors (r A B C R : ℝ) (hr : 0 ≤ r) (hr1 : r ≤ 1) + (hR : 0 ≤ R) (hA : |A| ≤ R) (hB : |B| ≤ R) (hC : |C| ≤ R) : + |A*C-r*B^2| ≤ 2*R^2 := by + have hAC : |A*C| ≤ R^2 := by + rw [abs_mul, pow_two] + exact mul_le_mul hA hC (abs_nonneg _) hR + have hBB : B^2 ≤ R^2 := by nlinarith [sq_abs B, abs_nonneg B] + have hrBB : |r*B^2| ≤ R^2 := by + rw [abs_of_nonneg (mul_nonneg hr (sq_nonneg B))] + nlinarith [sq_nonneg B] + exact (abs_sub _ _).trans (by linarith) + +/-- Exact untyped determinant domination; no jet-coordinate or type assumptions. -/ +theorem dm2_untyped (r : ℝ) (j : DM2Jet) (hr : 0 < r) (hr1 : r ≤ 1) : + ((dm2HM r j).det * (dm2HS r j).det)^2 ≤ 16*r^4*dm2Radius j^8 := by + let R := dm2Radius j + have hR : 0 ≤ R := by dsimp [R, dm2Radius]; positivity + have hsum : (∑ i, |j i|) = |j 0|+|j 1|+|j 2|+|j 3|+|j 4|+|j 5|+|j 6|+|j 7|+|j 8| := by + simp [Fin.sum_univ_succ]; ring + have hbudget : 1+|j 0|+|j 1|+|j 2|+|j 4|+|j 5|+|j 6| ≤ R := by + dsimp [R, dm2Radius]; rw [hsum] + linarith [abs_nonneg (j 3), abs_nonneg (j 7), abs_nonneg (j 8)] + have hscale (i : Fin 9) : r*|j i| ≤ |j i| := by nlinarith [abs_nonneg (j i)] + have hscale2 (i : Fin 9) : r^2*|j i| ≤ |j i| := by + exact mul_le_of_le_one_left (abs_nonneg _) (by nlinarith) + have haM : |-1+r*j 4/12| ≤ R := by + calc + _ ≤ 1+|j 4| := by + have := abs_add_le (-1 : ℝ) (r*j 4/12) + rw [abs_div] at this + norm_num [abs_of_pos hr] at this + linarith [hscale 4, abs_nonneg (j 4)] + _ ≤ R := by linarith [abs_nonneg (j 0), abs_nonneg (j 1), abs_nonneg (j 2), abs_nonneg (j 5), abs_nonneg (j 6)] + have haS : |1+r*j 4/12| ≤ R := by + have := abs_add_le (1 : ℝ) (r*j 4/12) + rw [abs_div] at this + norm_num [abs_of_pos hr] at this + linarith [hscale 4, abs_nonneg (j 0), abs_nonneg (j 1), abs_nonneg (j 2), abs_nonneg (j 4), abs_nonneg (j 5), abs_nonneg (j 6)] + have hbM : |-j 1/2+r*j 5/12| ≤ R := by + have := abs_add_le (-j 1/2) (r*j 5/12) + simp only [abs_div, abs_neg] at this + norm_num [abs_of_pos hr] at this + linarith [hscale 5, abs_nonneg (j 0), abs_nonneg (j 1), abs_nonneg (j 2), abs_nonneg (j 4), abs_nonneg (j 5), abs_nonneg (j 6)] + have hbS : |j 1/2+r*j 5/12| ≤ R := by + have := abs_add_le (j 1/2) (r*j 5/12) + simp only [abs_div] at this + norm_num [abs_of_pos hr] at this + linarith [hscale 5, abs_nonneg (j 0), abs_nonneg (j 1), abs_nonneg (j 2), abs_nonneg (j 4), abs_nonneg (j 5), abs_nonneg (j 6)] + have hcM : |j 0-r*j 2/2+r^2*j 6/8| ≤ R := by + have h1 := abs_sub (j 0) (r*j 2/2) + have h2 := abs_add_le (j 0-r*j 2/2) (r^2*j 6/8) + simp only [abs_div] at h1 h2 + norm_num [abs_of_pos hr] at h1 h2 + linarith [hscale 2, hscale2 6, abs_nonneg (j 1), abs_nonneg (j 2), abs_nonneg (j 4), abs_nonneg (j 5), abs_nonneg (j 6)] + have hcS : |j 0+r*j 2/2+r^2*j 6/8| ≤ R := by + have h1 := abs_add_le (j 0) (r*j 2/2) + have h2 := abs_add_le (j 0+r*j 2/2) (r^2*j 6/8) + simp only [abs_div] at h1 h2 + norm_num [abs_of_pos hr] at h1 h2 + linarith [hscale 2, hscale2 6, abs_nonneg (j 1), abs_nonneg (j 2), abs_nonneg (j 4), abs_nonneg (j 5), abs_nonneg (j 6)] + have hM := dm2_factors r _ _ _ R hr.le hr1 hR haM hbM hcM + have hS := dm2_factors r _ _ _ R hr.le hr1 hR haS hbS hcS + have hprod : |((-1+r*j 4/12)*(j 0-r*j 2/2+r^2*j 6/8)-r*(-j 1/2+r*j 5/12)^2) * + ((1+r*j 4/12)*(j 0+r*j 2/2+r^2*j 6/8)-r*(j 1/2+r*j 5/12)^2)| ≤ (2*R^2)^2 := by + rw [abs_mul] + calc + _ ≤ (2*R^2)*(2*R^2) := mul_le_mul hM hS (abs_nonneg _) (by positivity) + _ = (2*R^2)^2 := (pow_two _).symm + have hsquare : (((-1+r*j 4/12)*(j 0-r*j 2/2+r^2*j 6/8)-r*(-j 1/2+r*j 5/12)^2) * + ((1+r*j 4/12)*(j 0+r*j 2/2+r^2*j 6/8)-r*(j 1/2+r*j 5/12)^2))^2 ≤ ((2*R^2)^2)^2 := by + simpa only [sq_abs] using + (sq_le_sq₀ (abs_nonneg _) (sq_nonneg (2*R^2))).2 hprod + have hid : ((dm2HM r j).det*(dm2HS r j).det)^2 = r^4 * + (((-1+r*j 4/12)*(j 0-r*j 2/2+r^2*j 6/8)-r*(-j 1/2+r*j 5/12)^2) * + ((1+r*j 4/12)*(j 0+r*j 2/2+r^2*j 6/8)-r*(j 1/2+r*j 5/12)^2))^2 := by + simp [dm2HM, dm2HS, Matrix.det_fin_two]; ring + rw [hid] + calc + _ ≤ r^4*((2*R^2)^2)^2 := mul_le_mul_of_nonneg_left hsquare (by positivity) + _ = 16*r^4*dm2Radius j^8 := by dsimp [R]; ring + +/-- P02-LM-002 section 2, with the literal quadratic-form type indicator. -/ +theorem dm2_pointwise (r : ℝ) (j : DM2Jet) (hr : 0 < r) (hr1 : r ≤ 1) : + dm2Weight r j ^ 2 ≤ 16*r^4*dm2Radius j^8 := by + classical + unfold dm2Weight + split_ifs + · simpa only [sq_abs] using dm2_untyped r j hr hr1 + · simpa only [zero_pow (by decide : (2 : ℕ) ≠ 0)] using + (show 0 ≤ 16*r^4*dm2Radius j^8 by positivity) + +/-- Two-dimensional Sylvester criterion for the literal quadratic-form definition. -/ +theorem dm2_negative_iff (a b c : ℝ) : + dm2Negative !![a,b;b,c] ↔ a < 0 ∧ 0 < a*c-b^2 := by + constructor + · intro h + have ha := h ![1,0] (by simp) + simp [Fin.sum_univ_two] at ha + have hv : (![(-b),a] : Fin 2 → ℝ) ≠ 0 := by + intro heq + have := congrFun heq 1 + simp at this + linarith + have hd := h ![-b,a] hv + simp [Fin.sum_univ_two] at hd + refine ⟨ha, ?_⟩ + by_contra hn + have hh : 0 ≤ a*(a*c-b^2) := mul_nonneg_of_nonpos_of_nonpos ha.le (le_of_not_gt hn) + nlinarith + · rintro ⟨ha,hd⟩ v hv + have hvxy : v 0 ≠ 0 ∨ v 1 ≠ 0 := by + by_contra hn + push Not at hn + apply hv + ext i + fin_cases i <;> simp_all + have hq : 0 < (a*v 0+b*v 1)^2+(a*c-b^2)*(v 1)^2 := by + rcases eq_or_ne (v 1) 0 with hy|hy + · have hx : v 0 ≠ 0 := hvxy.resolve_right (not_ne_iff.mpr hy) + simpa [hy] using sq_pos_of_ne_zero (mul_ne_zero ha.ne hx) + · exact add_pos_of_nonneg_of_pos (sq_nonneg _) (mul_pos hd (sq_pos_of_ne_zero hy)) + simp [Fin.sum_univ_two] + nlinarith + +/-- Measurability of the exact typed weight, with no assumed type-null boundary. -/ +theorem dm2_weight_measurable (r : ℝ) : Measurable (dm2Weight r) := by + classical + have hm : Continuous (fun j : DM2Jet => (dm2HM r j).det) := by + simp only [dm2HM, Matrix.det_fin_two] + fun_prop + have hs : Continuous (fun j : DM2Jet => (dm2HS r j).det) := by + simp only [dm2HS, Matrix.det_fin_two] + fun_prop + have htype : MeasurableSet {j : DM2Jet | dm2Negative (dm2HM r j) ∧ (dm2HS r j).det < 0} := by + simp only [dm2HM, dm2_negative_iff] + apply MeasurableSet.inter + · apply MeasurableSet.inter + · have h : Continuous (fun j : DM2Jet => -r+r^2*j 4/12) := by fun_prop + exact (isOpen_lt h continuous_const).measurableSet + · have h : Continuous (fun j : DM2Jet => (-r+r^2*j 4/12)*(j 0-r*j 2/2+r^2*j 6/8)-(-j 1*r/2+r^2*j 5/12)^2) := by fun_prop + exact (isOpen_lt continuous_const h).measurableSet + · exact (isOpen_lt hs continuous_const).measurableSet + change Measurable (fun j : DM2Jet => if dm2Negative (dm2HM r j) ∧ (dm2HS r j).det < 0 then |(dm2HM r j).det*(dm2HS r j).det| else 0) + exact Measurable.ite htype (hm.measurable.mul hs.measurable).abs measurable_const + +/-- P02-LM-002's expectation step: eighth-moment control supplies integrability +and the second moment, rather than presupposing that second moment. -/ +theorem dm2_moment {Ω : Type*} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) + (J : Ω → DM2Jet) (hJ : Measurable J) (r M8 : ℝ) (hr : 0 < r) (hr1 : r ≤ 1) + (h8 : MeasureTheory.Integrable (fun x => dm2Radius (J x)^8) μ) + (hM8 : (∫ x, dm2Radius (J x)^8 ∂μ) ≤ M8) : + MeasureTheory.Integrable (fun x => dm2Weight r (J x)^2) μ ∧ + (∫ x, dm2Weight r (J x)^2 ∂μ) ≤ 16*M8*r^4 := by + have hmajor : MeasureTheory.Integrable (fun x => 16*r^4*dm2Radius (J x)^8) μ := + h8.const_mul _ + have hmeas : MeasureTheory.AEStronglyMeasurable (fun x => dm2Weight r (J x)^2) μ := + (((dm2_weight_measurable r).comp hJ).pow_const 2).aestronglyMeasurable + have hi : MeasureTheory.Integrable (fun x => dm2Weight r (J x)^2) μ := + hmajor.mono' hmeas (Filter.Eventually.of_forall (fun x => by + rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] + exact dm2_pointwise r (J x) hr hr1)) + refine ⟨hi, ?_⟩ + calc + _ ≤ ∫ x, 16*r^4*dm2Radius (J x)^8 ∂μ := + MeasureTheory.integral_mono hi hmajor (fun x => dm2_pointwise r (J x) hr hr1) + _ = 16*r^4*(∫ x, dm2Radius (J x)^8 ∂μ) := MeasureTheory.integral_const_mul _ _ + _ ≤ 16*r^4*M8 := mul_le_mul_of_nonneg_left hM8 (by positivity) + _ = 16*M8*r^4 := by ring + +end ResearchFormalCoreR1 + + diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/EndpointGaussianMoments.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/EndpointGaussianMoments.lean new file mode 100644 index 000000000..7683112b1 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/EndpointGaussianMoments.lean @@ -0,0 +1,139 @@ +import ResearchFormalCoreR1.GaussianJetMoments + +/-! Gaussian moments for the six actual normalized endpoint entries from +GP-DER-047 section 13. Coordinates are (fxx(M)/r,fxy(M)/r,fyy(M), +fxx(S)/r,fxy(S)/r,fyy(S)); no polynomial midpoint-jet identity is assumed. -/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory +open scoped BigOperators NNReal + +set_option autoImplicit false + +def egm_HM (r : ℝ) (j : Fin 6 → ℝ) : Matrix (Fin 2) (Fin 2) ℝ := + !![r * j 0, r * j 1; r * j 1, j 2] + +def egm_HS (r : ℝ) (j : Fin 6 → ℝ) : Matrix (Fin 2) (Fin 2) ℝ := + !![r * j 3, r * j 4; r * j 4, j 5] + +noncomputable def egm_weight (r : ℝ) (j : Fin 6 → ℝ) : ℝ := by + classical + exact if dm2Negative (egm_HM r j) ∧ (egm_HS r j).det < 0 then + |(egm_HM r j).det * (egm_HS r j).det| else 0 + +/-- The maximum condition is the original strict quadratic-form condition; +no null-boundary assumption is needed for measurable typing. -/ +theorem egm_weight_measurable (r : ℝ) : Measurable (egm_weight r) := by + classical + have hm : Continuous (fun j : Fin 6 → ℝ => (egm_HM r j).det) := by + simp only [egm_HM, Matrix.det_fin_two] + fun_prop + have hs : Continuous (fun j : Fin 6 → ℝ => (egm_HS r j).det) := by + simp only [egm_HS, Matrix.det_fin_two] + fun_prop + have htype : MeasurableSet {j : Fin 6 → ℝ | + dm2Negative (egm_HM r j) ∧ (egm_HS r j).det < 0} := by + simp only [egm_HM, dm2_negative_iff] + apply MeasurableSet.inter + · apply MeasurableSet.inter + · have h : Continuous (fun j : Fin 6 → ℝ => r * j 0) := by fun_prop + exact (isOpen_lt h continuous_const).measurableSet + · have h : Continuous (fun j : Fin 6 → ℝ => + (r * j 0) * j 2 - (r * j 1) ^ 2) := by fun_prop + exact (isOpen_lt continuous_const h).measurableSet + · exact (isOpen_lt hs continuous_const).measurableSet + change Measurable (fun j => if dm2Negative (egm_HM r j) ∧ (egm_HS r j).det < 0 + then |(egm_HM r j).det * (egm_HS r j).det| else 0) + exact Measurable.ite htype (hm.measurable.mul hs.measurable).abs measurable_const + +private theorem egm_coordinate_bound (j : Fin 6 → ℝ) (i : Fin 6) : + |j i| ≤ gm8Radius j := by + have hsum : |j i| ≤ ∑ k, |j k| := + Finset.single_le_sum (fun k _ => abs_nonneg (j k)) (Finset.mem_univ i) + unfold gm8Radius + linarith + +private theorem egm_factor_bound (r a b c R : ℝ) (hr : 0 ≤ r) (hr1 : r ≤ 1) + (hR : 0 ≤ R) (ha : |a| ≤ R) (hb : |b| ≤ R) (hc : |c| ≤ R) : + |a * c - r * b ^ 2| ≤ 2 * R ^ 2 := by + have hac : |a * c| ≤ R ^ 2 := by + rw [abs_mul, pow_two] + exact mul_le_mul ha hc (abs_nonneg c) hR + have hbb : b ^ 2 ≤ R ^ 2 := by nlinarith [sq_abs b, abs_nonneg b] + have hrbb : |r * b ^ 2| ≤ R ^ 2 := by + rw [abs_of_nonneg (mul_nonneg hr (sq_nonneg b))] + exact (mul_le_of_le_one_left (sq_nonneg b) hr1).trans hbb + exact (abs_sub _ _).trans (by linarith) + +private theorem egm_untyped (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) (hr1 : r ≤ 1) : + ((egm_HM r j).det * (egm_HS r j).det) ^ 2 ≤ + 16 * r ^ 4 * gm8Radius j ^ 8 := by + let R := gm8Radius j + let dM := j 0 * j 2 - r * j 1 ^ 2 + let dS := j 3 * j 5 - r * j 4 ^ 2 + have hR : 0 ≤ R := by dsimp [R, gm8Radius]; positivity + have hM : |dM| ≤ 2 * R ^ 2 := + egm_factor_bound r _ _ _ R hr.le hr1 hR + (egm_coordinate_bound j 0) (egm_coordinate_bound j 1) (egm_coordinate_bound j 2) + have hS : |dS| ≤ 2 * R ^ 2 := + egm_factor_bound r _ _ _ R hr.le hr1 hR + (egm_coordinate_bound j 3) (egm_coordinate_bound j 4) (egm_coordinate_bound j 5) + have hprod : |dM * dS| ≤ (2 * R ^ 2) ^ 2 := by + rw [abs_mul, pow_two] + exact mul_le_mul hM hS (abs_nonneg _) (by positivity) + have hsq : (dM * dS) ^ 2 ≤ ((2 * R ^ 2) ^ 2) ^ 2 := by + simpa only [sq_abs] using + (sq_le_sq₀ (abs_nonneg (dM * dS)) (sq_nonneg (2 * R ^ 2))).2 hprod + have hdet : ((egm_HM r j).det * (egm_HS r j).det) ^ 2 = + r ^ 4 * (dM * dS) ^ 2 := by + simp [egm_HM, egm_HS, Matrix.det_fin_two, dM, dS] + ring + calc + _ = r ^ 4 * (dM * dS) ^ 2 := hdet + _ ≤ r ^ 4 * ((2 * R ^ 2) ^ 2) ^ 2 := + mul_le_mul_of_nonneg_left hsq (by positivity) + _ = 16 * r ^ 4 * gm8Radius j ^ 8 := by dsimp [R]; ring + +/-- Exact endpoint determinant domination in the actual normalized six-vector. +The type indicators can only reduce the nonnegative determinant product. -/ +theorem egm_pointwise (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) (hr1 : r ≤ 1) : + egm_weight r j ^ 2 ≤ 16 * r ^ 4 * gm8Radius j ^ 8 := by + classical + unfold egm_weight + split_ifs + · simpa only [sq_abs] using egm_untyped r j hr hr1 + · simpa only [zero_pow (by decide : (2 : ℕ) ≠ 0)] using + (show 0 ≤ 16 * r ^ 4 * gm8Radius j ^ 8 by positivity) + +/-- Genuine Gaussian marginal laws of the actual normalized endpoint vector +supply both integrability and its determinant second moment. The derivative-map, +conditioning law and uniform parameter bounds remain explicit model inputs. -/ +theorem egm_gaussian_moment {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsProbabilityMeasure μ] (J : Ω → Fin 6 → ℝ) + (hJ : Measurable J) (m : Fin 6 → ℝ) (v : Fin 6 → ℝ≥0) + (B V r : ℝ) (hB : 0 ≤ B) (hV : 0 ≤ V) (hr : 0 < r) (hr1 : r ≤ 1) + (hmean : ∀ i, |m i| ≤ B) (hvar : ∀ i, (v i : ℝ) ≤ V) + (hlaw : ∀ i, HasLaw (fun x => J x i) (gaussianReal (m i) (v i)) μ) : + Integrable (fun x => egm_weight r (J x) ^ 2) μ ∧ + (∫ x, egm_weight r (J x) ^ 2 ∂μ) ≤ 16 * gm8FiniteM8 6 B V * r ^ 4 := by + obtain ⟨h8, hbound⟩ := gm8_finite_integrable_bound 6 μ J hJ m v B V hB hV + hmean hvar hlaw + have hmajor : Integrable (fun x => 16 * r ^ 4 * gm8Radius (J x) ^ 8) μ := + h8.const_mul _ + have hmeas : AEStronglyMeasurable (fun x => egm_weight r (J x) ^ 2) μ := + (((egm_weight_measurable r).comp hJ).pow_const 2).aestronglyMeasurable + have hi : Integrable (fun x => egm_weight r (J x) ^ 2) μ := + hmajor.mono' hmeas (Filter.Eventually.of_forall (fun x => by + rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] + exact egm_pointwise r (J x) hr hr1)) + refine ⟨hi, ?_⟩ + calc + _ ≤ ∫ x, 16 * r ^ 4 * gm8Radius (J x) ^ 8 ∂μ := + integral_mono hi hmajor (fun x => egm_pointwise r (J x) hr hr1) + _ = 16 * r ^ 4 * (∫ x, gm8Radius (J x) ^ 8 ∂μ) := integral_const_mul _ _ + _ ≤ 16 * r ^ 4 * gm8FiniteM8 6 B V := + mul_le_mul_of_nonneg_left hbound (by positivity) + _ = 16 * gm8FiniteM8 6 B V * r ^ 4 := by ring + +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/EndpointNormalizerBridge.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/EndpointNormalizerBridge.lean new file mode 100644 index 000000000..73d861b7e --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/EndpointNormalizerBridge.lean @@ -0,0 +1,191 @@ +import ResearchFormalCoreR1.EndpointGaussianMoments +import ResearchFormalCoreR1.NormalizerFloor + +/-! +# Literal endpoint weight and continuous normalized weight + +The actual endpoint vector has order `(aM,bM,cM,aS,bS,cS)`, while the +continuous normalizer uses `(aM,aS,bM,bS,cM,cS)`. The permutation below +and the identity hold pointwise, including all determinant/type boundaries. +See ../SOURCE_ENDPOINT_NORMALIZER_BRIDGE.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory Filter +open scoped Topology NNReal + +set_option autoImplicit false + +/-- Convert the actual six endpoint entries to the continuous weight's order. -/ +def enb_permutation (j : Fin 6 → ℝ) : Fin 6 → ℝ := + ![j 0, j 3, j 1, j 4, j 2, j 5] + +theorem enb_permutation_continuous : Continuous enb_permutation := by + unfold enb_permutation + fun_prop + +private theorem enb_maximum_factor (r a b c : ℝ) (hr : 0 < r) : + max (max (-a) 0 * max (-c) 0 - r * b ^ 2) 0 = + if a < 0 ∧ 0 < a * c - r * b ^ 2 then a * c - r * b ^ 2 else 0 := by + by_cases h : a < 0 ∧ 0 < a * c - r * b ^ 2 + · have hc : c < 0 := by + by_contra hn + have hac : a * c ≤ 0 := mul_nonpos_of_nonpos_of_nonneg h.1.le (le_of_not_gt hn) + have hrb : 0 ≤ r * b ^ 2 := mul_nonneg hr.le (sq_nonneg b) + linarith [h.2] + rw [ite_eq_left h, max_eq_left (neg_nonneg.mpr h.1.le), + max_eq_left (neg_nonneg.mpr hc.le)] + have heq : -a * -c - r * b ^ 2 = a * c - r * b ^ 2 := by ring + rw [heq, max_eq_left h.2.le] + · rw [ite_eq_right h] + by_cases ha : a < 0 + · have hd : a * c - r * b ^ 2 ≤ 0 := + le_of_not_gt (fun hp => h ⟨ha, hp⟩) + by_cases hc : c ≤ 0 + · rw [max_eq_left (neg_nonneg.mpr ha.le), max_eq_left (neg_nonneg.mpr hc)] + have heq : -a * -c - r * b ^ 2 = a * c - r * b ^ 2 := by ring + rw [heq, max_eq_right hd] + · rw [max_eq_right (by linarith : -c ≤ 0)] + simpa using (max_eq_right (neg_nonpos.mpr (mul_nonneg hr.le (sq_nonneg b)))) + · rw [max_eq_right (by linarith : -a ≤ 0)] + simpa using (max_eq_right (neg_nonpos.mpr (mul_nonneg hr.le (sq_nonneg b)))) + +/-- Strict negative definiteness in actual endpoint coordinates, with no +assumed sign of the other diagonal entry or omission of singular boundaries. -/ +theorem enb_maximum_iff (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) : + dm2Negative (egm_HM r j) ↔ j 0 < 0 ∧ 0 < j 0 * j 2 - r * j 1 ^ 2 := by + rw [egm_HM, dm2_negative_iff] + have heq : r * j 0 * j 2 - (r * j 1) ^ 2 = + r * (j 0 * j 2 - r * j 1 ^ 2) := by ring + have hmul (x : ℝ) : r * x < 0 ↔ x < 0 := by + constructor <;> intro hx <;> nlinarith + rw [heq, hmul, mul_pos_iff_of_pos_left hr] + +/-- The literal strict saddle condition is the sign of its factored determinant. -/ +theorem enb_saddle_iff (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) : + (egm_HS r j).det < 0 ↔ j 3 * j 5 - r * j 4 ^ 2 < 0 := by + have heq : (egm_HS r j).det = r * (j 3 * j 5 - r * j 4 ^ 2) := by + simp [egm_HS, Matrix.det_fin_two] + ring + rw [heq] + constructor <;> intro h <;> nlinarith + +/-- Exact bridge from the literal typed physical endpoint weight to the +continuous expression. All excluded types and all zero boundaries give zero. -/ +theorem enb_weight_identity (r : ℝ) (j : Fin 6 → ℝ) (hr : 0 < r) : + egm_weight r j = r ^ 2 * gn_normalizedWeight r (enb_permutation j) := by + classical + have hM := enb_maximum_iff r j hr + have hS := enb_saddle_iff r j hr + have hdetM : (egm_HM r j).det = r * (j 0 * j 2 - r * j 1 ^ 2) := by + simp [egm_HM, Matrix.det_fin_two] + ring + have hdetS : (egm_HS r j).det = r * (j 3 * j 5 - r * j 4 ^ 2) := by + simp [egm_HS, Matrix.det_fin_two] + ring + unfold egm_weight gn_normalizedWeight + change (if dm2Negative (egm_HM r j) ∧ (egm_HS r j).det < 0 then + |(egm_HM r j).det * (egm_HS r j).det| else 0) = + r ^ 2 * (max (max (-j 0) 0 * max (-j 2) 0 - r * j 1 ^ 2) 0 * + max (r * j 4 ^ 2 - j 3 * j 5) 0) + rw [enb_maximum_factor r (j 0) (j 1) (j 2) hr] + by_cases hmax : j 0 < 0 ∧ 0 < j 0 * j 2 - r * j 1 ^ 2 + · rw [ite_eq_left hmax] + by_cases hsad : j 3 * j 5 - r * j 4 ^ 2 < 0 + · rw [ite_eq_left ⟨hM.mpr hmax, hS.mpr hsad⟩, hdetM, hdetS] + rw [max_eq_left (by linarith : 0 ≤ r * j 4 ^ 2 - j 3 * j 5)] + have hp : r * (j 0 * j 2 - r * j 1 ^ 2) * + (r * (j 3 * j 5 - r * j 4 ^ 2)) ≤ 0 := + mul_nonpos_of_nonneg_of_nonpos (mul_nonneg hr.le hmax.2.le) + (mul_nonpos_of_nonneg_of_nonpos hr.le hsad.le) + rw [abs_of_nonpos hp] + ring + · rw [ite_eq_right (fun h => hsad (hS.mp h.2)), + max_eq_right (by linarith : r * j 4 ^ 2 - j 3 * j 5 ≤ 0)] + ring + · rw [ite_eq_right hmax, ite_eq_right (fun h => hmax (hM.mp h.1))] + ring + +/-- The physical and continuous normalizers are the same integral up to the +exact radius factor, for any common random six-vector and any measure. -/ +theorem enb_integral_identity {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (J : Ω → Fin 6 → ℝ) (r : ℝ) (hr : 0 < r) : + (∫ ω, egm_weight r (J ω) ∂P) = + r ^ 2 * ∫ ω, gn_normalizedWeight r (enb_permutation (J ω)) ∂P := by + calc + _ = ∫ ω, r ^ 2 * gn_normalizedWeight r (enb_permutation (J ω)) ∂P := + integral_congr_ae (Eventually.of_forall (fun ω => enb_weight_identity r (J ω) hr)) + _ = _ := integral_const_mul _ _ + +/-- Positive radius makes the exact scaling nonzero, so either weight is +integrable precisely when the other is. -/ +theorem enb_integrable_iff {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (J : Ω → Fin 6 → ℝ) (r : ℝ) (hr : 0 < r) : + Integrable (fun ω => egm_weight r (J ω)) P ↔ + Integrable (fun ω => gn_normalizedWeight r (enb_permutation (J ω))) P := by + have heq : (fun ω => egm_weight r (J ω)) = + (fun ω => r ^ 2 * gn_normalizedWeight r (enb_permutation (J ω))) := + funext (fun ω => enb_weight_identity r (J ω) hr) + rw [heq, integrable_const_mul_iff (isUnit_iff_ne_zero.mpr (pow_ne_zero 2 hr.ne'))] + +/-- On a finite measure, the actual endpoint second-moment theorem supplies +the first-moment hypotheses needed by the normalizer join. -/ +theorem enb_integrable_of_integrable_sq {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) [IsFiniteMeasure P] + (J : Ω → Fin 6 → ℝ) (hJ : Measurable J) (r : ℝ) (hr : 0 < r) + (hsq : Integrable (fun ω => egm_weight r (J ω) ^ 2) P) : + Integrable (fun ω => egm_weight r (J ω)) P ∧ + Integrable (fun ω => gn_normalizedWeight r (enb_permutation (J ω))) P := by + have hmajor : Integrable (fun ω => 1 + egm_weight r (J ω) ^ 2) P := + (integrable_const 1).add hsq + have hphysical : Integrable (fun ω => egm_weight r (J ω)) P := by + apply hmajor.mono' (((egm_weight_measurable r).comp hJ).aestronglyMeasurable) + filter_upwards [] with ω + rw [Real.norm_eq_abs] + change |egm_weight r (J ω)| ≤ 1 + egm_weight r (J ω) ^ 2 + nlinarith [sq_nonneg (|egm_weight r (J ω)| - 1/2), sq_abs (egm_weight r (J ω))] + exact ⟨hphysical, (enb_integrable_iff P J r hr).mp hphysical⟩ + +/-- The proved common-coupling normalizer floor transfers to the literal +physical endpoint weight, with its own actual coordinate order. Current +physical integrability and the conditioned-field convergence remain inputs. -/ +theorem enb_normalizer_floor_of_common_coupling + {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) [IsProbabilityMeasure P] + (J : ℝ → Ω → Fin 6 → ℝ) (hJ : ∀ r, Measurable (J r)) + (A Q : Ω → ℝ) (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) + (hQ : HasLaw Q (gaussianReal m v) P) + (hconv : ∀ᵐ ω ∂P, Tendsto (fun r => J r ω) (𝓝[>] (0 : ℝ)) + (𝓝 ![-1,-A ω/2,Q ω,1,A ω/2,Q ω])) + (hint : ∀ᶠ r in 𝓝[>] (0 : ℝ), + Integrable (fun ω => egm_weight r (J r ω)) P) : + ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → + c * r ^ 2 ≤ ∫ ω, egm_weight r (J r ω) ∂P := by + let X : ℝ → Ω → Fin 6 → ℝ := fun r ω => enb_permutation (J r ω) + have hX (r : ℝ) : Measurable (X r) := + enb_permutation_continuous.measurable.comp (hJ r) + have hXconv : ∀ᵐ ω ∂P, Tendsto (fun r => X r ω) (𝓝[>] (0 : ℝ)) + (𝓝 ![-1,1,-A ω/2,A ω/2,Q ω,Q ω]) := by + filter_upwards [hconv] with ω hω + have hp := (enb_permutation_continuous.tendsto + ![-1,-A ω/2,Q ω,1,A ω/2,Q ω]).comp hω + have hcontact : enb_permutation ![-1,-A ω/2,Q ω,1,A ω/2,Q ω] = + ![-1,1,-A ω/2,A ω/2,Q ω,Q ω] := rfl + simpa only [Function.comp_def, hcontact, X] using hp + have hXint : ∀ᶠ r in 𝓝[>] (0 : ℝ), + Integrable (fun ω => gn_normalizedWeight r (X r ω)) P := by + filter_upwards [hint, self_mem_nhdsWithin] with r hri hr + exact (enb_integrable_iff P (J r) r hr).mp hri + obtain ⟨c, hc, δ, hδ, hfloor⟩ := + gn_normalizer_floor_of_common_coupling P X hX A Q m v hv hQ hXconv hXint + refine ⟨c, hc, δ, hδ, ?_⟩ + intro r hr hrδ + have hbound := hfloor r hr hrδ + calc + c * r ^ 2 ≤ ∫ ω, r ^ 2 * gn_normalizedWeight r (X r ω) ∂P := hbound + _ = ∫ ω, egm_weight r (J r ω) ∂P := + integral_congr_ae (Eventually.of_forall (fun ω => (enb_weight_identity r (J r ω) hr).symm)) + +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianCoupling.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianCoupling.lean new file mode 100644 index 000000000..143c6253a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianCoupling.lean @@ -0,0 +1,65 @@ +import Mathlib.Probability.Distributions.Gaussian.Multivariate +import Mathlib.Probability.HasLaw +import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric + +/-! +# One continuous standard-Gaussian coupling of a finite Gaussian family + +PSD square-root continuity allows singular covariance and arbitrary varying +means. The affine maps have the literal pinned multivariateGaussian laws. +Actual source endpoint means/covariances and their limits remain application +joins; see ../SOURCE_GAUSSIAN_COUPLING.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter Matrix WithLp +open scoped Topology ENNReal NNReal MatrixOrder Matrix.Norms.L2Operator +set_option autoImplicit false + +private theorem gc_matrix_eval_tendsto {ι R : Type*} [Fintype ι] [DecidableEq ι] + {u : Filter R} {S : R → Matrix ι ι ℝ} {S0 : Matrix ι ι ℝ} + (hS : Tendsto S u (𝓝 S0)) (z : EuclideanSpace ℝ ι) : + Tendsto (fun r => Matrix.toEuclideanCLM (𝕜 := ℝ) (S r) z) u + (𝓝 (Matrix.toEuclideanCLM (𝕜 := ℝ) S0 z)) := by + have hcoord (i j : ι) : Tendsto (fun r => S r i j) u (𝓝 (S0 i j)) := + ((continuous_apply j).tendsto (S0 i)).comp + (((continuous_apply i).tendsto S0).comp hS) + have hvec : Tendsto (fun r => (S r) *ᵥ ofLp z) u (𝓝 (S0 *ᵥ ofLp z)) := by + apply tendsto_pi_nhds.mpr + intro i + exact tendsto_finsetSum Finset.univ (fun j _ => + (hcoord i j).mul tendsto_const_nhds) + change Tendsto (fun r => toLp 2 ((S r) *ᵥ ofLp z)) u + (𝓝 (toLp 2 (S0 *ᵥ ofLp z))) + exact ((PiLp.continuous_toLp 2 (fun _ : ι => ℝ)).tendsto _).comp hvec + +/-- The canonical affine vector on one common standard Gaussian space. -/ +noncomputable def gc_coupledVector {ι : Type*} [Fintype ι] [DecidableEq ι] + (m : EuclideanSpace ℝ ι) (S : Matrix ι ι ℝ) (z : EuclideanSpace ℝ ι) : + EuclideanSpace ℝ ι := m + Matrix.toEuclideanCLM (𝕜 := ℝ) (CFC.sqrt S) z + +/-- Exact marginal law; this is the pinned multivariateGaussian definition. -/ +theorem gc_coupledVector_hasLaw {ι : Type*} [Fintype ι] [DecidableEq ι] + (m : EuclideanSpace ℝ ι) (S : Matrix ι ι ℝ) : + HasLaw (gc_coupledVector m S) (multivariateGaussian m S) + (stdGaussian (EuclideanSpace ℝ ι)) where + aemeasurable := by unfold gc_coupledVector; fun_prop + map_eq := rfl + +/-- Every common Gaussian coordinate converges when means and PSD covariances +converge. No invertibility, fixed rank, or square-root Lipschitz estimate. -/ +theorem gc_coupledVector_tendsto {ι κ : Type*} [Fintype ι] [DecidableEq ι] + (u : Filter κ) (m : κ → EuclideanSpace ℝ ι) (S : κ → Matrix ι ι ℝ) + (m0 : EuclideanSpace ℝ ι) (S0 : Matrix ι ι ℝ) + (hm : Tendsto m u (𝓝 m0)) (hS : Tendsto S u (𝓝 S0)) + (hPSD : ∀ᶠ t in u, (S t).PosSemidef) (hPSD0 : S0.PosSemidef) + (z : EuclideanSpace ℝ ι) : + Tendsto (fun t => gc_coupledVector (m t) (S t) z) u + (𝓝 (gc_coupledVector m0 S0 z)) := by + have hwithin : Tendsto S u (𝓝[{B : Matrix ι ι ℝ | 0 ≤ B}] S0) := + tendsto_nhdsWithin_iff.mpr ⟨hS, hPSD.mono (fun t ht => ht.nonneg)⟩ + have hsqrt : Tendsto (fun t => CFC.sqrt (S t)) u (𝓝 (CFC.sqrt S0)) := + (CFC.continuousOn_sqrt S0 hPSD0.nonneg).tendsto.comp hwithin + have heval := gc_matrix_eval_tendsto hsqrt z + exact hm.add heval +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianJetMoments.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianJetMoments.lean new file mode 100644 index 000000000..52498daf0 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianJetMoments.lean @@ -0,0 +1,124 @@ +import ResearchFormalCoreR1.DeterminantMoment +import ResearchFormalCoreR1.GaussianScalarMoments +import Mathlib.Probability.HasLaw +import Mathlib.Algebra.Order.Chebyshev + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory +open scoped BigOperators NNReal + +set_option autoImplicit false + +/-- Radius for a finite vector, including its constant one. -/ +def gm8Radius {n : ℕ} (j : Fin n → ℝ) : ℝ := 1 + ∑ i, |j i| + +/-- One finite-dimensional Gaussian radius moment constant. -/ +noncomputable def gm8FiniteM8 (n : ℕ) (B V : ℝ) : ℝ := + ((n+1 : ℕ) : ℝ)^7 * (1 + (n : ℝ)*128*(B^8 + V^4*gm8C8)) + +theorem gm8FiniteM8_nonneg (n : ℕ) (B V : ℝ) : 0 ≤ gm8FiniteM8 n B V := by + unfold gm8FiniteM8 + have hC := gm8_standard_nonneg + positivity + +/-- The source's exact fixed nine-coordinate constant. -/ +noncomputable def gm8M8 (B V : ℝ) : ℝ := + (10 : ℝ)^7 * (1 + 9*128*(B^8 + V^4*gm8C8)) + +theorem gm8M8_nonneg (B V : ℝ) : 0 ≤ gm8M8 B V := by + simpa [gm8M8, gm8FiniteM8] using gm8FiniteM8_nonneg 9 B V + +theorem gm8_finite_radius_power_bound (n : ℕ) (j : Fin n → ℝ) : + gm8Radius j ^ 8 ≤ ((n+1 : ℕ) : ℝ)^7 * (1 + ∑ i, |j i|^8) := by + let f : Fin (n+1) → ℝ := Fin.cases 1 (fun i : Fin n => |j i|) + have h := pow_sum_le_card_mul_sum_pow (s := Finset.univ) (f := f) + (fun i _ => by + refine Fin.cases ?_ (fun k => ?_) i + · norm_num [f] + · exact abs_nonneg (j k)) 7 + simpa [f, gm8Radius, Fin.sum_univ_succ] using h + +theorem gm8_finite_integrable_bound (n : ℕ) {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsProbabilityMeasure μ] (J : Ω → Fin n → ℝ) + (hJ : Measurable J) (m : Fin n → ℝ) (v : Fin n → ℝ≥0) + (B V : ℝ) (hB : 0 ≤ B) (hV : 0 ≤ V) + (hmean : ∀ i, |m i| ≤ B) (hvar : ∀ i, (v i : ℝ) ≤ V) + (hlaw : ∀ i, HasLaw (fun x => J x i) (gaussianReal (m i) (v i)) μ) : + Integrable (fun x => gm8Radius (J x)^8) μ ∧ + (∫ x, gm8Radius (J x)^8 ∂μ) ≤ gm8FiniteM8 n B V := by + have hi (i : Fin n) : Integrable (fun x => |J x i|^8) μ := by + exact (hlaw i).integrable_comp + (gm8_scalar_integrable_bound (m i) B V (v i) hB hV (hmean i) (hvar i)).1 + have hsum : Integrable (fun x => ∑ i, |J x i|^8) μ := + integrable_finsetSum _ (fun i _ => hi i) + have hmajor : Integrable (fun x => ((n+1 : ℕ) : ℝ)^7*(1+∑ i, |J x i|^8)) μ := + ((integrable_const 1).add hsum).const_mul _ + have hc : Continuous (gm8Radius (n := n)) := by unfold gm8Radius; fun_prop + have hmeas : AEStronglyMeasurable (fun x => gm8Radius (J x)^8) μ := + ((hc.measurable.comp hJ).pow_const 8).aestronglyMeasurable + have hR : Integrable (fun x => gm8Radius (J x)^8) μ := + hmajor.mono' hmeas (Filter.Eventually.of_forall (fun x => by + rw [Real.norm_eq_abs, abs_of_nonneg (pow_nonneg (by unfold gm8Radius; positivity) 8)] + exact gm8_finite_radius_power_bound n (J x))) + have hcoord (i : Fin n) : + (∫ x, |J x i|^8 ∂μ) ≤ 128*(B^8+V^4*gm8C8) := by + have heq := (hlaw i).integral_comp + (show AEStronglyMeasurable (fun y : ℝ => |y|^8) (gaussianReal (m i) (v i)) by fun_prop) + exact heq.trans_le (gm8_scalar_integrable_bound (m i) B V (v i) hB hV + (hmean i) (hvar i)).2 + refine ⟨hR, ?_⟩ + calc + _ ≤ ∫ x, ((n+1 : ℕ) : ℝ)^7*(1+∑ i, |J x i|^8) ∂μ := + integral_mono hR hmajor (fun x => gm8_finite_radius_power_bound n (J x)) + _ = ((n+1 : ℕ) : ℝ)^7*(1+∑ i, ∫ x, |J x i|^8 ∂μ) := by + rw [integral_const_mul, integral_add (integrable_const _) hsum, + integral_finsetSum _ (fun i _ => hi i)] + simp + _ ≤ ((n+1 : ℕ) : ℝ)^7*(1+∑ _i : Fin n, 128*(B^8+V^4*gm8C8)) := by + apply mul_le_mul_of_nonneg_left _ (by positivity) + exact add_le_add le_rfl (Finset.sum_le_sum (fun i _ => hcoord i)) + _ = gm8FiniteM8 n B V := by simp [gm8FiniteM8, mul_assoc] + +/-- The exact DM2 radius is the nine-dimensional specialization. -/ +theorem gm8_radius_power_bound (j : DM2Jet) : + dm2Radius j ^ 8 ≤ (10 : ℝ)^7 * (1 + ∑ i, |j i|^8) := by + simpa [gm8Radius, dm2Radius] using gm8_finite_radius_power_bound 9 j + +/-- The nine-jet Gaussian marginal moment interface, with no independence or +moment conclusion assumed. -/ +theorem gm8_jet_integrable_bound {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsProbabilityMeasure μ] (J : Ω → DM2Jet) + (hJ : Measurable J) (m : Fin 9 → ℝ) (v : Fin 9 → ℝ≥0) + (B V : ℝ) (hB : 0 ≤ B) (hV : 0 ≤ V) + (hmean : ∀ i, |m i| ≤ B) (hvar : ∀ i, (v i : ℝ) ≤ V) + (hlaw : ∀ i, HasLaw (fun x => J x i) (gaussianReal (m i) (v i)) μ) : + Integrable (fun x => dm2Radius (J x)^8) μ ∧ + (∫ x, dm2Radius (J x)^8 ∂μ) ≤ gm8M8 B V := by + simpa [gm8Radius, dm2Radius, gm8M8, gm8FiniteM8] using + gm8_finite_integrable_bound 9 μ J hJ m v B V hB hV hmean hvar hlaw + +/-- The same B,V give one determinant second-moment coefficient throughout +0 J r x i) (gaussianReal (m r i) (v r i)) (μ r)) : + 0 ≤ 16*gm8M8 B V ∧ ∀ r, 0 < r → r ≤ r0 → + Integrable (fun x => dm2Weight r (J r x)^2) (μ r) ∧ + (∫ x, dm2Weight r (J r x)^2 ∂μ r) ≤ (16*gm8M8 B V)*r^4 := by + refine ⟨mul_nonneg (by norm_num) (gm8M8_nonneg B V), ?_⟩ + intro r hr hr0 + have : IsProbabilityMeasure (μ r) := hprob r hr hr0 + obtain ⟨h8, hbound⟩ := gm8_jet_integrable_bound (μ r) (J r) (hJ r hr hr0) + (m r) (v r) B V hB hV (hmean r hr hr0) (hvar r hr hr0) (hlaw r hr hr0) + exact dm2_moment (μ r) (J r) (hJ r hr hr0) r (gm8M8 B V) hr (hr0.trans hr01) + h8 hbound + +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianNormalizer.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianNormalizer.lean new file mode 100644 index 000000000..a9657cfb8 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianNormalizer.lean @@ -0,0 +1,110 @@ +import Mathlib.Probability.Distributions.Gaussian.Real +import Mathlib.MeasureTheory.Integral.Bochner.Basic +import Mathlib.MeasureTheory.Measure.Lebesgue.Basic +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum +import Mathlib.Tactic.FunProp +import Mathlib.Tactic.Positivity + +/-! +# Gaussian reference positivity and an endpoint-box weight floor + +The literal continuous normalized endpoint weight is from LM006 stability, +sections 1–2. These components do not construct the conditioned field law, +its endpoint convergence, a positive box mass, or the full normalizer floor. +See ../SOURCE_GAUSSIAN_NORMALIZER.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory +open scoped ENNReal NNReal + +set_option autoImplicit false + +/-- The six coordinates are aM,aS,bM,bS,cM,cS. For physical positive radius, +the source identifies this continuous expression with its typed W_r/r². -/ +noncomputable def gn_normalizedWeight (r : ℝ) (v : Fin 6 → ℝ) : ℝ := + max (max (-v 0) 0 * max (-v 4) 0 - r * (v 2)^2) 0 * + max (r * (v 3)^2 - v 1 * v 5) 0 + +/-- The contact negative-curvature reference has an integrable, strictly +positive squared negative part, with arbitrary mean and nonzero variance. -/ +theorem gn_gaussian_reference_integrable_pos (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) : + Integrable (fun q : ℝ => (max (-q) 0)^2) (gaussianReal m v) ∧ + 0 < ∫ q : ℝ, (max (-q) 0)^2 ∂gaussianReal m v := by + have hsq : Integrable (fun q : ℝ => q^2) (gaussianReal m v) := by + simpa using (memLp_id_gaussianReal (μ := m) (v := v) 2).integrable_sq + have hint : Integrable (fun q : ℝ => (max (-q) 0)^2) (gaussianReal m v) := by + apply hsq.mono' (by fun_prop) + filter_upwards [] with q + rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] + rcases le_total 0 q with hq | hq + · rw [max_eq_right (neg_nonpos.mpr hq)] + simpa using sq_nonneg q + · rw [max_eq_left (neg_nonneg.mpr hq)] + nlinarith + have hsupp : Function.support (fun q : ℝ => (max (-q) 0)^2) = Set.Iio 0 := by + ext q + simp only [Function.mem_support, Set.mem_Iio] + constructor + · intro h + by_contra hn + have hq : 0 ≤ q := le_of_not_gt hn + apply h + simp [max_eq_right (neg_nonpos.mpr hq)] + · intro hq + have hp : 0 < (max (-q) 0)^2 := sq_pos_of_pos (lt_max_of_lt_left (neg_pos.mpr hq)) + exact hp.ne' + have hmass : 0 < gaussianReal m v (Set.Iio 0) := by + apply pos_iff_ne_zero.mpr + intro hzero + have hvol := (gaussianReal_absolutelyContinuous' m hv) hzero + simp at hvol + refine ⟨hint, ?_⟩ + apply (integral_pos_iff_support_of_nonneg (fun q => sq_nonneg _) hint).mpr + simpa only [hsupp] using hmass + +/-- An explicit open endpoint box has a uniform positive normalized weight. +This is a pointwise component, with no claimed probability mass for the box. -/ +theorem gn_open_box_floor + (r K : ℝ) (v : Fin 6 → ℝ) + (hr : 0 ≤ r) (hK : 0 < K) (hsmall : r * K^2 ≤ 1/4) + (haM : v 0 ∈ Set.Ioo (-5/4) (-3/4)) + (haS : v 1 ∈ Set.Ioo (3/4) (5/4)) + (hbM : |v 2| < K) (_hbS : |v 3| < K) + (hcM : v 4 ∈ Set.Ioo (-2) (-1)) + (hcS : v 5 ∈ Set.Ioo (-2) (-1)) : + (3 : ℝ)/8 ≤ gn_normalizedWeight r v := by + have ha0 : 3/4 ≤ -v 0 := by linarith [haM.2] + have hc4 : 1 ≤ -v 4 := by linarith [hcM.2] + have ha1 : 3/4 ≤ v 1 := haS.1.le + have hc5 : 1 ≤ -v 5 := by linarith [hcS.2] + have hAM : max (-v 0) 0 = -v 0 := max_eq_left (by linarith) + have hCM : max (-v 4) 0 = -v 4 := max_eq_left (by linarith) + have hb2 : (v 2)^2 ≤ K^2 := by + simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg (v 2)) hK.le).2 hbM.le + have hrb : r * (v 2)^2 ≤ 1/4 := + (mul_le_mul_of_nonneg_left hb2 hr).trans hsmall + have hprodM : (3 : ℝ)/4 ≤ (-v 0) * (-v 4) := by + have h := mul_le_mul ha0 hc4 (by norm_num : (0 : ℝ) ≤ 1) (by linarith : 0 ≤ -v 0) + norm_num at h ⊢ + exact h + have hprodS : (3 : ℝ)/4 ≤ -(v 1 * v 5) := by + have h := mul_le_mul ha1 hc5 (by norm_num : (0 : ℝ) ≤ 1) (by linarith : 0 ≤ v 1) + norm_num at h + nlinarith + have hF : (1 : ℝ)/2 ≤ max (max (-v 0) 0 * max (-v 4) 0 - r * (v 2)^2) 0 := by + rw [hAM, hCM] + exact (show (1 : ℝ)/2 ≤ (-v 0) * (-v 4) - r * (v 2)^2 by linarith).trans + (le_max_left _ _) + have hG : (3 : ℝ)/4 ≤ max (r * (v 3)^2 - v 1 * v 5) 0 := by + have hp : 0 ≤ r * (v 3)^2 := mul_nonneg hr (sq_nonneg _) + exact (show (3 : ℝ)/4 ≤ r * (v 3)^2 - v 1 * v 5 by linarith).trans + (le_max_left _ _) + have h := mul_le_mul hF hG (by norm_num : (0 : ℝ) ≤ 3/4) + (le_max_right _ _) + norm_num at h + exact h + +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianNormalizerLimit.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianNormalizerLimit.lean new file mode 100644 index 000000000..be08ef92c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianNormalizerLimit.lean @@ -0,0 +1,210 @@ +import ResearchFormalCoreR1.GaussianCoupling +import ResearchFormalCoreR1.NormalizerFloor +import ResearchFormalCoreR1.EndpointNormalizerBridge +import Mathlib.Probability.Moments.Variance +import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp + +/-! +# A full normalizer floor from finite Gaussian parameter limits + +Contact identities follow from the displayed zero variances. The common +coupling, noncentered nondegenerate scalar reference and weight integrability +are derived. No positive normalizer, coupling limit or moment bound is assumed. +Actual torus conditional means/covariances remain a separate source join. +See ../SOURCE_GAUSSIAN_NORMALIZER_LIMIT.