From 80e80243da9de0bd81703bbc2dd19782196fbefe Mon Sep 17 00:00:00 2001 From: Claude Date: Sun, 11 Oct 2026 19:32:18 +0000 Subject: [PATCH 1/3] Proof packet round 4: theorems of 13 registered Side24/RN Props (Side24Analysis, Side24Interfaces, RNCluster) Three new modules of specifications/lean_statement_proofs_20261011, each a theorem whose type is exactly a registered Prop of the statement packet applied to its own binders, with helpers under Aux: SpecificationProofs/Side24Analysis.lean (9): thetaSummable (d), DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, ReferenceOddAndPinBlocks, ReferencePinDensityProduct, IntegrandRatioBounds, LatticeImageSumBound, DensityComparison. Routes: operator-norm Leibniz bound plus the Hermite-type recursion (no closed form for the iterated Frechet derivative), peeling of iteratedFDeriv in frame directions, HurwitzKernelBounds one-dimensional summability lifted to Fin d -> Z by a piFinset product bound, Loewner inverse/determinant monotonicity through CFC.sqrt. SpecificationProofs/Side24Interfaces.lean (3): coneMomentD1Identity (I), traceMoments (I), rayleighSquareLaw (I), from the interface's law fields alone (a.e.-measurability by the Dirac-image argument; polar-coordinate disk integral for the Rayleigh law). SpecificationProofs/RNCluster.lean (1): rareClusterPoissonObstructionO (mu a A). Root SpecificationProofs.lean imports the three modules after the existing eight. check.py --allow-dirty: SOURCE_CHECK_PASS, theorem_count 73 (was 60), 72 Props covered (was 59); test_check.py 70/70 in both Python modes; lake build clean with -DwarningAsError; every one of the 156 theorems of the new modules depends on exactly {propext, Classical.choice, Quot.sound}; environment audit PASS. No hosted receipt yet: the first lean-specification-proofs.yml run on this branch is it. Statement packet untouched; no claim-level label moves; alignment PENDING_INDEPENDENT_REVIEW; organizational-independence credit 0. Scientific effect NONE. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01Cz7WZybv8znP64SpPj6sWY --- .../SpecificationProofs.lean | 3 + .../SpecificationProofs/RNCluster.lean | 335 ++++ .../SpecificationProofs/Side24Analysis.lean | 1639 +++++++++++++++++ .../SpecificationProofs/Side24Interfaces.lean | 625 +++++++ 4 files changed, 2602 insertions(+) create mode 100644 specifications/lean_statement_proofs_20261011/SpecificationProofs/RNCluster.lean create mode 100644 specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Analysis.lean create mode 100644 specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Interfaces.lean diff --git a/specifications/lean_statement_proofs_20261011/SpecificationProofs.lean b/specifications/lean_statement_proofs_20261011/SpecificationProofs.lean index 43685b383..32f0e0c1f 100644 --- a/specifications/lean_statement_proofs_20261011/SpecificationProofs.lean +++ b/specifications/lean_statement_proofs_20261011/SpecificationProofs.lean @@ -6,3 +6,6 @@ import SpecificationProofs.P15Budget import SpecificationProofs.Side24Elementary import SpecificationProofs.RNProbability import SpecificationProofs.P15Interpolation +import SpecificationProofs.Side24Analysis +import SpecificationProofs.Side24Interfaces +import SpecificationProofs.RNCluster diff --git a/specifications/lean_statement_proofs_20261011/SpecificationProofs/RNCluster.lean b/specifications/lean_statement_proofs_20261011/SpecificationProofs/RNCluster.lean new file mode 100644 index 000000000..405e6071c --- /dev/null +++ b/specifications/lean_statement_proofs_20261011/SpecificationProofs/RNCluster.lean @@ -0,0 +1,335 @@ +import Specifications.RN + +/-! +# RN: kernel-checked proof of the abstract rare-cluster Poisson obstruction +(`RareClusterPoissonObstructionO`, statement module `Specifications.RN`) + +Scientific effect: NONE. Organizational-independence credit: 0 (Anthropic / Claude). + +The theorem outside the `Aux` namespace has as its type exactly one registered `Prop` of +`Specifications/RN.lean` (statement packet `specifications/lean_statements_20261010`, which is +not modified here), applied to its own parameters `μ a A`. A kernel-checked proof of such a +`Prop` is evidence about that `Prop` only: it does not make the informal Layer 0 note +(OA-C6-RARE-CLUSTER-20260929-v1, Theorem O of c6_rare_cluster_laws_20260929/PROOF.md) reviewed, +accepted or "verified", it does not identify the formal objects (`pmfTV`, `poissonPMF`, the +family `μ`) with the source's (that alignment is PENDING_INDEPENDENT_REVIEW), it says nothing +about whether (H1) or (H2) hold for any Gaussian window count, and it promotes, closes, +discharges or reclassifies nothing. The Radon-Nikodym uniformity lemma `D3-LEMMA-RN-UNIF` and the +five validity premises of Theorem D1 v2.2(2) are untouched and remain OPEN. + +Provenance: this `Prop` was selected by the round-4 triage (workflow `wf_a3531fd8-891`) of the +registered `Prop`s still without a theorem; the proof was written and verified in its own lane +and is placed here unchanged. Helper lemmas live in `UniversalLaw.Proofs.RNCluster.Aux` (prefix +`rc_`); the gate inventories only the one theorem outside that namespace. + +The route is the source's, with the thresholds rearranged so that no `A^2` appears: +* the antecedents at `r = r0` give `2 a r0^3 ≤ 2 (1 - μ 0 - μ 1) ≤ ∑ n μ n ≤ A r0^3`, hence + `2a ≤ A` and in particular `A > 0`; +* `rStar := min r0 (min (a/2) (min (1/(a + 2A)) (1/2)))`; +* upper half, witness `lam = 0`: `poissonPMF 0 = δ_0`, so `pmfTV (μ r) (poissonPMF 0) = 1 - μ r 0 + = ∑_{n ≥ 1} μ r n ≤ ∑ n μ r n ≤ A r^3`; +* lower half, `lam ≥ 0`: half the `l^1` distance dominates the difference of the masses of every + tail event `{n ≥ k}` (for two summable families of total mass one). For `lam ≤ r^2` the tail + `{n ≥ 2}` gives `pmfTV ≥ (1 - μ 0 - μ 1) - (1 - e^{-lam} - lam e^{-lam}) ≥ a r^3 - lam^2 ≥ + a r^3 - r^4 ≥ (a/2) r^3`; for `lam > r^2` the tail `{n ≥ 1}` gives `pmfTV ≥ (1 - e^{-lam}) - + (1 - μ 0) ≥ (1 - e^{-r^2}) - A r^3 ≥ r^2/2 - A r^3 ≥ (a/2) r^3`, using `x/2 ≤ 1 - e^{-x}` on + `[0, 1/2]` and `r (a + 2A) ≤ 1`. +Total mass one of `poissonPMF lam` for `lam ≥ 0` is Mathlib's +`ProbabilityTheory.hasSum_one_poissonMeasure`. + +Remark recorded by the lane, for the maintainers: a theorem of this `Prop` kernel-checks only the +abstract implication (H1) ∧ (H2) ⇒ (3.1), which is what the source claims for Theorem O; the +hypotheses imply `2a ≤ A`, so the `Prop` is not vacuous. The claim row keeps its register labels. +-/ + +namespace UniversalLaw.Proofs.RNCluster + +open UniversalLaw.Spec.RN + +namespace Aux + +/-! ### (a) Tails of a probability mass function on `ℕ` -/ + +/-- `∑_{n ≥ 1} μ n = 1 - μ 0` for a summable family of total mass one. -/ +theorem rc_tail_one_eq (μ : ℕ → ℝ) (hs : Summable μ) (h1 : ∑' n, μ n = 1) : + ∑' n : ℕ, μ (n + 1) = 1 - μ 0 := by + have h := Summable.sum_add_tsum_nat_add 1 hs + rw [h1, Finset.sum_range_one] at h + linarith + +/-- `∑_{n ≥ 2} μ n = 1 - μ 0 - μ 1` for a summable family of total mass one. -/ +theorem rc_tail_two_eq (μ : ℕ → ℝ) (hs : Summable μ) (h1 : ∑' n, μ n = 1) : + ∑' n : ℕ, μ (n + 2) = 1 - μ 0 - μ 1 := by + have h := Summable.sum_add_tsum_nat_add 2 hs + rw [h1] at h + simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add] at h + linarith + +/-- Markov for the mean: `∑_{n ≥ 1} μ n ≤ ∑ n μ n` for a nonnegative family with summable +mean. -/ +theorem rc_tail_one_le_mean (μ : ℕ → ℝ) (h0 : ∀ n, 0 ≤ μ n) (hs : Summable μ) + (hm : Summable (fun n : ℕ => (n : ℝ) * μ n)) : + ∑' n : ℕ, μ (n + 1) ≤ ∑' n : ℕ, (n : ℝ) * μ n := by + have hs1 : Summable (fun n : ℕ => μ (n + 1)) := (summable_nat_add_iff 1).mpr hs + have hm1 : Summable (fun n : ℕ => ((n + 1 : ℕ) : ℝ) * μ (n + 1)) := + (summable_nat_add_iff 1).mpr hm + calc ∑' n : ℕ, μ (n + 1) ≤ ∑' n : ℕ, ((n + 1 : ℕ) : ℝ) * μ (n + 1) := by + refine hs1.tsum_le_tsum (fun n => ?_) hm1 + have : (1 : ℝ) ≤ ((n + 1 : ℕ) : ℝ) := by exact_mod_cast Nat.succ_le_succ (Nat.zero_le n) + nlinarith [h0 (n + 1)] + _ ≤ ∑' n : ℕ, (n : ℝ) * μ n := by + have h2 := Summable.sum_add_tsum_nat_add 1 hm + simp only [Finset.sum_range_one, Nat.cast_zero, zero_mul, zero_add] at h2 + exact le_of_eq h2 + +/-- `1 - μ 0 ≤ ∑ n μ n` for a probability mass function with summable mean. -/ +theorem rc_one_sub_head_le_mean (μ : ℕ → ℝ) (h0 : ∀ n, 0 ≤ μ n) (hs : Summable μ) + (h1 : ∑' n, μ n = 1) (hm : Summable (fun n : ℕ => (n : ℝ) * μ n)) : + 1 - μ 0 ≤ ∑' n : ℕ, (n : ℝ) * μ n := by + have h := rc_tail_one_le_mean μ h0 hs hm + rwa [rc_tail_one_eq μ hs h1] at h + +/-- `2 (1 - μ 0 - μ 1) ≤ ∑ n μ n`: the mean dominates twice the mass of `{n ≥ 2}` (this is the +source's remark `2a ≤ A`, before dividing by `r^3`). -/ +theorem rc_two_mul_tail_two_le_mean (μ : ℕ → ℝ) (h0 : ∀ n, 0 ≤ μ n) (hs : Summable μ) + (h1 : ∑' n, μ n = 1) (hm : Summable (fun n : ℕ => (n : ℝ) * μ n)) : + 2 * (1 - μ 0 - μ 1) ≤ ∑' n : ℕ, (n : ℝ) * μ n := by + have hs2 : Summable (fun n : ℕ => μ (n + 2)) := (summable_nat_add_iff 2).mpr hs + have hm2 : Summable (fun n : ℕ => ((n + 2 : ℕ) : ℝ) * μ (n + 2)) := + (summable_nat_add_iff 2).mpr hm + have htail : ∑' n : ℕ, μ (n + 2) = 1 - μ 0 - μ 1 := rc_tail_two_eq μ hs h1 + have hsplit := Summable.sum_add_tsum_nat_add 2 hm + have hhead : ∑ i ∈ Finset.range 2, (i : ℝ) * μ i = μ 1 := by + simp [Finset.sum_range_succ] + rw [hhead] at hsplit + have hle : ∑' n : ℕ, 2 * μ (n + 2) ≤ ∑' n : ℕ, ((n + 2 : ℕ) : ℝ) * μ (n + 2) := by + refine (hs2.mul_left 2).tsum_le_tsum (fun n => ?_) hm2 + have : (2 : ℝ) ≤ ((n + 2 : ℕ) : ℝ) := by exact_mod_cast Nat.le_add_left 2 n + exact mul_le_mul_of_nonneg_right this (h0 _) + rw [tsum_mul_left, htail] at hle + linarith [h0 1] + +/-! ### (b) The Poisson family -/ + +/-- `poissonPMF lam` has total mass one for `lam ≥ 0` (Mathlib's +`ProbabilityTheory.hasSum_one_poissonMeasure`, whose summand is definitionally +`poissonPMF lam n`). -/ +theorem rc_hasSum_poissonPMF (lam : ℝ) (hl : 0 ≤ lam) : HasSum (poissonPMF lam) 1 := by + have h : HasSum (fun n : ℕ => Real.exp (-lam) * lam ^ n / (Nat.factorial n : ℝ)) 1 := + ProbabilityTheory.hasSum_one_poissonMeasure ⟨lam, hl⟩ + exact h + +/-- `poissonPMF lam 0 = e^{-lam}`. -/ +theorem rc_poissonPMF_zero (lam : ℝ) : poissonPMF lam 0 = Real.exp (-lam) := by + simp [poissonPMF] + +/-- `poissonPMF lam 1 = e^{-lam} lam`. -/ +theorem rc_poissonPMF_one (lam : ℝ) : poissonPMF lam 1 = Real.exp (-lam) * lam := by + simp [poissonPMF] + +/-- The Poisson tail `∑_{n ≥ 1} Pois(lam)(n) = 1 - e^{-lam}` for `lam ≥ 0`. -/ +theorem rc_poisson_tail_one (lam : ℝ) (hl : 0 ≤ lam) : + ∑' n : ℕ, poissonPMF lam (n + 1) = 1 - Real.exp (-lam) := by + have hsum := rc_hasSum_poissonPMF lam hl + have h := Summable.sum_add_tsum_nat_add 1 hsum.summable + rw [hsum.tsum_eq, Finset.sum_range_one, rc_poissonPMF_zero] at h + linarith + +/-- The Poisson tail `∑_{n ≥ 2} Pois(lam)(n) = 1 - e^{-lam} - lam e^{-lam}` for `lam ≥ 0`. -/ +theorem rc_poisson_tail_two (lam : ℝ) (hl : 0 ≤ lam) : + ∑' n : ℕ, poissonPMF lam (n + 2) = 1 - Real.exp (-lam) - lam * Real.exp (-lam) := by + have hsum := rc_hasSum_poissonPMF lam hl + have h := Summable.sum_add_tsum_nat_add 2 hsum.summable + rw [hsum.tsum_eq] at h + simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add] at h + rw [rc_poissonPMF_zero, rc_poissonPMF_one] at h + linarith + +/-- `x/2 ≤ 1 - e^{-x}` for `0 ≤ x ≤ 1/2`, from `|e^y - 1 - y| ≤ y^2` on `|y| ≤ 1`. -/ +theorem rc_half_le_one_sub_exp (x : ℝ) (h0 : 0 ≤ x) (h1 : x ≤ 1 / 2) : + x / 2 ≤ 1 - Real.exp (-x) := by + have h := Real.abs_exp_sub_one_sub_id_le (x := -x) (by rw [abs_neg, abs_of_nonneg h0]; linarith) + rw [abs_le] at h + nlinarith [h.1, h.2] + +/-- `Pois(lam)(n ≥ 2) = 1 - e^{-lam} - lam e^{-lam} ≤ lam^2` for `lam ≥ 0`, from +`1 + y ≤ e^y` multiplied by `1 + lam ≥ 0`. -/ +theorem rc_poisson_tail_two_le_sq (lam : ℝ) (h0 : 0 ≤ lam) : + 1 - Real.exp (-lam) - lam * Real.exp (-lam) ≤ lam ^ 2 := by + have h := Real.add_one_le_exp (-lam) + have hprod := mul_nonneg (sub_nonneg.mpr h) (show (0 : ℝ) ≤ 1 + lam by linarith) + nlinarith [hprod, Real.exp_pos (-lam)] + +/-! ### (c) Total variation against `δ_0` and against tail events -/ + +/-- The upper half's witness: `pmfTV μ (poissonPMF 0) = 1 - μ 0` for a probability mass +function `μ` (`poissonPMF 0 = δ_0`). -/ +theorem rc_pmfTV_poissonPMF_zero (μ : ℕ → ℝ) (h0 : ∀ n, 0 ≤ μ n) (hs : Summable μ) + (h1 : ∑' n, μ n = 1) : pmfTV μ (poissonPMF 0) = 1 - μ 0 := by + unfold pmfTV + have hμ1 : μ 0 ≤ 1 := by + rw [← h1]; exact hs.le_tsum 0 (fun j _ => h0 j) + have hterm : ∀ n, |μ n - poissonPMF 0 n| = if n = 0 then 1 - μ 0 else μ n := by + intro n + unfold poissonPMF + rcases n with _ | n + · simp + rw [abs_sub_comm, abs_of_nonneg (by linarith)] + · simp [abs_of_nonneg (h0 _)] + simp_rw [hterm] + have hsum : Summable (fun n : ℕ => if n = 0 then 1 - μ 0 else μ n) := by + rw [← summable_nat_add_iff 1] + simpa using (summable_nat_add_iff 1).mpr hs + rw [hsum.tsum_eq_zero_add] + have htail : ∑' n : ℕ, (if n + 1 = 0 then 1 - μ 0 else μ (n + 1)) = ∑' n : ℕ, μ (n + 1) := + tsum_congr (fun n => by simp) + rw [htail, ite_eq_left rfl, rc_tail_one_eq μ hs h1] + ring + +/-- Half the `l^1` distance dominates the difference of the masses of every tail event +`{n ≥ k}`, for two summable families of total mass one: the head and the tail of `μ - ν` sum to +zero, so the tail difference is bounded by the head's `l^1` mass and by its own. -/ +theorem rc_abs_tail_sub_le_pmfTV (μ ν : ℕ → ℝ) (hμ : Summable μ) (hν : Summable ν) + (h1 : ∑' n, μ n = 1) (h1' : ∑' n, ν n = 1) (k : ℕ) : + |(∑' n : ℕ, μ (n + k)) - ∑' n : ℕ, ν (n + k)| ≤ pmfTV μ ν := by + have hμk : Summable (fun n : ℕ => μ (n + k)) := (summable_nat_add_iff k).mpr hμ + have hνk : Summable (fun n : ℕ => ν (n + k)) := (summable_nat_add_iff k).mpr hν + rw [← hμk.tsum_sub hνk] + unfold pmfTV + have hfs : Summable (fun n => μ n - ν n) := hμ.sub hν + have hfabs : Summable (fun n => |μ n - ν n|) := hfs.abs + have htot : ∑' n, (μ n - ν n) = 0 := by rw [hμ.tsum_sub hν, h1, h1']; ring + have hsplit := Summable.sum_add_tsum_nat_add k hfs + rw [htot] at hsplit + have hsplit_abs := Summable.sum_add_tsum_nat_add k hfabs + have hhead : |∑ i ∈ Finset.range k, (μ i - ν i)| ≤ ∑ i ∈ Finset.range k, |μ i - ν i| := + Finset.abs_sum_le_sum_abs _ _ + have htail : |∑' i : ℕ, (μ (i + k) - ν (i + k))| ≤ ∑' i : ℕ, |μ (i + k) - ν (i + k)| := by + have := norm_tsum_le_tsum_norm (f := fun i : ℕ => μ (i + k) - ν (i + k)) ?