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 +open MeasureTheory ProbabilityTheory Filter Matrix WithLp +open scoped Topology ENNReal NNReal MatrixOrder Matrix.Norms.L2Operator +set_option autoImplicit false + +private theorem gnl_contact_relations (m : EuclideanSpace ℝ (Fin 6)) (S : Matrix (Fin 6) (Fin 6) ℝ) + (hS : S.PosSemidef) + (hm0 : m 0 = -1) (hm1 : m 1 = 1) + (hm23 : m 2 + m 3 = 0) (hm45 : m 4 - m 5 = 0) + (hS0 : S 0 0 = 0) (hS1 : S 1 1 = 0) + (hS23 : S 2 2 + 2*S 2 3 + S 3 3 = 0) + (hS45 : S 4 4 - 2*S 4 5 + S 5 5 = 0) : + ∀ᵐ x ∂multivariateGaussian m S, + x 0 = -1 ∧ x 1 = 1 ∧ x 2 + x 3 = 0 ∧ x 4 - x 5 = 0 := by + let P := multivariateGaussian m S + have hLp (i : Fin 6) : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => x i) 2 P := by + exact (IsGaussian.memLp_two_id (μ := P)).continuousLinearMap_comp (EuclideanSpace.proj (𝕜 := ℝ) i) + have hi (i : Fin 6) : ∫ x : EuclideanSpace ℝ (Fin 6), x i ∂P = m i := by + change ∫ x, (EuclideanSpace.proj (𝕜 := ℝ) i) x ∂P = m i + rw [ContinuousLinearMap.integral_comp_id_comm] + · simp [P] + · exact IsGaussian.integrable_id + have hvar0 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 0; P] = 0 := by + exact (variance_eval_multivariateGaussian hS 0).trans hS0 + have hvar1 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 1; P] = 0 := by + exact (variance_eval_multivariateGaussian hS 1).trans hS1 + have hvar23 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 2 + x 3; P] = 0 := by + rw [variance_fun_add (hLp 2) (hLp 3)] + simpa [P, variance_eval_multivariateGaussian hS, + covariance_eval_multivariateGaussian hS] using hS23 + have hvar45 : Var[fun x : EuclideanSpace ℝ (Fin 6) => x 4 - x 5; P] = 0 := by + rw [variance_fun_sub (hLp 4) (hLp 5)] + simpa [P, variance_eval_multivariateGaussian hS, + covariance_eval_multivariateGaussian hS] using hS45 + have hi23 : ∫ x : EuclideanSpace ℝ (Fin 6), (x 2 + x 3) ∂P = 0 := by + rw [integral_add ((hLp 2).integrable (by norm_num)) ((hLp 3).integrable (by norm_num)), hi, hi] + exact hm23 + have hi45 : ∫ x : EuclideanSpace ℝ (Fin 6), (x 4 - x 5) ∂P = 0 := by + rw [integral_sub ((hLp 4).integrable (by norm_num)) ((hLp 5).integrable (by norm_num)), hi, hi] + exact hm45 + filter_upwards [ae_eq_integral_of_variance_eq_zero (hLp 0) hvar0, + ae_eq_integral_of_variance_eq_zero (hLp 1) hvar1, + ae_eq_integral_of_variance_eq_zero ((hLp 2).add (hLp 3)) hvar23, + ae_eq_integral_of_variance_eq_zero ((hLp 4).sub (hLp 5)) hvar45] + with x hx0 hx1 hx23 hx45 + exact ⟨hx0.trans ((hi 0).trans hm0), hx1.trans ((hi 1).trans hm1), + hx23.trans hi23, hx45.trans hi45⟩ + +private lemma gnl_positivePart_memLp {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {f : Ω → ℝ} {p : ℝ≥0∞} (hf : MemLp f p P) : + MemLp (fun x => max (f x) 0) p P := by + apply hf.norm.mono' ((hf.aestronglyMeasurable.aemeasurable.sup_const 0).aestronglyMeasurable) + filter_upwards [] with x + rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_nonneg (le_max_right _ _)] + exact max_le (le_abs_self _) (abs_nonneg _) + +private lemma gnl_gaussian_weight_integrable (r : ℝ) (m : EuclideanSpace ℝ (Fin 6)) + (S : Matrix (Fin 6) (Fin 6) ℝ) (hS : S.PosSemidef) : + Integrable (fun x : EuclideanSpace ℝ (Fin 6) => gn_normalizedWeight r (ofLp x)) + (multivariateGaussian m S) := by + have : ENNReal.HolderTriple 4 4 2 := ⟨by + rw [← two_mul, show (4 : ℝ≥0∞) = 2*2 by norm_num, ENNReal.mul_inv (by simp) (by simp), + ← mul_assoc, ENNReal.mul_inv_cancel two_ne_zero ENNReal.ofNat_ne_top, one_mul]⟩ + let P := multivariateGaussian m S + have hc (i : Fin 6) : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => x i) 4 P := by + have hp := measurePreserving_eval_multivariateGaussian (μ := m) hS (i := i) + have hlaw : HasLaw (fun x : EuclideanSpace ℝ (Fin 6) => x i) + (gaussianReal (m i) (S i i).toNNReal) P := ⟨hp.measurable.aemeasurable, hp.map_eq⟩ + exact hlaw.memLp (memLp_id_gaussianReal 4) + have hpos (i : Fin 6) : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => max (-x i) 0) 4 P := + gnl_positivePart_memLp (hc i).neg + have hM : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => + max (-x 0) 0 * max (-x 4) 0 - r*(x 2)^2) 2 P := by + have h1 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => max (-x 0) 0 * max (-x 4) 0) 2 P := + (hpos 0).fun_mul (hpos 4) + have h2 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => (x 2)^2) 2 P := by + simpa only [pow_two] using (hc 2).fun_mul (hc 2) + exact h1.sub (h2.const_mul r) + have hSadd : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => + r*(x 3)^2 - x 1*x 5) 2 P := by + have h1 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => (x 3)^2) 2 P := by + simpa only [pow_two] using (hc 3).fun_mul (hc 3) + have h2 : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => x 1*x 5) 2 P := + (hc 1).fun_mul (hc 5) + exact (h1.const_mul r).sub h2 + have hT : MemLp (fun x : EuclideanSpace ℝ (Fin 6) => gn_normalizedWeight r (ofLp x)) 1 P := + (gnl_positivePart_memLp hM).fun_mul (gnl_positivePart_memLp hSadd) + exact memLp_one_iff_integrable.mp hT + + +/-- Gaussian endpoint parameter convergence and concrete contact moments supply +one positive physical normalizer coefficient at every sufficiently small radius. +The covariance may be singular; the transverse reference mean is unrestricted. -/ +theorem gnl_normalizer_floor_of_parameter_limits + (m : ℝ → EuclideanSpace ℝ (Fin 6)) (S : ℝ → Matrix (Fin 6) (Fin 6) ℝ) + (m0 : EuclideanSpace ℝ (Fin 6)) (S0 : Matrix (Fin 6) (Fin 6) ℝ) + (hm : Tendsto m (𝓝[>] (0 : ℝ)) (𝓝 m0)) + (hS : Tendsto S (𝓝[>] (0 : ℝ)) (𝓝 S0)) + (hPSD : ∀ᶠ r in 𝓝[>] (0 : ℝ), (S r).PosSemidef) (hPSD0 : S0.PosSemidef) + (hm0 : m0 0 = -1) (hm1 : m0 1 = 1) + (hm23 : m0 2 + m0 3 = 0) (hm45 : m0 4 - m0 5 = 0) + (hS0 : S0 0 0 = 0) (hS1 : S0 1 1 = 0) + (hS23 : S0 2 2 + 2*S0 2 3 + S0 3 3 = 0) + (hS45 : S0 4 4 - 2*S0 4 5 + S0 5 5 = 0) (hQpos : 0 < S0 4 4) : + ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → c*r^2 ≤ + ∫ x : EuclideanSpace ℝ (Fin 6), r^2 * gn_normalizedWeight r (ofLp x) + ∂multivariateGaussian (m r) (S r) := by + let P := stdGaussian (EuclideanSpace ℝ (Fin 6)) + let X : ℝ → EuclideanSpace ℝ (Fin 6) → Fin 6 → ℝ := + fun r z => ofLp (gc_coupledVector (m r) (S r) z) + let A : EuclideanSpace ℝ (Fin 6) → ℝ := fun z => 2 * gc_coupledVector m0 S0 z 3 + let Q : EuclideanSpace ℝ (Fin 6) → ℝ := fun z => gc_coupledVector m0 S0 z 4 + have hX (r : ℝ) : Measurable (X r) := by + dsimp [X, gc_coupledVector] + fun_prop + have hLaw0 : HasLaw (gc_coupledVector m0 S0) (multivariateGaussian m0 S0) P := + gc_coupledVector_hasLaw m0 S0 + have hcontacts : ∀ᵐ z ∂P, + gc_coupledVector m0 S0 z 0 = -1 ∧ gc_coupledVector m0 S0 z 1 = 1 ∧ + gc_coupledVector m0 S0 z 2 + gc_coupledVector m0 S0 z 3 = 0 ∧ + gc_coupledVector m0 S0 z 4 - gc_coupledVector m0 S0 z 5 = 0 := by + have hb : ∀ᵐ x ∂P.map (gc_coupledVector m0 S0), + x 0 = -1 ∧ x 1 = 1 ∧ x 2 + x 3 = 0 ∧ x 4 - x 5 = 0 := by + rw [hLaw0.map_eq] + exact gnl_contact_relations m0 S0 hPSD0 hm0 hm1 hm23 hm45 hS0 hS1 hS23 hS45 + exact ae_of_ae_map hLaw0.aemeasurable hb + have hconv : ∀ᵐ z ∂P, Tendsto (fun r => X r z) (𝓝[>] (0 : ℝ)) + (𝓝 ![-1,1,-A z/2,A z/2,Q z,Q z]) := by + filter_upwards [hcontacts] with z hz + have hshape : ofLp (gc_coupledVector m0 S0 z) = ![-1,1,-A z/2,A z/2,Q z,Q z] := by + funext i + fin_cases i <;> simp [A, Q] <;> linarith [hz.1, hz.2.1, hz.2.2.1, hz.2.2.2] + have ht := ((PiLp.continuous_ofLp 2 (fun _ : Fin 6 => ℝ)).tendsto + (gc_coupledVector m0 S0 z)).comp + (gc_coupledVector_tendsto (𝓝[>] (0 : ℝ)) m S m0 S0 hm hS hPSD hPSD0 z) + simpa only [Function.comp_def, X, hshape] using ht + have hv : (S0 4 4).toNNReal ≠ 0 := ne_of_gt (Real.toNNReal_pos.mpr hQpos) + have hQ : HasLaw Q (gaussianReal (m0 4) (S0 4 4).toNNReal) P := by + exact (measurePreserving_eval_multivariateGaussian (μ := m0) hPSD0 (i := 4)).hasLaw.fun_comp hLaw0 + have hint : ∀ᶠ r in 𝓝[>] (0 : ℝ), + Integrable (fun z => gn_normalizedWeight r (X r z)) P := by + filter_upwards [hPSD] with r hr + exact (gc_coupledVector_hasLaw (m r) (S r)).integrable_comp + (gnl_gaussian_weight_integrable r (m r) (S r) hr) + obtain ⟨c, hc, δ, hδ, hfloor⟩ := + gn_normalizer_floor_of_common_coupling P X hX A Q (m0 4) (S0 4 4).toNNReal hv hQ hconv hint + refine ⟨c, hc, δ, hδ, ?_⟩ + intro r hr hrδ + calc + c*r^2 ≤ ∫ z, r^2 * gn_normalizedWeight r (X r z) ∂P := hfloor r hr hrδ + _ = ∫ x : EuclideanSpace ℝ (Fin 6), r^2 * gn_normalizedWeight r (ofLp x) + ∂multivariateGaussian (m r) (S r) := by + simpa only [Function.comp_def, X] using + (gc_coupledVector_hasLaw (m r) (S r)).integral_comp + (f := fun x : EuclideanSpace ℝ (Fin 6) => r^2 * gn_normalizedWeight r (ofLp x)) + (by unfold gn_normalizedWeight; fun_prop) + + +/-- The same Gaussian parameter limit gives the full literal typed endpoint +normalizer, after the explicit permutation back to physical endpoint order. -/ +theorem gnl_physical_normalizer_floor_of_parameter_limits + (m : ℝ → EuclideanSpace ℝ (Fin 6)) (S : ℝ → Matrix (Fin 6) (Fin 6) ℝ) + (m0 : EuclideanSpace ℝ (Fin 6)) (S0 : Matrix (Fin 6) (Fin 6) ℝ) + (hm : Tendsto m (𝓝[>] (0 : ℝ)) (𝓝 m0)) + (hS : Tendsto S (𝓝[>] (0 : ℝ)) (𝓝 S0)) + (hPSD : ∀ᶠ r in 𝓝[>] (0 : ℝ), (S r).PosSemidef) (hPSD0 : S0.PosSemidef) + (hm0 : m0 0 = -1) (hm1 : m0 1 = 1) + (hm23 : m0 2 + m0 3 = 0) (hm45 : m0 4 - m0 5 = 0) + (hS0 : S0 0 0 = 0) (hS1 : S0 1 1 = 0) + (hS23 : S0 2 2 + 2*S0 2 3 + S0 3 3 = 0) + (hS45 : S0 4 4 - 2*S0 4 5 + S0 5 5 = 0) (hQpos : 0 < S0 4 4) : + ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → c*r^2 ≤ + ∫ x : EuclideanSpace ℝ (Fin 6), egm_weight r ![x 0,x 2,x 4,x 1,x 3,x 5] + ∂multivariateGaussian (m r) (S r) := by + obtain ⟨c, hc, δ, hδ, hfloor⟩ := gnl_normalizer_floor_of_parameter_limits + m S m0 S0 hm hS hPSD hPSD0 hm0 hm1 hm23 hm45 hS0 hS1 hS23 hS45 hQpos + refine ⟨c, hc, δ, hδ, ?_⟩ + intro r hr hrδ + calc + c*r^2 ≤ ∫ x : EuclideanSpace ℝ (Fin 6), r^2 * gn_normalizedWeight r (ofLp x) + ∂multivariateGaussian (m r) (S r) := hfloor r hr hrδ + _ = ∫ x : EuclideanSpace ℝ (Fin 6), egm_weight r ![x 0,x 2,x 4,x 1,x 3,x 5] + ∂multivariateGaussian (m r) (S r) := by + apply integral_congr_ae + filter_upwards [] with x + have hp : enb_permutation ![x 0,x 2,x 4,x 1,x 3,x 5] = ofLp x := by + funext i + fin_cases i <;> rfl + simpa only [hp] using + (enb_weight_identity r ![x 0,x 2,x 4,x 1,x 3,x 5] hr).symm + +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianRegressionKernel.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianRegressionKernel.lean new file mode 100644 index 000000000..face5de81 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianRegressionKernel.lean @@ -0,0 +1,270 @@ +import Mathlib.Probability.HasCondDistrib +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence + +/-! +# Canonical Gaussian regression kernel + +For a residual law `ν`, `regressionKernel ν B mX mY x` is the literal affine pushforward +`ν.map (fun r => r + mY + B (x - mX))` at every observation `x`. The kernel is constructed +measurably from a deterministic observation and a constant residual kernel, so it has a +specified value even where the observation has probability zero. + +The Gaussian theorem derives observation/residual independence from joint Gaussianity and +zero coordinate cross-covariance. No nondegeneracy or residual covariance inverse is needed. +The direct observed/target theorem derives those residual hypotheses from the coordinate +normal equations for `B`. The construction does not identify concrete derivative observations, +regression or Schur covariance blocks, a Palm weight, or a conditional law of a literal smooth field. + +Source contract: `SOURCE_REGRESSION_KERNEL.md` in this packet. Author-side Lean verification +is separate from source alignment and the required review by another model family. +-/ + +open MeasureTheory ProbabilityTheory +open scoped ProbabilityTheory BigOperators + +set_option autoImplicit false + +namespace ResearchFormalCoreR1 + +/-- Copying the first coordinate and retaining an independent second coordinate. -/ +lemma regression_compProd_id_const {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] + (μ : Measure α) (ν : Measure β) [SFinite μ] [SFinite ν] : + μ ⊗ₘ (Kernel.id ×ₖ Kernel.const α ν) = + (μ.prod ν).map (fun p : α × β => (p.1, p)) := by + ext s hs + rw [Measure.compProd_apply hs, Measure.map_apply (by fun_prop) hs, + Measure.prod_apply (s := (fun p : α × β => (p.1, p)) ⁻¹' s) (hs.preimage (by fun_prop))] + simp_rw [Kernel.prod_apply, Kernel.id_apply, Kernel.const_apply, Measure.dirac_prod] + congr with x + rw [Measure.map_apply (by fun_prop) (measurable_prodMk_left hs)] + rfl + +/-- An independent residual paired with the observation has the explicit product kernel. -/ +lemma regression_pair_hasCondDistrib {Ω α β : Type*} [MeasurableSpace Ω] + [MeasurableSpace α] [MeasurableSpace β] {P : Measure Ω} [IsProbabilityMeasure P] + {X : Ω → α} {R : Ω → β} (hX : AEMeasurable X P) (hR : AEMeasurable R P) + (hXR : IndepFun X R P) : + HasCondDistrib (fun ω => (X ω, R ω)) X + (Kernel.id ×ₖ Kernel.const α (P.map R)) P := by + refine ⟨hX.prodMk (hX.prodMk hR), ?_⟩ + calc + P.map (fun ω => (X ω, (X ω, R ω))) = + (P.map (fun ω => (X ω, R ω))).map (fun p : α × β => (p.1, p)) := by + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) (hX.prodMk hR)] + rfl + _ = ((P.map X).prod (P.map R)).map (fun p : α × β => (p.1, p)) := by + rw [hXR.map_prod_eq_prod_map_map hX hR] + _ = (P.map X) ⊗ₘ (Kernel.id ×ₖ Kernel.const α (P.map R)) := + (regression_compProd_id_const (P.map X) (P.map R)).symm + + +/-- Affine reconstruction of the target from the observation and the regression residual. -/ +def regressionReconstruction {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (p : (Fin n → ℝ) × (Fin k → ℝ)) : Fin k → ℝ := + p.2 + mY + B (p.1 - mX) + +lemma regressionReconstruction_measurable {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Measurable (regressionReconstruction B mX mY) := by + unfold regressionReconstruction + fun_prop + +/-- The canonical regression law is specified at every observation, including null pins. -/ +noncomputable def regressionKernel {n k : ℕ} (ν : Measure (Fin k → ℝ)) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Kernel (Fin n → ℝ) (Fin k → ℝ) := + ((Kernel.id : Kernel (Fin n → ℝ) (Fin n → ℝ)) ×ₖ + Kernel.const (Fin n → ℝ) ν).map (regressionReconstruction B mX mY) + +instance regressionKernel_isMarkov {n k : ℕ} (ν : Measure (Fin k → ℝ)) + [IsProbabilityMeasure ν] (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) + (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + IsMarkovKernel (regressionKernel ν B mX mY) := by + unfold regressionKernel + exact Kernel.IsMarkovKernel.map _ (regressionReconstruction_measurable B mX mY) + +lemma regressionKernel_apply {n k : ℕ} (ν : Measure (Fin k → ℝ)) [SFinite ν] + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (x : Fin n → ℝ) : + regressionKernel ν B mX mY x = ν.map (fun r => r + mY + B (x - mX)) := by + rw [regressionKernel, Kernel.map_apply _ (regressionReconstruction_measurable B mX mY), + Kernel.prod_apply, Kernel.id_apply, Kernel.const_apply, Measure.dirac_prod, + Measure.map_map (regressionReconstruction_measurable B mX mY) (by fun_prop)] + rfl + +/-- Affine reconstruction from an independent residual has the canonical conditional kernel. -/ +lemma independentResidual_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {n k : ℕ} + {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hX : AEMeasurable X P) (hR : AEMeasurable R P) (hXR : IndepFun X R P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + exact (regression_pair_hasCondDistrib hX hR hXR).comp_left + (regressionReconstruction_measurable B mX mY) + +/-- Joint Gaussianity and vanishing observation/residual cross-covariance give the canonical +conditional law, without choosing an almost-everywhere version at a prescribed observation. -/ +theorem gaussianRegression_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hXR : HasGaussianLaw (fun ω => (X ω, R ω)) P) + (hzero : ∀ i j, covariance (fun ω => X ω i) (fun ω => R ω j) P = 0) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + have : IsProbabilityMeasure P := hXR.isProbabilityMeasure + exact independentResidual_hasCondDistrib hXR.fst.aemeasurable hXR.snd.aemeasurable + (hXR.indepFun_of_covariance_eval hzero) B mX mY + +/-- Kernel measurability is pointwise in the observation and does not require a Gaussian law. -/ +lemma regressionKernel_measurable {n k : ℕ} (ν : Measure (Fin k → ℝ)) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + Measurable (regressionKernel ν B mX mY) := + Kernel.measurable _ + +/-- All parts of the canonical Gaussian conditional-law construction, including the literal +pushforward at every prescribed observation. Zero-dimensional and degenerate laws are allowed. -/ +theorem gaussianRegressionKernel_spec {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {R : Ω → Fin k → ℝ} + (hXR : HasGaussianLaw (fun ω => (X ω, R ω)) P) + (hzero : ∀ i j, covariance (fun ω => X ω i) (fun ω => R ω j) P = 0) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + IsMarkovKernel (regressionKernel (P.map R) B mX mY) ∧ + Measurable (regressionKernel (P.map R) B mX mY) ∧ + (∀ x, regressionKernel (P.map R) B mX mY x = + (P.map R).map (fun r => r + mY + B (x - mX))) ∧ + HasCondDistrib (fun ω => R ω + mY + B (X ω - mX)) X + (regressionKernel (P.map R) B mX mY) P := by + have : IsProbabilityMeasure P := hXR.isProbabilityMeasure + exact ⟨inferInstance, regressionKernel_measurable _ B mX mY, + regressionKernel_apply _ B mX mY, gaussianRegression_hasCondDistrib hXR hzero B mX mY⟩ + +/-- Coordinates of a finite-vector continuous linear map in the standard basis. -/ +lemma regression_map_coordinate {n k : ℕ} + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (v : Fin n → ℝ) (j : Fin k) : + B v j = ∑ l, B (Pi.single l 1) j * v l := by + calc + B v j = B (∑ l, v l • (Pi.single l 1 : Fin n → ℝ)) j := + congrArg (fun w => B w j) (pi_eq_sum_univ' v) + _ = ∑ l, B (Pi.single l 1) j * v l := by + simp [map_sum, mul_comm] + +/-- The covariance of a centered finite-vector linear predictor is its normal-equation sum. -/ +lemma regression_covariance_map {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {n k : ℕ} + {X : Ω → Fin n → ℝ} (hX : HasGaussianLaw X P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) + (i : Fin n) (j : Fin k) : + covariance (fun ω => X ω i) (fun ω => B (X ω - mX) j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P := by + have hcoord : (fun ω => B (X ω - mX) j) = + (fun ω => ∑ l, B (Pi.single l 1) j * (X ω l - mX l)) := by + funext ω + exact regression_map_coordinate B (X ω - mX) j + rw [hcoord, covariance_fun_sum_right] + · apply Finset.sum_congr rfl + intro l _ + rw [covariance_const_mul_right, covariance_sub_const_right (hX.eval l).integrable] + · intro l + exact ((hX.eval l).memLp_two.sub (memLp_const _)).const_mul _ + · exact (hX.eval i).memLp_two + +/-- The actual regression residual has zero observation cross-covariance by the normal equations. -/ +lemma regression_residual_zero_covariance {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (hnormal : ∀ i j, covariance (fun ω => X ω i) (fun ω => Y ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P) : + ∀ i j, covariance (fun ω => X ω i) + (fun ω => (Y ω - mY - B (X ω - mX)) j) P = 0 := by + have : IsProbabilityMeasure P := hXY.isProbabilityMeasure + intro i j + have hXi := (hXY.fst.eval i).memLp_two + have hYj := (hXY.snd.eval j).memLp_two + have hBj : MemLp (fun ω => B (X ω - mX) j) 2 P := by + have hcoord : (fun ω => B (X ω - mX) j) = + (fun ω => ∑ l, B (Pi.single l 1) j * (X ω l - mX l)) := by + funext ω + exact regression_map_coordinate B (X ω - mX) j + rw [hcoord] + exact memLp_finsetSum _ (fun l _ => + ((hXY.fst.eval l).memLp_two.sub (memLp_const _)).const_mul _) + simp only [Pi.sub_apply] + rw [covariance_fun_sub_right (Y := fun ω => Y ω j - mY j) hXi (hYj.sub (memLp_const _)) hBj, + covariance_sub_const_right (hXY.snd.eval j).integrable, + regression_covariance_map hXY.fst B mX i j, hnormal i j, sub_self] + +/-- The observation and the actual affine residual are jointly Gaussian. -/ +lemma regression_residual_jointGaussian {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {n k : ℕ} + {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) : + HasGaussianLaw (fun ω => (X ω, Y ω - mY - B (X ω - mX))) P := by + let L : ((Fin n → ℝ) × (Fin k → ℝ)) →L[ℝ] + ((Fin n → ℝ) × (Fin k → ℝ)) := + (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ)).prod + (ContinuousLinearMap.snd ℝ (Fin n → ℝ) (Fin k → ℝ) - + B.comp (ContinuousLinearMap.fst ℝ (Fin n → ℝ) (Fin k → ℝ))) + let c : (Fin n → ℝ) × (Fin k → ℝ) := (0, B mX - mY) + have hL : HasGaussianLaw (fun ω => L (X ω, Y ω)) P := hXY.map_fun L + have hshift : HasGaussianLaw (fun ω => L (X ω, Y ω) + c) P := { + aemeasurable := hL.aemeasurable.add aemeasurable_const + isGaussian_map := by + have : IsGaussian (P.map (fun ω => L (X ω, Y ω))) := hL.isGaussian_map + convert (inferInstance : IsGaussian + ((P.map (fun ω => L (X ω, Y ω))).map (fun p => p + c))) using 1 + rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hL.aemeasurable] + rfl } + apply hshift.congr + filter_upwards [] with ω + apply Prod.ext + · simp [L, c] + · change Y ω - B (X ω) + (B mX - mY) = Y ω - mY - B (X ω - mX) + rw [map_sub] + abel + + + +/-- The actual target has the explicit regression conditional law when the coefficient map +satisfies the coordinate normal equations. The residual hypotheses are derived. -/ +theorem gaussianRegression_observed_hasCondDistrib {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (hnormal : ∀ i j, covariance (fun ω => X ω i) (fun ω => Y ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P) : + HasCondDistrib Y X + (regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY) P := by + have hXR := regression_residual_jointGaussian hXY B mX mY + have hzero := regression_residual_zero_covariance hXY B mX mY hnormal + have hcond := gaussianRegression_hasCondDistrib hXR hzero B mX mY + convert hcond using 1 + funext ω + abel + +/-- The canonical conditional-law specification for the actual jointly Gaussian target. +The residual Gaussianity and its zero cross-covariance are conclusions of the construction. -/ +theorem gaussianRegression_observed_spec {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {n k : ℕ} {X : Ω → Fin n → ℝ} {Y : Ω → Fin k → ℝ} + (hXY : HasGaussianLaw (fun ω => (X ω, Y ω)) P) + (B : (Fin n → ℝ) →L[ℝ] (Fin k → ℝ)) (mX : Fin n → ℝ) (mY : Fin k → ℝ) + (hnormal : ∀ i j, covariance (fun ω => X ω i) (fun ω => Y ω j) P = + ∑ l, B (Pi.single l 1) j * covariance (fun ω => X ω i) (fun ω => X ω l) P) : + IsMarkovKernel + (regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY) ∧ + Measurable + (regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY) ∧ + (∀ x, regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY x = + (P.map (fun ω => Y ω - mY - B (X ω - mX))).map + (fun r => r + mY + B (x - mX))) ∧ + HasCondDistrib Y X + (regressionKernel (P.map (fun ω => Y ω - mY - B (X ω - mX))) B mX mY) P := by + have hXR := regression_residual_jointGaussian hXY B mX mY + have hzero := regression_residual_zero_covariance hXY B mX mY hnormal + have hspec := gaussianRegressionKernel_spec hXR hzero B mX mY + exact ⟨hspec.1, hspec.2.1, hspec.2.2.1, + gaussianRegression_observed_hasCondDistrib hXY B mX mY hnormal⟩ + +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianScalarMoments.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianScalarMoments.lean new file mode 100644 index 000000000..ce432cb71 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/GaussianScalarMoments.lean @@ -0,0 +1,99 @@ +import Mathlib.Probability.Distributions.Gaussian.Real +import Mathlib.Algebra.Order.Ring.Basic +import Mathlib.Tactic.FunProp +import Mathlib.Tactic.NormNum +import Mathlib.Tactic.Ring + +/-! Scalar eighth moments for noncentered Gaussian laws. The finite standard +Gaussian moment is proved integrable; it is not an assumed model bound. -/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory +open scoped NNReal + +set_option autoImplicit false + +noncomputable def gm8C8 : ℝ := ∫ z : ℝ, |z| ^ 8 ∂gaussianReal 0 1 + +theorem gm8_standard_integrable : + Integrable (fun z : ℝ => |z| ^ 8) (gaussianReal 0 1) := by + simpa only [id_eq, Real.norm_eq_abs] using + (memLp_id_gaussianReal' (μ := 0) (v := 1) 8 (by norm_num)).integrable_norm_pow + (by norm_num : (8 : ℕ) ≠ 0) + +theorem gm8_standard_nonneg : 0 ≤ gm8C8 := by + exact integral_nonneg (fun z => pow_nonneg (abs_nonneg z) 8) + +private theorem gm8_integrable (m : ℝ) (v : ℝ≥0) : + Integrable (fun x : ℝ => |x| ^ 8) (gaussianReal m v) := by + simpa only [id_eq, Real.norm_eq_abs] using + (memLp_id_gaussianReal' (μ := m) (v := v) 8 (by norm_num)).integrable_norm_pow + (by norm_num : (8 : ℕ) ≠ 0) + +private theorem gm8_sqrt_pow (V : ℝ) (hV : 0 ≤ V) : + (Real.sqrt V) ^ 8 = V ^ 4 := by + calc + _ = ((Real.sqrt V) ^ 2) ^ 4 := by ring + _ = V ^ 4 := by rw [Real.sq_sqrt hV] + +private theorem gm8_affine_map (m : ℝ) (v : ℝ≥0) : + (gaussianReal 0 1).map (fun z : ℝ => m + Real.sqrt (v : ℝ) * z) = + gaussianReal m v := by + calc + _ = ((gaussianReal 0 1).map (fun z : ℝ => Real.sqrt (v : ℝ) * z)).map + (fun z : ℝ => m + z) := by + rw [Measure.map_map (by fun_prop) (by fun_prop)] + rfl + _ = gaussianReal m v := by + rw [gaussianReal_map_const_mul, gaussianReal_map_const_add] + simp only [mul_zero, zero_add, mul_one] + congr 1 + apply NNReal.eq + exact Real.sq_sqrt v.property + +private theorem gm8_affine_bound (m B V : ℝ) (v : ℝ≥0) (hB : 0 ≤ B) (hV : 0 ≤ V) + (hm : |m| ≤ B) (hv : (v : ℝ) ≤ V) (z : ℝ) : + |m + Real.sqrt (v : ℝ) * z| ^ 8 ≤ 128 * (B ^ 8 + V ^ 4 * |z| ^ 8) := by + have habs : |m + Real.sqrt (v : ℝ) * z| ≤ B + Real.sqrt V * |z| := by + calc + _ ≤ |m| + |Real.sqrt (v : ℝ) * z| := abs_add_le _ _ + _ = |m| + Real.sqrt (v : ℝ) * |z| := by + rw [abs_mul, abs_of_nonneg (Real.sqrt_nonneg _)] + _ ≤ B + Real.sqrt V * |z| := + add_le_add hm (mul_le_mul_of_nonneg_right (Real.sqrt_le_sqrt hv) (abs_nonneg z)) + calc + _ ≤ (B + Real.sqrt V * |z|) ^ 8 := pow_le_pow_left₀ (abs_nonneg _) habs 8 + _ ≤ 128 * (B ^ 8 + (Real.sqrt V * |z|) ^ 8) := by + convert add_pow_le hB (mul_nonneg (Real.sqrt_nonneg V) (abs_nonneg z)) 8 using 1 + norm_num + _ = 128 * (B ^ 8 + V ^ 4 * |z| ^ 8) := by + rw [mul_pow, gm8_sqrt_pow V hV] + +/-- Bounded means and variances give a uniform eighth-moment bound, including +zero variance and nonzero means. No eighth-moment hypothesis is assumed. -/ +theorem gm8_scalar_integrable_bound (m B V : ℝ) (v : ℝ≥0) + (hB : 0 ≤ B) (hV : 0 ≤ V) (hm : |m| ≤ B) (hv : (v : ℝ) ≤ V) : + Integrable (fun x : ℝ => |x| ^ 8) (gaussianReal m v) ∧ + (∫ x : ℝ, |x| ^ 8 ∂gaussianReal m v) ≤ + 128 * (B ^ 8 + V ^ 4 * gm8C8) := by + have hi := gm8_integrable m v + have hmajor : Integrable (fun z : ℝ => 128 * (B ^ 8 + V ^ 4 * |z| ^ 8)) + (gaussianReal 0 1) := + ((integrable_const (B ^ 8)).add (gm8_standard_integrable.const_mul (V ^ 4))).const_mul 128 + have hiaff : Integrable (fun z : ℝ => |m + Real.sqrt (v : ℝ) * z| ^ 8) + (gaussianReal 0 1) := by + rw [← gm8_affine_map m v] at hi + exact (integrable_map_measure (by fun_prop) (by fun_prop)).mp hi + refine ⟨hi, ?_⟩ + calc + _ = ∫ z : ℝ, |m + Real.sqrt (v : ℝ) * z| ^ 8 ∂gaussianReal 0 1 := by + rw [← gm8_affine_map m v, integral_map (by fun_prop) (by fun_prop)] + _ ≤ ∫ z : ℝ, 128 * (B ^ 8 + V ^ 4 * |z| ^ 8) ∂gaussianReal 0 1 := + integral_mono hiaff hmajor (gm8_affine_bound m B V v hB hV hm hv) + _ = 128 * (B ^ 8 + V ^ 4 * gm8C8) := by + rw [integral_const_mul, integral_add (integrable_const _) + (gm8_standard_integrable.const_mul _), integral_const_mul] + simp only [integral_const, probReal_univ, smul_eq_mul, one_mul, gm8C8] + +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/NormalizerFloor.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/NormalizerFloor.lean new file mode 100644 index 000000000..3635695ef --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/NormalizerFloor.lean @@ -0,0 +1,120 @@ +import ResearchFormalCoreR1.GaussianNormalizer +import Mathlib.Probability.HasLaw +import Mathlib.MeasureTheory.Integral.Lebesgue.Add +import Mathlib.Topology.Instances.ENNReal.Lemmas +import Mathlib.Topology.Order.LeftRightNhds + +/-! +# A full small-radius lower normalizer from an explicit common coupling + +Nonnegative Fatou and the proved positive Gaussian reference give one lower +constant for every sufficiently small positive radius. Neither uniform +integrability nor convergence of normalizers is assumed. The literal +conditioned-field construction of the coupling remains an application join. +See ../SOURCE_GAUSSIAN_NORMALIZER.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory ProbabilityTheory Filter +open scoped Topology ENNReal NNReal + +set_option autoImplicit false + +private theorem nfloor_eventually_lower_of_fatou {Ω ι : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (u : Filter ι) [IsCountablyGenerated u] [NeBot u] + (f : ι → Ω → ℝ) (g : Ω → ℝ) + (hmeas : ∀ r, Measurable (f r)) + (hnonneg : ∀ r ω, 0 ≤ f r ω) + (hg : Integrable g P) (hgpos : 0 < ∫ ω, g ω ∂P) + (hconv : ∀ᵐ ω ∂P, Tendsto (fun r => f r ω) u (𝓝 (g ω))) + (hfint : ∀ᶠ r in u, Integrable (f r) P) : + ∀ᶠ r in u, (∫ ω, g ω ∂P)/2 ≤ ∫ ω, f r ω ∂P := by + have hg0 : 0 ≤ᵐ[P] g := hconv.mono (fun ω hω => + ge_of_tendsto hω (Eventually.of_forall (fun r => hnonneg r ω))) + have hlim : (fun ω => liminf (fun r => ENNReal.ofReal (f r ω)) u) =ᵐ[P] + (fun ω => ENNReal.ofReal (g ω)) := by + filter_upwards [hconv] with ω hω + exact ((ENNReal.continuous_ofReal.tendsto (g ω)).comp hω).liminf_eq + have hfatou : ENNReal.ofReal (∫ ω, g ω ∂P) ≤ + liminf (fun r => ∫⁻ ω, ENNReal.ofReal (f r ω) ∂P) u := by + rw [ofReal_integral_eq_lintegral_ofReal hg hg0] + rw [← lintegral_congr_ae hlim] + exact lintegral_liminf_le (fun r => (hmeas r).ennreal_ofReal) + have hhalf : ENNReal.ofReal ((∫ ω, g ω ∂P)/2) < + ENNReal.ofReal (∫ ω, g ω ∂P) := by + exact (ENNReal.ofReal_lt_ofReal_iff hgpos).mpr (by linarith) + have hevent := eventually_lt_of_lt_liminf (hhalf.trans_le hfatou) + filter_upwards [hevent, hfint] with r hr hri + rw [← ofReal_integral_eq_lintegral_ofReal hri + (Eventually.of_forall (fun ω => hnonneg r ω))] at hr + exact (ENNReal.ofReal_lt_ofReal_iff'.mp hr).1.le + +private theorem nfloor_weight_continuous : + Continuous (fun p : ℝ × (Fin 6 → ℝ) => gn_normalizedWeight p.1 p.2) := by + unfold gn_normalizedWeight + fun_prop +private theorem nfloor_contact_value (a q : ℝ) : + gn_normalizedWeight 0 ![-1,1,-a/2,a/2,q,q] = (max (-q) 0)^2 := by + simp [gn_normalizedWeight, pow_two] +private theorem nfloor_weight_limit (X : ℝ → Fin 6 → ℝ) (a q : ℝ) + (hX : Tendsto X (𝓝[>] (0:ℝ)) (𝓝 ![-1,1,-a/2,a/2,q,q])) : + Tendsto (fun r => gn_normalizedWeight r (X r)) (𝓝[>] (0:ℝ)) + (𝓝 ((max (-q) 0)^2)) := by + have hr : Tendsto (fun r : ℝ => r) (𝓝[>] (0:ℝ)) (𝓝 0) := + continuousWithinAt_id + simpa only [nfloor_contact_value, Function.comp_def] using + (nfloor_weight_continuous.tendsto (0, ![-1,1,-a/2,a/2,q,q])).comp (hr.prodMk_nhds hX) +private theorem nfloor_reference {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (Q : Ω → ℝ) (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) + (hlaw : HasLaw Q (gaussianReal m v) P) : + Integrable (fun ω => (max (-Q ω) 0)^2) P ∧ + 0 < ∫ ω, (max (-Q ω) 0)^2 ∂P := by + obtain ⟨hInt,hPos⟩ := gn_gaussian_reference_integrable_pos m v hv + refine ⟨hlaw.integrable_comp hInt, ?_⟩ + have heq := hlaw.integral_comp + (show AEStronglyMeasurable (fun q : ℝ => (max (-q) 0)^2) (gaussianReal m v) by fun_prop) + exact heq.trans_gt hPos + +/-- The actual integral of the physical r²-scaled continuous endpoint weight +has a positive lower constant at every sufficiently small positive radius. +The common coupling and current integrability are explicit premises; its +positive reference integral and the lower bound are proved, not premises. -/ +theorem gn_normalizer_floor_of_common_coupling + {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) [IsProbabilityMeasure P] + (X : ℝ → Ω → Fin 6 → ℝ) (hX : ∀ r, Measurable (X r)) + (A Q : Ω → ℝ) (m : ℝ) (v : ℝ≥0) (hv : v ≠ 0) + (hQ : HasLaw Q (gaussianReal m v) P) + (hconv : ∀ᵐ ω ∂P, Tendsto (fun r => X r ω) (𝓝[>] (0 : ℝ)) + (𝓝 ![-1,1,-A ω/2,A ω/2,Q ω,Q ω])) + (hint : ∀ᶠ r in 𝓝[>] (0 : ℝ), + Integrable (fun ω => gn_normalizedWeight r (X r ω)) P) : + ∃ c : ℝ, 0 < c ∧ ∃ δ : ℝ, 0 < δ ∧ + ∀ r : ℝ, 0 < r → r < δ → + c * r^2 ≤ ∫ ω, r^2 * gn_normalizedWeight r (X r ω) ∂P := by + let g : Ω → ℝ := fun ω => (max (-Q ω) 0)^2 + obtain ⟨hg, hgpos⟩ := nfloor_reference P Q m v hv hQ + have hmeas (r : ℝ) : Measurable (fun ω => gn_normalizedWeight r (X r ω)) := by + have hc : Continuous (gn_normalizedWeight r) := by + unfold gn_normalizedWeight + fun_prop + exact hc.measurable.comp (hX r) + have hnonneg (r : ℝ) (ω : Ω) : 0 ≤ gn_normalizedWeight r (X r ω) := + mul_nonneg (le_max_right _ _) (le_max_right _ _) + have hweightconv : ∀ᵐ ω ∂P, + Tendsto (fun r => gn_normalizedWeight r (X r ω)) (𝓝[>] (0 : ℝ)) (𝓝 (g ω)) := by + filter_upwards [hconv] with ω hω + exact nfloor_weight_limit (fun r => X r ω) (A ω) (Q ω) hω + have hfloor : ∀ᶠ r in 𝓝[>] (0 : ℝ), + (∫ ω, g ω ∂P)/2 ≤ ∫ ω, gn_normalizedWeight r (X r ω) ∂P := + nfloor_eventually_lower_of_fatou P (𝓝[>] (0 : ℝ)) + (fun r ω => gn_normalizedWeight r (X r ω)) g hmeas hnonneg hg hgpos hweightconv hint + obtain ⟨δ, hδ, hinterval⟩ := mem_nhdsGT_iff_exists_Ioo_subset.mp hfloor + refine ⟨(∫ ω, g ω ∂P)/2, by positivity, δ, hδ, ?_⟩ + intro r hr hrδ + have hbound := hinterval ⟨hr, hrδ⟩ + rw [integral_const_mul] + simpa only [mul_comm] using mul_le_mul_of_nonneg_left hbound (sq_nonneg r) + +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/PalmReweighting.lean b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/PalmReweighting.lean new file mode 100644 index 000000000..f0d6abafc --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/ResearchFormalCoreR1/PalmReweighting.lean @@ -0,0 +1,139 @@ +import Mathlib.MeasureTheory.Function.L2Space +import Mathlib.MeasureTheory.Measure.WithDensity +import Mathlib.MeasureTheory.Integral.Bochner.Set +import Mathlib.MeasureTheory.Measure.Real +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum +import Mathlib.Tactic.Positivity +import Mathlib.Tactic.FieldSimp + +/-! +# A canonical weighted probability and its square-root event bound + +This constructs the literal measure with density `W / integral W`, proves its +event formula, and applies integral Cauchy–Schwarz. The determinant second +moment and normalizer estimates cancel their exact radius powers. Identifying +this measure with a geometric Kac–Rice Palm law remains an application join. +See ../SOURCE_PALM_REWEIGHTING.md for the frozen source contract. +-/ + +namespace ResearchFormalCoreR1 + +open MeasureTheory +open scoped ENNReal + +set_option autoImplicit false + +/-- The literal normalized nonnegative real density, defined for every input. -/ +noncomputable def prwMeasure {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) : Measure Ω := + μ.withDensity (fun ω => ENNReal.ofReal (W ω / ∫ x, W x ∂μ)) + +/-- On a finite measure, a measurable square-integrable real weight is integrable. -/ +theorem prw_integrable_of_integrable_sq {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsFiniteMeasure μ] (W : Ω → ℝ) (hW : Measurable W) + (hsq : Integrable (fun ω => W ω ^ 2) μ) : Integrable W μ := by + have hmajor : Integrable (fun ω => 1 + W ω ^ 2) μ := + (integrable_const 1).add hsq + apply hmajor.mono' hW.aestronglyMeasurable + filter_upwards [] with ω + rw [Real.norm_eq_abs] + nlinarith [sq_nonneg (|W ω| - 1 / 2), sq_abs (W ω)] + +/-- The actual event mass is the normalized set integral, in extended reals. -/ +theorem prw_measure_apply {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + prwMeasure μ W E = ENNReal.ofReal ((∫ x in E, W x ∂μ) / ∫ x, W x ∂μ) := by + unfold prwMeasure + rw [withDensity_apply _ hE] + rw [← ofReal_integral_eq_lintegral_ofReal + (hW.integrableOn.div_const _) (Filter.Eventually.of_forall (fun ω => + div_nonneg (hW0 ω) hZ.le))] + rw [integral_div] + +/-- Positive finite normalizer makes the literal density a probability law. -/ +theorem prw_isProbabilityMeasure {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) : + IsProbabilityMeasure (prwMeasure μ W) := by + constructor + rw [prw_measure_apply μ W hW hW0 hZ Set.univ MeasurableSet.univ] + simp [hZ.ne'] + +/-- The event identity is proved for the constructed law, not assumed. -/ +theorem prw_measure_real {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) (W : Ω → ℝ) (hW : Integrable W μ) + (hW0 : ∀ ω, 0 ≤ W ω) (hZ : 0 < ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + (prwMeasure μ W).real E = (∫ x in E, W x ∂μ) / ∫ x, W x ∂μ := by + rw [measureReal_def, prw_measure_apply μ W hW hW0 hZ E hE, + ENNReal.toReal_ofReal (div_nonneg (integral_nonneg hW0) hZ.le)] + +/-- Integral Cauchy–Schwarz against the event indicator, including null events. -/ +theorem prw_numerator_bound {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsFiniteMeasure μ] (W : Ω → ℝ) (hW : Measurable W) + (hW0 : ∀ ω, 0 ≤ W ω) (hsq : Integrable (fun ω => W ω ^ 2) μ) + (E : Set Ω) (hE : MeasurableSet E) : + (∫ x in E, W x ∂μ) ≤ + Real.sqrt (∫ x, W x ^ 2 ∂μ) * Real.sqrt (μ.real E) := by + classical + let I : Ω → ℝ := E.indicator (fun _ => 1) + have hI0 : ∀ ω, 0 ≤ I ω := by + intro ω + simp only [I, Set.indicator_apply] + split_ifs <;> norm_num + have hWLp : MemLp W (ENNReal.ofReal 2) μ := by + simpa using (memLp_two_iff_integrable_sq hW.aestronglyMeasurable).mpr hsq + have hILp : MemLp I (ENNReal.ofReal 2) μ := + (memLp_const (1 : ℝ)).indicator hE + have hCS := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two + (Filter.Eventually.of_forall hW0) (Filter.Eventually.of_forall hI0) hWLp hILp + have hleft : (fun ω => W ω * I ω) = E.indicator W := by + funext ω + by_cases hω : ω ∈ E <;> simp [I, hω] + have hright : (fun ω => I ω ^ (2 : ℝ)) = E.indicator (fun _ => (1 : ℝ)) := by + funext ω + by_cases hω : ω ∈ E <;> simp [I, hω] + have hIint : (∫ ω, I ω ^ (2 : ℝ) ∂μ) = μ.real E := by + rw [hright] + change (∫ ω, E.indicator (1 : Ω → ℝ) ω ∂μ) = μ.real E + exact integral_indicator_one hE + rw [hleft, integral_indicator hE, hIint] at hCS + simpa only [Real.rpow_two, ← Real.sqrt_eq_rpow] using hCS + +/-- The exact `r^4` second moment and `r^2` normalizer cancel in the literal +weighted probability. No event probability is assumed positive. -/ +theorem prw_square_root_event_bound {Ω : Type*} [MeasurableSpace Ω] + (μ : Measure Ω) [IsProbabilityMeasure μ] (W : Ω → ℝ) (hW : Measurable W) + (hW0 : ∀ ω, 0 ≤ W ω) (hsq : Integrable (fun ω => W ω ^ 2) μ) + (C c r : ℝ) (hC : 0 ≤ C) (hc : 0 < c) (hr : 0 < r) + (hmoment : (∫ x, W x ^ 2 ∂μ) ≤ C * r ^ 4) + (hfloor : c * r ^ 2 ≤ ∫ x, W x ∂μ) + (E : Set Ω) (hE : MeasurableSet E) : + IsProbabilityMeasure (prwMeasure μ W) ∧ + (prwMeasure μ W).real E ≤ (Real.sqrt C / c) * Real.sqrt (μ.real E) := by + have hZ : 0 < ∫ x, W x ∂μ := + (mul_pos hc (sq_pos_of_pos hr)).trans_le hfloor + have hW1 : Integrable W μ := prw_integrable_of_integrable_sq μ W hW hsq + refine ⟨prw_isProbabilityMeasure μ W hW1 hW0 hZ, ?_⟩ + rw [prw_measure_real μ W hW1 hW0 hZ E hE] + have hnum := prw_numerator_bound μ W hW hW0 hsq E hE + have hsqrt : Real.sqrt (∫ x, W x ^ 2 ∂μ) ≤ Real.sqrt C * r ^ 2 := by + calc + _ ≤ Real.sqrt (C * r ^ 4) := Real.sqrt_le_sqrt hmoment + _ = Real.sqrt C * r ^ 2 := by + rw [Real.sqrt_mul hC] + congr 1 + have heq : r ^ 4 = (r ^ 2) ^ 2 := by ring + rw [heq, Real.sqrt_sq (sq_nonneg r)] + calc + _ ≤ (Real.sqrt C * r ^ 2 * Real.sqrt (μ.real E)) / (c * r ^ 2) := by + apply div_le_div₀ (by positivity) + (hnum.trans (mul_le_mul_of_nonneg_right hsqrt (Real.sqrt_nonneg _))) + (mul_pos hc (sq_pos_of_pos hr)) hfloor + _ = (Real.sqrt C / c) * Real.sqrt (μ.real E) := by + field_simp [hr.ne'] + +end ResearchFormalCoreR1 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceC4.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceC4.lean new file mode 100644 index 000000000..f5188d51c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceC4.lean @@ -0,0 +1,141 @@ +import SourceFourierSeries +import Mathlib.Analysis.Calculus.SmoothSeries +import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv + +/-! Actual same-sample C4 of the canonical normalized Fourier lift. +Covariance identification, Gaussian field/jet laws and pin regression are separate. -/ + +noncomputable section +open I4Foundation I4Weights I4Source MeasureTheory ProbabilityTheory +open scoped BigOperators + +namespace I4C4 + +def phaseCLM (d : ℕ) (L : ℝ) (k : Fin d → ℤ) : (Fin d → ℝ) →L[ℝ] ℝ := + ∑ i, (sourceC L * (k i : ℝ)) • ContinuousLinearMap.proj i + +theorem phaseCLM_apply (d : ℕ) (L : ℝ) (k : Fin d → ℤ) (x : Fin d → ℝ) : + phaseCLM d L k x = phase d L k x := by + simp [phaseCLM, phase] + +theorem phaseCLM_norm_le (d : ℕ) (L : ℝ) (k : Fin d → ℤ) : + ‖phaseCLM d L k‖ ≤ ∑ i, |sourceC L * (k i : ℝ)| := by + apply ContinuousLinearMap.opNorm_le_bound _ + (Finset.sum_nonneg (fun i _ => abs_nonneg _)) + intro x + rw [phaseCLM_apply] + calc + ‖phase d L k x‖ ≤ ∑ i, ‖sourceC L * (k i : ℝ) * x i‖ := norm_sum_le _ _ + _ ≤ ∑ i, |sourceC L * (k i : ℝ)| * ‖x‖ := by + apply Finset.sum_le_sum + intro i _ + rw [norm_mul, Real.norm_eq_abs] + exact mul_le_mul_of_nonneg_left (norm_le_pi_norm x i) (abs_nonneg _) + _ = (∑ i, |sourceC L * (k i : ℝ)|) * ‖x‖ := by rw [Finset.sum_mul] + +theorem modeCoefficient_iteratedFDeriv_bound (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (m : SourceMode d) (j : ℕ) (hj : j ≤ 4) (x : Fin d → ℝ) : + ‖iteratedFDeriv ℝ j (modeCoefficient d L m) x‖ ≤ modeMajorant d L m := by + classical + cases m with + | none => + cases j with + | zero => + simpa only [norm_iteratedFDeriv_zero, Real.norm_eq_abs] using + modeCoefficient_bound d hL none x + | succ j => + change ‖iteratedFDeriv ℝ (j + 1) + (fun _ : Fin d → ℝ => Real.sqrt (1 / sourceZ d L)) x‖ ≤ Real.sqrt (1 / sourceZ d L) + rw [iteratedFDeriv_succ_const] + simp only [Pi.zero_apply, norm_zero] + exact Real.sqrt_nonneg _ + | some p => + rcases p with ⟨k, b⟩ + let A := Real.sqrt (2 * sourceWeight d L k / sourceZ d L) + let P := ∏ i, (1 + |sourceC L * (k.1 i : ℝ)|) + let g : ℝ → ℝ := if b then Real.sin else Real.cos + let ell := phaseCLM d L k + have hg : ContDiff ℝ j g := by + cases b <;> simp only [g, Bool.false_eq_true, ite_false, ite_true] <;> fun_prop + have hgc : ContDiff ℝ j (g ∘ ell) := hg.comp_continuousLinearMap + have hgn : ‖iteratedFDeriv ℝ j g (ell x)‖ ≤ 1 := by + cases b with + | false => + simpa only [g, Bool.false_eq_true, ite_false, + norm_iteratedFDeriv_eq_norm_iteratedDeriv, Real.norm_eq_abs] using + Real.abs_iteratedDeriv_cos_le_one j (ell x) + | true => + simpa only [g, ite_true, norm_iteratedFDeriv_eq_norm_iteratedDeriv, + Real.norm_eq_abs] using Real.abs_iteratedDeriv_sin_le_one j (ell x) + have hP : 1 ≤ P := by + apply Finset.one_le_prod₀ + intro i _ + linarith [abs_nonneg (sourceC L * (k.1 i : ℝ))] + have hl : ‖ell‖ ≤ P := by + calc + ‖ell‖ ≤ ∑ i, |sourceC L * (k.1 i : ℝ)| := phaseCLM_norm_le d L k + _ ≤ 1 + ∑ i, |sourceC L * (k.1 i : ℝ)| := by linarith + _ ≤ P := frequency_l1_le_product d _ (fun i => abs_nonneg _) + have hpow : ‖ell‖ ^ j ≤ P ^ 4 := by + calc + ‖ell‖ ^ j ≤ P ^ j := by gcongr + _ ≤ P ^ 4 := pow_le_pow_right₀ hP hj + have hder : ‖iteratedFDeriv ℝ j (g ∘ ell) x‖ ≤ ‖ell‖ ^ j := by + rw [ell.iteratedFDeriv_comp_right hg x le_rfl] + calc + _ ≤ ‖iteratedFDeriv ℝ j g (ell x)‖ * ∏ _i : Fin j, ‖ell‖ := + (iteratedFDeriv ℝ j g (ell x)).norm_compContinuousLinearMap_le (fun _ => ell) + _ ≤ 1 * ∏ _i : Fin j, ‖ell‖ := + mul_le_mul_of_nonneg_right hgn (Finset.prod_nonneg (fun _ _ => norm_nonneg _)) + _ = ‖ell‖ ^ j := by simp + have heq : modeCoefficient d L (some (k, b)) = A • (g ∘ ell) := by + funext y + cases b <;> simp [modeCoefficient, A, g, Function.comp_apply, Pi.smul_apply, + smul_eq_mul, ell, phaseCLM_apply] + rw [heq, iteratedFDeriv_const_smul_apply hgc.contDiffAt, norm_smul, + Real.norm_eq_abs, abs_of_nonneg (Real.sqrt_nonneg _)] + calc + _ ≤ A * P ^ 4 := mul_le_mul_of_nonneg_left (hder.trans hpow) (Real.sqrt_nonneg _) + _ = modeMajorant d L (some (k, b)) := by + rw [modeMajorant, pairedBound_eq_amplitude d hL k] + simp only [A, P, Finset.prod_pow] + +theorem sourceField_contDiff_ae (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + ∀ᵐ ω ∂sourceLaw d, ContDiff ℝ 4 (sourceField d L ω) := by + have hc (m : SourceMode d) : ContDiff ℝ 4 (modeCoefficient d L m) := by + cases m with + | none => + change ContDiff ℝ 4 (fun _ : Fin d → ℝ => Real.sqrt (1 / sourceZ d L)) + fun_prop + | some p => + rcases p with ⟨k, b⟩ + change ContDiff ℝ 4 (fun x : Fin d → ℝ => + Real.sqrt (2 * sourceWeight d L k / sourceZ d L) * + (if b then Real.sin (phase d L k x) else Real.cos (phase d L k x))) + cases b <;> simp [phase] <;> fun_prop + filter_upwards [sourceMajorant_ae d hL] with ω hω + change ContDiff ℝ 4 (fun x => ∑' m, modeCoefficient d L m x * ω m) + apply contDiff_tsum (v := fun _ m => modeMajorant d L m * |ω m|) + · intro m + exact (hc m).mul contDiff_const + · intro j hj + exact hω + · intro j m x hj + have hj' : j ≤ 4 := by exact_mod_cast hj + have hcj : ContDiff ℝ j (modeCoefficient d L m) := + (hc m).of_le (by exact_mod_cast hj') + have heq : (fun y => modeCoefficient d L m y * ω m) = ω m • modeCoefficient d L m := by + funext y + simp [smul_eq_mul, mul_comm] + rw [heq, iteratedFDeriv_const_smul_apply hcj.contDiffAt, norm_smul, Real.norm_eq_abs] + simpa only [mul_comm] using mul_le_mul_of_nonneg_left + (modeCoefficient_iteratedFDeriv_bound d hL m j hj' x) (abs_nonneg (ω m)) + +theorem sourceField_contDiff_affine_ae (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + ∀ᵐ ω ∂sourceLaw d, ∀ (T : (Fin d → ℝ) →L[ℝ] (Fin d → ℝ)) (a : Fin d → ℝ), + ContDiff ℝ 4 (fun x => sourceField d L ω (a + T x)) := by + filter_upwards [sourceField_contDiff_ae d hL] with ω hω + intro T a + exact hω.comp (contDiff_const.add T.contDiff) + +end I4C4 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceCovariance.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceCovariance.lean new file mode 100644 index 000000000..50aa85f30 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceCovariance.lean @@ -0,0 +1,74 @@ +import SourceKernel +import SourceGaussianLaw +import Mathlib.Probability.Moments.Covariance +import Mathlib.Probability.Moments.Variance + +noncomputable section +open I4Foundation I4Weights I4Source I4Kernel I4SourceGaussian MeasureTheory ProbabilityTheory +open scoped BigOperators ENNReal NNReal +namespace I4SourceCovariance + +variable {ι : Type*} [Countable ι] + +theorem scalarSeries_covariance (a b : ι → ℝ) (ha : Summable a) (hb : Summable b) : + cov[scalarSeries a, scalarSeries b; coefficientLaw ι] = ∑' i, a i * b i := by + have hsq (f : ι → ℝ) (hf : Summable f) : Summable (fun i => f i ^ 2) := by + have hp := summable_mul_of_summable_norm hf.norm hf.norm + simpa only [Function.comp_def, pow_two] using hp.comp_injective + (show Function.Injective (fun i : ι => (i,i)) from + fun i j h => congrArg Prod.fst h) + have hprod : Summable (fun i => a i * b i) := by + have hp := summable_mul_of_summable_norm ha.norm hb.norm + simpa only [Function.comp_def] using hp.comp_injective + (show Function.Injective (fun i : ι => (i,i)) from + fun i j h => congrArg Prod.fst h) + have hLp (f : ι → ℝ) (hf : Summable f) : + MemLp (scalarSeries f) 2 (coefficientLaw ι) := by + have h : MemLp id 2 ((coefficientLaw ι).map (scalarSeries f)) := by + rw [scalarSeries_map_gaussian f hf] + exact memLp_id_gaussianReal' 2 (by norm_num) + simpa only [Function.comp_def, Function.id_def] using + h.comp_of_map (scalarSeries_measurable f).aemeasurable + have heq : scalarSeries (a + b) =ᵐ[coefficientLaw ι] + (fun ω => scalarSeries a ω + scalarSeries b ω) := by + filter_upwards [scalarSeries_hasSum_ae a ha, scalarSeries_hasSum_ae b hb, + scalarSeries_hasSum_ae (a+b) (ha.add hb)] with ω haω hbω habω + exact habω.unique (by simpa only [Pi.add_apply, add_mul] using haω.add hbω) + have hts : (∑' i, (a i + b i)^2) = + (∑' i, a i ^ 2) + 2 * (∑' i, a i * b i) + (∑' i, b i ^ 2) := by + calc + _ = ∑' i, (a i ^ 2 + 2 * (a i * b i) + b i ^ 2) := by + exact tsum_congr (fun i => by ring) + _ = _ := by + rw [((hsq a ha).add (hprod.mul_left 2)).tsum_add (hsq b hb), + (hsq a ha).tsum_add (hprod.mul_left 2), tsum_mul_left] + have hv := variance_fun_add (hLp a ha) (hLp b hb) + rw [← variance_congr heq, scalarSeries_variance (a+b) (ha.add hb), + scalarSeries_variance a ha, scalarSeries_variance b hb] at hv + change (∑' i, (a i+b i)^2) = _ at hv + rw [hts] at hv + linarith + +theorem sourceField_integral_zero (d : ℕ) {L : ℝ} (hL : L ≠ 0) (x : Fin d → ℝ) : + ∫ ω, sourceField d L ω x ∂sourceLaw d = 0 := by + have hc : Summable (fun m => modeCoefficient d L m x) := + (modeMajorant_summable d hL).of_norm_bounded (fun m => by + simpa only [Real.norm_eq_abs] using modeCoefficient_bound d hL m x) + exact scalarSeries_integral_zero (fun m => modeCoefficient d L m x) hc + +theorem sourceField_covariance (d : ℕ) {L : ℝ} (hL : L ≠ 0) (x y : Fin d → ℝ) : + cov[fun ω => sourceField d L ω x, fun ω => sourceField d L ω y; sourceLaw d] = + sourceSpectralKernel d L (x - y) := by + have hc (z : Fin d → ℝ) : Summable (fun m => modeCoefficient d L m z) := + (modeMajorant_summable d hL).of_norm_bounded (fun m => by + simpa only [Real.norm_eq_abs] using modeCoefficient_bound d hL m z) + change cov[scalarSeries (fun m => modeCoefficient d L m x), + scalarSeries (fun m => modeCoefficient d L m y); coefficientLaw (SourceMode d)] = _ + rw [scalarSeries_covariance _ _ (hc x) (hc y), coefficientKernel_eq_fullLattice d hL x y] + +theorem sourceField_variance_one (d : ℕ) {L : ℝ} (hL : L ≠ 0) (x : Fin d → ℝ) : + Var[fun ω => sourceField d L ω x; sourceLaw d] = 1 := by + rw [← covariance_self (sourceField_eval_measurable d L x).aemeasurable, + sourceField_covariance d hL x x, sub_self, spectralKernel_zero d hL] + +end I4SourceCovariance diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceFourierSeries.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceFourierSeries.lean new file mode 100644 index 000000000..e4cdec9e6 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceFourierSeries.lean @@ -0,0 +1,147 @@ +import GaussianSeriesFoundation +import SpectralWeights +import Mathlib.Logic.Equiv.Option +import Mathlib.Topology.Algebra.InfiniteSum.TsumUniformlyOn + +/-! Actual canonical real Fourier series and a shared-event uniform convergence bridge. +The half-lattice partition, covariance, sample C4 and finite-jet law require separate proofs. -/ + +noncomputable section +open MeasureTheory ProbabilityTheory I4Foundation I4Weights +open scoped BigOperators ENNReal + +namespace I4Source + +/-- The representative whose first nonzero coordinate is positive. -/ +def halfLattice (d : ℕ) : Set (Fin d → ℤ) := + {k | ∃ i, 0 < k i ∧ ∀ j, j < i → k j = 0} + +abbrev SourceMode (d : ℕ) := Option (↥(halfLattice d) × Bool) + +/-- The actual common law, with independent variance-one real coefficients. -/ +abbrev sourceLaw (d : ℕ) : Measure (SourceMode d → ℝ) := + coefficientLaw (SourceMode d) + +def phase (d : ℕ) (L : ℝ) (k : Fin d → ℤ) (x : Fin d → ℝ) : ℝ := + ∑ i, sourceC L * (k i : ℝ) * x i + +/-- False is the cosine coefficient and true the independent sine coefficient. -/ +def modeCoefficient (d : ℕ) (L : ℝ) (m : SourceMode d) (x : Fin d → ℝ) : ℝ := + match m with + | none => Real.sqrt (1 / sourceZ d L) + | some (k, b) => Real.sqrt (2 * sourceWeight d L k / sourceZ d L) * + if b then Real.sin (phase d L k x) else Real.cos (phase d L k x) + +def modeMajorant (d : ℕ) (L : ℝ) (m : SourceMode d) : ℝ := + match m with + | none => Real.sqrt (1 / sourceZ d L) + | some (k, _) => pairedBound d L k + +/-- Literal normalized real Fourier sum on the common coefficient samples. -/ +def sourceField (d : ℕ) (L : ℝ) (ω : SourceMode d → ℝ) (x : Fin d → ℝ) : ℝ := + scalarSeries (fun m => modeCoefficient d L m x) ω + +theorem modeMajorant_nonneg (d : ℕ) (L : ℝ) (m : SourceMode d) : + 0 ≤ modeMajorant d L m := by + cases m with + | none => exact Real.sqrt_nonneg _ + | some p => + change 0 ≤ pairedBound d L p.1 + unfold pairedBound productBound coordinateBound decay + positivity + +theorem modeMajorant_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + Summable (modeMajorant d L) := by + classical + have hs := pairedBound_subtype_summable d hL (halfLattice d) + have hb : Summable (fun _ : Bool => (1 : ℝ)) := (hasSum_fintype _).summable + have hp : Summable (fun p : ↥(halfLattice d) × Bool => pairedBound d L p.1) := by + simpa only [mul_one] using hs.mul_of_nonneg hb + (fun k => modeMajorant_nonneg d L (some (k, false))) (fun _ => zero_le_one) + let e := Equiv.optionEquivSumPUnit.{0, 0} (↥(halfLattice d) × Bool) + apply e.symm.summable_iff.mp + apply Summable.sum (fun q => modeMajorant d L (e.symm q)) + · simpa [e, Function.comp_def, modeMajorant] using hp + · exact (hasSum_fintype _).summable + +theorem modeCoefficient_bound (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (m : SourceMode d) (x : Fin d → ℝ) : + |modeCoefficient d L m x| ≤ modeMajorant d L m := by + cases m with + | none => + exact (abs_of_nonneg (Real.sqrt_nonneg _)).le + | some p => + rcases p with ⟨k, b⟩ + have hprod : 1 ≤ ∏ i, (1 + |sourceC L * (k.val i : ℝ)|) ^ 4 := by + apply Finset.one_le_prod₀ + intro i _ + apply one_le_pow₀ + linarith [abs_nonneg (sourceC L * (k.val i : ℝ))] + have hamp : Real.sqrt (2 * sourceWeight d L k / sourceZ d L) ≤ + pairedBound d L k := by + rw [pairedBound_eq_amplitude d hL k] + simpa only [mul_one] using mul_le_mul_of_nonneg_left hprod + (Real.sqrt_nonneg (2 * sourceWeight d L k / sourceZ d L)) + change |Real.sqrt (2 * sourceWeight d L k / sourceZ d L) * + (if b then Real.sin (phase d L k x) else Real.cos (phase d L k x))| ≤ _ + rw [abs_mul, abs_of_nonneg (Real.sqrt_nonneg _)] + have ht : |if b then Real.sin (phase d L k x) else Real.cos (phase d L k x)| ≤ 1 := by + cases b with + | false => exact Real.abs_cos_le_one _ + | true => exact Real.abs_sin_le_one _ + have hbase : Real.sqrt (2 * sourceWeight d L k / sourceZ d L) * + |if b then Real.sin (phase d L k x) else Real.cos (phase d L k x)| ≤ + Real.sqrt (2 * sourceWeight d L k / sourceZ d L) := by + simpa only [mul_one] using mul_le_mul_of_nonneg_left ht + (Real.sqrt_nonneg (2 * sourceWeight d L k / sourceZ d L)) + exact hbase.trans hamp + +theorem sourceField_eval_measurable (d : ℕ) (L : ℝ) (x : Fin d → ℝ) : + Measurable (fun ω => sourceField d L ω x) := by + exact scalarSeries_measurable _ + +theorem sourceMajorant_ae (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + ∀ᵐ ω ∂sourceLaw d, Summable (fun m => modeMajorant d L m * |ω m|) := by + exact weighted_abs_summable_ae _ (modeMajorant_nonneg d L) (modeMajorant_summable d hL) + +theorem sourceAbs_uniform_ae (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + ∀ᵐ ω ∂sourceLaw d, + HasSumUniformly (fun m x => |modeCoefficient d L m x * ω m|) + (fun x => ∑' m, |modeCoefficient d L m x * ω m|) := by + filter_upwards [sourceMajorant_ae d hL] with ω hω + apply hasSumUniformly_iff_tendstoUniformly.mpr + apply tendstoUniformly_tsum hω + intro m x + simp only [Real.norm_eq_abs, abs_abs, abs_mul] + exact mul_le_mul_of_nonneg_right (modeCoefficient_bound d hL m x) (abs_nonneg _) + +theorem sourceField_uniform_ae (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + ∀ᵐ ω ∂sourceLaw d, + HasSumUniformly (fun m x => modeCoefficient d L m x * ω m) + (sourceField d L ω) := by + filter_upwards [sourceMajorant_ae d hL] with ω hω + apply hasSumUniformly_iff_tendstoUniformly.mpr + apply tendstoUniformly_tsum hω + intro m x + simp only [Real.norm_eq_abs, abs_mul] + exact mul_le_mul_of_nonneg_right (modeCoefficient_bound d hL m x) (abs_nonneg _) + +theorem sourceField_continuous_ae (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + ∀ᵐ ω ∂sourceLaw d, Continuous (sourceField d L ω) := by + filter_upwards [sourceMajorant_ae d hL] with ω hω + change Continuous (fun x => ∑' m, modeCoefficient d L m x * ω m) + have hcont (m : SourceMode d) : + Continuous (fun x : Fin d → ℝ => modeCoefficient d L m x * ω m) := by + cases m with + | none => + change Continuous (fun _ : Fin d → ℝ => Real.sqrt (1 / sourceZ d L) * ω none) + exact continuous_const + | some p => + rcases p with ⟨k, b⟩ + cases b <;> simp [modeCoefficient, phase] <;> fun_prop + apply continuous_tsum hcont hω + intro m x + simp only [Real.norm_eq_abs, abs_mul] + exact mul_le_mul_of_nonneg_right (modeCoefficient_bound d hL m x) (abs_nonneg _) + +end I4Source diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceGaussianLaw.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceGaussianLaw.lean new file mode 100644 index 000000000..bb8450b5a --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceGaussianLaw.lean @@ -0,0 +1,189 @@ +import SourceFourierSeries +import Mathlib.MeasureTheory.Integral.DominatedConvergence +import Mathlib.Probability.Independence.CharacteristicFunction +import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence +import Mathlib.Analysis.Normed.Ring.InfiniteSum +import Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Def + +/-! Actual Gaussian laws of summable coefficient series and finite evaluation combinations. +No covariance/source-kernel, torus, derivative-jet, or pinned-law conclusion is assumed. -/ + +noncomputable section +open I4Foundation I4Weights I4Source MeasureTheory ProbabilityTheory +open scoped BigOperators ENNReal NNReal Topology + +namespace I4SourceGaussian + +variable {ι : Type*} + +private theorem summable_square (a : ι → ℝ) (ha : Summable a) : + Summable (fun i => a i ^ 2) := by + have h := summable_mul_of_summable_norm ha.norm ha.norm + simpa only [Function.comp_def, pow_two] using h.comp_injective + (show Function.Injective (fun i : ι => (i, i)) from + fun i j hij => congrArg Prod.fst hij) + +variable [Countable ι] + +omit [Countable ι] in +private theorem partial_charFun (a : ι → ℝ) (s : Finset ι) (t : ℝ) : + charFun ((coefficientLaw ι).map (fun ω => ∑ i ∈ s, a i * ω i)) t = + Complex.exp (-((∑ i ∈ s, a i ^ 2 : ℝ) : ℂ) * (t : ℂ) ^ 2 / 2) := by + classical + have hi : iIndepFun (fun i (ω : ι → ℝ) => a i * ω i) (coefficientLaw ι) := + (coordinates_independent ι).comp (fun i y => a i * y) (fun _ => by fun_prop) + have hm (i : ι) : Measurable (fun ω : ι → ℝ => a i * ω i) := + measurable_const.mul (measurable_pi_apply i) + have hchar (i : ι) : charFun ((coefficientLaw ι).map + (fun ω : ι → ℝ => a i * ω i)) t = + Complex.exp (-((a i : ℂ) * (t : ℂ)) ^ 2 / 2) := by + rw [charFun_map_mul_comp (f := fun ω : ι → ℝ => ω i) + (measurable_pi_apply i).aemeasurable, coordinate_marginal, charFun_gaussianReal] + simp only [Complex.ofReal_mul, mul_zero, Complex.ofReal_zero, zero_mul, + NNReal.coe_one, Complex.ofReal_one, one_mul, zero_sub] + congr 1 + ring + rw [(hi.restrict s).charFun_map_fun_finsetSum_eq_prod (fun i _ => (hm i).aemeasurable)] + simp only [Finset.prod_apply] + simp_rw [hchar] + rw [← Complex.exp_sum] + congr 1 + push_cast + simp only [mul_pow] + rw [← Finset.sum_div, Finset.sum_neg_distrib, ← Finset.sum_mul] + rw [neg_mul] + +private theorem charFun_partial_tendsto (a : ι → ℝ) (ha : Summable a) (t : ℝ) : + Filter.Tendsto + (fun s : Finset ι => charFun ((coefficientLaw ι).map + (fun ω => ∑ i ∈ s, a i * ω i)) t) + Filter.atTop (𝓝 (charFun ((coefficientLaw ι).map (scalarSeries a)) t)) := by + classical + have hm (s : Finset ι) : Measurable (fun ω : ι → ℝ => ∑ i ∈ s, a i * ω i) := + Finset.measurable_sum s (fun i _ => measurable_const.mul (measurable_pi_apply i)) + have heq (s : Finset ι) : + charFun ((coefficientLaw ι).map (fun ω => ∑ i ∈ s, a i * ω i)) t = + ∫ ω, Complex.exp (((t * ∑ i ∈ s, a i * ω i : ℝ) : ℂ) * Complex.I) + ∂coefficientLaw ι := by + rw [charFun_apply_real, integral_map (hm s).aemeasurable (by fun_prop)] + push_cast + rfl + have heq' : charFun ((coefficientLaw ι).map (scalarSeries a)) t = + ∫ ω, Complex.exp (((t * scalarSeries a ω : ℝ) : ℂ) * Complex.I) + ∂coefficientLaw ι := by + rw [charFun_apply_real, integral_map (scalarSeries_measurable a).aemeasurable + (by fun_prop)] + push_cast + rfl + simp_rw [heq, heq'] + apply tendsto_integral_filter_of_norm_le_const + · filter_upwards with s + have hc : Continuous (fun q : ℝ => Complex.exp (((t * q : ℝ) : ℂ) * Complex.I)) := + by fun_prop + exact (hc.measurable.comp (hm s)).aestronglyMeasurable + · refine ⟨1, Filter.Eventually.of_forall (fun s => ?