_ + · simpa only [Real.norm_eq_abs] using this + · simpa only [Real.norm_eq_abs] using (summable_nat_add_iff k).mpr hfabs + have heq : ∑ i ∈ Finset.range k, (μ i - ν i) = -∑' i : ℕ, (μ (i + k) - ν (i + k)) := by + linarith + rw [heq, abs_neg] at hhead + linarith + +/-- The tail `{n ≥ 2}` against `Pois(lam)`: +`|(1 - μ 0 - μ 1) - (1 - e^{-lam} - lam e^{-lam})| ≤ pmfTV μ (poissonPMF lam)`. -/ +theorem rc_tail_two_bound (μ : ℕ → ℝ) (hs : Summable μ) (h1 : ∑' n, μ n = 1) (lam : ℝ) + (hl : 0 ≤ lam) : + |(1 - μ 0 - μ 1) - (1 - Real.exp (-lam) - lam * Real.exp (-lam))| ≤ + pmfTV μ (poissonPMF lam) := by + have hP := rc_hasSum_poissonPMF lam hl + have h := rc_abs_tail_sub_le_pmfTV μ (poissonPMF lam) hs hP.summable h1 hP.tsum_eq 2 + rwa [rc_tail_two_eq μ hs h1, rc_poisson_tail_two lam hl] at h + +/-- The tail `{n ≥ 1}` against `Pois(lam)`: +`|(1 - μ 0) - (1 - e^{-lam})| ≤ pmfTV μ (poissonPMF lam)`. -/ +theorem rc_tail_one_bound (μ : ℕ → ℝ) (hs : Summable μ) (h1 : ∑' n, μ n = 1) (lam : ℝ) + (hl : 0 ≤ lam) : + |(1 - μ 0) - (1 - Real.exp (-lam))| ≤ pmfTV μ (poissonPMF lam) := by + have hP := rc_hasSum_poissonPMF lam hl + have h := rc_abs_tail_sub_le_pmfTV μ (poissonPMF lam) hs hP.summable h1 hP.tsum_eq 1 + rwa [rc_tail_one_eq μ hs h1, rc_poisson_tail_one lam hl] at h + +/-! ### (d) The two halves at a fixed `r` -/ + +/-- The lower half at a fixed `r` with `0 < r ≤ min (a/2) (min (1/(a + 2A)) (1/2))`, for a +probability mass function `μ` with summable mean satisfying (H1) and (H2) at `r`: every +`lam ≥ 0` is at distance at least `(a/2) r^3`. -/ +theorem rc_lower_half (μ : ℕ → ℝ) (a A r : ℝ) (hA : 0 < A) (hr : 0 < r) + (hr_a : r ≤ a / 2) (hr_A : r ≤ 1 / (a + 2 * A)) (hr_half : r ≤ 1 / 2) + (h0 : ∀ n, 0 ≤ μ n) (hs : Summable μ) (h1 : ∑' n, μ n = 1) + (hm : Summable (fun n : ℕ => (n : ℝ) * μ n)) (hmean : ∑' n : ℕ, (n : ℝ) * μ n ≤ A * r ^ 3) + (hH2 : a * r ^ 3 ≤ 1 - μ 0 - μ 1) (lam : ℝ) (hlam : 0 ≤ lam) : + (a / 2) * r ^ 3 ≤ pmfTV μ (poissonPMF lam) := by + have hr3 : 0 ≤ r ^ 3 := pow_nonneg hr.le 3 + have hr2 : r ^ 2 ≤ 1 / 2 := by + have h := pow_le_pow_left₀ hr.le hr_half 2 + norm_num at h + linarith + rcases le_or_gt lam (r ^ 2) with hcase | hcase + · -- `lam ≤ r^2`: test the tail `{n ≥ 2}` + have hk2 := abs_le.mp (rc_tail_two_bound μ hs h1 lam hlam) + have hsq := rc_poisson_tail_two_le_sq lam hlam + have hlam2 : lam ^ 2 ≤ (a / 2) * r ^ 3 := by + calc lam ^ 2 ≤ (r ^ 2) ^ 2 := pow_le_pow_left₀ hlam hcase 2 + _ = r * r ^ 3 := by ring + _ ≤ (a / 2) * r ^ 3 := mul_le_mul_of_nonneg_right hr_a hr3 + linarith [hk2.2] + · -- `lam > r^2`: test the tail `{n ≥ 1}` + have hk1 := abs_le.mp (rc_tail_one_bound μ hs h1 lam hlam) + have hexp : Real.exp (-lam) ≤ Real.exp (-(r ^ 2)) := + Real.exp_le_exp.mpr (neg_le_neg hcase.le) + have hhalf := rc_half_le_one_sub_exp (r ^ 2) (sq_nonneg r) hr2 + have hμ0 : 1 - μ 0 ≤ A * r ^ 3 := (rc_one_sub_head_le_mean μ h0 hs h1 hm).trans hmean + have hrA : r * (a + 2 * A) ≤ 1 := by + have hpos : 0 < a + 2 * A := by linarith + exact (le_div_iff₀ hpos).mp hr_A + have hkey : (a / 2) * r ^ 3 ≤ r ^ 2 / 2 - A * r ^ 3 := by + have hprod := mul_nonneg (div_nonneg (sq_nonneg r) (show (0 : ℝ) ≤ 2 by norm_num)) + (sub_nonneg.mpr hrA) + nlinarith [hprod] + linarith [hk1.1] + +/-- The upper half at a fixed `r`, witness `lam = 0`: `pmfTV μ (poissonPMF 0) = 1 - μ 0 ≤ +∑ n μ n ≤ A r^3`. -/ +theorem rc_upper_half (μ : ℕ → ℝ) (A r : ℝ) (h0 : ∀ n, 0 ≤ μ n) (hs : Summable μ) + (h1 : ∑' n, μ n = 1) (hm : Summable (fun n : ℕ => (n : ℝ) * μ n)) + (hmean : ∑' n : ℕ, (n : ℝ) * μ n ≤ A * r ^ 3) : + pmfTV μ (poissonPMF 0) ≤ A * r ^ 3 := by + rw [rc_pmfTV_poissonPMF_zero μ h0 hs h1] + exact (rc_one_sub_head_le_mean μ h0 hs h1 hm).trans hmean + +end Aux + +/-! ## Theorem O's shape: the abstract implication (H1) ∧ (H2) ⇒ (3.1) -/ + +/-- Proves: `UniversalLaw.Spec.RN.RareClusterPoissonObstructionO` (claim `OA-C6-RARE-CLUSTER-20260929-v1`). +Method: the antecedents at `r0` give `2a ≤ A` (`Aux.rc_two_mul_tail_two_le_mean`, +`le_of_mul_le_mul_right`), so `rStar := min r0 (min (a/2) (min (1/(a + 2A)) (1/2)))` is positive; +the upper half is the witness `lam = 0` with `pmfTV (μ r) (poissonPMF 0) = 1 - μ r 0 ≤ ∑ n μ r n` +(`Aux.rc_upper_half`, from `Aux.rc_pmfTV_poissonPMF_zero` and `Aux.rc_one_sub_head_le_mean`); +the lower half tests the tail `{n ≥ 2}` when `lam ≤ r^2` and the tail `{n ≥ 1}` when +`lam > r^2` (`Aux.rc_lower_half`, from `Aux.rc_abs_tail_sub_le_pmfTV`, the Poisson tails via +`ProbabilityTheory.hasSum_one_poissonMeasure`, `Real.add_one_le_exp` and +`Real.abs_exp_sub_one_sub_id_le`). +Does not establish: the informal Theorem O (3.1) of the source as the source's statement (the +alignment of `pmfTV`, `poissonPMF` and the family `μ` with the source's `dTV`, `Pois(lambda)` and +`Law N_r` is PENDING_INDEPENDENT_REVIEW); the result (Theorem O) or its proof as a reviewed fact; +`2a ≤ A` as a registered statement (it is derived internally from the antecedents at `r0`, as in +the source, and is neither a hypothesis nor a conclusion of the Prop); that (H1) or (H2) hold for +the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary +Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment +matching or the independent-replica limit; the conditional-law liminf remark; that the upper half +is the source's infimum bound verbatim (the Prop's existential form with witness `lam = 0` is +what is proved); the statement module's own `Does not claim:` text (the statement packet is not +edited here); the disposition of `OA-C6-RARE-CLUSTER-20260929-v1`, for which the register row +transcribes none (`disposition_transcribed: null`; title: PROOF_INDEX reviewed scoped result, +conditional packet) and which is in any case transcribed there, not derived here; nonauthor +review or acceptance of OA-C6-RARE-CLUSTER-20260929-v1. -/ +theorem rareClusterPoissonObstructionO (μ : ℝ → ℕ → ℝ) (a A : ℝ) : + UniversalLaw.Spec.RN.RareClusterPoissonObstructionO μ a A := by + intro ha hex + obtain ⟨r0, hr0, H⟩ := hex + -- the antecedents at `r0` force `2a ≤ A`, hence `A > 0` + have hA2 : 2 * a ≤ A := by + obtain ⟨h0, hs, h1, hm, hmean, hH2⟩ := H r0 hr0 le_rfl + have hmid := Aux.rc_two_mul_tail_two_le_mean (μ r0) h0 hs h1 hm + have hr3 : 0 < r0 ^ 3 := pow_pos hr0 3 + have hmul : 2 * a * r0 ^ 3 ≤ A * r0 ^ 3 := by linarith + exact le_of_mul_le_mul_right hmul hr3 + have hA : 0 < A := by linarith + have hpos : 0 < a + 2 * A := by linarith + refine ⟨min r0 (min (a / 2) (min (1 / (a + 2 * A)) (1 / 2))), ?_, ?_⟩ + · exact lt_min hr0 (lt_min (by linarith) (lt_min (div_pos one_pos hpos) (by norm_num))) + · intro r hr hrle + rw [le_min_iff, le_min_iff, le_min_iff] at hrle + obtain ⟨hr_r0, hr_a, hr_A, hr_half⟩ := hrle + obtain ⟨h0, hs, h1, hm, hmean, hH2⟩ := H r hr hr_r0 + refine ⟨?_, ⟨0, le_rfl, ?_⟩⟩ + · intro lam hlam + exact Aux.rc_lower_half (μ r) a A r hA hr hr_a hr_A hr_half h0 hs h1 hm hmean hH2 lam hlam + · exact Aux.rc_upper_half (μ r) A r h0 hs h1 hm hmean + +end UniversalLaw.Proofs.RNCluster diff --git a/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Analysis.lean b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Analysis.lean new file mode 100644 index 000000000..64caafbc8 --- /dev/null +++ b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Analysis.lean @@ -0,0 +1,1639 @@ +import Specifications.Side24 +import SpecificationProofs.Side24Elementary + +/-! +# SIDE24 analysis proofs: the theta family, the Section 2 derivative bounds and lattice sum, +the Section 1 reference blocks and pin density, the Section 4 ratio and density comparison +(statement module `Specifications.Side24`) + +Scientific effect: NONE. Organizational-independence credit: 0 (Anthropic / Claude). + +Every theorem outside the `Aux` namespace has as its type exactly one registered `Prop` of +`Specifications/Side24.lean` (statement packet `specifications/lean_statements_20261010`, which +is not modified here). A kernel-checked proof of such a `Prop` is evidence about that `Prop` +only: it does not make the informal Layer 0 note (SIDE24-COEFFICIENT-D23-20260924-v1, register +row `side24-coefficient`) reviewed, accepted or "verified", it does not identify the formal +definitions with the informal objects (that alignment is PENDING_INDEPENDENT_REVIEW), it does +not enclose the coefficient `c_{d,24}`, and it promotes, closes, discharges or reclassifies +nothing. The disposition of `side24-coefficient` (REVIEWED_SCOPED) is transcribed in the +register, not derived here. + +Provenance: the nine `Prop`s proved here were selected by the round-4 triage (workflow +`wf_a3531fd8-891`) of the registered `Prop`s still without a theorem; each proof was written and +verified in its own lane (one file per `Prop`) and the lane files were merged into this module, +with the developments the lanes reproduced from one another removed, call sites repointed to the +sibling inventory theorems, and the lanes' helper tags kept. Helper lemmas live in +`UniversalLaw.Proofs.Side24Analysis.Aux`, tagged by the `Prop` they serve (`th_`, `dp_`, `d1_`, +`d0_`, `rb_`, `rp_`, `ir_`, `li_`, `dc_`); the gate inventories only the nine theorems outside +that namespace. The mathematics, in the order of the module: + +* `ThetaSummable d` (`th_`): `‖24 n‖^2 = 576 ∑ n_i^2`, the shift bound + `exp(-‖z + v‖^2/2) ≤ exp(‖z‖^2/2) exp(-‖v‖^2/4)`, `Real.exp_sum`, the one-dimensional + summability of `m ↦ exp(-c m^2)` on `ℤ` (`HurwitzKernelBounds.summable_f_int`) and a + `Fintype.piFinset` product bound for `n ↦ ∏ i, g (n i)` over `Fin d → ℤ`; + `Summable.of_nonneg_of_le` assembles the `Prop` for every `d`. +* `DerivativePairingBound` (`dp_`): no closed form for the iterated Fréchet derivative of the + Gaussian is used; the operator norms `b_q = ‖iteratedFDeriv ℝ q (Kinf d) x‖` satisfy the + Hermite-type recursion `b_{q+2} ≤ ‖x‖ b_{q+1} + (q+1) b_q` (`norm_iteratedFDeriv_fderiv`, + `fderiv ℝ (Kinf d) y = Kinf d y • (-(innerSL ℝ y))`, the Leibniz bound + `norm_iteratedFDeriv_smul_le`), `b_0 = Kinf d x`, `b_1 = Kinf d x ‖x‖`, and the orders `q ≤ 6` + match the unfolded `pairingSum`; `ContinuousMultilinearMap.le_opNorm` at a unit tuple. +* `DerivativeBoundAtLeastOne` (`d1_`) and `DerivativeBoundAtZero` (`d0_`): consequences of the + pairing bound through hypothesis-form helpers, with `pairingSum q r ≤ 76 r^6` for `r ≥ 1` and + `pairingSum q 0 ≤ 15`. +* `ReferenceOddAndPinBlocks` (`rb_`): the contractions of `Kinf d` at `0` are evaluated by + peeling one direction at a time (`iteratedFDeriv_succ_apply_right`, + `iteratedFDeriv_clm_apply_const_apply`) with explicit `HasFDerivAt` closed forms of the + intermediate functions, giving `-⟪a, b⟫`, the three pairings at order four and `-15 ⟪u, u⟫^3` + at order six; in an orthonormal frame the five blocks follow. +* `ReferencePinDensityProduct` (`rp_`): the two block identities of the previous `Prop` and the + density algebra `(2π)^(-d/2) (2π)^(-d/2) 3^(-1/2) = (2π)^(-d) / √3`. +* `IntegrandRatioBounds` (`ir_`): pure real arithmetic in `epsilon = 10^(-108)`; Bernoulli's + inequality, exponent monotonicity of the real power and two exact rational inequalities of + degree five decided by `norm_num`. +* `LatticeImageSumBound` (`li_`): a uniform bound on the finite partial sums via a power-mean + inequality, `Real.exp_sum`, the `Fintype.piFinset` factorisation and one-dimensional sums over + `ℤ` compared with the geometric series through the registered `SuccessiveTermRatio` + (`SpecificationProofs.Side24Elementary.successiveTermRatio`, whence the import); the bound + obtained is `432 exp(-288) ≤ 1458 exp(-288)`. +* `DensityComparison` (`dc_`): real matrix analysis; Loewner antitonicity of the inverse and + monotonicity of the determinant (through `CFC.sqrt`, `Matrix.PosSemidef.conjTranspose_mul_mul_same`, + `Matrix.IsHermitian.det_eq_prod_eigenvalues`), determinant scaling and real-power bookkeeping. + +Remarks recorded by the lanes, for the maintainers (nothing below is edited here): the +proof-packet README rows "Side24 derivative bounds" ("Mathlib has no closed form for the iterated +Fréchet derivatives of a Gaussian"), "Side24 lattice sums" ("Mathlib's Jacobi-theta results are +one-dimensional") and "Side24 covariance and density" ("the evaluation of the Fréchet derivatives +is not available"; `IntegrandRatioBounds` grouped as "built on the derivative contractions") give +informal reasons that these proofs show are not load-bearing for the `Prop`s proved here; the +statement module's docstring for `ThetaSummable` reads "Does not claim: that the family is summable +(true, but not shown here)". Each is a remark about an informal reason or a statement-packet text, +not about any claim; no register label changes. +-/ + +namespace UniversalLaw.Proofs.Side24Analysis + +open UniversalLaw.Spec.Side24 +open scoped ContDiff +open scoped RealInnerProductSpace +open scoped MatrixOrder +open scoped Pointwise + +namespace Aux + +/-! ## `th_`: the theta family (`ThetaSummable`) -/ + +/-! ### (a) Coordinates and squared norm of the lattice vector `24 n` -/ + +/-- The `i`-th coordinate of `latticeVec d n` is `24 n_i`. -/ +theorem th_latticeVec_apply (d : ℕ) (n : Fin d → ℤ) (i : Fin d) : + latticeVec d n i = 24 * (n i : ℝ) := by + simp [latticeVec] + +/-- `‖24 n‖^2 = 576 * ∑ i, n_i^2` (`EuclideanSpace.norm_sq_eq`, coordinatewise). -/ +theorem th_norm_latticeVec_sq (d : ℕ) (n : Fin d → ℤ) : + ‖latticeVec d n‖ ^ 2 = 576 * intNormSq d n := by + rw [EuclideanSpace.norm_sq_eq, intNormSq, Finset.mul_sum] + refine Finset.sum_congr rfl fun i _ => ?_ + rw [th_latticeVec_apply, Real.norm_eq_abs, sq_abs] + ring + +/-- `exp(-‖24 n‖^2/4) = ∏ i, exp(-144 n_i^2)`, by `Real.exp_sum` and the previous lemma. -/ +theorem th_exp_neg_norm_latticeVec_sq (d : ℕ) (n : Fin d → ℤ) : + Real.exp (-(‖latticeVec d n‖ ^ 2) / 4) = ∏ i, Real.exp (-144 * ((n i : ℤ) : ℝ) ^ 2) := by + rw [← Real.exp_sum, th_norm_latticeVec_sq, intNormSq, Finset.mul_sum, neg_div, + Finset.sum_div, ← Finset.sum_neg_distrib] + congr 1 + refine Finset.sum_congr rfl fun i _ => ?_ + ring + +/-! ### (b) One-dimensional Gaussian summability on `ℤ` -/ + +/-- `m ↦ exp(-c m^2)` is summable on `ℤ` for `c > 0`: Mathlib's +`HurwitzKernelBounds.summable_f_int 0 0` at `t = c/π` after `-π m^2 (c/π) = -c m^2`. -/ +theorem th_summable_gauss_int (c : ℝ) (hc : 0 < c) : + Summable (fun m : ℤ => Real.exp (-c * (m : ℝ) ^ 2)) := by + have h := HurwitzKernelBounds.summable_f_int 0 0 (t := c / Real.pi) (by positivity) + refine h.congr fun m => ?_ + simp only [HurwitzKernelBounds.f_int, add_zero, pow_zero, one_mul] + congr 1 + field_simp + +/-! ### (c) Products of a summable nonnegative function over `Fin d → ℤ` -/ + +/-- For a nonnegative summable `g : ℤ → ℝ`, the product `n ↦ ∏ i, g (n i)` is summable over +`Fin d → ℤ`: a finite set `s` of lattice points lies in the box `Fintype.piFinset t` with +`t i = s.image (· i)`, over which the sum factors as `∏ i, ∑ m ∈ t i, g m ≤ (∑' m, g m)^d`. -/ +theorem th_summable_pi_prod (d : ℕ) (g : ℤ → ℝ) (hg0 : ∀ m, 0 ≤ g m) (hg : Summable g) : + Summable (fun n : Fin d → ℤ => ∏ i, g (n i)) := by + refine summable_of_sum_le (c := (∑' m, g m) ^ d) + (fun n => Finset.prod_nonneg fun i _ => hg0 _) fun s => ?