_)⟩ + filter_upwards with ω + rw [Complex.norm_exp_ofReal_mul_I] + · filter_upwards [scalarSeries_hasSum_ae a ha] with ω hω + exact ((by fun_prop : Continuous (fun q : ℝ => + Complex.exp (((t * q : ℝ) : ℂ) * Complex.I))).tendsto _).comp hω + +theorem scalarSeries_map_gaussian (a : ι → ℝ) (ha : Summable a) : + (coefficientLaw ι).map (scalarSeries a) = + gaussianReal 0 (∑' i, a i ^ 2).toNNReal := by + apply Measure.ext_of_charFun + funext t + have h1 := charFun_partial_tendsto a ha t + have h2 : Filter.Tendsto (fun s : Finset ι => + charFun ((coefficientLaw ι).map (fun ω => ∑ i ∈ s, a i * ω i)) t) + Filter.atTop (𝓝 (Complex.exp (-((∑' i, a i ^ 2 : ℝ) : ℂ) * (t : ℂ)^2 / 2))) := by + simp_rw [partial_charFun] + exact ((by fun_prop : Continuous (fun q : ℝ => + Complex.exp (-(q : ℂ) * (t : ℂ)^2 / 2))).tendsto _).comp + (summable_square a ha).hasSum + rw [tendsto_nhds_unique h1 h2, charFun_gaussianReal] + rw [Real.coe_toNNReal _ (tsum_nonneg (fun i => sq_nonneg (a i)))] + congr 1 + push_cast + ring + +theorem scalarSeries_integral_zero (a : ι → ℝ) (ha : Summable a) : + ∫ ω, scalarSeries a ω ∂coefficientLaw ι = 0 := by + have h := integral_id_gaussianReal (μ := 0) (v := (∑' i, a i ^ 2).toNNReal) + rw [← scalarSeries_map_gaussian a ha, + integral_map (scalarSeries_measurable a).aemeasurable (by fun_prop)] at h + exact h + +theorem scalarSeries_variance (a : ι → ℝ) (ha : Summable a) : + Var[scalarSeries a; coefficientLaw ι] = ∑' i, a i ^ 2 := by + rw [← variance_id_map (scalarSeries_measurable a).aemeasurable, + scalarSeries_map_gaussian a ha, variance_id_gaussianReal] + exact Real.coe_toNNReal _ (tsum_nonneg (fun i => sq_nonneg (a i))) + +theorem source_linearCombination_hasGaussianLaw (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (n : ℕ) (x : Fin n → (Fin d → ℝ)) (c : Fin n → ℝ) : + HasGaussianLaw (fun ω => ∑ j, c j * sourceField d L ω (x j)) (sourceLaw d) := by + classical + have hc (j : Fin n) : Summable (fun m => modeCoefficient d L m (x j)) := + (modeMajorant_summable d hL).of_norm_bounded (fun m => by + simpa only [Real.norm_eq_abs] using modeCoefficient_bound d hL m (x j)) + let a : SourceMode d → ℝ := fun m => ∑ j, c j * modeCoefficient d L m (x j) + have ha : Summable a := by + exact summable_sum (fun j _ => (hc j).mul_left (c j)) + have hG : HasGaussianLaw (scalarSeries a) (sourceLaw d) := by + constructor + · exact (scalarSeries_measurable a).aemeasurable + · change IsGaussian ((coefficientLaw (SourceMode d)).map (scalarSeries a)) + rw [scalarSeries_map_gaussian a ha] + infer_instance + apply hG.congr + filter_upwards [ae_all_iff.2 (fun j : Fin n => + scalarSeries_hasSum_ae (fun m => modeCoefficient d L m (x j)) (hc j)), + scalarSeries_hasSum_ae a ha] with ω hω hsum + have h : HasSum (fun m => ∑ j, c j * modeCoefficient d L m (x j) * ω m) + (∑ j, c j * sourceField d L ω (x j)) := by + simpa only [sourceField, mul_assoc] using hasSum_sum (fun j _ => (hω j).mul_left (c j)) + exact hsum.unique (by simpa only [a, Finset.sum_mul] using h) + +theorem source_finiteVector_hasGaussianLaw (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (n : ℕ) (x : Fin n → (Fin d → ℝ)) : + HasGaussianLaw (fun ω j => sourceField d L ω (x j)) (sourceLaw d) := by + classical + have hm : Measurable (fun ω j => sourceField d L ω (x j)) := + Measurable.of_eval (fun j => sourceField_eval_measurable d L (x j)) + refine ⟨hm.aemeasurable, ?_⟩ + apply isGaussian_of_map_eq_gaussianReal + intro ell + let c : Fin n → ℝ := fun j => ell (Pi.single j 1) + have hrep (v : Fin n → ℝ) : ell v = ∑ j, c j * v j := by + rw [← LinearMap.sum_single_apply (fun _ : Fin n => ℝ) v, map_sum] + apply Finset.sum_congr rfl + intro j _ + have hs : Pi.single j (v j) = v j • Pi.single j (1 : ℝ) := by + rw [← Pi.single_smul, smul_eq_mul, mul_one] + rw [hs, map_smul] + simp [c, smul_eq_mul, mul_comm] + have hg : HasGaussianLaw (fun ω => ell (fun j => sourceField d L ω (x j))) + (sourceLaw d) := + (source_linearCombination_hasGaussianLaw d hL n x c).congr + (Filter.Eventually.of_forall (fun ω => (hrep _).symm)) + refine ⟨(sourceLaw d)[fun ω => ell (fun j => sourceField d L ω (x j))], + Var[fun ω => ell (fun j => sourceField d L ω (x j)); sourceLaw d].toNNReal, ?_⟩ + rw [Measure.map_map ell.continuous.measurable hm] + exact hg.map_eq_gaussianReal + +theorem sourceField_isGaussianProcess (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + IsGaussianProcess (fun x ω => sourceField d L ω x) (sourceLaw d) := by + classical + constructor + intro I + let e : ↥I ≃ Fin (Fintype.card ↥I) := Fintype.equivFin ↥I + let x : Fin (Fintype.card ↥I) → (Fin d → ℝ) := fun j => (e.symm j).1 + let R : (Fin (Fintype.card ↥I) → ℝ) →L[ℝ] (↥I → ℝ) := + ContinuousLinearMap.pi (fun i => ContinuousLinearMap.proj (e i)) + apply ((source_finiteVector_hasGaussianLaw d hL _ x).map_fun R).congr + filter_upwards with ω + funext i + simp [R, x] + +end I4SourceGaussian diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceKernel.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceKernel.lean new file mode 100644 index 000000000..5d5602ac7 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceKernel.lean @@ -0,0 +1,112 @@ +import SourceLattice +import Mathlib.Analysis.Normed.Ring.InfiniteSum + +noncomputable section +open I4Foundation I4Weights I4Source I4Lattice MeasureTheory +open scoped BigOperators +namespace I4Kernel + +def frequencyEquiv (d : ℕ) : SourceMode d ≃ (Fin d → ℤ) := + Equiv.ofBijective (latticeFrequency d) (latticeFrequency_bijective d) + +/-- The prescribed normalized full-lattice real Fourier kernel; image/Poisson linkage is separate. -/ +def sourceSpectralKernel (d : ℕ) (L : ℝ) (z : Fin d → ℝ) : ℝ := + ∑' k : Fin d → ℤ, sourceWeight d L k / sourceZ d L * Real.cos (phase d L k z) + +private theorem tsum_mode (d : ℕ) (f : SourceMode d → ℝ) (hf : Summable f) : + (∑' m, f m) = f none + ∑' k : ↥(halfLattice d), + (f (some (k,false)) + f (some (k,true))) := by + classical + let e := Equiv.optionEquivSumPUnit.{0,0} (↥(halfLattice d) × Bool) + have hp : Summable (fun p : ↥(halfLattice d) × Bool => f (some p)) := + hf.comp_injective (Option.some_injective (↥(halfLattice d) × Bool)) + calc + (∑' m, f m) = ∑' q, f (e.symm q) := (e.symm.tsum_eq f).symm + _ = (∑' p : ↥(halfLattice d) × Bool, f (some p)) + f none := by + rw [Summable.tsum_sum (by simpa [e, Function.comp_def] using hp) + (hasSum_fintype _).summable] + simp [e] + _ = _ := by + rw [hp.tsum_prod] + simp_rw [tsum_bool] + ring + +theorem modeProduct_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) (x y : Fin d → ℝ) : + Summable (fun m => modeCoefficient d L m x * modeCoefficient d L m y) := by + have hc (z : Fin d → ℝ) : Summable (fun m => modeCoefficient d L m z) := by + apply Summable.of_norm_bounded (modeMajorant_summable d hL) + intro m + simpa only [Real.norm_eq_abs] using modeCoefficient_bound d hL m z + have hp : Summable (fun p : SourceMode d × SourceMode d => + modeCoefficient d L p.1 x * modeCoefficient d L p.2 y) := + summable_mul_of_summable_norm (hc x).norm (hc y).norm + simpa only [Function.comp_def] using hp.comp_injective + (show Function.Injective (fun m : SourceMode d => (m,m)) from + fun a b h => congrArg Prod.fst h) + +theorem spectralKernel_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) (z : Fin d → ℝ) : + Summable (fun k : Fin d → ℤ => + sourceWeight d L k / sourceZ d L * Real.cos (phase d L k z)) := by + have hz := sourceZ_pos d hL + apply Summable.of_norm_bounded ((sourceWeight_summable d hL).div_const (sourceZ d L)) + intro k + have hq : 0 ≤ sourceWeight d L k / sourceZ d L := by + rw [sourceWeight_eq_physical] + exact div_nonneg (Real.exp_pos _).le hz.le + rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg hq, Real.norm_eq_abs] + simpa only [mul_one] using mul_le_mul_of_nonneg_left (Real.abs_cos_le_one _) hq + +theorem coefficientKernel_eq_fullLattice (d : ℕ) {L : ℝ} (hL : L ≠ 0) (x y : Fin d → ℝ) : + (∑' m, modeCoefficient d L m x * modeCoefficient d L m y) = + sourceSpectralKernel d L (x - y) := by + classical + let g : (Fin d → ℤ) → ℝ := fun k => + sourceWeight d L k / sourceZ d L * Real.cos (phase d L k (x - y)) + have hg : Summable g := spectralKernel_summable d hL (x - y) + have hneg (k : Fin d → ℤ) : g (-k) = g k := by + have hp : phase d L (-k) (x - y) = -phase d L k (x - y) := by + simp [phase, Finset.sum_neg_distrib] + simp only [g, sourceWeight_neg, hp, Real.cos_neg] + have hzero : g 0 = 1 / sourceZ d L := by + simp [g, sourceWeight_eq_physical, phase] + have hc0 : modeCoefficient d L none x * modeCoefficient d L none y = 1 / sourceZ d L := by + simp only [modeCoefficient] + exact Real.mul_self_sqrt (one_div_nonneg.mpr (sourceZ_pos d hL).le) + have hpair (k : ↥(halfLattice d)) : + modeCoefficient d L (some (k,false)) x * modeCoefficient d L (some (k,false)) y + + modeCoefficient d L (some (k,true)) x * modeCoefficient d L (some (k,true)) y = 2 * g k := by + have h := pairedCoefficient_kernel d hL k x y + rw [Fintype.sum_bool] at h + calc + _ = modeCoefficient d L (some (k,true)) x * modeCoefficient d L (some (k,true)) y + + modeCoefficient d L (some (k,false)) x * modeCoefficient d L (some (k,false)) y := add_comm _ _ + _ = _ := h + _ = 2 * g k := by dsimp [g]; ring + have hleft : (∑' m, modeCoefficient d L m x * modeCoefficient d L m y) = + 1 / sourceZ d L + ∑' k : ↥(halfLattice d), 2 * g k := by + rw [tsum_mode d _ (modeProduct_summable d hL x y), hc0] + congr 1 + exact tsum_congr hpair + have hm : Summable (fun m => g (frequencyEquiv d m)) := (frequencyEquiv d).summable_iff.mpr hg + have hright : sourceSpectralKernel d L (x - y) = + 1 / sourceZ d L + ∑' k : ↥(halfLattice d), 2 * g k := by + change (∑' k, g k) = _ + rw [← (frequencyEquiv d).tsum_eq g, tsum_mode d _ hm] + change g 0 + (∑' k : ↥(halfLattice d), (g k.1 + g (-k.1))) = _ + rw [hzero] + congr 1 + apply tsum_congr + intro k + rw [hneg] + ring + exact hleft.trans hright.symm + +theorem spectralKernel_zero (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + sourceSpectralKernel d L 0 = 1 := by + unfold sourceSpectralKernel + simp only [phase, Pi.zero_apply, mul_zero, Finset.sum_const_zero, Real.cos_zero, mul_one] + rw [tsum_div_const] + change sourceZ d L / sourceZ d L = 1 + exact div_self (sourceZ_pos d hL).ne' + +end I4Kernel diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceLattice.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceLattice.lean new file mode 100644 index 000000000..9c1e0557c --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceLattice.lean @@ -0,0 +1,159 @@ +import SourceFourierSeries +import Mathlib.Data.Finset.Max + +noncomputable section +open I4Foundation I4Weights I4Source MeasureTheory +open scoped BigOperators +namespace I4Lattice + +def latticeFrequency (d : ℕ) (m : SourceMode d) : Fin d → ℤ := + match m with + | none => 0 + | some (k, b) => if b then -k.1 else k.1 + +theorem halfLattice_ne_zero (d : ℕ) {k : Fin d → ℤ} (hk : k ∈ halfLattice d) : + k ≠ 0 := by + rcases hk with ⟨i, hi, _⟩ + intro h + have : k i = 0 := congrFun h i + omega + +theorem halfLattice_partition (d : ℕ) {k : Fin d → ℤ} (hk : k ≠ 0) : + (k ∈ halfLattice d ∨ -k ∈ halfLattice d) ∧ + ¬ (k ∈ halfLattice d ∧ -k ∈ halfLattice d) := by + classical + have hex : ∃ i, k i ≠ 0 := by + by_contra h + push Not at h + exact hk (funext h) + let s : Finset (Fin d) := Finset.univ.filter (fun i => k i ≠ 0) + have hs : s.Nonempty := by + obtain ⟨i, hi⟩ := hex + exact ⟨i, Finset.mem_filter.mpr ⟨Finset.mem_univ i, hi⟩⟩ + let i := s.min' hs + have hi : k i ≠ 0 := (Finset.mem_filter.mp (Finset.min'_mem s hs)).2 + have hbefore (j : Fin d) (hj : j < i) : k j = 0 := by + by_contra hn + have hle := Finset.min'_le s j (Finset.mem_filter.mpr ⟨Finset.mem_univ j, hn⟩) + exact (not_le_of_gt hj) hle + constructor + · rcases lt_or_gt_of_ne hi with hneg | hpos + · right + refine ⟨i, ?_, ?_⟩ + · change 0 < -k i + omega + · intro j hj + change -k j = 0 + rw [hbefore j hj, neg_zero] + · exact Or.inl ⟨i, hpos, hbefore⟩ + · rintro ⟨⟨i, hi, hib⟩, ⟨j, hj, hjb⟩⟩ + change 0 < -k j at hj + rcases lt_trichotomy i j with hij | heq | hji + · have h0 := hjb i hij + change -k i = 0 at h0 + omega + · subst j + omega + · have h0 := hib j hji + omega + +theorem sourceWeight_neg (d : ℕ) (L : ℝ) (k : Fin d → ℤ) : + sourceWeight d L (-k) = sourceWeight d L k := by + rw [sourceWeight_eq_physical, sourceWeight_eq_physical] + simp only [Pi.neg_apply, Int.cast_neg, mul_neg, neg_sq] + +theorem phase_lattice_shift (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (k n : Fin d → ℤ) (x : Fin d → ℝ) : + phase d L k (fun i => x i + L * (n i : ℝ)) = + phase d L k x + ((∑ i, k i * n i : ℤ) : ℝ) * (2 * Real.pi) := by + have he (i : Fin d) : sourceC L * (k i : ℝ) * (x i + L * (n i : ℝ)) = + sourceC L * (k i : ℝ) * x i + ((k i * n i : ℤ) : ℝ) * (2 * Real.pi) := by + simp only [Int.cast_mul] + unfold sourceC + field_simp + unfold phase + simp_rw [he] + rw [Finset.sum_add_distrib, ← Finset.sum_mul] + simp only [Int.cast_sum] + +theorem modeCoefficient_lattice_shift (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (m : SourceMode d) (n : Fin d → ℤ) (x : Fin d → ℝ) : + modeCoefficient d L m (fun i => x i + L * (n i : ℝ)) = + modeCoefficient d L m x := by + cases m with + | none => rfl + | some p => + rcases p with ⟨k,b⟩ + cases b <;> simp only [modeCoefficient, Bool.false_eq_true, ite_false, ite_true] + · rw [phase_lattice_shift d hL, Real.cos_add_int_mul_two_pi] + · rw [phase_lattice_shift d hL, Real.sin_add_int_mul_two_pi] + +theorem sourceField_lattice_shift (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (ω : SourceMode d → ℝ) (n : Fin d → ℤ) (x : Fin d → ℝ) : + sourceField d L ω (fun i => x i + L * (n i : ℝ)) = sourceField d L ω x := by + unfold sourceField scalarSeries + apply tsum_congr + intro m + change modeCoefficient d L m (fun i => x i + L * (n i : ℝ)) * ω m = + modeCoefficient d L m x * ω m + rw [modeCoefficient_lattice_shift d hL] + +theorem pairedCoefficient_kernel (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (k : ↥(halfLattice d)) (x y : Fin d → ℝ) : + (∑ b : Bool, modeCoefficient d L (some (k,b)) x * + modeCoefficient d L (some (k,b)) y) = + (2 * sourceWeight d L k / sourceZ d L) * Real.cos (phase d L k (x - y)) := by + have hphase : phase d L k (x - y) = phase d L k x - phase d L k y := by + simp [phase, mul_sub, Finset.sum_sub_distrib] + have hz : 0 ≤ 2 * sourceWeight d L k / sourceZ d L := by + rw [sourceWeight_eq_physical] + exact div_nonneg (mul_nonneg (by norm_num) (Real.exp_pos _).le) (sourceZ_pos d hL).le + rw [Fintype.sum_bool] + simp only [modeCoefficient, ite_true, Bool.false_eq_true, ite_false] + rw [hphase, Real.cos_sub] + calc + _ = (Real.sqrt (2 * sourceWeight d L k / sourceZ d L) * + Real.sqrt (2 * sourceWeight d L k / sourceZ d L)) * + (Real.cos (phase d L k x) * Real.cos (phase d L k y) + + Real.sin (phase d L k x) * Real.sin (phase d L k y)) := by ring + _ = _ := by rw [Real.mul_self_sqrt hz] + +theorem latticeFrequency_bijective (d : ℕ) : Function.Bijective (latticeFrequency d) := by + classical + constructor + · intro m n he + cases m with + | none => + cases n with + | none => rfl + | some p => + rcases p with ⟨k,b⟩ + have hk := halfLattice_ne_zero d k.property + cases b <;> simp [latticeFrequency] at he <;> exact (hk (by first | exact he | exact he.symm)).elim + | some p => + rcases p with ⟨k,b⟩ + cases n with + | none => + have hk := halfLattice_ne_zero d k.property + cases b <;> simp [latticeFrequency] at he <;> exact (hk he).elim + | some q => + rcases q with ⟨l,c⟩ + cases b <;> cases c <;> simp only [latticeFrequency, Bool.false_eq_true, ite_false, ite_true] at he + · have hkl : k = l := Subtype.ext he + subst l + rfl + · have hn : -k.1 ∈ halfLattice d := by simpa only [he, neg_neg] using l.property + exact ((halfLattice_partition d (halfLattice_ne_zero d k.property)).2 ⟨k.property,hn⟩).elim + · have hn : -k.1 ∈ halfLattice d := he.symm ▸ l.property + exact ((halfLattice_partition d (halfLattice_ne_zero d k.property)).2 ⟨k.property,hn⟩).elim + · have hkl : k = l := Subtype.ext (neg_inj.mp he) + subst l + rfl + · intro k + by_cases hk : k = 0 + · exact ⟨none, by simpa [latticeFrequency] using hk.symm⟩ + · rcases (halfLattice_partition d hk).1 with hp | hn + · exact ⟨some (⟨k,hp⟩,false), by simp [latticeFrequency]⟩ + · exact ⟨some (⟨-k,hn⟩,true), by simp [latticeFrequency]⟩ + +end I4Lattice diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceSmooth.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceSmooth.lean new file mode 100644 index 000000000..5b1518449 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceSmooth.lean @@ -0,0 +1,254 @@ +import SourceC4 +import Mathlib.Data.Nat.Choose.Sum + +/-! +# All-order smoothness of the actual infinite Gaussian Fourier source + +This module extends the source-derived polynomial majorant from order four to +every natural order. It uses the same SourceMode, sourceLaw and sourceField as +the qualified F7 dependency; no finite-mode approximation or new field is used. +The good sample predicate is a single countable intersection of explicit +coefficient summability conditions. Its measurability and probability-law +restriction are proved in SourceSmoothCarrier. + +This is a C261 Stage A prerequisite. It does not identify the spectral covariance +with Lifetime's theta sum, construct a conditional kernel, or discharge a +weighted elder, Morse, count/density, or universality conclusion. +-/ + +noncomputable section +open I4Foundation I4Weights I4Source I4C4 MeasureTheory ProbabilityTheory +open scoped BigOperators ENNReal + +namespace I4Smooth + +def orderCoordinateBound (n : ℕ) (a c : ℝ) (k : ℤ) : ℝ := + decay a k * (1 + |c * (k : ℝ)|) ^ n + +def orderProductBound (n d : ℕ) (a c : ℝ) (k : Fin d → ℤ) : ℝ := + ∏ i, orderCoordinateBound n a c (k i) + +def orderPairedBound (n d : ℕ) (L : ℝ) (k : Fin d → ℤ) : ℝ := + Real.sqrt (2 / sourceZ d L) * orderProductBound n d (sourceA L) (sourceC L) k + +def orderModeBound (n d : ℕ) (L : ℝ) (m : SourceMode d) : ℝ := + match m with + | none => Real.sqrt (1 / sourceZ d L) + | some (k, _) => orderPairedBound n d L k + +theorem orderCoordinateBound_summable (n : ℕ) {a c : ℝ} (ha : 0 < a) : + Summable (orderCoordinateBound n a c) := by + classical + let f (r : ℕ) (k : ℤ) : ℝ := + ((n.choose r : ℝ) * |c| ^ (n - r)) * (decay a k * |(k : ℝ)| ^ (n - r)) + have hf (r : ℕ) : Summable (f r) := + (summable_decay_pow ha (n - r)).mul_left _ + have hsum (s : Finset ℕ) : Summable (fun k : ℤ => ∑ r ∈ s, f r k) := by + induction s using Finset.induction_on with + | empty => simpa using (summable_zero : Summable (fun _ : ℤ => (0 : ℝ))) + | @insert r s hr ih => + simpa only [Finset.sum_insert hr] using (hf r).add ih + convert hsum (Finset.range (n + 1)) using 1 + funext k + unfold orderCoordinateBound + rw [add_pow, Finset.mul_sum] + apply Finset.sum_congr rfl + intro r _ + simp only [one_pow, one_mul, abs_mul, mul_pow, f] + ring + +theorem orderProductBound_summable (n d : ℕ) {a c : ℝ} (ha : 0 < a) : + Summable (orderProductBound n d a c) := by + exact finite_product_summable (orderCoordinateBound_summable n ha) + (fun k => by unfold orderCoordinateBound decay; positivity) d + +theorem orderPairedBound_summable (n d : ℕ) {L : ℝ} (hL : L ≠ 0) : + Summable (orderPairedBound n d L) := by + exact (orderProductBound_summable n d (sourceA_pos hL)).mul_left _ + +theorem orderPairedBound_eq_amplitude (n d : ℕ) {L : ℝ} (hL : L ≠ 0) + (k : Fin d → ℤ) : + orderPairedBound n d L k = + Real.sqrt (2 * sourceWeight d L k / sourceZ d L) * + ∏ i, (1 + |sourceC L * (k i : ℝ)|) ^ n := by + have hsqrt : Real.sqrt (sourceWeight d L k) = productDecay d (sourceA L) k := by + have hw : sourceWeight d L k = (productDecay d (sourceA L) k) ^ 2 := by + rw [sourceWeight, productDecay_eq_exp, productDecay_eq_exp, pow_two, ← Real.exp_add] + congr 1 + ring + rw [hw, Real.sqrt_sq_eq_abs, abs_of_nonneg] + unfold productDecay decay + positivity + have hamp : Real.sqrt (2 * sourceWeight d L k / sourceZ d L) = + Real.sqrt (2 / sourceZ d L) * productDecay d (sourceA L) k := by + have he : 2 * sourceWeight d L k / sourceZ d L = + (2 / sourceZ d L) * sourceWeight d L k := by ring + rw [he, Real.sqrt_mul (by have := sourceZ_pos d hL; positivity), hsqrt] + have hp : orderProductBound n d (sourceA L) (sourceC L) k = + productDecay d (sourceA L) k * ∏ i, (1 + |sourceC L * (k i : ℝ)|) ^ n := by + exact Finset.prod_mul_distrib + rw [orderPairedBound, hp, hamp] + ring + +theorem orderModeBound_nonneg (n d : ℕ) (L : ℝ) (m : SourceMode d) : + 0 ≤ orderModeBound n d L m := by + cases m with + | none => exact Real.sqrt_nonneg _ + | some p => + unfold orderModeBound orderPairedBound orderProductBound orderCoordinateBound decay + positivity + +theorem orderModeBound_summable (n d : ℕ) {L : ℝ} (hL : L ≠ 0) : + Summable (orderModeBound n d L) := by + classical + have hs : Summable (fun k : ↥(halfLattice d) => orderPairedBound n d L k) := + (orderPairedBound_summable n d hL).subtype (halfLattice d) + have hb : Summable (fun _ : Bool => (1 : ℝ)) := (hasSum_fintype _).summable + have hp : Summable (fun p : ↥(halfLattice d) × Bool => orderPairedBound n d L p.1) := by + simpa only [mul_one, Function.comp_def] using hs.mul_of_nonneg hb + (fun k => orderModeBound_nonneg n d L (some (k, false))) (fun _ => zero_le_one) + let e := Equiv.optionEquivSumPUnit.