_ + set t : Fin d → Finset ℤ := fun i => s.image (fun n => n i) + have hsub : s ⊆ Fintype.piFinset t := fun n hn => by + rw [Fintype.mem_piFinset] + intro i + exact Finset.mem_image_of_mem _ hn + have hle : ∀ i, ∑ m ∈ t i, g m ≤ ∑' m, g m := fun i => + hg.sum_le_tsum (t i) (fun m _ => hg0 m) + calc ∑ n ∈ s, ∏ i, g (n i) ≤ ∑ n ∈ Fintype.piFinset t, ∏ i, g (n i) := + Finset.sum_le_sum_of_subset_of_nonneg hsub fun n _ _ => Finset.prod_nonneg fun i _ => hg0 _ + _ = ∏ i, ∑ m ∈ t i, g m := (Finset.prod_univ_sum t fun _ m => g m).symm + _ ≤ ∏ _i : Fin d, ∑' m, g m := + Finset.prod_le_prod₀ (fun i _ => Finset.sum_nonneg fun m _ => hg0 m) (fun i _ => hle i) + _ = (∑' m, g m) ^ d := by simp + +/-! ### (d) The pointwise shift bound -/ + +/-- `exp(-‖z + v‖^2/2) ≤ exp(‖z‖^2/2) exp(-‖v‖^2/4)`: from `‖v‖ ≤ ‖z + v‖ + ‖z‖` one has +`‖v‖^2/4 ≤ (‖z + v‖ + ‖z‖)^2/4 ≤ ‖z + v‖^2/2 + ‖z‖^2/2`. -/ +theorem th_exp_shift_le (d : ℕ) (z v : E d) : + Real.exp (-(‖z + v‖ ^ 2) / 2) ≤ Real.exp (‖z‖ ^ 2 / 2) * Real.exp (-(‖v‖ ^ 2) / 4) := by + rw [← Real.exp_add, Real.exp_le_exp] + have h1 : ‖v‖ ≤ ‖z + v‖ + ‖z‖ := by + calc ‖v‖ = ‖(z + v) - z‖ := by rw [add_sub_cancel_left] + _ ≤ ‖z + v‖ + ‖z‖ := norm_sub_le _ _ + have h2 : 0 ≤ ‖z + v‖ := norm_nonneg _ + have h3 : 0 ≤ ‖z‖ := norm_nonneg _ + have h4 : 0 ≤ ‖v‖ := norm_nonneg _ + nlinarith [sq_nonneg (‖z + v‖ - ‖z‖)] + +/-- The dominating family `n ↦ exp(‖z‖^2/2) ∏ i, exp(-144 n_i^2)` is summable over +`Fin d → ℤ`. -/ +theorem th_summable_dominant (d : ℕ) (z : E d) : + Summable (fun n : Fin d → ℤ => + Real.exp (‖z‖ ^ 2 / 2) * ∏ i, Real.exp (-144 * ((n i : ℤ) : ℝ) ^ 2)) := + (th_summable_pi_prod d _ (fun m => (Real.exp_pos _).le) + (th_summable_gauss_int 144 (by norm_num))).mul_left _ + +/-- Each theta term is bounded by the corresponding dominating term. -/ +theorem th_term_le_dominant (d : ℕ) (z : E d) (n : Fin d → ℤ) : + Real.exp (-(‖z + latticeVec d n‖ ^ 2) / 2) ≤ + Real.exp (‖z‖ ^ 2 / 2) * ∏ i, Real.exp (-144 * ((n i : ℤ) : ℝ) ^ 2) := by + calc Real.exp (-(‖z + latticeVec d n‖ ^ 2) / 2) + ≤ Real.exp (‖z‖ ^ 2 / 2) * Real.exp (-(‖latticeVec d n‖ ^ 2) / 4) := th_exp_shift_le d z _ + _ = Real.exp (‖z‖ ^ 2 / 2) * ∏ i, Real.exp (-144 * ((n i : ℤ) : ℝ) ^ 2) := by + rw [th_exp_neg_norm_latticeVec_sq] + +/-! ## `dp_`: the derivative pairing bound (`DerivativePairingBound`) -/ + +/-! ### Smoothness and first derivative of the reference kernel `Kinf` -/ + +/-- `Kinf d = exp (-‖·‖^2 / 2)` is `C^∞` on `E d`. -/ +theorem dp_contDiff_Kinf (d : ℕ) : ContDiff ℝ ∞ (Kinf d) := by + unfold Kinf + have h : ContDiff ℝ ∞ (fun x : E d => -(‖x‖ ^ 2) / 2) := + (contDiff_norm_sq ℝ (E := E d)).neg.div_const 2 + exact Real.contDiff_exp.comp h + +/-- The first derivative of `Kinf d` at `x` is `Kinf d x • (-(innerSL ℝ x))`, i.e. +`v ↦ -⟪x, v⟫ Kinf d x`. -/ +theorem dp_hasFDerivAt_Kinf (d : ℕ) (x : E d) : + HasFDerivAt (Kinf d) (Kinf d x • (-(innerSL ℝ x))) x := by + have h0 : HasFDerivAt (fun y : E d => ‖y‖ ^ 2) ((2 : ℕ) • innerSL ℝ x) x := + (hasStrictFDerivAt_norm_sq x).hasFDerivAt + have h1 := (h0.const_smul (-(1 / 2 : ℝ))).exp + convert h1 using 1 + · funext y + simp only [Kinf, Pi.smul_apply, smul_eq_mul] + ring_nf + · ext v + simp [Kinf] + left + ring + +/-- The derivative function of `Kinf d`. -/ +theorem dp_fderiv_Kinf (d : ℕ) : + fderiv ℝ (Kinf d) = fun y => Kinf d y • (-(innerSL ℝ y)) := + funext fun y => (dp_hasFDerivAt_Kinf d y).fderiv + +/-! ### The operator-valued factor `g y = -(innerSL ℝ y)` -/ + +/-- `y ↦ -(innerSL ℝ y)` is `C^∞` (it is a continuous linear map). -/ +theorem dp_contDiff_g (d : ℕ) : ContDiff ℝ ∞ (fun y : E d => -(innerSL ℝ y)) := + (ContinuousLinearMap.contDiff (innerSL ℝ)).neg + +/-- The factor as the coercion of the continuous linear map `-(innerSL ℝ)`. -/ +theorem dp_g_eq (d : ℕ) : + (fun y : E d => -(innerSL ℝ y)) = ⇑(-(innerSL ℝ (E := E d))) := by + funext y + simp + +/-- Order zero: `‖g x‖ = ‖x‖` (`innerSL_apply_norm`). -/ +theorem dp_norm_g_zero (d : ℕ) (x : E d) : + ‖iteratedFDeriv ℝ 0 (fun y : E d => -(innerSL ℝ y)) x‖ = ‖x‖ := by + rw [norm_iteratedFDeriv_zero, norm_neg, innerSL_apply_norm] + +/-- Order one: `‖D g x‖ = ‖innerSL ℝ‖ ≤ 1` (`norm_innerSL_le`). -/ +theorem dp_norm_g_one (d : ℕ) (x : E d) : + ‖iteratedFDeriv ℝ 1 (fun y : E d => -(innerSL ℝ y)) x‖ ≤ 1 := by + rw [norm_iteratedFDeriv_one, dp_g_eq, ContinuousLinearMap.fderiv, norm_neg] + exact norm_innerSL_le ℝ + +/-- Orders `k ≥ 2` of the linear factor vanish. -/ +theorem dp_g_higher (d : ℕ) (x : E d) (k : ℕ) (hk : 2 ≤ k) : + iteratedFDeriv ℝ k (fun y : E d => -(innerSL ℝ y)) x = 0 := by + obtain ⟨j, rfl⟩ : ∃ j, k = j + 1 := ⟨k - 1, by omega⟩ + rw [dp_g_eq, iteratedFDeriv_succ_eq_comp_right] + have hconst : fderiv ℝ ⇑(-(innerSL ℝ (E := E d))) = fun _ => -(innerSL ℝ (E := E d)) := + funext fun y => (-(innerSL ℝ (E := E d))).fderiv + simp only [Function.comp_apply, hconst, iteratedFDeriv_const_of_ne (show j ≠ 0 by omega), + Pi.zero_apply] + exact LinearIsometryEquiv.map_zero _ + +/-! ### The Hermite-type recursion for the operator norms -/ + +/-- Hermite-type recursion for `b_q = ‖D^q Kinf (x)‖`: +`b_{q+2} ≤ ‖x‖ b_{q+1} + (q+1) b_q`, from `norm_iteratedFDeriv_fderiv`, the explicit first +derivative, and the Leibniz bound `norm_iteratedFDeriv_smul_le` in which only the terms +`i = q` and `i = q + 1` of the binomial sum survive. -/ +theorem dp_recursion (d : ℕ) (x : E d) (q : ℕ) : + ‖iteratedFDeriv ℝ (q + 2) (Kinf d) x‖ ≤ + ‖x‖ * ‖iteratedFDeriv ℝ (q + 1) (Kinf d) x‖ + + (q + 1) * ‖iteratedFDeriv ℝ q (Kinf d) x‖ := by + rw [← norm_iteratedFDeriv_fderiv, dp_fderiv_Kinf] + have h := norm_iteratedFDeriv_smul_le (dp_contDiff_Kinf d) (dp_contDiff_g d) x + (n := q + 1) (WithTop.coe_le_coe.mpr le_top) + refine h.trans ?_ + rw [Finset.sum_range_succ, Finset.sum_range_succ] + have hz : ∑ i ∈ Finset.range q, ((q + 1).choose i : ℝ) * ‖iteratedFDeriv ℝ i (Kinf d) x‖ * + ‖iteratedFDeriv ℝ (q + 1 - i) (fun y : E d => -(innerSL ℝ y)) x‖ = 0 := by + refine Finset.sum_eq_zero fun i hi => ?_ + rw [Finset.mem_range] at hi + rw [dp_g_higher d x _ (by omega), norm_zero, mul_zero] + rw [hz, zero_add, Nat.choose_self, Nat.choose_succ_self_right, Nat.add_sub_cancel_left, + Nat.sub_self, dp_norm_g_zero] + have h1 := dp_norm_g_one d x + have hb : 0 ≤ ‖iteratedFDeriv ℝ q (Kinf d) x‖ := norm_nonneg _ + push_cast + nlinarith [mul_le_mul_of_nonneg_left h1 (mul_nonneg (by positivity : (0 : ℝ) ≤ (q : ℝ) + 1) hb)] + +/-! ### The seven explicit orders -/ + +/-- `Kinf d x > 0`. -/ +theorem dp_Kinf_pos (d : ℕ) (x : E d) : 0 < Kinf d x := Real.exp_pos _ + +/-- `b_0 = Kinf d x`. -/ +theorem dp_b0 (d : ℕ) (x : E d) : ‖iteratedFDeriv ℝ 0 (Kinf d) x‖ = Kinf d x := by + rw [norm_iteratedFDeriv_zero, Real.norm_eq_abs, abs_of_pos (dp_Kinf_pos d x)] + +/-- `b_1 = Kinf d x * ‖x‖`. -/ +theorem dp_b1 (d : ℕ) (x : E d) : ‖iteratedFDeriv ℝ 1 (Kinf d) x‖ = Kinf d x * ‖x‖ := by + rw [norm_iteratedFDeriv_one, dp_fderiv_Kinf, norm_smul, norm_neg, innerSL_apply_norm, + Real.norm_eq_abs, abs_of_pos (dp_Kinf_pos d x)] + +/-- `b_2 ≤ Kinf d x * (‖x‖^2 + 1)`. -/ +theorem dp_b2 (d : ℕ) (x : E d) : + ‖iteratedFDeriv ℝ 2 (Kinf d) x‖ ≤ Kinf d x * (‖x‖ ^ 2 + 1) := by + have h := dp_recursion d x 0 + rw [dp_b1, dp_b0] at h + push_cast at h + nlinarith [h] + +/-- `b_3 ≤ Kinf d x * (‖x‖^3 + 3 ‖x‖)`. -/ +theorem dp_b3 (d : ℕ) (x : E d) : + ‖iteratedFDeriv ℝ 3 (Kinf d) x‖ ≤ Kinf d x * (‖x‖ ^ 3 + 3 * ‖x‖) := by + have h := dp_recursion d x 1 + have h2 := dp_b2 d x + have h1 := dp_b1 d x + have hr : 0 ≤ ‖x‖ := norm_nonneg _ + push_cast at h + nlinarith [mul_le_mul_of_nonneg_left h2 hr] + +/-- `b_4 ≤ Kinf d x * (‖x‖^4 + 6 ‖x‖^2 + 3)`. -/ +theorem dp_b4 (d : ℕ) (x : E d) : + ‖iteratedFDeriv ℝ 4 (Kinf d) x‖ ≤ Kinf d x * (‖x‖ ^ 4 + 6 * ‖x‖ ^ 2 + 3) := by + have h := dp_recursion d x 2 + have h3 := dp_b3 d x + have h2 := dp_b2 d x + have hr : 0 ≤ ‖x‖ := norm_nonneg _ + push_cast at h + nlinarith [mul_le_mul_of_nonneg_left h3 hr] + +/-- `b_5 ≤ Kinf d x * (‖x‖^5 + 10 ‖x‖^3 + 15 ‖x‖)`. -/ +theorem dp_b5 (d : ℕ) (x : E d) : + ‖iteratedFDeriv ℝ 5 (Kinf d) x‖ ≤ Kinf d x * (‖x‖ ^ 5 + 10 * ‖x‖ ^ 3 + 15 * ‖x‖) := by + have h := dp_recursion d x 3 + have h4 := dp_b4 d x + have h3 := dp_b3 d x + have hr : 0 ≤ ‖x‖ := norm_nonneg _ + push_cast at h + nlinarith [mul_le_mul_of_nonneg_left h4 hr] + +/-- `b_6 ≤ Kinf d x * (‖x‖^6 + 15 ‖x‖^4 + 45 ‖x‖^2 + 15)`. -/ +theorem dp_b6 (d : ℕ) (x : E d) : + ‖iteratedFDeriv ℝ 6 (Kinf d) x‖ ≤ + Kinf d x * (‖x‖ ^ 6 + 15 * ‖x‖ ^ 4 + 45 * ‖x‖ ^ 2 + 15) := by + have h := dp_recursion d x 4 + have h5 := dp_b5 d x + have h4 := dp_b4 d x + have hr : 0 ≤ ‖x‖ := norm_nonneg _ + push_cast at h + nlinarith [mul_le_mul_of_nonneg_left h5 hr] + +/-- For `q ≤ 6`, `b_q ≤ Kinf d x * pairingSum q ‖x‖`: the explicit bounds `dp_b0` to `dp_b6` +match the unfolded pairing sums (`Finset.sum_range_succ`, `Nat.factorial`). -/ +theorem dp_pairing_all (d : ℕ) (x : E d) (q : ℕ) (hq : q ≤ 6) : + ‖iteratedFDeriv ℝ q (Kinf d) x‖ ≤ Kinf d x * pairingSum q ‖x‖ := by + interval_cases q + · rw [dp_b0] + simp [pairingSum] + · rw [dp_b1] + simp [pairingSum] + · refine (dp_b2 d x).trans (le_of_eq ?_) + congr 1 + simp [pairingSum, Finset.sum_range_succ, Nat.factorial] + try ring + · refine (dp_b3 d x).trans (le_of_eq ?_) + congr 1 + simp [pairingSum, Finset.sum_range_succ, Nat.factorial] + try ring + · refine (dp_b4 d x).trans (le_of_eq ?_) + congr 1 + simp [pairingSum, Finset.sum_range_succ, Nat.factorial] + try ring + · refine (dp_b5 d x).trans (le_of_eq ?_) + congr 1 + simp [pairingSum, Finset.sum_range_succ, Nat.factorial] + try ring + · refine (dp_b6 d x).trans (le_of_eq ?_) + congr 1 + simp [pairingSum, Finset.sum_range_succ, Nat.factorial] + try ring + +/-! ### Contraction at a unit tuple -/ + +/-- At a unit tuple the contraction is bounded by the operator norm +(`ContinuousMultilinearMap.le_opNorm` with `∏ ‖u i‖ = 1`). -/ +theorem dp_abs_le_opNorm (d q : ℕ) (x : E d) (u : Fin q → E d) (hu : IsUnitTuple d q u) : + |iteratedFDeriv ℝ q (Kinf d) x u| ≤ ‖iteratedFDeriv ℝ q (Kinf d) x‖ := by + have h := (iteratedFDeriv ℝ q (Kinf d) x).le_opNorm u + have hp : ∏ i, ‖u i‖ = 1 := Finset.prod_eq_one (fun i _ => hu i) + rw [hp, mul_one] at h + simpa [Real.norm_eq_abs] using h + +/-! ## `d1_`: the derivative bound at `|x| ≥ 1` (`DerivativeBoundAtLeastOne`) -/ + +/-! ### The elementary pairing-sum bound at `r ≥ 1` -/ + +/-- For `q ≤ 6` and `1 ≤ r`, `pairingSum q r ≤ 76 r^6`. Each pairing sum unfolds +(`Finset.sum_range_succ`, `Nat.factorial`) to a polynomial with nonnegative coefficients summing +to `1, 1, 2, 4, 10, 26, 76` for `q = 0, …, 6`; every monomial `r^k`, `k ≤ 6`, is at most `r^6` +(`pow_le_pow_right₀`, `one_le_pow₀`). -/ +theorem d1_pairingSum_le_of_one_le (q : ℕ) (hq : q ≤ 6) (r : ℝ) (hr : 1 ≤ r) : + pairingSum q r ≤ 76 * r ^ 6 := by + have h0 : 1 ≤ r ^ 6 := one_le_pow₀ hr + have h1 : r ≤ r ^ 6 := by + simpa using pow_le_pow_right₀ hr (show 1 ≤ 6 by norm_num) + have h2 : r ^ 2 ≤ r ^ 6 := pow_le_pow_right₀ hr (by norm_num) + have h3 : r ^ 3 ≤ r ^ 6 := pow_le_pow_right₀ hr (by norm_num) + have h4 : r ^ 4 ≤ r ^ 6 := pow_le_pow_right₀ hr (by norm_num) + have h5 : r ^ 5 ≤ r ^ 6 := pow_le_pow_right₀ hr (by norm_num) + interval_cases q <;> simp only [pairingSum, Finset.sum_range_succ, Finset.sum_range_zero, + Nat.factorial] <;> norm_num <;> nlinarith [h0, h1, h2, h3, h4, h5, hr] + +/-! ### Hypothesis form: the Prop from `DerivativePairingBound` -/ + +/-- `DerivativeBoundAtLeastOne` from `DerivativePairingBound`: at `1 ≤ ‖x‖` the pairing bound +`|D^q Kinf (x) u| ≤ exp(-‖x‖²/2) · pairingSum q ‖x‖` and `pairingSum q ‖x‖ ≤ 76 ‖x‖^6` +(`d1_pairingSum_le_of_one_le`) give the Prop by `mul_le_mul_of_nonneg_left` with +`0 ≤ exp(-‖x‖²/2)` (`Real.exp_pos`). -/ +theorem d1_derivativeBoundAtLeastOne_of_derivativePairingBound + (h : UniversalLaw.Spec.Side24.DerivativePairingBound) : + UniversalLaw.Spec.Side24.DerivativeBoundAtLeastOne := by + intro d q hq x hx u hu + have hpair : |iteratedFDeriv ℝ q (Kinf d) x u| ≤ + Real.exp (-(‖x‖ ^ 2) / 2) * pairingSum q ‖x‖ := h d q hq x u hu + have hps : pairingSum q ‖x‖ ≤ 76 * ‖x‖ ^ 6 := d1_pairingSum_le_of_one_le q hq ‖x‖ hx + have hexp : 0 ≤ Real.exp (-(‖x‖ ^ 2) / 2) := (Real.exp_pos _).le + calc |iteratedFDeriv ℝ q (Kinf d) x u| + ≤ Real.exp (-(‖x‖ ^ 2) / 2) * pairingSum q ‖x‖ := hpair + _ ≤ Real.exp (-(‖x‖ ^ 2) / 2) * (76 * ‖x‖ ^ 6) := mul_le_mul_of_nonneg_left hps hexp + _ = 76 * ‖x‖ ^ 6 * Real.exp (-(‖x‖ ^ 2) / 2) := by ring + +/-! ## `d0_`: the derivative bound at the origin (`DerivativeBoundAtZero`) -/ + +/-! ### The pairing sums at the origin -/ + +/-- At `r = 0` the pairing sums of order `q ≤ 6` are `1, 0, 1, 0, 3, 0, 15` (only the `j = q/2` +term of an even order survives, `0 ^ 0 = 1`), hence at most `15`. -/ +theorem d0_pairingSum_zero_le (q : ℕ) (hq : q ≤ 6) : pairingSum q 0 ≤ 15 := by + interval_cases q <;> simp [pairingSum, Finset.sum_range_succ, Nat.factorial] <;> norm_num + +/-! ### The hypothesis form: the bound at the origin from the pairing bound -/ + +/-- `DerivativeBoundAtZero` from `DerivativePairingBound` taken as a hypothesis: specialise the +pairing bound to `x = 0`, where `‖0‖ = 0` and `exp (-(0 ^ 2) / 2) = 1`, and bound +`pairingSum q 0 ≤ 15` by `d0_pairingSum_zero_le`. -/ +theorem d0_derivativeBoundAtZero_of_derivativePairingBound + (h : UniversalLaw.Spec.Side24.DerivativePairingBound) : + UniversalLaw.Spec.Side24.DerivativeBoundAtZero := by + intro d q hq u hu + have h0 := h d q hq 0 u hu + have hx : Real.exp (-(‖(0 : E d)‖ ^ 2) / 2) * pairingSum q ‖(0 : E d)‖ = pairingSum q 0 := by + simp + rw [hx] at h0 + exact h0.trans (d0_pairingSum_zero_le q hq) + +/-! ## `rb_`: the Section 1 reference blocks (`ReferenceOddAndPinBlocks`) -/ + +/-! ### (a) Smoothness and first derivative of the reference kernel -/ + +/-- `Kinf d = exp (-‖·‖^2 / 2)` is `C^∞` (`contDiff_norm_sq`, `ContDiff.exp`). -/ +theorem rb_kinf_contDiff {d : ℕ} : ContDiff ℝ (⊤ : ℕ∞) (Kinf d) := by + unfold Kinf + exact ContDiff.exp (ContDiff.div_const (ContDiff.neg (contDiff_norm_sq ℝ)) 2) + +/-- The Fréchet derivative of `Kinf d` at `y`, as `HasFDerivAt` (chain rule through +`hasStrictFDerivAt_norm_sq` and `HasFDerivAt.exp`). -/ +theorem rb_hasFDerivAt_kinf {d : ℕ} (y : E d) : + HasFDerivAt (Kinf d) + (Real.exp (-(‖y‖ ^ 2) / 2) • ((2 : ℝ)⁻¹ • -((2 : ℕ) • innerSL ℝ y))) y := by + have h1 : HasFDerivAt (fun z : E d => -(‖z‖ ^ 2) / 2) + ((2 : ℝ)⁻¹ • -((2 : ℕ) • innerSL ℝ y)) y := by + have := (hasStrictFDerivAt_norm_sq y).hasFDerivAt.neg.mul_const (2 : ℝ)⁻¹ + refine this.congr_of_eventuallyEq (Filter.Eventually.of_forall fun z => ?_) + simp [div_eq_mul_inv] + exact h1.exp + +/-- `fderiv ℝ (Kinf d) y v = -(⟪y, v⟫ * Kinf d y)`. -/ +theorem rb_fderiv_kinf {d : ℕ} (y v : E d) : + fderiv ℝ (Kinf d) y v = -(⟪y, v⟫ * Kinf d y) := by + rw [(rb_hasFDerivAt_kinf y).fderiv] + simp [Kinf, two_smul, innerSL_apply_apply] + ring + +/-! ### (b) Peeling one direction off an iterated derivative -/ + +/-- For a `C^∞` real function `f`, `D^{n+1} f x [m] = D^n (y ↦ D f y (m (Fin.last n))) x [Fin.init m]` +(`iteratedFDeriv_succ_apply_right` with `iteratedFDeriv_clm_apply_const_apply`). -/ +theorem rb_peel {d : ℕ} (f : E d → ℝ) (hf : ContDiff ℝ (⊤ : ℕ∞) f) (n : ℕ) (x : E d) + (m : Fin (n + 1) → E d) : + iteratedFDeriv ℝ (n + 1) f x m = + iteratedFDeriv ℝ n (fun y => fderiv ℝ f y (m (Fin.last n))) x (Fin.init m) := by + rw [iteratedFDeriv_succ_apply_right] + rw [iteratedFDeriv_clm_apply_const_apply (contDiff_infty_iff_fderiv.mp hf).2 + (mod_cast le_top)] + +/-- `y ↦ ⟪y, v⟫` is `C^∞`. -/ +theorem rb_inner_contDiff {d : ℕ} (v : E d) : ContDiff ℝ (⊤ : ℕ∞) (fun y : E d => ⟪y, v⟫) := + contDiff_id.inner ℝ contDiff_const + +/-! ### (c) Level one: `g₁ y = -(⟪y, v⟫ Kinf d y)` -/ + +/-- Smoothness of the level-one function. -/ +theorem rb_g1_contDiff {d : ℕ} (v : E d) : + ContDiff ℝ (⊤ : ℕ∞) (fun y : E d => -(⟪y, v⟫ * Kinf d y)) := + ((rb_inner_contDiff v).mul rb_kinf_contDiff).neg + +/-- Derivative of the level-one function in direction `w`. -/ +theorem rb_fderiv_g1 {d : ℕ} (v y w : E d) : + fderiv ℝ (fun y : E d => -(⟪y, v⟫ * Kinf d y)) y w = + -(⟪w, v⟫ * Kinf d y) + ⟪y, v⟫ * ⟪y, w⟫ * Kinf d y := by + have h : HasFDerivAt (fun y : E d => -(⟪y, v⟫ * Kinf d y)) _ y := + (((hasFDerivAt_id y).inner ℝ (hasFDerivAt_const v y)).mul (rb_hasFDerivAt_kinf y)).neg + rw [h.fderiv] + simp [Kinf, two_smul, innerSL_apply_apply, fderivInnerCLM_apply] + ring + +/-! ### (d) Level two: `g₂ y = -(⟪w, v⟫ Kinf d y) + ⟪y, v⟫ ⟪y, w⟫ Kinf d y` -/ + +/-- Smoothness of the level-two function. -/ +theorem rb_g2_contDiff {d : ℕ} (v w : E d) : + ContDiff ℝ (⊤ : ℕ∞) (fun y : E d => -(⟪w, v⟫ * Kinf d y) + ⟪y, v⟫ * ⟪y, w⟫ * Kinf d y) := + ((contDiff_const.mul rb_kinf_contDiff).neg.add + (((rb_inner_contDiff v).mul (rb_inner_contDiff w)).mul rb_kinf_contDiff)) + +/-- Derivative of the level-two function in direction `a`. -/ +theorem rb_fderiv_g2 {d : ℕ} (v w y a : E d) : + fderiv ℝ (fun y : E d => -(⟪w, v⟫ * Kinf d y) + ⟪y, v⟫ * ⟪y, w⟫ * Kinf d y) y a = + ⟪w, v⟫ * ⟪y, a⟫ * Kinf d y + (⟪a, v⟫ * ⟪y, w⟫ + ⟪y, v⟫ * ⟪a, w⟫) * Kinf d y + - ⟪y, v⟫ * ⟪y, w⟫ * ⟪y, a⟫ * Kinf d y := by + have h : HasFDerivAt + (fun y : E d => -(⟪w, v⟫ * Kinf d y) + ⟪y, v⟫ * ⟪y, w⟫ * Kinf d y) _ y := + (((rb_hasFDerivAt_kinf y).const_mul _).neg).add + ((((hasFDerivAt_id y).inner ℝ (hasFDerivAt_const v y)).mul + ((hasFDerivAt_id y).inner ℝ (hasFDerivAt_const w y))).mul (rb_hasFDerivAt_kinf y)) + rw [h.fderiv] + simp [Kinf, two_smul, innerSL_apply_apply, fderivInnerCLM_apply] + ring + +/-! ### (e) Level three at the origin, general directions -/ + +/-- Derivative at `0` of the level-three function in direction `c`: the three pairings. -/ +theorem rb_fderiv_g3_zero {d : ℕ} (v w a c : E d) : + fderiv ℝ (fun y : E d => ⟪w, v⟫ * ⟪y, a⟫ * Kinf d y + + (⟪a, v⟫ * ⟪y, w⟫ + ⟪y, v⟫ * ⟪a, w⟫) * Kinf d y + - ⟪y, v⟫ * ⟪y, w⟫ * ⟪y, a⟫ * Kinf d y) 0 c = + ⟪w, v⟫ * ⟪c, a⟫ + ⟪a, v⟫ * ⟪c, w⟫ + ⟪c, v⟫ * ⟪a, w⟫ := by + have h : HasFDerivAt (fun y : E d => ⟪w, v⟫ * ⟪y, a⟫ * Kinf d y + + (⟪a, v⟫ * ⟪y, w⟫ + ⟪y, v⟫ * ⟪a, w⟫) * Kinf d y + - ⟪y, v⟫ * ⟪y, w⟫ * ⟪y, a⟫ * Kinf d y) _ 0 := + (((((hasFDerivAt_id (0 : E d)).inner ℝ (hasFDerivAt_const a 0)).const_mul _).mul + (rb_hasFDerivAt_kinf 0)).add + ((((((hasFDerivAt_id (0 : E d)).inner ℝ (hasFDerivAt_const w 0)).const_mul _).add + (((hasFDerivAt_id (0 : E d)).inner ℝ (hasFDerivAt_const v 0)).mul_const _))).mul + (rb_hasFDerivAt_kinf 0))).sub + (((((hasFDerivAt_id (0 : E d)).inner ℝ (hasFDerivAt_const v 0)).mul + ((hasFDerivAt_id (0 : E d)).inner ℝ (hasFDerivAt_const w 0))).mul + ((hasFDerivAt_id (0 : E d)).inner ℝ (hasFDerivAt_const a 0))).mul + (rb_hasFDerivAt_kinf 0)) + rw [h.fderiv] + simp [Kinf, fderivInnerCLM_apply] + ring + +/-- Fourth-order contraction of `Kinf d` at `0`: `D^4 Kinf 0 [c, a, w, v]` is the sum of the +three pairings `⟪w, v⟫⟪c, a⟫ + ⟪a, v⟫⟪c, w⟫ + ⟪c, v⟫⟪a, w⟫`. -/ +theorem rb_order_four {d : ℕ} (c a w v : E d) : + derivContraction d (Kinf d) 0 [c, a, w, v] = + ⟪w, v⟫ * ⟪c, a⟫ + ⟪a, v⟫ * ⟪c, w⟫ + ⟪c, v⟫ * ⟪a, w⟫ := by + show iteratedFDeriv ℝ 4 (Kinf d) 0 (fun i => [c, a, w, v].get i) = _ + rw [rb_peel _ rb_kinf_contDiff] + simp only [rb_fderiv_kinf] + show iteratedFDeriv ℝ 3 (fun y => -(⟪y, v⟫ * Kinf d y)) 0 + (Fin.init fun i => [c, a, w, v].get i) = _ + rw [rb_peel _ (rb_g1_contDiff v)] + simp only [rb_fderiv_g1] + show iteratedFDeriv ℝ 2 (fun y => -(⟪w, v⟫ * Kinf d y) + ⟪y, v⟫ * ⟪y, w⟫ * Kinf d y) 0 + (Fin.init (Fin.init fun i => [c, a, w, v].get i)) = _ + rw [rb_peel _ (rb_g2_contDiff v w)] + simp only [rb_fderiv_g2] + show iteratedFDeriv ℝ 1 (fun y => ⟪w, v⟫ * ⟪y, a⟫ * Kinf d y + + (⟪a, v⟫ * ⟪y, w⟫ + ⟪y, v⟫ * ⟪a, w⟫) * Kinf d y + - ⟪y, v⟫ * ⟪y, w⟫ * ⟪y, a⟫ * Kinf d y) 0 + (Fin.init (Fin.init (Fin.init fun i => [c, a, w, v].get i))) = _ + rw [iteratedFDeriv_one_apply] + show fderiv ℝ (fun y : E d => ⟪w, v⟫ * ⟪y, a⟫ * Kinf d y + + (⟪a, v⟫ * ⟪y, w⟫ + ⟪y, v⟫ * ⟪a, w⟫) * Kinf d y + - ⟪y, v⟫ * ⟪y, w⟫ * ⟪y, a⟫ * Kinf d y) 0 c = _ + exact rb_fderiv_g3_zero v w a c + +/-! ### (f) Levels three to five along one direction `u` (Hermite forms) -/ + +/-- Smoothness of `(3 ⟪u,u⟫ ⟪y,u⟫ - ⟪y,u⟫^3) Kinf d y`. -/ +theorem rb_P3_contDiff {d : ℕ} (u : E d) : ContDiff ℝ (⊤ : ℕ∞) + (fun y : E d => (3 * ⟪u, u⟫ * ⟪y, u⟫ - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) := + ((contDiff_const.mul (rb_inner_contDiff u)).sub + (((rb_inner_contDiff u).mul (rb_inner_contDiff u)).mul (rb_inner_contDiff u))).mul + rb_kinf_contDiff + +/-- Derivative along `u` of the level-three Hermite form. -/ +theorem rb_fderiv_P3 {d : ℕ} (u y : E d) : + fderiv ℝ (fun y : E d => (3 * ⟪u, u⟫ * ⟪y, u⟫ - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) y u = + (3 * ⟪u, u⟫ * ⟪u, u⟫ - 6 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫) + + ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y := by + have hin : HasFDerivAt (fun y : E d => ⟪y, u⟫) _ y := + (hasFDerivAt_id y).inner ℝ (hasFDerivAt_const u y) + have h : HasFDerivAt + (fun y : E d => (3 * ⟪u, u⟫ * ⟪y, u⟫ - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) _ y := + ((hin.const_mul _).sub ((hin.mul hin).mul hin)).mul (rb_hasFDerivAt_kinf y) + rw [h.fderiv] + simp [Kinf, two_smul, innerSL_apply_apply, fderivInnerCLM_apply] + ring + +/-- Smoothness of the level-four Hermite form. -/ +theorem rb_P4_contDiff {d : ℕ} (u : E d) : ContDiff ℝ (⊤ : ℕ∞) + (fun y : E d => (3 * ⟪u, u⟫ * ⟪u, u⟫ - 6 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫) + + ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) := + ((contDiff_const.sub + (contDiff_const.mul ((rb_inner_contDiff u).mul (rb_inner_contDiff u)))).add + ((((rb_inner_contDiff u).mul (rb_inner_contDiff u)).mul (rb_inner_contDiff u)).mul + (rb_inner_contDiff u))).mul rb_kinf_contDiff + +/-- Derivative along `u` of the level-four Hermite form. -/ +theorem rb_fderiv_P4 {d : ℕ} (u y : E d) : + fderiv ℝ (fun y : E d => (3 * ⟪u, u⟫ * ⟪u, u⟫ - 6 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫) + + ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) y u = + (-15 * (⟪u, u⟫ * ⟪u, u⟫) * ⟪y, u⟫ + 10 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) + - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y := by + have hin : HasFDerivAt (fun y : E d => ⟪y, u⟫) _ y := + (hasFDerivAt_id y).inner ℝ (hasFDerivAt_const u y) + have h : HasFDerivAt (fun y : E d => (3 * ⟪u, u⟫ * ⟪u, u⟫ - 6 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫) + + ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) _ y := + (((hasFDerivAt_const _ y).sub ((hin.mul hin).const_mul _)).add + (((hin.mul hin).mul hin).mul hin)).mul (rb_hasFDerivAt_kinf y) + rw [h.fderiv] + simp [Kinf, two_smul, innerSL_apply_apply, fderivInnerCLM_apply] + ring + +/-- Derivative at `0` along `u` of the level-five Hermite form: `-15 ⟪u, u⟫^3`. -/ +theorem rb_fderiv_P5_zero {d : ℕ} (u : E d) : + fderiv ℝ (fun y : E d => (-15 * (⟪u, u⟫ * ⟪u, u⟫) * ⟪y, u⟫ + + 10 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) + - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) 0 u = + -15 * (⟪u, u⟫ * ⟪u, u⟫ * ⟪u, u⟫) := by + have hin : HasFDerivAt (fun y : E d => ⟪y, u⟫) _ (0 : E d) := + (hasFDerivAt_id (0 : E d)).inner ℝ (hasFDerivAt_const u 0) + have h : HasFDerivAt (fun y : E d => (-15 * (⟪u, u⟫ * ⟪u, u⟫) * ⟪y, u⟫ + + 10 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) + - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) _ 0 := + (((hin.const_mul _).add (((hin.mul hin).mul hin).const_mul _)).sub + ((((hin.mul hin).mul hin).mul hin).mul hin)).mul (rb_hasFDerivAt_kinf 0) + rw [h.fderiv] + simp [Kinf, fderivInnerCLM_apply] + ring + +/-- Sixth-order contraction of `Kinf d` at `0` along one direction: `-15 ⟪u, u⟫^3`. -/ +theorem rb_order_six {d : ℕ} (u : E d) : + derivContraction d (Kinf d) 0 [u, u, u, u, u, u] = -15 * (⟪u, u⟫ * ⟪u, u⟫ * ⟪u, u⟫) := by + show iteratedFDeriv ℝ 6 (Kinf d) 0 (fun i => [u, u, u, u, u, u].get i) = _ + rw [rb_peel _ rb_kinf_contDiff] + simp only [rb_fderiv_kinf] + show iteratedFDeriv ℝ 5 (fun y => -(⟪y, u⟫ * Kinf d y)) 0 _ = _ + rw [rb_peel _ (rb_g1_contDiff u)] + simp only [rb_fderiv_g1] + show iteratedFDeriv ℝ 4 (fun y => -(⟪u, u⟫ * Kinf d y) + ⟪y, u⟫ * ⟪y, u⟫ * Kinf d y) 0 _ = _ + rw [rb_peel _ (rb_g2_contDiff u u)] + simp only [rb_fderiv_g2] + show iteratedFDeriv ℝ 3 (fun y => ⟪u, u⟫ * ⟪y, u⟫ * Kinf d y + + (⟪u, u⟫ * ⟪y, u⟫ + ⟪y, u⟫ * ⟪u, u⟫) * Kinf d y + - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * Kinf d y) 0 _ = _ + rw [show (fun y : E d => ⟪u, u⟫ * ⟪y, u⟫ * Kinf d y + + (⟪u, u⟫ * ⟪y, u⟫ + ⟪y, u⟫ * ⟪u, u⟫) * Kinf d y + - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * Kinf d y) = + fun y : E d => (3 * ⟪u, u⟫ * ⟪y, u⟫ - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y from + funext fun y => by ring] + rw [rb_peel _ (rb_P3_contDiff u)] + show iteratedFDeriv ℝ 2 (fun y => fderiv ℝ (fun y : E d => + (3 * ⟪u, u⟫ * ⟪y, u⟫ - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) y u) 0 _ = _ + simp only [rb_fderiv_P3] + show iteratedFDeriv ℝ 2 (fun y : E d => (3 * ⟪u, u⟫ * ⟪u, u⟫ - 6 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫) + + ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) 0 _ = _ + rw [rb_peel _ (rb_P4_contDiff u)] + show iteratedFDeriv ℝ 1 (fun y => fderiv ℝ (fun y : E d => + (3 * ⟪u, u⟫ * ⟪u, u⟫ - 6 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫) + + ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) y u) 0 _ = _ + simp only [rb_fderiv_P4] + show iteratedFDeriv ℝ 1 (fun y : E d => (-15 * (⟪u, u⟫ * ⟪u, u⟫) * ⟪y, u⟫ + + 10 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) + - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) 0 _ = _ + rw [iteratedFDeriv_one_apply] + show fderiv ℝ (fun y : E d => (-15 * (⟪u, u⟫ * ⟪u, u⟫) * ⟪y, u⟫ + + 10 * ⟪u, u⟫ * (⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) + - ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫ * ⟪y, u⟫) * Kinf d y) 0 u = _ + exact rb_fderiv_P5_zero u + +/-! ### (g) Second-order contraction and the blocks in an orthonormal frame -/ + +/-- Second-order contraction of `Kinf d` at `0`: `D^2 Kinf 0 [a, b] = -⟪a, b⟫`. -/ +theorem rb_order_two {d : ℕ} (a b : E d) : derivContraction d (Kinf d) 0 [a, b] = -⟪a, b⟫ := by + show iteratedFDeriv ℝ 2 (Kinf d) 0 (fun i => [a, b].get i) = -⟪a, b⟫ + rw [rb_peel _ rb_kinf_contDiff, iteratedFDeriv_one_apply] + simp only [rb_fderiv_kinf] + show fderiv ℝ (fun y : E d => -(⟪y, b⟫ * Kinf d y)) 0 a = -⟪a, b⟫ + rw [rb_fderiv_g1] + simp [Kinf] + +/-- The gradient block of `Kinf d` in an orthonormal frame is the identity. -/ +theorem rb_gradCov {d : ℕ} (R : Fin d → E d) (hR : Orthonormal ℝ R) : + gradCov d (Kinf d) R = 1 := by + ext i j + simp only [gradCov, Matrix.of_apply, rb_order_two, neg_neg, Matrix.one_apply] + exact orthonormal_iff_ite.mp hR i j + +/-- The pin block of `Kinf d` with `u = R 0` in an orthonormal frame is `diagonal (3, 1, …, 1)`. -/ +theorem rb_pinCov {d : ℕ} (hd : 0 < d) (R : Fin d → E d) (hR : Orthonormal ℝ R) : + pinCov d (Kinf d) (R ⟨0, hd⟩) R = + Matrix.diagonal (fun i => if i = ⟨0, hd⟩ then 3 else 1) := by + have hite := orthonormal_iff_ite.mp hR + ext i j + simp only [pinCov, Matrix.of_apply, rb_order_four, hite, Matrix.diagonal_apply] + by_cases hi : i = ⟨0, hd⟩ + · subst hi + by_cases hj : j = ⟨0, hd⟩ + · subst hj + norm_num + · simp [Ne.symm hj] + · by_cases hj : j = ⟨0, hd⟩ + · subst hj + simp [hi, Ne.symm hi] + · simp [hi, Ne.symm hi, Ne.symm hj] + +/-- The third-derivative/gradient covariance vector of `Kinf d` with `u = R 0` is +`(-3, 0, …, 0)`. -/ +theorem rb_thirdGradCov {d : ℕ} (hd : 0 < d) (R : Fin d → E d) (hR : Orthonormal ℝ R) : + thirdGradCov d (Kinf d) (R ⟨0, hd⟩) R = fun j => if j = ⟨0, hd⟩ then -3 else 0 := by + have hite := orthonormal_iff_ite.mp hR + funext j + simp only [thirdGradCov, rb_order_four, hite] + by_cases hj : j = ⟨0, hd⟩ + · simp [hj]; norm_num + · have hj' : ¬ ((⟨0, hd⟩ : Fin d) = j) := fun h => hj h.symm + simp [hj, hj'] + +/-- The third-derivative variance of `Kinf d` along a unit vector `u = R 0` is `15`. -/ +theorem rb_thirdVar {d : ℕ} (hd : 0 < d) (R : Fin d → E d) (hR : Orthonormal ℝ R) : + thirdVar d (Kinf d) (R ⟨0, hd⟩) = 15 := by + have hite := orthonormal_iff_ite.mp hR + simp only [thirdVar, rb_order_six, hite] + simp + +/-- The Schur complement `condThirdVariance` of `Kinf d` in an orthonormal frame is `15 - 9 = 6`. -/ +theorem rb_condThirdVariance {d : ℕ} (hd : 0 < d) (R : Fin d → E d) (hR : Orthonormal ℝ R) : + condThirdVariance d (Kinf d) (R ⟨0, hd⟩) R = 6 := by + unfold condThirdVariance + rw [rb_thirdVar hd R hR, rb_gradCov R hR, rb_thirdGradCov hd R hR, inv_one, + Matrix.one_mulVec] + simp [dotProduct, ite_mul, Finset.sum_ite_eq'] + norm_num + +/-! ## `rp_`: the reference pin-density product (`ReferencePinDensityProduct`) -/ + +/-! ### The density algebra at the reference blocks -/ + +/-- `det (diagonal (3, 1, …, 1)) = 3` (the single `3` sits at the coordinate `0`). -/ +theorem rp_det_pinDiagonal (d : ℕ) (hd : 0 < d) : + (Matrix.diagonal fun i : Fin d => if i = ⟨0, hd⟩ then (3 : ℝ) else 1).det = 3 := by + rw [Matrix.det_diagonal, Finset.prod_ite_eq'] + simp + +/-- The product of the two centred Gaussian densities at `0`, for the covariance blocks `1` and +`diagonal (3, 1, …, 1)`: `(2π)^(-d/2) · (2π)^(-d/2) · 3^(-1/2) = (2π)^(-d) / √3`. -/ +theorem rp_density_algebra (d : ℕ) (hd : 0 < d) : + gaussDensity d 1 0 * + gaussDensity d (Matrix.diagonal fun i : Fin d => if i = ⟨0, hd⟩ then (3 : ℝ) else 1) 0 = + (2 * Real.pi) ^ (-(d : ℝ)) / Real.sqrt 3 := by + have hpi : (0 : ℝ) < 2 * Real.pi := by positivity + simp only [gaussDensity, rp_det_pinDiagonal d hd, Matrix.det_one, Matrix.mulVec_zero, + dotProduct_zero, neg_zero, zero_div, Real.exp_zero, mul_one, Real.one_rpow] + rw [← mul_assoc, ← Real.rpow_add hpi, show -(d : ℝ) / 2 + -(d : ℝ) / 2 = -(d : ℝ) by ring, + show (-1 : ℝ) / 2 = -(1 / 2) by norm_num, Real.rpow_neg (by norm_num : (0 : ℝ) ≤ 3), + Real.sqrt_eq_rpow] + exact (div_eq_mul_inv _ _).symm + +/-! ### The common core: the Prop from the two block identities -/ + +/-- `ReferencePinDensityProduct` from the two block identities `gradCov d (Kinf d) R = 1` and +`pinCov d (Kinf d) (R 0) R = diagonal (3, 1, …, 1)` (for `d ∈ {2, 3}` and orthonormal `R`), by +rewriting the blocks and `rp_density_algebra`. -/ +theorem rp_referencePinDensityProduct_of_blockIdentities + (hblk : ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ (hd : 0 < d) (R : Fin d → E d), Orthonormal ℝ R → + gradCov d (Kinf d) R = 1 ∧ + pinCov d (Kinf d) (R ⟨0, hd⟩) R = Matrix.diagonal (fun i => if i = ⟨0, hd⟩ then 3 else 1)) : + UniversalLaw.Spec.Side24.ReferencePinDensityProduct := by + intro d hd23 hd R hR + obtain ⟨hG, hV⟩ := hblk d hd23 hd R hR + rw [hG, hV] + exact rp_density_algebra d hd + +/-! ### The hypothesis form: the Prop from `ReferenceOddAndPinBlocks` -/ + +/-- `ReferencePinDensityProduct` from `ReferenceOddAndPinBlocks` taken as a hypothesis: only its +first conjunct (`gradCov d (Kinf d) R = 1`) and fourth conjunct +(`pinCov d (Kinf d) (R 0) R = diagonal (3, 1, …, 1)`) are used. -/ +theorem rp_referencePinDensityProduct_of_referenceOddAndPinBlocks + (h : UniversalLaw.Spec.Side24.ReferenceOddAndPinBlocks) : + UniversalLaw.Spec.Side24.ReferencePinDensityProduct := by + apply rp_referencePinDensityProduct_of_blockIdentities + intro d hd23 hd R hR + obtain ⟨hG, -, -, hV, -⟩ := h d hd23 hd R hR + exact ⟨hG, hV⟩ + +/-! ## `ir_`: the Section 4 integrand-ratio bounds (`IntegrandRatioBounds`) -/ + +/-! ### (a) Elementary facts about `epsilon = 10^(-108)` -/ + +/-- `0 < epsilon`. -/ +theorem ir_epsilon_pos : (0 : ℝ) < epsilon := by + unfold epsilon + positivity + +/-- `epsilon < 1/100`: a crude upper bound sufficient for every sign condition below. -/ +theorem ir_epsilon_lt : epsilon < (1 / 100 : ℝ) := by + unfold epsilon + norm_num + +/-- `0 < 1 - epsilon`. -/ +theorem ir_one_sub_epsilon_pos : (0 : ℝ) < 1 - epsilon := by + have := ir_epsilon_lt + linarith + +/-- `1 ≤ 1 + epsilon`. -/ +theorem ir_one_le_one_add_epsilon : (1 : ℝ) ≤ 1 + epsilon := by + have := ir_epsilon_pos + linarith + +/-! ### (b) The two one-sided power estimates -/ + +/-- For a real exponent `b ≤ 5`, `(1 + epsilon)^b ≤ (1 + epsilon)^5`: monotonicity of the real +power in the exponent for a base `≥ 1` (`Real.rpow_le_rpow_of_exponent_le`), with the natural +power on the right via `Real.rpow_natCast`. -/ +theorem ir_rpow_le_pow_five (b : ℝ) (hb : b ≤ 5) : + (1 + epsilon) ^ b ≤ (1 + epsilon) ^ 5 := by + have h := Real.rpow_le_rpow_of_exponent_le ir_one_le_one_add_epsilon hb + rwa [show (5 : ℝ) = ((5 : ℕ) : ℝ) by norm_num, Real.rpow_natCast] at h + +/-- Bernoulli's inequality at `s = -epsilon`: for a real exponent `a ≥ 1`, +`1 - a epsilon ≤ (1 - epsilon)^a` (`one_add_mul_self_le_rpow_one_add`). -/ +theorem ir_bernoulli (a : ℝ) (ha : 1 ≤ a) : 1 - a * epsilon ≤ (1 - epsilon) ^ a := by + have hε := ir_epsilon_lt + have h := one_add_mul_self_le_rpow_one_add (show (-1 : ℝ) ≤ -epsilon by linarith) ha + rw [show (1 : ℝ) + -epsilon = 1 - epsilon by ring] at h + linarith + +/-! ### (c) The two exact rational inequalities (degree five in `epsilon`) -/ + +/-- `(1 - 13 epsilon)(1 + epsilon)^5 ≤ 1 - 5 epsilon`, exact rational arithmetic. -/ +theorem ir_lower_rational : (1 - 13 * epsilon) * (1 + epsilon) ^ 5 ≤ 1 - 5 * epsilon := by + unfold epsilon + norm_num + +/-- `(1 + epsilon)^5 ≤ (1 + 28 epsilon)(1 - 5 epsilon)`, exact rational arithmetic. -/ +theorem ir_upper_rational : (1 + epsilon) ^ 5 ≤ (1 + 28 * epsilon) * (1 - 5 * epsilon) := by + unfold epsilon + norm_num + +/-! ### (d) The ratio bounds for arbitrary exponents in `[1, 5]` -/ + +/-- Lower bound: for `1 ≤ a ≤ 5` and `1 ≤ b ≤ 5`, +`1 - 13 epsilon ≤ (1 - epsilon)^a / (1 + epsilon)^b`. -/ +theorem ir_lower_bound (a b : ℝ) (ha1 : 1 ≤ a) (ha5 : a ≤ 5) (hb5 : b ≤ 5) : + 1 - 13 * epsilon ≤ (1 - epsilon) ^ a / (1 + epsilon) ^ b := by + have hε := ir_epsilon_pos + have hε1 := ir_epsilon_lt + have hpb : 0 < (1 + epsilon) ^ b := Real.rpow_pos_of_pos (by linarith) _ + have hae : a * epsilon ≤ 5 * epsilon := mul_le_mul_of_nonneg_right ha5 hε.le + rw [le_div_iff₀ hpb] + calc (1 - 13 * epsilon) * (1 + epsilon) ^ b + ≤ (1 - 13 * epsilon) * (1 + epsilon) ^ 5 := + mul_le_mul_of_nonneg_left (ir_rpow_le_pow_five b hb5) (by linarith) + _ ≤ 1 - 5 * epsilon := ir_lower_rational + _ ≤ 1 - a * epsilon := by linarith + _ ≤ (1 - epsilon) ^ a := ir_bernoulli a ha1 + +/-- Upper bound: for `a ≤ 5` and `1 ≤ b ≤ 5`, +`(1 + epsilon)^a / (1 - epsilon)^b ≤ 1 + 28 epsilon`. -/ +theorem ir_upper_bound (a b : ℝ) (ha5 : a ≤ 5) (hb1 : 1 ≤ b) (hb5 : b ≤ 5) : + (1 + epsilon) ^ a / (1 - epsilon) ^ b ≤ 1 + 28 * epsilon := by + have hε := ir_epsilon_pos + have hqb : 0 < (1 - epsilon) ^ b := Real.rpow_pos_of_pos ir_one_sub_epsilon_pos _ + have hbe : b * epsilon ≤ 5 * epsilon := mul_le_mul_of_nonneg_right hb5 hε.le + rw [div_le_iff₀ hqb] + calc (1 + epsilon) ^ a ≤ (1 + epsilon) ^ 5 := ir_rpow_le_pow_five a ha5 + _ ≤ (1 + 28 * epsilon) * (1 - 5 * epsilon) := ir_upper_rational + _ ≤ (1 + 28 * epsilon) * (1 - b * epsilon) := + mul_le_mul_of_nonneg_left (by linarith) (by linarith) + _ ≤ (1 + 28 * epsilon) * (1 - epsilon) ^ b := + mul_le_mul_of_nonneg_left (ir_bernoulli b hb1) (by linarith) + +/-- Both ratio bounds together, for `1 ≤ a ≤ 5` and `1 ≤ b ≤ 5`. -/ +theorem ir_ratio_bounds (a b : ℝ) (ha1 : 1 ≤ a) (ha5 : a ≤ 5) (hb1 : 1 ≤ b) (hb5 : b ≤ 5) : + 1 - 13 * epsilon ≤ (1 - epsilon) ^ a / (1 + epsilon) ^ b ∧ + (1 + epsilon) ^ a / (1 - epsilon) ^ b ≤ 1 + 28 * epsilon := + ⟨ir_lower_bound a b ha1 ha5 hb5, ir_upper_bound a b ha5 hb1 hb5⟩ + +/-! ### (e) The exponents in dimensions two and three -/ + +/-- `exponentA 2 = 13/6` (`m = 1`, `n = 1`: `1 + 2/3 + 1/2`). -/ +theorem ir_exponentA_two : exponentA 2 = 13 / 6 := by + simp [exponentA, symCoordDim] + norm_num + +/-- `exponentB 2 = 5/2` (`d = 2`, `n = 1`: `2 + 1/2`). -/ +theorem ir_exponentB_two : exponentB 2 = 5 / 2 := by + simp [exponentB, symCoordDim] + norm_num + +/-- `exponentA 3 = 25/6` (`m = 2`, `n = 3`: `2 + 2/3 + 3/2`). -/ +theorem ir_exponentA_three : exponentA 3 = 25 / 6 := by + simp [exponentA, symCoordDim] + norm_num + +/-- `exponentB 3 = 9/2` (`d = 3`, `n = 3`: `3 + 3/2`). -/ +theorem ir_exponentB_three : exponentB 3 = 9 / 2 := by + simp [exponentB, symCoordDim] + norm_num + +/-! ## `li_`: the lattice-image sum bound (`LatticeImageSumBound`) -/ + +/-! ### (a) The geometric tail `(n+1)^9 exp(-288 (n+1)^2) ≤ exp(-288) (1/2)^n` -/ + +/-- Induction on `n` from the registered successive-term ratio: the ratio of consecutive terms +`j^9 exp(-288 j^2)` is at most `512 exp(-864) < 1/2` +(`UniversalLaw.Proofs.Side24Elementary.successiveTermRatio`). -/ +theorem li_geom_tail (n : ℕ) : + ((n : ℝ) + 1) ^ 9 * Real.exp (-288 * ((n : ℝ) + 1) ^ 2) ≤ Real.exp (-288) * (1 / 2) ^ n := by + induction n with + | zero => norm_num + | succ n ih => + have h1 := (UniversalLaw.Proofs.Side24Elementary.successiveTermRatio).1 (n + 1) (by omega) + have h2 := UniversalLaw.Proofs.Side24Elementary.Aux.ratio_const_lt_half + have hnn : 0 ≤ ((n : ℝ) + 1) ^ 9 * Real.exp (-288 * ((n : ℝ) + 1) ^ 2) := by positivity + push_cast at h1 ⊢ + rw [pow_succ] + calc _ ≤ 512 * Real.exp (-864) * (((n : ℝ) + 1) ^ 9 * Real.exp (-288 * ((n : ℝ) + 1) ^ 2)) := + h1 + _ ≤ (1 / 2) * (Real.exp (-288) * (1 / 2) ^ n) := mul_le_mul h2.le ih hnn (by norm_num) + _ = Real.exp (-288) * ((1 / 2) ^ n * (1 / 2)) := by ring + +/-- `(n+1)^6 exp(-288 (n+1)^2) ≤ exp(-288) (1/2)^n`: the sixth power is at most the ninth. -/ +theorem li_pow6_succ_le (n : ℕ) : + ((n : ℝ) + 1) ^ 6 * Real.exp (-288 * ((n : ℝ) + 1) ^ 2) ≤ Real.exp (-288) * (1 / 2) ^ n := by + refine le_trans ?_ (li_geom_tail n) + have h0 : (0 : ℝ) ≤ n := by positivity + have h1 : (1 : ℝ) ≤ (n : ℝ) + 1 := by linarith + exact mul_le_mul_of_nonneg_right (pow_le_pow_right₀ h1 (by norm_num)) (Real.exp_pos _).le + +/-- `exp(-288 (n+1)^2) ≤ exp(-288) (1/2)^n`: `1 ≤ (n+1)^9`. -/ +theorem li_exp_succ_le (n : ℕ) : + Real.exp (-288 * ((n : ℝ) + 1) ^ 2) ≤ Real.exp (-288) * (1 / 2) ^ n := by + refine le_trans ?_ (li_geom_tail n) + have h0 : (0 : ℝ) ≤ n := by positivity + have h1 : (1 : ℝ) ≤ ((n : ℝ) + 1) ^ 9 := one_le_pow₀ (by linarith) + exact le_mul_of_one_le_left (Real.exp_pos _).le h1 + +/-! ### (b) One-dimensional sums over `ℤ` -/ + +/-- A nonnegative `φ : ℤ → ℝ` whose values at `n+1` and at `-(n+1)` are at most +`exp(-288) (1/2)^n` is summable over `ℤ`, with `∑' φ ≤ φ 0 + 4 exp(-288)`: split `ℤ` into `ℕ` +and the negatives (`tsum_of_nat_of_neg_add_one`), peel off `φ 0` (`Summable.tsum_eq_zero_add`) +and compare both tails with the geometric series (`tsum_geometric_two`). -/ +theorem li_summable_int_tsum_le (φ : ℤ → ℝ) (h0 : ∀ m, 0 ≤ φ m) + (hpos : ∀ n : ℕ, φ ((n : ℤ) + 1) ≤ Real.exp (-288) * (1 / 2) ^ n) + (hneg : ∀ n : ℕ, φ (-((n : ℤ) + 1)) ≤ Real.exp (-288) * (1 / 2) ^ n) : + Summable φ ∧ ∑' m : ℤ, φ m ≤ φ 0 + 4 * Real.exp (-288) := by + have hgeo : Summable (fun n : ℕ => Real.exp (-288) * (1 / 2 : ℝ) ^ n) := + summable_geometric_two.mul_left _ + have hgeo_sum : ∑' n : ℕ, Real.exp (-288) * (1 / 2 : ℝ) ^ n = 2 * Real.exp (-288) := by + rw [tsum_mul_left, tsum_geometric_two] + ring + have hs1 : Summable (fun n : ℕ => φ ((n : ℤ) + 1)) := + Summable.of_nonneg_of_le (fun n => h0 _) hpos hgeo + have hs2 : Summable (fun n : ℕ => φ (-((n : ℤ) + 1))) := + Summable.of_nonneg_of_le (fun n => h0 _) hneg hgeo + have hs0 : Summable (fun n : ℕ => φ (n : ℤ)) := by + refine (summable_nat_add_iff 1).mp ?_ + refine hs1.congr fun n => ?_ + norm_cast + have hsum0 : ∑' n : ℕ, φ (n : ℤ) ≤ φ 0 + 2 * Real.exp (-288) := by + rw [hs0.tsum_eq_zero_add, Nat.cast_zero] + have htail : ∑' n : ℕ, φ (((n + 1 : ℕ) : ℤ)) ≤ 2 * Real.exp (-288) := by + rw [← hgeo_sum] + refine ((summable_nat_add_iff 1).mpr hs0).tsum_le_tsum (fun n => ?_) hgeo + rw [Nat.cast_succ] + exact hpos n + linarith + have hsum2 : ∑' n : ℕ, φ (-((n : ℤ) + 1)) ≤ 2 * Real.exp (-288) := by + rw [← hgeo_sum] + exact hs2.tsum_le_tsum hneg hgeo + refine ⟨Summable.of_nat_of_neg_add_one hs0 hs2, ?_⟩ + rw [tsum_of_nat_of_neg_add_one hs0 hs2] + linarith + +/-- The Gaussian weight `m ↦ exp(-288 m^2)` is summable over `ℤ` with sum `A ≤ 2` +(`A ≤ 1 + 4 exp(-288)` and `exp(-288) < 10^(-125)`). -/ +theorem li_gauss_int : + Summable (fun m : ℤ => Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2)) ∧ + ∑' m : ℤ, Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2) ≤ 2 := by + have h := li_summable_int_tsum_le (fun m : ℤ => Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2)) + (fun m => (Real.exp_pos _).le) + (fun n => by + push_cast + exact li_exp_succ_le n) + (fun n => by + push_cast + rw [neg_sq] + exact li_exp_succ_le n) + refine ⟨h.1, h.2.trans ?_⟩ + have he := UniversalLaw.Proofs.Side24Elementary.Aux.exp_neg_lt + have hp : (1 : ℝ) / 10 ^ 125 ≤ 1 / 4 := by norm_num + have h0 : Real.exp (-288 * ((0 : ℤ) : ℝ) ^ 2) = 1 := by simp + simp only [h0] + linarith + +/-- The weight `m ↦ m^6 exp(-288 m^2)` is summable over `ℤ` with sum `B ≤ 4 exp(-288)`. -/ +theorem li_pow_gauss_int : + Summable (fun m : ℤ => ((m : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2)) ∧ + ∑' m : ℤ, ((m : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2) ≤ + 4 * Real.exp (-288) := by + have h := li_summable_int_tsum_le + (fun m : ℤ => ((m : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2)) + (fun m => mul_nonneg (by positivity) (Real.exp_pos _).le) + (fun n => by + push_cast + exact li_pow6_succ_le n) + (fun n => by + push_cast + rw [neg_sq, Even.neg_pow (by decide : Even 6)] + exact li_pow6_succ_le n) + refine ⟨h.1, h.2.trans ?_⟩ + simp + +/-! ### (c) Finite sums of products over `Fin d → ℤ` -/ + +/-- For `h : Fin d → ℤ → ℝ` nonnegative and summable in each coordinate, the sum of +`∏ k, h k (n k)` over a finite set `s` of lattice points is at most `∏ k, ∑' m, h k m`: `s` lies +in the box `Fintype.piFinset (fun k => s.image (· k))`, over which the sum factors +(`Finset.prod_univ_sum`), and each factor is a partial sum of a nonnegative summable family +(`Summable.sum_le_tsum`). -/ +theorem li_sum_prod_le (d : ℕ) (h : Fin d → ℤ → ℝ) (h0 : ∀ k m, 0 ≤ h k m) + (hs : ∀ k, Summable (h k)) (s : Finset (Fin d → ℤ)) : + ∑ n ∈ s, ∏ k, h k (n k) ≤ ∏ k, ∑' m, h k m := by + set t : Fin d → Finset ℤ := fun k => s.image (fun n => n k) + have hsub : s ⊆ Fintype.piFinset t := fun n hn => by + rw [Fintype.mem_piFinset] + intro k + exact Finset.mem_image_of_mem _ hn + have hle : ∀ k, ∑ m ∈ t k, h k m ≤ ∑' m, h k m := fun k => + (hs k).sum_le_tsum (t k) (fun m _ => h0 k m) + calc ∑ n ∈ s, ∏ k, h k (n k) ≤ ∑ n ∈ Fintype.piFinset t, ∏ k, h k (n k) := + Finset.sum_le_sum_of_subset_of_nonneg hsub fun n _ _ => Finset.prod_nonneg fun k _ => h0 _ _ + _ = ∏ k, ∑ m ∈ t k, h k m := (Finset.prod_univ_sum t fun k m => h k m).symm + _ ≤ ∏ k, ∑' m, h k m := + Finset.prod_le_prod₀ (fun k _ => Finset.sum_nonneg fun m _ => h0 k m) (fun k _ => hle k) + +/-- A product over `Fin d` with one distinguished factor: +`∏ k, (if k = i then F k else G k) = F i * ∏ k ∈ univ.erase i, G k`. -/ +theorem li_prod_ite_eq (d : ℕ) (i : Fin d) (F G : Fin d → ℝ) : + (∏ k : Fin d, if k = i then F k else G k) = F i * ∏ k ∈ Finset.univ.erase i, G k := by + rw [← Finset.mul_prod_erase Finset.univ (fun k => if k = i then F k else G k) + (Finset.mem_univ i), ite_eq_left rfl] + congr 1 + exact Finset.prod_congr rfl fun k hk => ite_eq_right (Finset.ne_of_mem_erase hk) + +/-- `∏ k, exp(-288 n_k^2) = exp(-288 |n|^2)` (`Real.exp_sum`). -/ +theorem li_prod_exp (d : ℕ) (n : Fin d → ℤ) : + ∏ k : Fin d, Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2) = Real.exp (-288 * intNormSq d n) := by + rw [← Real.exp_sum, intNormSq, Finset.mul_sum] + +/-! ### (d) The pointwise domination for `d ≤ 3` -/ + +/-- Power mean for at most three nonnegative terms: `(∑ a_i)^3 ≤ 9 ∑ a_i^3` for `d ≤ 3` +(Jensen's `pow_sum_div_card_le_sum_pow` with `d^2 ≤ 9`; the case `d = 0` is `0 ≤ 0`). -/ +theorem li_sum_cube_le (d : ℕ) (hd : d ≤ 3) (a : Fin d → ℝ) (ha : ∀ i, 0 ≤ a i) : + (∑ i, a i) ^ 3 ≤ 9 * ∑ i, a i ^ 3 := by + rcases Nat.eq_zero_or_pos d with hd0 | hdpos + · subst hd0 + simp + · have hdR : (0 : ℝ) < d := by exact_mod_cast hdpos + have h := pow_sum_div_card_le_sum_pow (s := Finset.univ) (f := a) (fun i _ => ha i) 2 + rw [Finset.card_univ, Fintype.card_fin, div_le_iff₀ (pow_pos hdR 2)] at h + have hd' : (d : ℝ) ≤ 3 := by exact_mod_cast hd + have hsum : 0 ≤ ∑ i, a i ^ 3 := Finset.sum_nonneg fun i _ => pow_nonneg (ha i) 3 + calc (∑ i, a i) ^ 3 ≤ (∑ i, a i ^ 3) * (d : ℝ) ^ 2 := h + _ ≤ (∑ i, a i ^ 3) * 9 := by + apply mul_le_mul_of_nonneg_left _ hsum + nlinarith + _ = 9 * ∑ i, a i ^ 3 := by ring + +/-- `imageTerm d n ≥ 0`. -/ +theorem li_imageTerm_nonneg (d : ℕ) (n : Fin d → ℤ) : 0 ≤ imageTerm d n := by + unfold imageTerm + split_ifs + · exact le_rfl + · exact mul_nonneg (pow_nonneg (Finset.sum_nonneg fun i _ => sq_nonneg _) 3) (Real.exp_pos _).le + +/-- Pointwise domination for `d ≤ 3`: `imageTerm d n ≤ 9 ∑ i, ∏ k, h_i k (n k)`, where +`h_i i m = m^6 exp(-288 m^2)` and `h_i k m = exp(-288 m^2)` for `k ≠ i`. Each product equals +`(n_i^2)^3 exp(-288 |n|^2)` (`li_prod_ite_eq`, `li_prod_exp`), and `(∑ n_k^2)^3 ≤ 9 ∑ (n_i^2)^3` +is `li_sum_cube_le`. -/ +theorem li_imageTerm_le (d : ℕ) (hd : d ≤ 3) (n : Fin d → ℤ) : + imageTerm d n ≤ 9 * ∑ i : Fin d, ∏ k : Fin d, + (if k = i then ((n k : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2) + else Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2)) := by + have hprod : ∀ i : Fin d, (∏ k : Fin d, + (if k = i then ((n k : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2) + else Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2))) = + (((n i : ℤ) : ℝ) ^ 2) ^ 3 * Real.exp (-288 * intNormSq d n) := by + intro i + rw [li_prod_ite_eq d i (fun k => ((n k : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2)) + (fun k => Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2)), ← li_prod_exp d n, + ← Finset.mul_prod_erase Finset.univ (fun k => Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2)) + (Finset.mem_univ i)] + ring + rw [Finset.sum_congr rfl (fun i _ => hprod i), ← Finset.sum_mul] + unfold imageTerm + split_ifs + · positivity + · have h := li_sum_cube_le d hd (fun i => ((n i : ℤ) : ℝ) ^ 2) (fun i => sq_nonneg _) + unfold intNormSq + calc (∑ i, ((n i : ℤ) : ℝ) ^ 2) ^ 3 * Real.exp (-288 * ∑ i, ((n i : ℤ) : ℝ) ^ 2) + ≤ (9 * ∑ i, (((n i : ℤ) : ℝ) ^ 2) ^ 3) * Real.exp (-288 * ∑ i, ((n i : ℤ) : ℝ) ^ 2) := + mul_le_mul_of_nonneg_right h (Real.exp_pos _).le + _ = 9 * ((∑ i, (((n i : ℤ) : ℝ) ^ 2) ^ 3) * Real.exp (-288 * ∑ i, ((n i : ℤ) : ℝ) ^ 2)) := by + ring + +/-! ### (e) The uniform partial-sum bound -/ + +/-- For `d ≤ 3` and every finite `s`, `∑ n ∈ s, imageTerm d n ≤ 1458 exp(-288)`: the pointwise +bound `li_imageTerm_le`, the box bound `li_sum_prod_le` for each `i`, the one-dimensional sums +`A ≤ 2` and `B ≤ 4 exp(-288)` (`li_gauss_int`, `li_pow_gauss_int`), and +`9 · d · B · A^(d-1) ≤ 9 · 3 · 4 exp(-288) · 4 = 432 exp(-288)`. -/ +theorem li_partial_sum_le (d : ℕ) (hd : d ≤ 3) (s : Finset (Fin d → ℤ)) : + ∑ n ∈ s, imageTerm d n ≤ 1458 * Real.exp (-288) := by + obtain ⟨hgs, hgle⟩ := li_gauss_int + obtain ⟨hfs, hfle⟩ := li_pow_gauss_int + have hA0 : 0 ≤ ∑' m : ℤ, Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2) := + tsum_nonneg fun m => (Real.exp_pos _).le + have hexp : 0 ≤ Real.exp (-288) := (Real.exp_pos _).le + set A : ℝ := ∑' m : ℤ, Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2) with hA + set B : ℝ := ∑' m : ℤ, ((m : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2) with hB + have hidx : ∀ i : Fin d, ∑ n ∈ s, ∏ k : Fin d, + (if k = i then ((n k : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2) + else Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2)) ≤ 16 * Real.exp (-288) := by + intro i + have h1 := li_sum_prod_le d + (fun k m => if k = i then ((m : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2) + else Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2)) + (fun k m => by + split_ifs + · positivity + · positivity) + (fun k => by + by_cases hk : k = i + · simp only [hk, ite_true] + exact hfs + · simp only [hk, ite_false] + exact hgs) + s + have h2 : ∀ k : Fin d, ∑' m : ℤ, + (if k = i then ((m : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2) + else Real.exp (-288 * ((m : ℤ) : ℝ) ^ 2)) = if k = i then B else A := by + intro k + by_cases hk : k = i + · simp only [hk, ite_true, hB] + · simp only [hk, ite_false, hA] + calc _ ≤ _ := h1 + _ = ∏ k : Fin d, (if k = i then B else A) := Finset.prod_congr rfl fun k _ => h2 k + _ = B * ∏ _k ∈ Finset.univ.erase i, A := li_prod_ite_eq d i (fun _ => B) (fun _ => A) + _ = B * A ^ (d - 1) := by + rw [Finset.prod_const, Finset.card_erase_of_mem (Finset.mem_univ i), Finset.card_univ, + Fintype.card_fin] + _ ≤ (4 * Real.exp (-288)) * 2 ^ (d - 1) := + mul_le_mul hfle (pow_le_pow_left₀ hA0 hgle _) (pow_nonneg hA0 _) (by positivity) + _ ≤ (4 * Real.exp (-288)) * 2 ^ 2 := by + apply mul_le_mul_of_nonneg_left _ (by positivity) + exact pow_le_pow_right₀ (by norm_num) (by omega) + _ = 16 * Real.exp (-288) := by ring + calc ∑ n ∈ s, imageTerm d n + ≤ ∑ n ∈ s, 9 * ∑ i : Fin d, ∏ k : Fin d, + (if k = i then ((n k : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2) + else Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2)) := + Finset.sum_le_sum fun n _ => li_imageTerm_le d hd n + _ = 9 * ∑ i : Fin d, ∑ n ∈ s, ∏ k : Fin d, + (if k = i then ((n k : ℤ) : ℝ) ^ 6 * Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2) + else Real.exp (-288 * ((n k : ℤ) : ℝ) ^ 2)) := by + rw [← Finset.mul_sum, Finset.sum_comm] + _ ≤ 9 * ∑ _i : Fin d, 16 * Real.exp (-288) := + mul_le_mul_of_nonneg_left (Finset.sum_le_sum fun i _ => hidx i) (by norm_num) + _ = 9 * ((d : ℝ) * (16 * Real.exp (-288))) := by + rw [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] + _ ≤ 9 * (3 * (16 * Real.exp (-288))) := by + have hd' : (d : ℝ) ≤ 3 := by exact_mod_cast hd + gcongr + _ ≤ 1458 * Real.exp (-288) := by linarith + +/-! ## `dc_`: the Section 4 Gaussian density comparison (`DensityComparison`) -/ + +/-! ### (a) Elementary facts about `epsilon = 10^(-108)` + +`0 < epsilon` and `0 < 1 - epsilon` are `ir_epsilon_pos` and `ir_one_sub_epsilon_pos` above +(the `dc_` copies were removed at integration); only `0 < 1 + epsilon` is added here. -/ + +/-- `0 < 1 + epsilon`. -/ +theorem dc_one_add_epsilon_pos : (0 : ℝ) < 1 + epsilon := by + have := ir_epsilon_pos + linarith + +/-! ### (b) Loewner order: antitonicity of the inverse (L1) -/ + +/-- The expansion identity `(C - A) A⁻¹ (C - A) + (C - A) = C A⁻¹ C - C` for an invertible `A`. -/ +theorem dc_expand_identity {k : ℕ} (A C : Matrix (Fin k) (Fin k) ℝ) (hA : IsUnit A.det) : + (C - A) * A⁻¹ * (C - A) + (C - A) = C * A⁻¹ * C - C := by + have h1 : A * A⁻¹ = 1 := Matrix.mul_nonsing_inv A hA + have h2 : A⁻¹ * A = 1 := Matrix.nonsing_inv_mul A hA + calc (C - A) * A⁻¹ * (C - A) + (C - A) + = C * A⁻¹ * C - C * (A⁻¹ * A) - (A * A⁻¹) * C + (A * A⁻¹) * A + (C - A) := by + noncomm_ring + _ = C * A⁻¹ * C - C := by + rw [h1, h2] + simp only [Matrix.mul_one, Matrix.one_mul] + abel + +/-- The conjugation identity `C⁻¹ (C A⁻¹ C - C) C⁻¹ = A⁻¹ - C⁻¹` for an invertible `C`. -/ +theorem dc_conj_identity {k : ℕ} (A C : Matrix (Fin k) (Fin k) ℝ) (hC : IsUnit C.det) : + C⁻¹ * (C * A⁻¹ * C - C) * C⁻¹ = A⁻¹ - C⁻¹ := by + have h1 : C * C⁻¹ = 1 := Matrix.mul_nonsing_inv C hC + have h2 : C⁻¹ * C = 1 := Matrix.nonsing_inv_mul C hC + calc C⁻¹ * (C * A⁻¹ * C - C) * C⁻¹ + = (C⁻¹ * C) * A⁻¹ * (C * C⁻¹) - C⁻¹ * (C * C⁻¹) := by + noncomm_ring + _ = A⁻¹ - C⁻¹ := by + rw [h1, h2] + simp only [Matrix.mul_one, Matrix.one_mul] + +/-- (L1) For real matrices: `A` positive definite, `C` positive definite and `C - A` positive +semidefinite give `A⁻¹ - C⁻¹` positive semidefinite, i.e. `C⁻¹ ≤ A⁻¹` in the Loewner order. -/ +theorem dc_inv_sub_inv_posSemidef {k : ℕ} {A C : Matrix (Fin k) (Fin k) ℝ} + (hA : A.PosDef) (hC : C.PosDef) (hAC : (C - A).PosSemidef) : + (A⁻¹ - C⁻¹).PosSemidef := by + have hAu : IsUnit A.det := isUnit_iff_ne_zero.mpr hA.det_pos.ne' + have hCu : IsUnit C.det := isUnit_iff_ne_zero.mpr hC.det_pos.ne' + have hP : (C * A⁻¹ * C - C).PosSemidef := by + have h1 : ((C - A).conjTranspose * A⁻¹ * (C - A)).PosSemidef := + hA.inv.posSemidef.conjTranspose_mul_mul_same (C - A) + rw [hAC.isHermitian.eq] at h1 + have h2 := h1.add hAC + rwa [dc_expand_identity A C hAu] at h2 + have h3 : ((C⁻¹).conjTranspose * (C * A⁻¹ * C - C) * C⁻¹).PosSemidef := + hP.conjTranspose_mul_mul_same C⁻¹ + rw [hC.inv.isHermitian.eq] at h3 + rwa [dc_conj_identity A C hCu] at h3 + +/-- The quadratic-form consequence of (L1): `x ⬝ C⁻¹ x ≤ x ⬝ A⁻¹ x`. -/ +theorem dc_quad_le {k : ℕ} {A C : Matrix (Fin k) (Fin k) ℝ} + (hA : A.PosDef) (hC : C.PosDef) (hAC : (C - A).PosSemidef) (x : Fin k → ℝ) : + dotProduct x (Matrix.mulVec C⁻¹ x) ≤ dotProduct x (Matrix.mulVec A⁻¹ x) := by + have h := (dc_inv_sub_inv_posSemidef hA hC hAC).dotProduct_mulVec_nonneg x + simp only [star_trivial, Matrix.sub_mulVec, dotProduct_sub] at h + linarith + +/-! ### (c) Loewner order: monotonicity of the determinant (L2) -/ + +/-- A real matrix `M` with `M - 1` positive semidefinite has determinant at least one: every +eigenvalue of `M` lies in `spectrum ℝ M`, so every eigenvalue minus one lies in +`spectrum ℝ (M - 1) ⊆ [0, ∞)`, and `det M` is the product of the eigenvalues. -/ +theorem dc_one_le_det {k : ℕ} {M : Matrix (Fin k) (Fin k) ℝ} (hM : (M - 1).PosSemidef) : + 1 ≤ M.det := by + have hherm : M.IsHermitian := by + have h := hM.isHermitian.add (Matrix.isHermitian_one (n := Fin k) (α := ℝ)) + simpa using h + rw [hherm.det_eq_prod_eigenvalues] + apply Finset.one_le_prod₀ + intro i _ + have hspec := (Matrix.posSemidef_iff_isHermitian_and_spectrum_nonneg.mp hM).2 + have hmem : hherm.eigenvalues i ∈ spectrum ℝ M := hherm.eigenvalues_mem_spectrum_real i + have hmem' : hherm.eigenvalues i - 1 ∈ spectrum ℝ (M - 1) := by + have h : hherm.eigenvalues i - 1 ∈ spectrum ℝ M - ({1} : Set ℝ) := + Set.sub_mem_sub hmem (Set.mem_singleton (1 : ℝ)) + rw [spectrum.sub_singleton_eq] at h + simpa using h + have h0 : (0 : ℝ) ≤ hherm.eigenvalues i - 1 := hspec hmem' + simp only [RCLike.ofReal_real_eq_id, id] + linarith + +/-- (L2) For real matrices: `A` positive definite and `C - A` positive semidefinite give +`det A ≤ det C`, through the square root `S = CFC.sqrt A` and `M = S⁻¹ C S⁻¹ ≥ 1`. -/ +theorem dc_det_le_det {k : ℕ} {A C : Matrix (Fin k) (Fin k) ℝ} + (hA : A.PosDef) (hAC : (C - A).PosSemidef) : A.det ≤ C.det := by + obtain ⟨S, hSS, hSpsd⟩ : ∃ S : Matrix (Fin k) (Fin k) ℝ, S * S = A ∧ S.PosSemidef := + ⟨CFC.sqrt A, CFC.sqrt_mul_sqrt_self A hA.posSemidef.nonneg, + Matrix.nonneg_iff_posSemidef.mp (CFC.sqrt_nonneg A)⟩ + have hSherm : S.conjTranspose = S := hSpsd.isHermitian.eq + have hdetA : A.det = S.det * S.det := by + rw [← hSS, Matrix.det_mul] + have hSdet : S.det ≠ 0 := by + intro h + have h0 : A.det = 0 := by + rw [hdetA, h, mul_zero] + exact hA.det_pos.ne' h0 + have hSu : IsUnit S.det := isUnit_iff_ne_zero.mpr hSdet + have hSinv : (S⁻¹).conjTranspose = S⁻¹ := by + rw [Matrix.conjTranspose_nonsing_inv, hSherm] + have h1 : S * S⁻¹ = 1 := Matrix.mul_nonsing_inv S hSu + have h2 : S⁻¹ * S = 1 := Matrix.nonsing_inv_mul S hSu + have hM : (S⁻¹ * C * S⁻¹ - 1).PosSemidef := by + have h3 : ((S⁻¹).conjTranspose * (C - A) * S⁻¹).PosSemidef := + hAC.conjTranspose_mul_mul_same S⁻¹ + rw [hSinv] at h3 + have h4 : S⁻¹ * (C - A) * S⁻¹ = S⁻¹ * C * S⁻¹ - 1 := by + rw [← hSS] + calc S⁻¹ * (C - S * S) * S⁻¹ + = S⁻¹ * C * S⁻¹ - (S⁻¹ * S) * (S * S⁻¹) := by + noncomm_ring + _ = S⁻¹ * C * S⁻¹ - 1 := by + rw [h1, h2, Matrix.mul_one] + rwa [h4] at h3 + have hdetM : 1 ≤ (S⁻¹ * C * S⁻¹).det := dc_one_le_det hM + have hCeq : S * (S⁻¹ * C * S⁻¹) * S = C := by + calc S * (S⁻¹ * C * S⁻¹) * S = (S * S⁻¹) * C * (S⁻¹ * S) := by + noncomm_ring + _ = C := by + rw [h1, h2, Matrix.one_mul, Matrix.mul_one] + have hdetC : C.det = S.det * S.det * (S⁻¹ * C * S⁻¹).det := by + calc C.det = (S * (S⁻¹ * C * S⁻¹) * S).det := by + rw [hCeq] + _ = S.det * (S⁻¹ * C * S⁻¹).det * S.det := by + rw [Matrix.det_mul, Matrix.det_mul] + _ = S.det * S.det * (S⁻¹ * C * S⁻¹).det := by + ring + have hSS2 : 0 ≤ S.det * S.det := mul_self_nonneg _ + calc A.det = S.det * S.det := hdetA + _ = S.det * S.det * 1 := (mul_one _).symm + _ ≤ S.det * S.det * (S⁻¹ * C * S⁻¹).det := mul_le_mul_of_nonneg_left hdetM hSS2 + _ = C.det := hdetC.symm + +/-! ### (d) Determinant scaling and real-power bookkeeping -/ + +/-- `det (c • C) = c ^ k * det C` for a `k × k` real matrix. -/ +theorem dc_det_smul {k : ℕ} (C : Matrix (Fin k) (Fin k) ℝ) (c : ℝ) : + (c • C).det = c ^ k * C.det := by + rw [Matrix.det_smul, Fintype.card_fin] + +/-- The scaling identity `(a / b)^(k/2) · (a^k D)^(-1/2) = (b^k D)^(-1/2)` for `a, b, D > 0`. -/ +theorem dc_scale_identity (a b D : ℝ) (ha : 0 < a) (hb : 0 < b) (hD : 0 < D) (k : ℕ) : + (a / b) ^ ((k : ℝ) / 2) * (a ^ k * D) ^ (-(1 : ℝ) / 2) = (b ^ k * D) ^ (-(1 : ℝ) / 2) := by + have hak : (a ^ k) ^ (-(1 : ℝ) / 2) = (a ^ ((k : ℝ) / 2))⁻¹ := by + rw [← Real.rpow_natCast, ← Real.rpow_mul ha.le, ← Real.rpow_neg ha.le] + congr 1 + ring + have hbk : (b ^ k) ^ (-(1 : ℝ) / 2) = (b ^ ((k : ℝ) / 2))⁻¹ := by + rw [← Real.rpow_natCast, ← Real.rpow_mul hb.le, ← Real.rpow_neg hb.le] + congr 1 + ring + have hapos : a ^ ((k : ℝ) / 2) ≠ 0 := (Real.rpow_pos_of_pos ha _).ne' + have hbpos : b ^ ((k : ℝ) / 2) ≠ 0 := (Real.rpow_pos_of_pos hb _).ne' + rw [Real.div_rpow ha.le hb.le, Real.mul_rpow (pow_nonneg ha.le k) hD.le, + Real.mul_rpow (pow_nonneg hb.le k) hD.le, hak, hbk] + field_simp + +/-! ### (e) Assembly of the two pointwise inequalities from abstract real factors -/ + +/-- Lower assembly: `s * dA = dB`, `dB ≤ dC`, `eA ≤ eC` give `s (P dA eA) ≤ P dC eC`. -/ +theorem dc_assemble_lower (P s dA dB dC eA eC : ℝ) (hP : 0 ≤ P) (hs : s * dA = dB) + (hd : dB ≤ dC) (hdC : 0 ≤ dC) (he : eA ≤ eC) (heA : 0 ≤ eA) : + s * (P * dA * eA) ≤ P * dC * eC := by + calc s * (P * dA * eA) = P * (s * dA) * eA := by ring + _ = P * dB * eA := by rw [hs] + _ ≤ P * dC * eC := mul_le_mul (mul_le_mul_of_nonneg_left hd hP) he heA (mul_nonneg hP hdC) + +/-- Upper assembly: `s * dB = dA`, `dC ≤ dA`, `eC ≤ eB` give `P dC eC ≤ s (P dB eB)`. -/ +theorem dc_assemble_upper (P s dA dB dC eB eC : ℝ) (hP : 0 ≤ P) (hs : s * dB = dA) + (hd : dC ≤ dA) (hdA : 0 ≤ dA) (he : eC ≤ eB) (heC : 0 ≤ eC) : + P * dC * eC ≤ s * (P * dB * eB) := by + calc P * dC * eC ≤ P * dA * eB := + mul_le_mul (mul_le_mul_of_nonneg_left hd hP) he heC (mul_nonneg hP hdA) + _ = P * (s * dB) * eB := by rw [hs] + _ = s * (P * dB * eB) := by ring + +end Aux + +/-! ## The theta family is summable at every point, in every dimension -/ + +/-- Proves: `UniversalLaw.Spec.Side24.ThetaSummable` (claim `side24-coefficient`). +Method: comparison `Summable.of_nonneg_of_le` with the dominating family +`exp(‖z‖^2/2) ∏ i, exp(-144 n_i^2)` (`Aux.th_term_le_dominant`, from `‖24 n‖^2 = 576 ∑ n_i^2` +and `Real.exp_sum`); that family is summable (`Aux.th_summable_dominant`) because +`m ↦ exp(-144 m^2)` is summable on `ℤ` (`HurwitzKernelBounds.summable_f_int`) and products of +a nonnegative summable function are summable over `Fin d → ℤ` (`Aux.th_summable_pi_prod`, via +`Fintype.piFinset`, `Finset.prod_univ_sum` and `summable_of_sum_le`). +Does not establish: the informal convergence statement of the source as the source's object +(the alignment of `E d`, `latticeVec` and this family with the source's `R^d`, `24 Z^d` and +`K_24` is PENDING_INDEPENDENT_REVIEW); any value, positivity, smoothness or Poisson-summation +property of `thetaSum24` or `K24`; `ImageBound`, `LatticeImageSumBound` or the derivative +bounds of Section 2; any enclosure of the coefficient; the statement module's own +`Does not claim:` text (the statement packet is not edited here); the disposition of +`side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived +here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1. -/ +theorem thetaSummable (d : ℕ) : UniversalLaw.Spec.Side24.ThetaSummable d := by + intro z + exact Summable.of_nonneg_of_le (fun n => (Real.exp_pos _).le) + (fun n => Aux.th_term_le_dominant d z n) (Aux.th_summable_dominant d z) + +/-! ## Inventory theorem -/ + +/-- Proves: `UniversalLaw.Spec.Side24.DerivativePairingBound` (register row `side24-coefficient`, +Section 2 first display of SIDE24-COEFFICIENT-D23-20260924-v1). +Method: the contraction at a unit tuple is at most the operator norm `‖iteratedFDeriv ℝ q (Kinf d) x‖` +(`Aux.dp_abs_le_opNorm`, from `ContinuousMultilinearMap.le_opNorm`); the operator norms satisfy the +Hermite-type recursion `b_{q+2} ≤ ‖x‖ b_{q+1} + (q+1) b_q` (`Aux.dp_recursion`, from +`norm_iteratedFDeriv_fderiv`, the explicit derivative `fderiv ℝ (Kinf d) y = Kinf d y • (-(innerSL ℝ y))` +and the Leibniz bound `norm_iteratedFDeriv_smul_le`, the linear factor's derivatives of order `≥ 2` +vanishing), and with `b_0 = Kinf d x`, `b_1 = Kinf d x ‖x‖` the orders `q ≤ 6` are matched to the +unfolded `pairingSum q ‖x‖` (`Aux.dp_pairing_all`, `interval_cases`, `nlinarith`). +Does not establish: the informal Section 2 display or its review and acceptance; the alignment of the +formal objects (`Kinf`, `IsUnitTuple`, `pairingSum`, Mathlib's `iteratedFDeriv ℝ q`) with the source's +(PENDING_INDEPENDENT_REVIEW); validity for `q > 6` (the Prop restricts to `q ≤ 6`, as the source does, +and nothing beyond it is stated or proved here); any enclosure of the coefficient `c_{d,24}`; the +disposition of `side24-coefficient`, which is REVIEWED_SCOPED as transcribed in the register, not +derived here. -/ +theorem derivativePairingBound : UniversalLaw.Spec.Side24.DerivativePairingBound := by + intro d q hq x u hu + exact (Aux.dp_abs_le_opNorm d q x u hu).trans (Aux.dp_pairing_all d x q hq) + +/-! ## The Section 2 derivative bound at `|x| ≥ 1` -/ + +/-- Proves: `UniversalLaw.Spec.Side24.DerivativeBoundAtLeastOne` (register row `side24-coefficient`, +Section 2 of SIDE24-COEFFICIENT-D23-20260924-v1, "At |x|>=1 this is at most 76 |x|^6 e^(-|x|^2/2)"). +Method: the Section 2 pairing bound `|D^q Kinf (x) u| ≤ exp(-‖x‖²/2) · pairingSum q ‖x‖` is the +inventory theorem `derivativePairingBound` of this module (the Hermite-type recursion +`b_{q+2} ≤ ‖x‖ b_{q+1} + (q+1) b_q` for the operator norms, `Aux.dp_recursion`, with +`ContinuousMultilinearMap.le_opNorm` at a unit tuple); the hypothesis-form helper +`Aux.d1_derivativeBoundAtLeastOne_of_derivativePairingBound` combines it with the elementary +inequality `pairingSum q r ≤ 76 r^6` for `1 ≤ r`, `q ≤ 6` (`Aux.d1_pairingSum_le_of_one_le`: +`interval_cases`, `Finset.sum_range_succ`, `Nat.factorial`, `norm_num`, and the monomial bounds +`r^k ≤ r^6` from `pow_le_pow_right₀`, `one_le_pow₀`), by `mul_le_mul_of_nonneg_left` with +`0 ≤ exp(-‖x‖²/2)`. +Does not establish: the informal Section 2 display or its review and acceptance; the alignment of the +formal objects (`Kinf`, `IsUnitTuple`, `pairingSum`, Mathlib's `iteratedFDeriv ℝ q`) with the source's +(PENDING_INDEPENDENT_REVIEW); any identification with the main-branch targets +`UniversalLaw.Side24.pairing_sum_six` and `pairing_sum_le_six`, which kernel-check the coefficient +arithmetic `1+15+45+15=76` and `76` bounding the smaller orders there, not here; validity for `q > 6` or +for `‖x‖ < 1` (the Prop restricts to `q ≤ 6` and `1 ≤ ‖x‖`, as the source does); any enclosure of the +coefficient `c_{d,24}`; the disposition of `side24-coefficient`, which is REVIEWED_SCOPED as transcribed +in the register, not derived here. -/ +theorem derivativeBoundAtLeastOne : UniversalLaw.Spec.Side24.DerivativeBoundAtLeastOne := + Aux.d1_derivativeBoundAtLeastOne_of_derivativePairingBound derivativePairingBound + +/-! ## The Section 2 derivative bound at the origin -/ + +/-- Proves: `UniversalLaw.Spec.Side24.DerivativeBoundAtZero` (register