{0, 0} (↥(halfLattice d) × Bool) + apply e.symm.summable_iff.mp + apply Summable.sum (fun q => orderModeBound n d L (e.symm q)) + · simpa [e, Function.comp_def, orderModeBound] using hp + · exact (hasSum_fintype _).summable + +theorem orderModeBound_four (d : ℕ) (L : ℝ) (m : SourceMode d) : + orderModeBound 4 d L m = modeMajorant d L m := by + cases m <;> rfl + +theorem modeCoefficient_iteratedFDeriv_order_bound (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (m : SourceMode d) (j : ℕ) (x : Fin d → ℝ) : + ‖iteratedFDeriv ℝ j (modeCoefficient d L m) x‖ ≤ orderModeBound j d L m := by + classical + cases m with + | none => + cases j with + | zero => + simpa only [norm_iteratedFDeriv_zero, Real.norm_eq_abs, orderModeBound, modeMajorant] using + modeCoefficient_bound d hL none x + | succ j => + change ‖iteratedFDeriv ℝ (j + 1) + (fun _ : Fin d → ℝ => Real.sqrt (1 / sourceZ d L)) x‖ ≤ Real.sqrt (1 / sourceZ d L) + rw [iteratedFDeriv_succ_const] + simp only [Pi.zero_apply, norm_zero] + exact Real.sqrt_nonneg _ + | some p => + rcases p with ⟨k, b⟩ + let A := Real.sqrt (2 * sourceWeight d L k / sourceZ d L) + let P := ∏ i, (1 + |sourceC L * (k.1 i : ℝ)|) + let g : ℝ → ℝ := if b then Real.sin else Real.cos + let ell := phaseCLM d L k + have hg : ContDiff ℝ j g := by + cases b <;> simp only [g, Bool.false_eq_true, ite_false, ite_true] <;> fun_prop + have hgc : ContDiff ℝ j (g ∘ ell) := hg.comp_continuousLinearMap + have hgn : ‖iteratedFDeriv ℝ j g (ell x)‖ ≤ 1 := by + cases b with + | false => + simpa only [g, Bool.false_eq_true, ite_false, + norm_iteratedFDeriv_eq_norm_iteratedDeriv, Real.norm_eq_abs] using + Real.abs_iteratedDeriv_cos_le_one j (ell x) + | true => + simpa only [g, ite_true, norm_iteratedFDeriv_eq_norm_iteratedDeriv, + Real.norm_eq_abs] using Real.abs_iteratedDeriv_sin_le_one j (ell x) + have hl : ‖ell‖ ≤ P := by + calc + ‖ell‖ ≤ ∑ i, |sourceC L * (k.1 i : ℝ)| := phaseCLM_norm_le d L k + _ ≤ 1 + ∑ i, |sourceC L * (k.1 i : ℝ)| := by linarith + _ ≤ P := frequency_l1_le_product d _ (fun i => abs_nonneg _) + have hpow : ‖ell‖ ^ j ≤ P ^ j := by gcongr + have hder : ‖iteratedFDeriv ℝ j (g ∘ ell) x‖ ≤ ‖ell‖ ^ j := by + rw [ell.iteratedFDeriv_comp_right hg x le_rfl] + calc + _ ≤ ‖iteratedFDeriv ℝ j g (ell x)‖ * ∏ _i : Fin j, ‖ell‖ := + (iteratedFDeriv ℝ j g (ell x)).norm_compContinuousLinearMap_le (fun _ => ell) + _ ≤ 1 * ∏ _i : Fin j, ‖ell‖ := + mul_le_mul_of_nonneg_right hgn (Finset.prod_nonneg (fun _ _ => norm_nonneg _)) + _ = ‖ell‖ ^ j := by simp + have heq : modeCoefficient d L (some (k, b)) = A • (g ∘ ell) := by + funext y + cases b <;> simp [modeCoefficient, A, g, Function.comp_apply, Pi.smul_apply, + smul_eq_mul, ell, phaseCLM_apply] + rw [heq, iteratedFDeriv_const_smul_apply hgc.contDiffAt, norm_smul, + Real.norm_eq_abs, abs_of_nonneg (Real.sqrt_nonneg _)] + calc + _ ≤ A * P ^ j := mul_le_mul_of_nonneg_left (hder.trans hpow) (Real.sqrt_nonneg _) + _ = orderModeBound j d L (some (k, b)) := by + rw [orderModeBound, orderPairedBound_eq_amplitude j d hL k] + simp only [A, P, Finset.prod_pow] + +/-- One explicit coefficient event, shared by every derivative order. -/ +def goodSample (d : ℕ) (L : ℝ) (ω : SourceMode d → ℝ) : Prop := + ∀ n : ℕ, Summable (fun m => orderModeBound n d L m * |ω m|) + +theorem goodSample_ae (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + ∀ᵐ ω ∂sourceLaw d, goodSample d L ω := by + apply ae_all_iff.mpr + intro n + exact weighted_abs_summable_ae _ (orderModeBound_nonneg n d L) + (orderModeBound_summable n d hL) + +theorem sourceField_smooth_of_goodSample (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (ω : SourceMode d → ℝ) (hω : goodSample d L ω) : + ∀ n : ℕ, ContDiff ℝ n (sourceField d L ω) := by + intro n + have hc (m : SourceMode d) : ContDiff ℝ n (modeCoefficient d L m) := by + cases m with + | none => + change ContDiff ℝ n (fun _ : Fin d → ℝ => Real.sqrt (1 / sourceZ d L)) + fun_prop + | some p => + rcases p with ⟨k, b⟩ + change ContDiff ℝ n (fun x : Fin d → ℝ => + Real.sqrt (2 * sourceWeight d L k / sourceZ d L) * + (if b then Real.sin (phase d L k x) else Real.cos (phase d L k x))) + cases b <;> simp [phase] <;> fun_prop + change ContDiff ℝ n (fun x => ∑' m, modeCoefficient d L m x * ω m) + apply contDiff_tsum (v := fun j m => orderModeBound j d L m * |ω m|) + · intro m + exact (hc m).mul contDiff_const + · intro j _ + exact hω j + · intro j m x hj + have hcj : ContDiff ℝ j (modeCoefficient d L m) := + (hc m).of_le (by exact_mod_cast hj) + have heq : (fun y => modeCoefficient d L m y * ω m) = + ω m • modeCoefficient d L m := by + funext y + simp [smul_eq_mul, mul_comm] + rw [heq, iteratedFDeriv_const_smul_apply hcj.contDiffAt, norm_smul, Real.norm_eq_abs] + simpa only [mul_comm] using mul_le_mul_of_nonneg_left + (modeCoefficient_iteratedFDeriv_order_bound d hL m j x) (abs_nonneg (ω m)) + +theorem sourceField_smooth_ae (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + ∀ᵐ ω ∂sourceLaw d, ∀ n : ℕ, ContDiff ℝ n (sourceField d L ω) := by + filter_upwards [goodSample_ae d hL] with ω hω + exact sourceField_smooth_of_goodSample d hL ω hω + +/-- The carrier depends only on the coefficient tail; finite coordinate changes +cannot destroy any derivative-order summability condition. -/ +theorem goodSample_congr_cofinite (d : ℕ) (L : ℝ) + (ω η : SourceMode d → ℝ) (hωη : ω =ᶠ[Filter.cofinite] η) : + goodSample d L ω ↔ goodSample d L η := by + constructor + · intro hω n + apply (hω n).congr_cofinite + filter_upwards [hωη] with m hm + rw [hm] + · intro hη n + apply (hη n).congr_cofinite + filter_upwards [hωη.symm] with m hm + rw [hm] + +theorem goodSample_finite_change_iff (d : ℕ) (L : ℝ) + (ω η : SourceMode d → ℝ) (s : Finset (SourceMode d)) + (hωη : ∀ m ∉ s, ω m = η m) : + goodSample d L ω ↔ goodSample d L η := by + apply goodSample_congr_cofinite d L ω η + apply Filter.eventually_cofinite.mpr + apply s.finite_toSet.subset + intro m hm + by_contra hms + exact hm (hωη m hms) + +end I4Smooth diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceSmoothCarrier.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceSmoothCarrier.lean new file mode 100644 index 000000000..cb6f7b108 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceSmoothCarrier.lean @@ -0,0 +1,230 @@ +import SourceSmooth +import SourceGaussianLaw +import SourceCovariance +import SourceTorusC4 +import Mathlib.MeasureTheory.Measure.Restrict +import Mathlib.MeasureTheory.Measure.Typeclasses.Probability +import Mathlib.MeasureTheory.Constructions.BorelSpace.Real + +/-! +# A measurable smooth carrier for the actual source law + +The carrier consists of the coefficient samples satisfying every explicit +order-weighted absolute summability condition from SourceSmooth. Its measure is +the pullback of the existing product Gaussian sourceLaw under literal subtype +inclusion. Inclusion pushes this measure back to exactly sourceLaw, so finite +coefficient observations, field evaluation laws, and covariance are unchanged. + +The construction concerns the same literal Fourier source, for fixed d and L. +It does not identify the spectral kernel with a periodized theta kernel, define +a conditional kernel or a smooth-function-space law, or discharge a Lifetime +Field adapter, Morse, elder, counting, density, or universality conclusion. +-/ + +noncomputable section +open I4Foundation I4Source I4Smooth I4SourceGaussian I4SourceCovariance I4Kernel +open I4Torus I4TorusC4 MeasureTheory ProbabilityTheory +open scoped BigOperators ENNReal NNReal Manifold Topology + +namespace I4SmoothCarrier + +/-- The actual common coefficient event, with no smoothness premise added. -/ +def goodSet (d : ℕ) (L : ℝ) : Set (SourceMode d → ℝ) := + {ω | goodSample d L ω} + +private def orderAbsNN (n d : ℕ) (L : ℝ) (ω : SourceMode d → ℝ) + (m : SourceMode d) : ℝ≥0 := + ⟨orderModeBound n d L m * |ω m|, + mul_nonneg (orderModeBound_nonneg n d L m) (abs_nonneg _)⟩ + +private def orderAbsTotal (n d : ℕ) (L : ℝ) (ω : SourceMode d → ℝ) : ℝ≥0∞ := + ∑' m, (orderAbsNN n d L ω m : ℝ≥0∞) + +private theorem orderAbsTotal_ne_top_iff (n d : ℕ) (L : ℝ) + (ω : SourceMode d → ℝ) : + orderAbsTotal n d L ω ≠ ∞ ↔ + Summable (fun m => orderModeBound n d L m * |ω m|) := by + exact ENNReal.tsum_coe_ne_top_iff_summable_coe (f := orderAbsNN n d L ω) + +private theorem orderAbsTotal_measurable (n d : ℕ) (L : ℝ) : + Measurable (orderAbsTotal n d L) := by + unfold orderAbsTotal + simp_rw [ENNReal.tsum_eq_iSup_sum] + apply Measurable.iSup + intro s + apply s.measurable_fun_sum + intro m _ + have hm : Measurable (fun ω : SourceMode d → ℝ => orderAbsNN n d L ω m) := by + unfold orderAbsNN + exact (show Measurable (fun ω : SourceMode d → ℝ => orderModeBound n d L m * |ω m|) by + fun_prop).nnreal_mk + exact hm.coe_nnreal_ennreal + +theorem goodSet_measurable (d : ℕ) (L : ℝ) : MeasurableSet (goodSet d L) := by + have heq : goodSet d L = + ⋂ n : ℕ, {ω : SourceMode d → ℝ | orderAbsTotal n d L ω ≠ ∞} := by + ext ω + simp only [goodSet, Set.mem_setOf_eq, Set.mem_iInter, goodSample] + exact forall_congr' fun n => (orderAbsTotal_ne_top_iff n d L ω).symm + rw [heq] + apply MeasurableSet.iInter + intro n + convert ((orderAbsTotal_measurable n d L) (measurableSet_singleton ∞)).compl using 1 + ext ω + simp + +theorem goodSet_ae (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + ∀ᵐ ω ∂sourceLaw d, ω ∈ goodSet d L := + goodSample_ae d hL + +theorem goodSet_measure_one (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + sourceLaw d (goodSet d L) = 1 := by + exact (mem_ae_iff_prob_eq_one (goodSet_measurable d L)).mp (goodSet_ae d hL) + +/-- A coefficient sample on the explicit all-order summability carrier. -/ +abbrev SmoothSample (d : ℕ) (L : ℝ) := ↥(goodSet d L) + +/-- Literal inclusion, retaining every coefficient of the original sample. -/ +def sampleInclusion (d : ℕ) (L : ℝ) : SmoothSample d L → (SourceMode d → ℝ) := + Subtype.val + +theorem sampleInclusion_measurable (d : ℕ) (L : ℝ) : + Measurable (sampleInclusion d L) := + measurable_subtype_coe + +/-- Pull back the original product law; no normalization or new law is chosen. -/ +def smoothLaw (d : ℕ) (L : ℝ) : Measure (SmoothSample d L) := + (sourceLaw d).comap (sampleInclusion d L) + +theorem sampleInclusion_map (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + (smoothLaw d L).map (sampleInclusion d L) = sourceLaw d := by + change ((sourceLaw d).comap ((↑) : SmoothSample d L → (SourceMode d → ℝ))).map + ((↑) : SmoothSample d L → (SourceMode d → ℝ)) = sourceLaw d + rw [map_comap_subtype_coe (goodSet_measurable d L), + Measure.restrict_eq_self_of_ae_mem (goodSet_ae d hL)] + +theorem sampleInclusion_measurePreserving (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + MeasurePreserving (sampleInclusion d L) (smoothLaw d L) (sourceLaw d) := + ⟨sampleInclusion_measurable d L, sampleInclusion_map d hL⟩ + +theorem smoothLaw_isProbabilityMeasure (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + IsProbabilityMeasure (smoothLaw d L) := by + apply (MeasurableEmbedding.subtype_coe (goodSet_measurable d L)).isProbabilityMeasure_comap + simpa only [Subtype.range_coe] using goodSet_ae d hL + +/-- Every measurable observation has exactly its original pushforward law. -/ +theorem observation_map_eq (d : ℕ) {L : ℝ} (hL : L ≠ 0) + {E : Type*} [MeasurableSpace E] (X : (SourceMode d → ℝ) → E) (hX : Measurable X) : + (smoothLaw d L).map (fun ω => X (sampleInclusion d L ω)) = (sourceLaw d).map X := by + change (smoothLaw d L).map (X ∘ sampleInclusion d L) = (sourceLaw d).map X + rw [← Measure.map_map hX (sampleInclusion_measurable d L), sampleInclusion_map d hL] + +theorem coordinate_marginal (d : ℕ) {L : ℝ} (hL : L ≠ 0) (m : SourceMode d) : + (smoothLaw d L).map (fun ω => sampleInclusion d L ω m) = gaussianReal 0 1 := by + rw [observation_map_eq d hL (fun ω => ω m) (measurable_pi_apply m)] + exact I4Foundation.coordinate_marginal m + +theorem finiteCoordinate_map_eq (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (n : ℕ) (m : Fin n → SourceMode d) : + (smoothLaw d L).map (fun ω j => sampleInclusion d L ω (m j)) = + (sourceLaw d).map (fun ω j => ω (m j)) := by + exact observation_map_eq d hL _ (Measurable.of_eval (fun j => measurable_pi_apply (m j))) + +/-- The unchanged literal real-cover field, evaluated on the carrier. -/ +def smoothSourceField (d : ℕ) (L : ℝ) (ω : SmoothSample d L) (x : Fin d → ℝ) : ℝ := + sourceField d L (sampleInclusion d L ω) x + +theorem smoothSourceField_contDiff (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (ω : SmoothSample d L) (n : ℕ) : ContDiff ℝ n (smoothSourceField d L ω) := + sourceField_smooth_of_goodSample d hL (sampleInclusion d L ω) ω.property n + +theorem smoothSourceField_smooth (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (ω : SmoothSample d L) : ContDiff ℝ (⊤ : ℕ∞) (smoothSourceField d L ω) := + contDiff_infty.mpr (smoothSourceField_contDiff d hL ω) + +theorem smoothSourceField_finiteVector_map_eq (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (n : ℕ) (x : Fin n → (Fin d → ℝ)) : + (smoothLaw d L).map (fun ω j => smoothSourceField d L ω (x j)) = + (sourceLaw d).map (fun ω j => sourceField d L ω (x j)) := by + exact observation_map_eq d hL _ + (Measurable.of_eval (fun j => sourceField_eval_measurable d L (x j))) + +theorem smoothSourceField_finiteVector_hasGaussianLaw (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (n : ℕ) (x : Fin n → (Fin d → ℝ)) : + HasGaussianLaw (fun ω j => smoothSourceField d L ω (x j)) (smoothLaw d L) := by + have hm : Measurable (fun ω j => smoothSourceField d L ω (x j)) := + (Measurable.of_eval (fun j => sourceField_eval_measurable d L (x j))).comp + (sampleInclusion_measurable d L) + refine ⟨hm.aemeasurable, ?_⟩ + rw [smoothSourceField_finiteVector_map_eq d hL n x] + exact (source_finiteVector_hasGaussianLaw d hL n x).isGaussian_map + +theorem covariance_comp_inclusion (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (X Y : (SourceMode d → ℝ) → ℝ) (hX : Measurable X) (hY : Measurable Y) : + cov[fun ω => X (sampleInclusion d L ω), fun ω => Y (sampleInclusion d L ω); smoothLaw d L] = + cov[X, Y; sourceLaw d] := by + have h := covariance_map_fun (μ := smoothLaw d L) (Z := sampleInclusion d L) + (X := X) (Y := Y) hX.aestronglyMeasurable hY.aestronglyMeasurable + (sampleInclusion_measurable d L).aemeasurable + rw [sampleInclusion_map d hL] at h + exact h.symm + +theorem smoothSourceField_covariance (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (x y : Fin d → ℝ) : + cov[fun ω => smoothSourceField d L ω x, fun ω => smoothSourceField d L ω y; smoothLaw d L] = + sourceSpectralKernel d L (x - y) := by + calc + _ = cov[fun ω => sourceField d L ω x, fun ω => sourceField d L ω y; sourceLaw d] := + covariance_comp_inclusion d hL _ _ (sourceField_eval_measurable d L x) + (sourceField_eval_measurable d L y) + _ = _ := sourceField_covariance d hL x y + +/-- The unchanged literal torus descent of the same coefficient sample. -/ +def smoothTorusSourceField (d : ℕ) (L : ℝ) [Fact (0 < L)] + (ω : SmoothSample d L) (q : SourceTorus d L) : ℝ := + torusSourceField d L (sampleInclusion d L ω) q + +attribute [local instance] actualChartedSpace + +theorem smoothTorusSourceField_smooth (d : ℕ) (L : ℝ) [Fact (0 < L)] + (ω : SmoothSample d L) : + ContMDiff 𝓘(ℝ, Fin d → ℝ) 𝓘(ℝ) (⊤ : ℕ∞) (smoothTorusSourceField d L ω) := by + let _ := torus_isManifold d L + have hs := smoothSourceField_smooth d (Fact.out : 0 < L).ne' ω + intro q + rw [contMDiffAt_iff_source] + have heq : smoothTorusSourceField d L ω ∘ (extChartAt 𝓘(ℝ, Fin d → ℝ) q).symm = + smoothSourceField d L ω := by + funext x + change torusSourceField d L (sampleInclusion d L ω) + ((quotientChart d L (chartCenter d L q)).symm x) = _ + rw [quotientChart_symm] + exact torusSourceField_pullback d L (sampleInclusion d L ω) x + rw [heq, contMDiffWithinAt_iff_contDiffWithinAt] + exact hs.contDiffAt.contDiffWithinAt + +theorem smoothTorusSourceField_finiteVector_hasGaussianLaw (d : ℕ) (L : ℝ) + [Fact (0 < L)] (n : ℕ) (q : Fin n → SourceTorus d L) : + HasGaussianLaw (fun ω j => smoothTorusSourceField d L ω (q j)) (smoothLaw d L) := by + have hm : Measurable (fun ω j => smoothTorusSourceField d L ω (q j)) := + (Measurable.of_eval (fun j => torusSourceField_eval_measurable d L (q j))).comp + (sampleInclusion_measurable d L) + refine ⟨hm.aemeasurable, ?_⟩ + change IsGaussian ((smoothLaw d L).map + (fun ω j => torusSourceField d L (sampleInclusion d L ω) (q j))) + rw [observation_map_eq d (Fact.out : 0 < L).ne' _ + (Measurable.of_eval (fun j => torusSourceField_eval_measurable d L (q j)))] + exact (torusSourceField_finiteVector_hasGaussianLaw d L n q).isGaussian_map + +theorem smoothTorusSourceField_covariance (d : ℕ) (L : ℝ) [Fact (0 < L)] + (q p : SourceTorus d L) : + cov[fun ω => smoothTorusSourceField d L ω q, fun ω => smoothTorusSourceField d L ω p; + smoothLaw d L] = + sourceSpectralKernel d L (torusRepresentative d L q - torusRepresentative d L p) := by + calc + _ = cov[fun ω => torusSourceField d L ω q, fun ω => torusSourceField d L ω p; sourceLaw d] := + covariance_comp_inclusion d (Fact.out : 0 < L).ne' _ _ + (torusSourceField_eval_measurable d L q) (torusSourceField_eval_measurable d L p) + _ = _ := torusSourceField_covariance d L q p + +end I4SmoothCarrier diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceTorus.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceTorus.lean new file mode 100644 index 000000000..b9fe49752 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceTorus.lean @@ -0,0 +1,111 @@ +import SourceCovariance +import Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic +import Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic + +/-! Literal descent of the existing real-cover field to a product of additive circles. +The sample law remains sourceLaw. Smooth torus and Banach-valued laws are separate. -/ +noncomputable section +open I4Foundation I4Weights I4Source I4Lattice I4Kernel I4SourceGaussian I4SourceCovariance +open MeasureTheory ProbabilityTheory +open scoped BigOperators ENNReal NNReal + +namespace I4Torus + +abbrev SourceTorus (d : ℕ) (L : ℝ) := Fin d → AddCircle L + +def torusProjection (d : ℕ) (L : ℝ) (x : Fin d → ℝ) : SourceTorus d L := + fun i => (x i : AddCircle L) + +def torusRepresentative (d : ℕ) (L : ℝ) [Fact (0 < L)] + (q : SourceTorus d L) : Fin d → ℝ := + fun i => (AddCircle.equivIco L 0 (q i)).1 + +def torusSourceField (d : ℕ) (L : ℝ) [Fact (0 < L)] + (ω : SourceMode d → ℝ) (q : SourceTorus d L) : ℝ := + sourceField d L ω (torusRepresentative d L q) + +theorem torusProjection_measurable (d : ℕ) (L : ℝ) : Measurable (torusProjection d L) := by + exact Measurable.of_eval (fun i => AddCircle.measurable_mk'.comp (measurable_pi_apply i)) + +theorem torusRepresentative_measurable (d : ℕ) (L : ℝ) [Fact (0 < L)] : + Measurable (torusRepresentative d L) := by + exact Measurable.of_eval (fun i => measurable_subtype_coe.comp + ((AddCircle.measurableEquivIco L 0).measurable.comp (measurable_pi_apply i))) + +theorem torusProjection_representative (d : ℕ) (L : ℝ) [Fact (0 < L)] + (q : SourceTorus d L) : torusProjection d L (torusRepresentative d L q) = q := by + funext i + exact AddCircle.coe_equivIco (p := L) (a := 0) (y := q i) + +theorem sourceField_eq_of_torusProjection_eq (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (ω : SourceMode d → ℝ) (x y : Fin d → ℝ) + (h : torusProjection d L x = torusProjection d L y) : + sourceField d L ω x = sourceField d L ω y := by + classical + have hz (i : Fin d) : ∃ n : ℤ, n • L = y i - x i := by + apply (AddCircle.coe_eq_zero_iff (p := L)).mp + rw [AddCircle.coe_sub] + exact sub_eq_zero.mpr (congrFun h i).symm + choose n hn using hz + have he : y = fun i => x i + L * (n i : ℝ) := by + funext i + have hi := hn i + rw [zsmul_eq_mul] at hi + linarith + rw [he] + exact (sourceField_lattice_shift d hL ω n x).symm + +theorem torusSourceField_pullback (d : ℕ) (L : ℝ) [Fact (0 < L)] + (ω : SourceMode d → ℝ) (x : Fin d → ℝ) : + torusSourceField d L ω (torusProjection d L x) = sourceField d L ω x := by + exact sourceField_eq_of_torusProjection_eq d (Fact.out : 0 < L).ne' ω + (torusRepresentative d L (torusProjection d L x)) x + (torusProjection_representative d L (torusProjection d L x)) + +theorem torusSourceField_eval_measurable (d : ℕ) (L : ℝ) [Fact (0 < L)] + (q : SourceTorus d L) : Measurable (fun ω => torusSourceField d L ω q) := by + exact sourceField_eval_measurable d L (torusRepresentative d L q) + +theorem torusSourceField_joint_measurable (d : ℕ) (L : ℝ) [Fact (0 < L)] : + Measurable (fun p : (SourceMode d → ℝ) × SourceTorus d L => + torusSourceField d L p.1 p.2) := by + unfold torusSourceField sourceField scalarSeries + apply Measurable.tsum + intro m + have hm : Continuous (fun x : Fin d → ℝ => modeCoefficient d L m x) := by + cases m with + | none => simp only [modeCoefficient]; fun_prop + | some p => + rcases p with ⟨k,b⟩ + cases b <;> simp only [modeCoefficient, Bool.false_eq_true, ite_false, ite_true] <;> + unfold phase <;> fun_prop + exact (hm.measurable.comp ((torusRepresentative_measurable d L).comp measurable_snd)).mul + ((measurable_pi_apply m).comp measurable_fst) + +theorem torusSourceField_finiteVector_hasGaussianLaw (d : ℕ) (L : ℝ) [Fact (0 < L)] + (n : ℕ) (q : Fin n → SourceTorus d L) : + HasGaussianLaw (fun ω j => torusSourceField d L ω (q j)) (sourceLaw d) := by + exact source_finiteVector_hasGaussianLaw d (Fact.out : 0 < L).ne' n + (fun j => torusRepresentative d L (q j)) + +theorem torusSourceField_isGaussianProcess (d : ℕ) (L : ℝ) [Fact (0 < L)] : + IsGaussianProcess (fun q ω => torusSourceField d L ω q) (sourceLaw d) := by + exact (sourceField_isGaussianProcess d (Fact.out : 0 < L).ne').comp_right + (torusRepresentative d L) + +theorem torusSourceField_covariance (d : ℕ) (L : ℝ) [Fact (0 < L)] + (q p : SourceTorus d L) : + cov[fun ω => torusSourceField d L ω q, fun ω => torusSourceField d L ω p; sourceLaw d] = + sourceSpectralKernel d L (torusRepresentative d L q - torusRepresentative d L p) := by + exact sourceField_covariance d (Fact.out : 0 < L).ne' + (torusRepresentative d L q) (torusRepresentative d L p) + +theorem torusSourceField_covariance_pullback (d : ℕ) (L : ℝ) [Fact (0 < L)] + (x y : Fin d → ℝ) : + cov[fun ω => torusSourceField d L ω (torusProjection d L x), + fun ω => torusSourceField d L ω (torusProjection d L y); sourceLaw d] = + sourceSpectralKernel d L (x - y) := by + simp_rw [torusSourceField_pullback] + exact sourceField_covariance d (Fact.out : 0 < L).ne' x y + +end I4Torus diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SourceTorusC4.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SourceTorusC4.lean new file mode 100644 index 000000000..30cfb670f --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SourceTorusC4.lean @@ -0,0 +1,134 @@ +import SourceC4 +import SourceTorus +import Mathlib.Topology.OpenPartialHomeomorph.Constructions +import Mathlib.Geometry.Manifold.ContMDiff.Basic +import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace +import Mathlib.Topology.LocallyConstant.Basic + +noncomputable section +open I4Foundation I4Weights I4Source I4C4 I4Torus MeasureTheory +open scoped Manifold ContDiff Topology +namespace I4TorusC4 + +def quotientChart (d : ℕ) (L : ℝ) [Fact (0 < L)] (a : Fin d → ℝ) : + OpenPartialHomeomorph (SourceTorus d L) (Fin d → ℝ) := + (OpenPartialHomeomorph.pi (fun i : Fin d => AddCircle.openPartialHomeomorphCoe L (a i))).symm + +def chartCenter (d : ℕ) (L : ℝ) [Fact (0 < L)] (q : SourceTorus d L) : Fin d → ℝ := + fun i => torusRepresentative d L q i - L / 2 + +theorem quotientChart_symm (d : ℕ) (L : ℝ) [Fact (0 < L)] (a : Fin d → ℝ) : + (quotientChart d L a).symm = torusProjection d L := by + rfl + +theorem centered_chart_covers (d : ℕ) (L : ℝ) [Fact (0 < L)] (q : SourceTorus d L) : + q ∈ (quotientChart d L (chartCenter d L q)).source := by + let e := OpenPartialHomeomorph.pi + (fun i : Fin d => AddCircle.openPartialHomeomorphCoe L (chartCenter d L q i)) + have hs : torusRepresentative d L q ∈ e.source := by + change torusRepresentative d L q ∈ Set.pi Set.univ + (fun i => Set.Ioo (chartCenter d L q i) (chartCenter d L q i + L)) + intro i _ + change chartCenter d L q i < torusRepresentative d L q i ∧ + torusRepresentative d L q i < chartCenter d L q i + L + have hL : 0 < L := Fact.out + dsimp [chartCenter] + constructor <;> linarith + have he : e (torusRepresentative d L q) = q := + torusProjection_representative d L q + have ht := e.map_source hs + rw [he] at ht + exact ht + +@[instance_reducible] def actualChartedSpace (d : ℕ) (L : ℝ) [Fact (0 < L)] : + ChartedSpace (Fin d → ℝ) (SourceTorus d L) where + atlas := Set.range (quotientChart d L) + chartAt := fun q => quotientChart d L (chartCenter d L q) + mem_chart_source := centered_chart_covers d L + chart_mem_atlas := fun q => ⟨chartCenter d L q, rfl⟩ + +attribute [local instance] actualChartedSpace + +theorem chart_transition_contDiff (d : ℕ) (L : ℝ) [Fact (0 < L)] (a b : Fin d → ℝ) : + ContDiffOn ℝ ∞ ((quotientChart d L a).symm.trans (quotientChart d L b)) + ((quotientChart d L a).symm.trans (quotientChart d L b)).source := by + let ea := quotientChart d L a + let eb := quotientChart d L b + let T := ea.symm.trans eb + let S := T.source + have hproj (x : S) : torusProjection d L (T x) = torusProjection d L x := by + have hb : ea.symm (x : Fin d → ℝ) ∈ eb.source := x.property.2 + have hi := eb.left_inv hb + simpa only [T, ea, eb, OpenPartialHomeomorph.trans_apply, quotientChart_symm] using hi + have hlat (x : S) (i : Fin d) : T x i - (x : Fin d → ℝ) i ∈ + AddSubgroup.zmultiples L := by + apply AddSubgroup.mem_zmultiples_iff.mpr + apply (AddCircle.coe_eq_zero_iff (p := L)).mp + rw [AddCircle.coe_sub] + exact sub_eq_zero.mpr (congrFun (hproj x) i) + let delta : S → Fin d → AddSubgroup.zmultiples L := + fun x i => ⟨T x i - (x : Fin d → ℝ) i, hlat x i⟩ + have hd : Continuous delta := by + apply continuous_pi + intro i + apply Continuous.subtype_mk + exact ((continuous_apply i).comp T.continuousOn.domRestrict).sub + ((continuous_apply i).comp continuous_subtype_val) + have hlc : IsLocallyConstant delta := + (IsLocallyConstant.iff_continuous delta).mpr hd + apply T.open_source.contDiffOn_iff.mpr + intro x hx + let x0 : S := ⟨x, hx⟩ + let c : Fin d → ℝ := fun i => (delta x0 i : ℝ) + have hsub : ∀ᶠ y : S in 𝓝 x0, T y = (y : Fin d → ℝ) + c := by + filter_upwards [hlc.eventually_eq x0] with y hy + funext i + have hi := congrArg (fun z : Fin d → AddSubgroup.zmultiples L => (z i : ℝ)) hy + change T y i - (y : Fin d → ℝ) i = c i at hi + change T y i = (y : Fin d → ℝ) i + c i + linarith + have hamb : (fun y => T y) =ᶠ[𝓝 x] (fun y => y + c) := by + have hm : ∀ᶠ y in Filter.map ((↑) : S → (Fin d → ℝ)) (𝓝 x0), T y = y + c := + Filter.eventually_map.mpr hsub + have hmap : Filter.map ((↑) : S → (Fin d → ℝ)) (𝓝 x0) = 𝓝 x := + map_nhds_subtype_coe_eq_nhds hx (T.open_source.mem_nhds hx) + rw [hmap] at hm + exact hm + exact (contDiff_id.add contDiff_const).contDiffAt.congr_of_eventuallyEq hamb + +theorem torus_isManifold (d : ℕ) (L : ℝ) [Fact (0 < L)] : + IsManifold 𝓘(ℝ, Fin d → ℝ) ∞ (SourceTorus d L) := by + apply isManifold_of_contDiffOn + intro e e' he he' + change e ∈ Set.range (quotientChart d L) at he + change e' ∈ Set.range (quotientChart d L) at he' + rcases he with ⟨a, rfl⟩ + rcases he' with ⟨b, rfl⟩ + simpa only [modelWithCornersSelf_coe, modelWithCornersSelf_coe_symm, + Function.comp_id, Function.id_comp, Set.preimage_id, Set.range_id, Set.inter_univ] using + chart_transition_contDiff d L a b + +theorem torusSourceField_contMDiff_ae (d : ℕ) (L : ℝ) [Fact (0 < L)] : + ∀ᵐ sample ∂sourceLaw d, + ContMDiff 𝓘(ℝ, Fin d → ℝ) 𝓘(ℝ) 4 (torusSourceField d L sample) := by + let _ := torus_isManifold d L + filter_upwards [sourceField_contDiff_ae d (Fact.out : 0 < L).ne'] with sample hs + intro q + rw [contMDiffAt_iff_source] + have heq : torusSourceField d L sample ∘ (extChartAt 𝓘(ℝ, Fin d → ℝ) q).symm = + sourceField d L sample := by + funext x + change torusSourceField d L sample ((quotientChart d L (chartCenter d L q)).symm x) = _ + rw [quotientChart_symm] + exact torusSourceField_pullback d L sample x + rw [heq, contMDiffWithinAt_iff_contDiffWithinAt] + exact hs.contDiffAt.contDiffWithinAt + +theorem torusSourceField_weakSuperlevel_closed_ae (d : ℕ) (L : ℝ) [Fact (0 < L)] : + ∀ᵐ sample ∂sourceLaw d, ∀ h : ℝ, + IsClosed {q : SourceTorus d L | h ≤ torusSourceField d L sample q} := by + filter_upwards [torusSourceField_contMDiff_ae d L] with sample hs + intro h + exact isClosed_Ici.preimage hs.continuous + +end I4TorusC4 diff --git a/frontiers/source_derivative_gaussian_20261025/inputs/SpectralWeights.lean b/frontiers/source_derivative_gaussian_20261025/inputs/SpectralWeights.lean new file mode 100644 index 000000000..191a22f14 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/inputs/SpectralWeights.lean @@ -0,0 +1,160 @@ +import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable +import Mathlib.Analysis.Normed.Ring.InfiniteSum +import Mathlib.Data.Fin.Tuple.Basic +import Mathlib.Algebra.BigOperators.Fin + +/-! Concrete square-root Gaussian spectral majorants for the required I4 bridge. +The product majorant dominates the l1 frequency norm for the default Pi sup norm. +This file alone proves neither a Gaussian field law nor sample smoothness. -/ + +noncomputable section +open Real +open scoped BigOperators +namespace I4Weights + +def decay (a : ℝ) (k : ℤ) : ℝ := exp (-a * (k : ℝ) ^ 2) +def coordinateBound (a c : ℝ) (k : ℤ) : ℝ := + decay a k * (1 + |c * (k : ℝ)|) ^ 4 +def productDecay (d : ℕ) (a : ℝ) (k : Fin d → ℤ) : ℝ := + ∏ i, decay a (k i) +def productBound (d : ℕ) (a c : ℝ) (k : Fin d → ℤ) : ℝ := + ∏ i, coordinateBound a c (k i) +def sourceA (L : ℝ) : ℝ := π ^ 2 / L ^ 2 +def sourceC (L : ℝ) : ℝ := 2 * π / L +def sourceWeight (d : ℕ) (L : ℝ) (k : Fin d → ℤ) : ℝ := + productDecay d (2 * sourceA L) k +def sourceZ (d : ℕ) (L : ℝ) : ℝ := ∑' k, sourceWeight d L k +def pairedBound (d : ℕ) (L : ℝ) (k : Fin d → ℤ) : ℝ := + Real.sqrt (2 / sourceZ d L) * productBound d (sourceA L) (sourceC L) k + +theorem summable_decay_pow {a : ℝ} (ha : 0 < a) (n : ℕ) : + Summable (fun k : ℤ => decay a k * |(k : ℝ)| ^ n) := by + have hb := summable_pow_mul_jacobiTheta₂_term_bound 0 + (show 0 < a / π by positivity) n + have he (k : ℤ) : -π * (a / π * (k : ℝ) ^ 2 - 2 * 0 * |(k : ℝ)|) = + -a * (k : ℝ) ^ 2 := by + field_simp + ring + simp_rw [Int.cast_abs, he] at hb + simpa [decay, mul_comm] using hb + +theorem coordinateBound_summable {a c : ℝ} (ha : 0 < a) : + Summable (coordinateBound a c) := by + have hs := summable_decay_pow ha + convert ((((hs 0).add ((hs 1).mul_left (4 * |c|))).add + ((hs 2).mul_left (6 * |c| ^ 2))).add + ((hs 3).mul_left (4 * |c| ^ 3))).add ((hs 4).mul_left (|c| ^ 4)) using 1 + funext k + simp only [coordinateBound, abs_mul] + ring + +set_option maxHeartbeats 800000 in +theorem finite_product_summable {f : ℤ → ℝ} (hf : Summable f) (hn : ∀ k, 0 ≤ f k) + (d : ℕ) : Summable (fun k : Fin d → ℤ => ∏ i, f (k i)) := by + classical + induction d with + | zero => exact (hasSum_fintype _).summable + | succ d ih => + have hp : Summable (fun p : ℤ × (Fin d → ℤ) => f p.1 * ∏ i, f (p.2 i)) := + Summable.mul_of_nonneg (ι := ℤ) (ι' := Fin d → ℤ) + (f := f) (g := fun k : Fin d → ℤ => ∏ i, f (k i)) + hf ih hn (fun k => Finset.prod_nonneg (fun i _ => hn (k i))) + have heq : + ((fun k : Fin (d + 1) → ℤ => ∏ i, f (k i)) ∘ + (Fin.consEquiv (fun _ : Fin (d + 1) => ℤ))) = + (fun p : ℤ × (Fin d → ℤ) => f p.1 * ∏ i, f (p.2 i)) := by + funext p + simp only [Function.comp_apply, Fin.consEquiv_apply, Fin.prod_univ_succ, + Fin.cons_zero, Fin.cons_succ] + apply (Fin.consEquiv (fun _ : Fin (d + 1) => ℤ)).summable_iff.mp + rw [heq] + exact hp + +theorem productBound_summable {a c : ℝ} (ha : 0 < a) (d : ℕ) : + Summable (productBound d a c) := by + exact finite_product_summable (coordinateBound_summable ha) + (fun k => by unfold coordinateBound decay; positivity) d + +theorem productDecay_eq_exp (d : ℕ) (a : ℝ) (k : Fin d → ℤ) : + productDecay d a k = exp (-a * ∑ i, (k i : ℝ) ^ 2) := by + unfold productDecay decay + rw [← Real.exp_sum, ← Finset.mul_sum] + +theorem frequency_l1_le_product (d : ℕ) (v : Fin d → ℝ) (hv : ∀ i, 0 ≤ v i) : + 1 + ∑ i, v i ≤ ∏ i, (1 + v i) := by + classical + have H (s : Finset (Fin d)) : 1 + ∑ i ∈ s, v i ≤ ∏ i ∈ s, (1 + v i) := by + induction s using Finset.induction_on with + | empty => simp + | @insert i s hi ih => + rw [Finset.sum_insert hi, Finset.prod_insert hi] + have hsum : 0 ≤ ∑ j ∈ s, v j := Finset.sum_nonneg (fun j _ => hv j) + calc + 1 + (v i + ∑ j ∈ s, v j) ≤ (1 + v i) * (1 + ∑ j ∈ s, v j) := by + nlinarith [mul_nonneg (hv i) hsum] + _ ≤ (1 + v i) * ∏ j ∈ s, (1 + v j) := + mul_le_mul_of_nonneg_left ih (by linarith [hv i]) + exact H Finset.univ + +theorem sourceA_pos {L : ℝ} (hL : L ≠ 0) : 0 < sourceA L := by + unfold sourceA + positivity + +theorem sourceWeight_eq_physical (d : ℕ) (L : ℝ) (k : Fin d → ℤ) : + sourceWeight d L k = exp (-(∑ i, (sourceC L * (k i : ℝ)) ^ 2) / 2) := by + have hc : sourceC L ^ 2 = 4 * sourceA L := by + unfold sourceC sourceA + ring + have hs : (∑ i, (sourceC L * (k i : ℝ)) ^ 2) = + 4 * sourceA L * ∑ i, (k i : ℝ) ^ 2 := by + rw [Finset.mul_sum] + apply Finset.sum_congr rfl + intro i _ + rw [mul_pow, hc] + rw [sourceWeight, productDecay_eq_exp, hs] + congr 1 + ring + +theorem sourceWeight_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + Summable (sourceWeight d L) := by + have ha : 0 < 2 * sourceA L := mul_pos (by norm_num) (sourceA_pos hL) + have hs : Summable (decay (2 * sourceA L)) := by + simpa using summable_decay_pow ha 0 + exact finite_product_summable hs (fun k => (Real.exp_pos _).le) d + +theorem sourceZ_pos (d : ℕ) {L : ℝ} (hL : L ≠ 0) : 0 < sourceZ d L := by + apply (sourceWeight_summable d hL).tsum_pos + (fun k => by unfold sourceWeight productDecay decay; positivity) (fun _ => 0) + simp [sourceWeight, productDecay, decay] + +theorem pairedBound_eq_amplitude (d : ℕ) {L : ℝ} (hL : L ≠ 0) (k : Fin d → ℤ) : + pairedBound d L k = Real.sqrt (2 * sourceWeight d L k / sourceZ d L) * + ∏ i, (1 + |sourceC L * (k i : ℝ)|) ^ 4 := by + have hsqrt : Real.sqrt (sourceWeight d L k) = productDecay d (sourceA L) k := by + have hw : sourceWeight d L k = (productDecay d (sourceA L) k) ^ 2 := by + rw [sourceWeight, productDecay_eq_exp, productDecay_eq_exp, pow_two, ← Real.exp_add] + congr 1 + ring + rw [hw, Real.sqrt_sq_eq_abs, abs_of_nonneg] + unfold productDecay decay + positivity + have hamp : Real.sqrt (2 * sourceWeight d L k / sourceZ d L) = + Real.sqrt (2 / sourceZ d L) * productDecay d (sourceA L) k := by + have he : 2 * sourceWeight d L k / sourceZ d L = + (2 / sourceZ d L) * sourceWeight d L k := by ring + rw [he, Real.sqrt_mul (by have := sourceZ_pos d hL; positivity), hsqrt] + have hp : productBound d (sourceA L) (sourceC L) k = + productDecay d (sourceA L) k * ∏ i, (1 + |sourceC L * (k i : ℝ)|) ^ 4 := by + exact Finset.prod_mul_distrib + rw [pairedBound, hp, hamp] + ring + +theorem pairedBound_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) : + Summable (pairedBound d L) := by + exact (productBound_summable (sourceA_pos hL) d).mul_left _ + +theorem pairedBound_subtype_summable (d : ℕ) {L : ℝ} (hL : L ≠ 0) + (S : Set (Fin d → ℤ)) : Summable (fun k : S => pairedBound d L k) := by + exact (pairedBound_summable d hL).subtype S + +end I4Weights diff --git a/frontiers/source_derivative_gaussian_20261025/lake-manifest.json b/frontiers/source_derivative_gaussian_20261025/lake-manifest.json new file mode 100644 index 000000000..9fe133235 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/lake-manifest.json @@ -0,0 +1,117 @@ +{ + "version": "1.2.0", + "packagesDir": ".lake/packages", + "packages": [ + { + "name": "plausible", + "url": "https://github.com/leanprover-community/plausible", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "118aa17ee84656b8bd727fef7c458ee8c833385c", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "name": 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}, + { + "name": "Cli", + "url": "https://github.com/leanprover/lean4-cli", + "type": "git", + "subDir": null, + "scope": "leanprover", + "rev": "e92c9f15fdfacc8536f31cfb3b7ad26c3c8cd204", + "manifestFile": "lake-manifest.json", + "inputRev": "v4.34.0", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "name": "mathlib", + "url": "https://github.com/leanprover-community/mathlib4.git", + "type": "git", + "subDir": null, + "scope": "", + "rev": "d13f23b723b8a846827a245b89c10fc7d3f11612", + "manifestFile": "lake-manifest.json", + "inputRev": "d13f23b723b8a846827a245b89c10fc7d3f11612", + "inherited": false, + "configFile": "lakefile.lean" + } + ], + "name": "source_smooth_realization", + "lakeDir": ".lake", + "fixedToolchain": true +} diff --git a/frontiers/source_derivative_gaussian_20261025/lakefile.toml b/frontiers/source_derivative_gaussian_20261025/lakefile.toml new file mode 100644 index 000000000..c167cc827 --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/lakefile.toml @@ -0,0 +1,55 @@ +name = "source_derivative_gaussian" +version = "0.1.0" +defaultTargets = ["SourceDerivativeGaussian", "ContactJetCovariance", "SourceContactCovariance", "SourceConfluentPins", "SourceContactSchur", "SourceDerivativeContinuity", "ConfluentIntegralIdentities", "SourceEndpointRegression", "GaussianRegressionLimits", "SourceEndpointParameters", "SourceContactEndpointParameters", "SourceConfluentLimits", "SourcePhysicalEndpointKernel", "SourceEndpointEstimates"] + +[[require]] +name = "mathlib" +git = "https://github.com/leanprover-community/mathlib4.git" +rev = "d13f23b723b8a846827a245b89c10fc7d3f11612" + +[[lean_lib]] +name = "SourceDerivativeInputs" +srcDir = "inputs" +roots = ["GaussianSeriesFoundation", "SpectralWeights", "SourceFourierSeries", "SourceC4", "SourceGaussianLaw", "SourceLattice", "SourceKernel", "SourceCovariance", "SourceTorus", "SourceTorusC4", "SourceSmooth", "SourceSmoothCarrier", "ResearchFormalCoreR1.DeterminantMoment", "ResearchFormalCoreR1.GaussianScalarMoments", "ResearchFormalCoreR1.GaussianJetMoments", "ResearchFormalCoreR1.GaussianNormalizer", "ResearchFormalCoreR1.EndpointGaussianMoments", "ResearchFormalCoreR1.NormalizerFloor", "ResearchFormalCoreR1.EndpointNormalizerBridge", "ResearchFormalCoreR1.GaussianCoupling", "ResearchFormalCoreR1.GaussianNormalizerLimit", "ResearchFormalCoreR1.PalmReweighting", "ResearchFormalCoreR1.GaussianRegressionKernel"] + +[[lean_lib]] +name = "SourceDerivativeGaussian" + +[[lean_lib]] +name = "ContactJetCovariance" + +[[lean_lib]] +name = "SourceContactCovariance" + +[[lean_lib]] +name = "SourceConfluentPins" + +[[lean_lib]] +name = "SourceContactSchur" + +[[lean_lib]] +name = "SourceDerivativeContinuity" + +[[lean_lib]] +name = "ConfluentIntegralIdentities" + +[[lean_lib]] +name = "SourceEndpointRegression" + +[[lean_lib]] +name = "GaussianRegressionLimits" + +[[lean_lib]] +name = "SourceEndpointParameters" + +[[lean_lib]] +name = "SourceContactEndpointParameters" + +[[lean_lib]] +name = "SourceConfluentLimits" + +[[lean_lib]] +name = "SourcePhysicalEndpointKernel" + +[[lean_lib]] +name = "SourceEndpointEstimates" diff --git a/frontiers/source_derivative_gaussian_20261025/lean-toolchain b/frontiers/source_derivative_gaussian_20261025/lean-toolchain new file mode 100644 index 000000000..ba8ebf2db --- /dev/null +++ b/frontiers/source_derivative_gaussian_20261025/lean-toolchain @@ -0,0 +1 @@ +leanprover/lean4:v4.34.1