row `side24-coefficient`, +Section 2 of SIDE24-COEFFICIENT-D23-20260924-v1, "contraction has absolute value at most 15"). +Method: the hypothesis-form helper `Aux.d0_derivativeBoundAtZero_of_derivativePairingBound` +specialises the pairing bound `DerivativePairingBound` to `x = 0`, where `‖0‖ = 0`, +`exp (-(0 ^ 2) / 2) = 1` and `pairingSum q 0 ≤ 15` for `q ≤ 6` (`Aux.d0_pairingSum_zero_le`: +the values are `1, 0, 1, 0, 3, 0, 15`, by `interval_cases`, `Finset.sum_range_succ`, `norm_num`); +the pairing bound itself is the inventory theorem `derivativePairingBound` of this module +(`Aux.dp_recursion`, `Aux.dp_pairing_all`, `Aux.dp_abs_le_opNorm`). +Does not establish: the informal Section 2 display or its review and acceptance; the alignment of +the formal objects (`Kinf`, `IsUnitTuple`, `pairingSum`, Mathlib's `iteratedFDeriv ℝ q` at `0`) with +the source's (PENDING_INDEPENDENT_REVIEW); the double-factorial identity `5!! = 15` as such, which +is kernel-checked on main (`UniversalLaw.Side24.sixth_moment_double_factorial`) and is neither +re-proved nor cited here (only `pairingSum 6 0 = 15` is computed); validity for `q > 6` (the Prop +restricts to `q ≤ 6`, as the source does); any value of a contraction (only an upper bound on its +absolute value); any enclosure of the coefficient `c_{d,24}`; the disposition of +`side24-coefficient`, which is REVIEWED_SCOPED as transcribed in the register, not derived here. -/ +theorem derivativeBoundAtZero : UniversalLaw.Spec.Side24.DerivativeBoundAtZero := + Aux.d0_derivativeBoundAtZero_of_derivativePairingBound derivativePairingBound + +/-! ## The Section 1 reference blocks of `K_infty` in every orthonormal frame -/ + +/-- Proves: `UniversalLaw.Spec.Side24.ReferenceOddAndPinBlocks` (claim `side24-coefficient`). +Method: the derivative contractions of `Kinf d` at `0` are evaluated by peeling one direction +at a time (`Aux.rb_peel`, from `iteratedFDeriv_succ_apply_right` and +`iteratedFDeriv_clm_apply_const_apply`) with explicit `HasFDerivAt` closed forms of the +intermediate functions (`Aux.rb_fderiv_kinf`, `rb_fderiv_g1`, `rb_fderiv_g2`, +`rb_fderiv_g3_zero`, `rb_fderiv_P3`, `rb_fderiv_P4`, `rb_fderiv_P5_zero`), giving +`Aux.rb_order_two` (`-⟪a, b⟫`), `Aux.rb_order_four` (the three pairings) and `Aux.rb_order_six` +(`-15 ⟪u, u⟫^3`); in an orthonormal frame (`orthonormal_iff_ite`) the five blocks follow +(`Aux.rb_gradCov`, `rb_thirdVar`, `rb_thirdGradCov`, `rb_pinCov`, `rb_condThirdVariance`, +the last with `inv_one`, `Matrix.one_mulVec` and `Finset.sum_ite_eq'`). +Does not establish: the informal Section 1 block values of the source as the source's objects +(the alignment of `E d`, `Kinf`, `derivContraction`, `gradCov`, `pinCov`, `thirdGradCov`, +`thirdVar` and `condThirdVariance` with the source's `R^d`, `K_infty`, covariances and +conditional variance is PENDING_INDEPENDENT_REVIEW); that any of these matrices is the +covariance of a random vector or field (none is constructed in the statement module); the +statement module's own `Does not claim:` text ("the identities (statement only); `15 - 9 = 6` is +kernel-checked on main (`UniversalLaw.Side24.conditional_third_derivative_variance`), not +here"), which describes the statement packet, is not edited here, and main's theorem is neither +cited nor relied on; `ReferencePinDensityProduct`, `CovarianceOrdering`, +`EntrywiseCovarianceBound` or any other Prop of the module; any enclosure of the coefficient; +the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, +not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1. -/ +theorem referenceOddAndPinBlocks : UniversalLaw.Spec.Side24.ReferenceOddAndPinBlocks := by + intro d _ hd R hR + exact ⟨Aux.rb_gradCov R hR, Aux.rb_thirdVar hd R hR, Aux.rb_thirdGradCov hd R hR, + Aux.rb_pinCov hd R hR, Aux.rb_condThirdVariance hd R hR⟩ + +/-! ## The Section 1 reference pin-density product -/ + +/-- Proves: `UniversalLaw.Spec.Side24.ReferencePinDensityProduct` (register row `side24-coefficient`, +Section 1 of SIDE24-COEFFICIENT-D23-20260924-v1, "p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3)"). +Method: the hypothesis form `Aux.rp_referencePinDensityProduct_of_referenceOddAndPinBlocks` takes +the inventory theorem `referenceOddAndPinBlocks` of this module and uses only its first conjunct +`gradCov d (Kinf d) R = 1` and fourth conjunct `pinCov d (Kinf d) (R 0) R = diagonal (3, 1, …, 1)` +(both from the contractions `Aux.rb_order_two`, `Aux.rb_order_four` in an orthonormal frame); the +common core `Aux.rp_referencePinDensityProduct_of_blockIdentities` rewrites the two blocks and +finishes with the density algebra `Aux.rp_density_algebra` (`Matrix.det_one`, `Matrix.det_diagonal`, +`Finset.prod_ite_eq'`, `Matrix.mulVec_zero`, `Real.rpow_add`, `Real.rpow_neg`, `Real.sqrt_eq_rpow`). +Does not establish: the informal Section 1 display or its review and acceptance; the alignment of +the formal objects (`gaussDensity`, `gradCov`, `pinCov`, `derivContraction` through Mathlib's +`iteratedFDeriv ℝ` at `0`, `Orthonormal ℝ R`) with the source's (PENDING_INDEPENDENT_REVIEW); the +determinant identity `det diag(3,1,…,1) = 3` as the main-branch target +`UniversalLaw.Side24.pin_determinant_and_joint_dimension`, which is kernel-checked on main and is +neither cited nor re-used here (the determinant is recomputed by `Matrix.det_diagonal` in +`Aux.rp_det_pinDiagonal`); that `gaussDensity` is the density of any random vector or that any field +is constructed; the remaining conjuncts of `ReferenceOddAndPinBlocks` (`Var(t) = 15`, +`Cov(t, G) = (-3, 0, …, 0)`, `tau^2 = 6`), which this theorem does not use; the value for +`d ∉ {2, 3}` (the Prop restricts to `d = 2 ∨ d = 3`, as the source does); any enclosure of the +coefficient `c_{d,24}`; the disposition of `side24-coefficient`, which is REVIEWED_SCOPED as +transcribed in the register, not derived here. -/ +theorem referencePinDensityProduct : UniversalLaw.Spec.Side24.ReferencePinDensityProduct := + Aux.rp_referencePinDensityProduct_of_referenceOddAndPinBlocks referenceOddAndPinBlocks + +/-! ## The Section 4 integrand-ratio bounds in dimensions two and three -/ + +/-- Proves: `UniversalLaw.Spec.Side24.IntegrandRatioBounds` (claim `side24-coefficient`). +Method: case split `d = 2`, `d = 3`, where the exponents evaluate to `a = 13/6, b = 5/2` and +`a = 25/6, b = 9/2` (`Aux.ir_exponentA_two`, `Aux.ir_exponentB_two`, `Aux.ir_exponentA_three`, +`Aux.ir_exponentB_three`); for arbitrary `1 ≤ a, b ≤ 5` both bounds follow +(`Aux.ir_ratio_bounds`) from Bernoulli's inequality `1 - a epsilon ≤ (1 - epsilon)^a` +(`one_add_mul_self_le_rpow_one_add`, `Aux.ir_bernoulli`), exponent monotonicity +`(1 + epsilon)^b ≤ (1 + epsilon)^5` (`Real.rpow_le_rpow_of_exponent_le`, +`Aux.ir_rpow_le_pow_five`), `le_div_iff₀` / `div_le_iff₀`, and the two exact rational +inequalities `(1 - 13 epsilon)(1 + epsilon)^5 ≤ 1 - 5 epsilon` and +`(1 + epsilon)^5 ≤ (1 + 28 epsilon)(1 - 5 epsilon)` decided by `norm_num` +(`Aux.ir_lower_rational`, `Aux.ir_upper_rational`). +Does not establish: the informal Section 4 display of the source as the source's object (the +alignment of `epsilon`, `exponentA`, `exponentB` and these power quotients with the source's +`epsilon`, `a`, `b` and integrand ratio is PENDING_INDEPENDENT_REVIEW); that the integrand ratio +of the source equals these power quotients (not specified in the statement packet); the +arithmetic `13 ≤ 32`, `28 ≤ 32` and `32*10^(-108) < 10^(-106)` (kernel-checked on main in +`UniversalLaw.Side24.ratio_constant_ordering` and `density_comparison_below_reported`, not +here); `DensityComparison` or any covariance ordering; any enclosure of the coefficient; the +statement module's own `Does not claim:` text (the statement packet is not edited here); the +disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, +not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1. -/ +theorem integrandRatioBounds : UniversalLaw.Spec.Side24.IntegrandRatioBounds := by + intro d hd + rcases hd with rfl | rfl + · rw [Aux.ir_exponentA_two, Aux.ir_exponentB_two] + exact Aux.ir_ratio_bounds _ _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + · rw [Aux.ir_exponentA_three, Aux.ir_exponentB_three] + exact Aux.ir_ratio_bounds _ _ (by norm_num) (by norm_num) (by norm_num) (by norm_num) + +/-! ## The lattice-image sum bound for `d ≤ 3` -/ + +/-- Proves: `UniversalLaw.Spec.Side24.LatticeImageSumBound` (claim `side24-coefficient`). +Method: both conjuncts from the uniform partial-sum bound `Aux.li_partial_sum_le` via +`summable_of_sum_le` and `Real.tsum_le_of_sum_le` (terms nonnegative, `Aux.li_imageTerm_nonneg`); +the partial sums are dominated pointwise by `9 ∑ i, ∏ k, h_i k (n k)` (`Aux.li_imageTerm_le`, +from the power mean `pow_sum_div_card_le_sum_pow` and `Real.exp_sum`), each finite sum of +products sits in a `Fintype.piFinset` box and factors (`Aux.li_sum_prod_le`, via +`Finset.prod_univ_sum` and `Summable.sum_le_tsum`), and the one-dimensional sums over `ℤ` are +bounded by `2` and `4 exp(-288)` (`Aux.li_gauss_int`, `Aux.li_pow_gauss_int`) through +`tsum_of_nat_of_neg_add_one`, the geometric series `tsum_geometric_two` and the registered +`SuccessiveTermRatio` (`SpecificationProofs.Side24Elementary.successiveTermRatio`), giving +`432 exp(-288) ≤ 1458 exp(-288)`. +Does not establish: the informal Section 2 lattice display of the source as the source's object +(the alignment of `intNormSq`, `imageTerm` and this `tsum` over `Fin d → ℤ` with the source's +`sum_{n ≠ 0} |n|^6 e^(-288|n|^2)` is PENDING_INDEPENDENT_REVIEW); the intermediate shell count +`729 ∑_j j^9 e^(-288 j^2)` of the source, which the Prop does not specify and this proof does not +reproduce; the arithmetic `27·27 = 729`, `729·2 = 1458` kernel-checked on main, not here; +`ImageBound`, the derivative bounds of Section 2 or any enclosure of the coefficient; the +disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not +derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1. -/ +theorem latticeImageSumBound : UniversalLaw.Spec.Side24.LatticeImageSumBound := by + intro d hd + exact ⟨summable_of_sum_le (fun n => Aux.li_imageTerm_nonneg d n) (Aux.li_partial_sum_le d hd), + Real.tsum_le_of_sum_le (fun n => Aux.li_imageTerm_nonneg d n) (Aux.li_partial_sum_le d hd)⟩ + +/-! ## The Section 4 Gaussian density comparison -/ + +/-- Proves: `UniversalLaw.Spec.Side24.DensityComparison` (claim `side24-coefficient`). +Method: with `A = (1 - epsilon) • C0` and `B = (1 + epsilon) • C0` (positive definite by +`Matrix.PosDef.smul`), the Loewner lemmas `Aux.dc_det_le_det` (`det A ≤ det C ≤ det B`, via +`CFC.sqrt`, `Matrix.PosSemidef.conjTranspose_mul_mul_same`, +`Matrix.posSemidef_iff_isHermitian_and_spectrum_nonneg` and +`Matrix.IsHermitian.det_eq_prod_eigenvalues`) and `Aux.dc_quad_le` +(`x ⬝ B⁻¹ x ≤ x ⬝ C⁻¹ x ≤ x ⬝ A⁻¹ x`, via `Aux.dc_inv_sub_inv_posSemidef`) give the two +determinant and exponent comparisons; `Aux.dc_scale_identity` with `Matrix.det_smul` turns +`((1 ∓ epsilon)/(1 ± epsilon))^(n/2) · det((1 ∓ epsilon) • C0)^(-1/2)` into +`det((1 ± epsilon) • C0)^(-1/2)`; `Real.rpow_le_rpow_of_nonpos`, `Real.exp_le_exp` and +`Aux.dc_assemble_lower` / `Aux.dc_assemble_upper` finish after unfolding `gaussDensity`. +Does not establish: the informal Section 4 display of the source as the source's object (the +alignment of `gaussDensity`, `symCoordDim`, `epsilon` and the positive-semidefinite hypotheses +with the source's densities, `n = m(m+1)/2`, `epsilon` and ordering is +PENDING_INDEPENDENT_REVIEW); the cone-moment consequence `(1 ± epsilon)^m D0` and the scaling +`a^m` of `h(A) = det(A)^2 1{A<0}` (not specified in the statement packet); that +`(1 ± epsilon) • C0` or `C` are covariances of any random object; `CovarianceOrdering` or any +ordering of `C24` against `Cref`; any enclosure of the coefficient; the statement module's own +`Does not claim:` text (the statement packet is not edited here); the disposition of +`side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; +nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1. -/ +theorem densityComparison : UniversalLaw.Spec.Side24.DensityComparison := by + intro m C0 C hC0 hC hlow hup x + have ha : (0 : ℝ) < 1 - epsilon := Aux.ir_one_sub_epsilon_pos + have hb : (0 : ℝ) < 1 + epsilon := Aux.dc_one_add_epsilon_pos + have hA : ((1 - epsilon) • C0).PosDef := hC0.smul ha + have hB : ((1 + epsilon) • C0).PosDef := hC0.smul hb + have hD : 0 < C0.det := hC0.det_pos + have hdetC : 0 < C.det := hC.det_pos + have hdetApos : 0 < ((1 - epsilon) • C0).det := hA.det_pos + have hdetAC : ((1 - epsilon) • C0).det ≤ C.det := Aux.dc_det_le_det hA hlow + have hdetCB : C.det ≤ ((1 + epsilon) • C0).det := Aux.dc_det_le_det hC hup + have hqAC : dotProduct x (Matrix.mulVec C⁻¹ x) ≤ + dotProduct x (Matrix.mulVec ((1 - epsilon) • C0)⁻¹ x) := Aux.dc_quad_le hA hC hlow x + have hqCB : dotProduct x (Matrix.mulVec ((1 + epsilon) • C0)⁻¹ x) ≤ + dotProduct x (Matrix.mulVec C⁻¹ x) := Aux.dc_quad_le hC hB hup x + have hdetA : ((1 - epsilon) • C0).det = (1 - epsilon) ^ symCoordDim m * C0.det := + Aux.dc_det_smul C0 _ + have hdetB : ((1 + epsilon) • C0).det = (1 + epsilon) ^ symCoordDim m * C0.det := + Aux.dc_det_smul C0 _ + have hscale1 : ((1 - epsilon) / (1 + epsilon)) ^ ((symCoordDim m : ℝ) / 2) * + ((1 - epsilon) • C0).det ^ (-(1 : ℝ) / 2) = ((1 + epsilon) • C0).det ^ (-(1 : ℝ) / 2) := by + rw [hdetA, hdetB] + exact Aux.dc_scale_identity _ _ _ ha hb hD _ + have hscale2 : ((1 + epsilon) / (1 - epsilon)) ^ ((symCoordDim m : ℝ) / 2) * + ((1 + epsilon) • C0).det ^ (-(1 : ℝ) / 2) = ((1 - epsilon) • C0).det ^ (-(1 : ℝ) / 2) := by + rw [hdetA, hdetB] + exact Aux.dc_scale_identity _ _ _ hb ha hD _ + have hP : (0 : ℝ) ≤ (2 * Real.pi) ^ (-(symCoordDim m : ℝ) / 2) := + (Real.rpow_pos_of_pos (by positivity) _).le + have hrB : ((1 + epsilon) • C0).det ^ (-(1 : ℝ) / 2) ≤ C.det ^ (-(1 : ℝ) / 2) := + Real.rpow_le_rpow_of_nonpos hdetC hdetCB (by norm_num) + have hrA : C.det ^ (-(1 : ℝ) / 2) ≤ ((1 - epsilon) • C0).det ^ (-(1 : ℝ) / 2) := + Real.rpow_le_rpow_of_nonpos hdetApos hdetAC (by norm_num) + have heA : Real.exp (-(dotProduct x (Matrix.mulVec ((1 - epsilon) • C0)⁻¹ x)) / 2) ≤ + Real.exp (-(dotProduct x (Matrix.mulVec C⁻¹ x)) / 2) := + Real.exp_le_exp.mpr (by linarith) + have heB : Real.exp (-(dotProduct x (Matrix.mulVec C⁻¹ x)) / 2) ≤ + Real.exp (-(dotProduct x (Matrix.mulVec ((1 + epsilon) • C0)⁻¹ x)) / 2) := + Real.exp_le_exp.mpr (by linarith) + unfold gaussDensity + constructor + · exact Aux.dc_assemble_lower _ _ _ _ _ _ _ hP hscale1 hrB (Real.rpow_nonneg hdetC.le _) heA + (Real.exp_pos _).le + · exact Aux.dc_assemble_upper _ _ _ _ _ _ _ hP hscale2 hrA (Real.rpow_nonneg hdetApos.le _) heB + (Real.exp_pos _).le + +end UniversalLaw.Proofs.Side24Analysis diff --git a/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Interfaces.lean b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Interfaces.lean new file mode 100644 index 000000000..16085e2e3 --- /dev/null +++ b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Interfaces.lean @@ -0,0 +1,625 @@ +import Specifications.Side24 + +/-! +# SIDE24 interface proofs: the `m = 1` cone moment identity, the trace moments and the law of +`R^2 = x^2 + y^2` over `ConeMomentInterface` (statement module `Specifications.Side24`) + +Scientific effect: NONE. Organizational-independence credit: 0 (Anthropic / Claude). + +Every theorem outside the `Aux` namespace has as its type exactly one registered `Prop` of +`Specifications/Side24.lean` (statement packet `specifications/lean_statements_20261010`, which +is not modified here), applied to its own parameter `I : ConeMomentInterface`. A kernel-checked +proof of such a `Prop` is evidence about that `Prop` only: it does not make the informal Layer 0 +note (SIDE24-COEFFICIENT-D23-20260924-v1, register row `side24-coefficient`) reviewed, accepted +or "verified", it does not identify the formal definitions with the informal objects (that +alignment is PENDING_INDEPENDENT_REVIEW), it does not enclose the coefficient `c_{d,24}`, and it +promotes, closes, discharges or reclassifies nothing. Nothing here constructs an inhabitant of +`ConeMomentInterface` or derives its law fields from any field; the theorems hold for every +inhabitant and use only the law fields `A_law`, `sxy_law` and the instance `isProb`. + +Provenance: the three `Prop`s proved here were selected by the round-4 triage (workflow +`wf_a3531fd8-891`) of the registered `Prop`s still without a theorem; each proof was written and +verified in its own lane and the lane files were merged into this module, with the helpers the +lanes proved twice removed (one copy kept, call sites repointed) and the lanes' tags kept. Helper +lemmas live in `UniversalLaw.Proofs.Side24Interfaces.Aux`, tagged `c1_`, `tm_`, `rl_`; the gate +inventories only the three theorems outside that namespace. The mathematics: + +* The common step: a map whose image measure under a probability measure is a non-degenerate + Gaussian (or a product of them) is a.e.-measurable, because at this Mathlib revision the image + measure of a non-a.e.-measurable map under a nonzero measure is a Dirac measure + (`Measure.map_of_not_aemeasurable_of_ne_zero`), which charges a singleton, while the Gaussian + law has `NullSingletonClass` (`c1_aemeasurable_of_map_eq_gaussian` for `A`, + `tm_aemeasurable_triple` for `(s, x, y)`). Integrals are then transported along the laws by + `integral_map`, and probabilities by `Measure.map_apply_of_aemeasurable`. +* `ConeMomentD1Identity I` (`c1_`): the negative half of the second moment of + `gaussianReal 0 (8/3)` is `(8/3)/2 = 4/3` (`gaussianReal_map_neg`, `integral_add_compl`, + `integral_Ici_eq_integral_Ioi`, `variance_fun_id_gaussianReal`). +* `TraceMoments I` (`tm_`): the first marginal of `sxy_law` is `gaussianReal 0 (5/3)` + (`Measure.map_fst_prod`); `∫ t^2 = 5/3`, `∫ t^4 = 3 (5/3)^2 = 25/3` (`iteratedDeriv_mgf_zero` + with the explicit fourth derivative of `exp (c t^2/2)` at `0`), `∫ exp(-t^2/2) = √(3/8)` + (`integral_gaussianReal_eq_integral_smul`, `integral_gaussian`). +* `RayleighSquareLaw I` (`rl_`): the second marginal of `sxy_law` is + `gaussianReal 0 1 ⊗ gaussianReal 0 1` (`Measure.map_snd_prod`); the measure of the closed disk of + radius `√a` is computed by hand (Mathlib has no Rayleigh or chi-squared law): `prod_withDensity`, + `withDensity_apply`, `ofReal_integral_eq_lintegral_ofReal`, polar coordinates + (`integral_comp_polarCoord_symm`, `setIntegral_prod_mul`), the angular integral `2π` and the + radial integral by the fundamental theorem of calculus; the right-hand side + `∫_0^a exp(-z/2)/2 dz = 1 - exp(-a/2)` is the same elementary integral. + +Remark recorded by the lanes, for the maintainers (not edited here): the proof-packet README row +"Side24 interface statements" gives as its informal reason that the Gaussian moment identities +"need the conditional law `A_u | V_u = 0` built in Mathlib's Gaussian measure theory"; for the three +`Prop`s proved here the interface's law fields suffice. This is a remark about an informal reason, +not about any claim; no register label changes. +-/ + +namespace UniversalLaw.Proofs.Side24Interfaces + +open MeasureTheory ProbabilityTheory UniversalLaw.Spec.Side24 + +namespace Aux + +/-! ## `c1_`: the `m = 1` cone moment identity (`ConeMomentD1Identity`) -/ + +/-! ### (a) A map with a non-degenerate Gaussian law is a.e.-measurable -/ + +/-- If `Measure.map A P` is the Gaussian `gaussianReal 0 v` with `v ≠ 0`, then `A` is +a.e.-measurable: otherwise the image measure would be a Dirac measure +(`Measure.map_of_not_aemeasurable_of_ne_zero`), giving a singleton of measure one, whereas the +Gaussian charges no singleton (`nullSingletonClass_gaussianReal`). -/ +theorem c1_aemeasurable_of_map_eq_gaussian {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) + [IsProbabilityMeasure P] (A : Ω → ℝ) {v : NNReal} (hv : v ≠ 0) + (h : Measure.map A P = gaussianReal 0 v) : AEMeasurable A P := by + by_contra hA + obtain ⟨c, hc⟩ : ∃ c : ℝ, Measure.map A P = Measure.dirac c := + ⟨_, Measure.map_of_not_aemeasurable_of_ne_zero hA (IsProbabilityMeasure.ne_zero P)⟩ + rw [hc] at h + have : NullSingletonClass (gaussianReal 0 v) := nullSingletonClass_gaussianReal hv + have h1 : Measure.dirac c {c} = gaussianReal 0 v {c} := by rw [h] + rw [Measure.dirac_apply_of_mem (Set.mem_singleton c), measure_singleton] at h1 + exact one_ne_zero h1 + +/-! ### (b) Second moments of the centred Gaussian -/ + +/-- The integrand `a ↦ 1_{Iio 0}(a) a^2` is measurable. -/ +theorem c1_measurable_indicator_sq : + Measurable (fun a : ℝ => Set.indicator (Set.Iio (0 : ℝ)) (fun a => a ^ 2) a) := + (measurable_id.pow_const 2).indicator measurableSet_Iio + +/-- The second moment of `gaussianReal 0 v` is `v`, from Mathlib's +`variance_fun_id_gaussianReal` rewritten with `variance_eq_integral` (the mean is `0`). -/ +theorem c1_second_moment_gaussianReal (v : NNReal) : + ∫ t, t ^ 2 ∂(gaussianReal 0 v) = (v : ℝ) := by + have h := variance_fun_id_gaussianReal (μ := 0) (v := v) + rw [variance_eq_integral (by fun_prop)] at h + simpa using h + +/-- `a ↦ a^2` is integrable against `gaussianReal 0 v` (`memLp_id_gaussianReal` at `p = 2`). -/ +theorem c1_integrable_sq_gaussianReal (v : NNReal) : + Integrable (fun a : ℝ => a ^ 2) (gaussianReal 0 v) := by + have := (memLp_id_gaussianReal (μ := 0) (v := v) 2).integrable_sq + simpa using this + +/-- Symmetry of the centred Gaussian: the second moment over `Iio 0` equals the second moment +over `Ioi 0`, by `gaussianReal_map_neg` and a change of variables `a ↦ -a` (`integral_map`), +using `neg_lt_zero` and `neg_sq` pointwise. -/ +theorem c1_setIntegral_sq_Iio_eq_Ioi (v : NNReal) : + ∫ a in Set.Iio (0 : ℝ), a ^ 2 ∂(gaussianReal 0 v) = + ∫ a in Set.Ioi (0 : ℝ), a ^ 2 ∂(gaussianReal 0 v) := by + set γ := gaussianReal 0 v with hγ + have hneg : γ.map (fun x => -x) = γ := by rw [hγ, gaussianReal_map_neg, neg_zero] + calc ∫ a in Set.Iio (0 : ℝ), a ^ 2 ∂γ + = ∫ a, Set.indicator (Set.Iio (0 : ℝ)) (fun a => a ^ 2) a ∂γ := + (integral_indicator measurableSet_Iio).symm + _ = ∫ a, Set.indicator (Set.Iio (0 : ℝ)) (fun a => a ^ 2) a ∂(γ.map (fun x => -x)) := by + rw [hneg] + _ = ∫ a, Set.indicator (Set.Iio (0 : ℝ)) (fun a => a ^ 2) (-a) ∂γ := + integral_map measurable_neg.aemeasurable c1_measurable_indicator_sq.aestronglyMeasurable + _ = ∫ a, Set.indicator (Set.Ioi (0 : ℝ)) (fun a => a ^ 2) a ∂γ := by + congr 1; funext a + simp only [Set.indicator, Set.mem_Iio, Set.mem_Ioi, neg_lt_zero, neg_sq] + _ = ∫ a in Set.Ioi (0 : ℝ), a ^ 2 ∂γ := integral_indicator measurableSet_Ioi + +/-- The negative half of the second moment of `gaussianReal 0 v`, `v ≠ 0`, is `v / 2`: +`integral_add_compl` over `Iio 0` and its complement `Ici 0`, `integral_Ici_eq_integral_Ioi` +(the Gaussian charges no singleton), the symmetry lemma, and the full second moment `v`. -/ +theorem c1_setIntegral_sq_Iio (v : NNReal) (hv : v ≠ 0) : + ∫ a in Set.Iio (0 : ℝ), a ^ 2 ∂(gaussianReal 0 v) = (v : ℝ) / 2 := by + have : NullSingletonClass (gaussianReal 0 v) := nullSingletonClass_gaussianReal hv + have hsplit := integral_add_compl (μ := gaussianReal 0 v) (s := Set.Iio (0 : ℝ)) + measurableSet_Iio (c1_integrable_sq_gaussianReal v) + rw [Set.compl_Iio, integral_Ici_eq_integral_Ioi, ← c1_setIntegral_sq_Iio_eq_Ioi, + c1_second_moment_gaussianReal] at hsplit + linarith + +/-! ### (c) Transport along the law of `A` -/ + +/-- For a map `A` with law `gaussianReal 0 (8/3)` under a probability measure `P`, the integral +of `1_{Iio 0}(A) A^2` equals the set integral of `a ↦ a^2` over `Iio 0` against +`gaussianReal 0 (8/3)`: a.e.-measurability of `A` from its law, then `integral_map` and +`integral_indicator`. Stated for an abstract `(Ω, P, A)` so that the interface's own instance +fields `I.mΩ`, `I.isProb` are supplied explicitly at the point of use. -/ +theorem c1_integral_indicator_eq {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) + [IsProbabilityMeasure P] (A : Ω → ℝ) + (hlaw : Measure.map A P = gaussianReal 0 (8 / 3 : NNReal)) : + ∫ ω, Set.indicator (Set.Iio (0 : ℝ)) (fun a => a ^ 2) (A ω) ∂P = + ∫ a in Set.Iio (0 : ℝ), a ^ 2 ∂(gaussianReal 0 (8 / 3 : NNReal)) := by + have hv : ((8 : NNReal) / 3) ≠ 0 := by norm_num + have hA : AEMeasurable A P := c1_aemeasurable_of_map_eq_gaussian P A hv hlaw + have e : ∫ ω, Set.indicator (Set.Iio (0 : ℝ)) (fun a => a ^ 2) (A ω) ∂P = + ∫ a, Set.indicator (Set.Iio (0 : ℝ)) (fun a => a ^ 2) a ∂(Measure.map A P) := + (integral_map hA c1_measurable_indicator_sq.aestronglyMeasurable).symm + rw [e, hlaw, integral_indicator measurableSet_Iio] + +/-! ## `tm_`: the trace moments (`TraceMoments`) -/ + +/-! ### (a) Calculus: the fourth derivative of `exp (c t^2 / 2)` at `0` -/ + +/-- Derivative of `t ↦ exp (c t^2 / 2)`: `c t exp (c t^2 / 2)` (chain rule through +`HasDerivAt.exp`). -/ +theorem tm_hasDerivAt_exp_sq (c t : ℝ) : + HasDerivAt (fun t : ℝ => Real.exp (c * t ^ 2 / 2)) (c * t * Real.exp (c * t ^ 2 / 2)) t := by + have h1 : HasDerivAt (fun t : ℝ => c * t ^ 2 / 2) + (c * (((2 : ℕ) : ℝ) * t ^ (2 - 1) * 1) / 2) t := + (((hasDerivAt_id' t).fun_pow 2).const_mul c).div_const 2 + have h2 := h1.exp + convert h2 using 1 + push_cast + ring + +/-- The fourth derivative of `t ↦ exp (c t^2 / 2)` at `0` is `3 c^2`: four explicit +`HasDerivAt` steps, chained through `iteratedDeriv_succ` and `iteratedDeriv_one`. -/ +theorem tm_iteratedDeriv_four (c : ℝ) : + iteratedDeriv 4 (fun t : ℝ => Real.exp (c * t ^ 2 / 2)) 0 = 3 * c ^ 2 := by + have d1 : deriv (fun t : ℝ => Real.exp (c * t ^ 2 / 2)) = + fun t => c * t * Real.exp (c * t ^ 2 / 2) := by + funext t + exact (tm_hasDerivAt_exp_sq c t).deriv + have d2 : deriv (fun t : ℝ => c * t * Real.exp (c * t ^ 2 / 2)) = + fun t => (c + c ^ 2 * t ^ 2) * Real.exp (c * t ^ 2 / 2) := by + funext t + have h := ((hasDerivAt_id' t).const_mul c).mul (tm_hasDerivAt_exp_sq c t) + have h' : HasDerivAt (fun t : ℝ => c * t * Real.exp (c * t ^ 2 / 2)) _ t := h + rw [h'.deriv] + ring + have d3 : deriv (fun t : ℝ => (c + c ^ 2 * t ^ 2) * Real.exp (c * t ^ 2 / 2)) = + fun t => (3 * c ^ 2 * t + c ^ 3 * t ^ 3) * Real.exp (c * t ^ 2 / 2) := by + funext t + have h := ((((hasDerivAt_id' t).fun_pow 2).const_mul (c ^ 2)).const_add c).mul + (tm_hasDerivAt_exp_sq c t) + have h' : HasDerivAt (fun t : ℝ => (c + c ^ 2 * t ^ 2) * Real.exp (c * t ^ 2 / 2)) _ t := h + rw [h'.deriv] + push_cast + ring + have d4 : deriv (fun t : ℝ => (3 * c ^ 2 * t + c ^ 3 * t ^ 3) * Real.exp (c * t ^ 2 / 2)) = + fun t => (3 * c ^ 2 + 6 * c ^ 3 * t ^ 2 + c ^ 4 * t ^ 4) * Real.exp (c * t ^ 2 / 2) := by + funext t + have ha := ((hasDerivAt_id' t).const_mul (3 * c ^ 2)).add + (((hasDerivAt_id' t).fun_pow 3).const_mul (c ^ 3)) + have h := ha.mul (tm_hasDerivAt_exp_sq c t) + have h' : HasDerivAt (fun t : ℝ => (3 * c ^ 2 * t + c ^ 3 * t ^ 3) * Real.exp (c * t ^ 2 / 2)) + _ t := h + rw [h'.deriv] + simp only [Pi.add_apply] + push_cast + ring + have e4 : iteratedDeriv 4 (fun t : ℝ => Real.exp (c * t ^ 2 / 2)) = + deriv (iteratedDeriv 3 (fun t : ℝ => Real.exp (c * t ^ 2 / 2))) := iteratedDeriv_succ + have e3 : iteratedDeriv 3 (fun t : ℝ => Real.exp (c * t ^ 2 / 2)) = + deriv (iteratedDeriv 2 (fun t : ℝ => Real.exp (c * t ^ 2 / 2))) := iteratedDeriv_succ + have e2 : iteratedDeriv 2 (fun t : ℝ => Real.exp (c * t ^ 2 / 2)) = + deriv (iteratedDeriv 1 (fun t : ℝ => Real.exp (c * t ^ 2 / 2))) := iteratedDeriv_succ + rw [e4, e3, e2, iteratedDeriv_one, d1, d2, d3, d4] + simp + +/-! ### (b) Moments of the centred real Gaussian `gaussianReal 0 v` + +The second moment `∫ t^2 = v` is `c1_second_moment_gaussianReal` above (the `tm_` copy was +removed at integration). -/ + +/-- Fourth moment of `N(0, v)`: `∫ t^4 = 3 v^2`, from the fourth derivative at `0` of the +moment generating function `t ↦ exp (v t^2 / 2)` (`iteratedDeriv_mgf_zero`, +`mgf_fun_id_gaussianReal`, `tm_iteratedDeriv_four`). -/ +theorem tm_fourth_moment (v : NNReal) : + ∫ t, t ^ 4 ∂(gaussianReal 0 v) = 3 * (v : ℝ) ^ 2 := by + have h := iteratedDeriv_mgf_zero (X := fun x : ℝ => x) (μ := gaussianReal 0 v) (by simp) 4 + rw [mgf_fun_id_gaussianReal] at h + simp only [zero_mul, zero_add] at h + rw [tm_iteratedDeriv_four] at h + simpa using h.symm + +/-- Exponential moment of `N(0, 5/3)`: `∫ exp (-t^2/2) = sqrt (3/8)`. The density +`(sqrt (10 π / 3))⁻¹ exp (-3 t^2 / 10)` times `exp (-t^2/2)` is +`(sqrt (10 π / 3))⁻¹ exp (-(4/5) t^2)`, whose Lebesgue integral is `sqrt (π / (4/5))` +(`integral_gaussian`); the square roots combine to `sqrt (3/8)`. -/ +theorem tm_exp_moment : + ∫ t, Real.exp (-(t ^ 2) / 2) ∂(gaussianReal 0 (5 / 3 : NNReal)) = Real.sqrt (3 / 8) := by + rw [integral_gaussianReal_eq_integral_smul (by norm_num)] + have hpt : ∀ x : ℝ, gaussianPDFReal 0 (5 / 3 : NNReal) x • Real.exp (-(x ^ 2) / 2) = + (Real.sqrt (10 * Real.pi / 3))⁻¹ * Real.exp (-(4 / 5) * x ^ 2) := by + intro x + rw [gaussianPDFReal_def, smul_eq_mul, mul_assoc, ← Real.exp_add] + push_cast + congr 2 + · congr 1 + ring + · ring + simp_rw [hpt] + rw [integral_const_mul, integral_gaussian, ← Real.sqrt_inv, ← Real.sqrt_mul (by positivity)] + congr 1 + field_simp + ring + +/-! ### (c) The marginal law of `s` from the interface's product law -/ + +/-- The triple `(s, x, y)` is a.e.-measurable: its image measure under `P` is the product +Gaussian law (`sxy_law`); were the triple not a.e.-measurable, `Measure.map` would be a Dirac +mass (`Measure.map_of_not_aemeasurable_of_ne_zero`), which gives a singleton measure `1`, +whereas the product of non-degenerate Gaussians has `NullSingletonClass`. -/ +theorem tm_aemeasurable_triple (I : ConeMomentInterface) : + @AEMeasurable _ _ _ I.mΩ (fun ω => (I.s ω, I.x ω, I.y ω)) I.P := by + let := I.mΩ + have := I.isProb + have hv : ((5 : NNReal) / 3) ≠ 0 := by norm_num + have : NullSingletonClass (gaussianReal 0 (5 / 3 : NNReal)) := nullSingletonClass_gaussianReal hv + by_contra hT + obtain ⟨c, hc⟩ : ∃ c : ℝ × ℝ × ℝ, + Measure.map (fun ω => (I.s ω, I.x ω, I.y ω)) I.P = Measure.dirac c := + ⟨_, Measure.map_of_not_aemeasurable_of_ne_zero hT (IsProbabilityMeasure.ne_zero I.P)⟩ + have h := I.sxy_law + rw [hc] at h + have h1 : Measure.dirac c {c} = + ((gaussianReal 0 (5 / 3)).prod ((gaussianReal 0 1).prod (gaussianReal 0 1))) {c} := by + rw [h] + rw [Measure.dirac_apply_of_mem (Set.mem_singleton c), measure_singleton] at h1 + exact one_ne_zero h1 + +/-- `s` is a.e.-measurable, as the first component of the a.e.-measurable triple. -/ +theorem tm_aemeasurable_s (I : ConeMomentInterface) : @AEMeasurable _ _ _ I.mΩ I.s I.P := + (tm_aemeasurable_triple I).fst + +/-- The law of `s` is `N(0, 5/3)`: the first marginal of the product law `sxy_law` +(`AEMeasurable.map_map_of_aemeasurable`, `Measure.map_fst_prod`; the composite +`Prod.fst ∘ (s, x, y)` is `s`). -/ +theorem tm_law_s (I : ConeMomentInterface) : + @Measure.map _ _ I.mΩ _ I.s I.P = gaussianReal 0 (5 / 3 : NNReal) := by + let := I.mΩ + have := I.isProb + have hT := tm_aemeasurable_triple I + have h := AEMeasurable.map_map_of_aemeasurable (measurable_fst.aemeasurable) hT + rw [I.sxy_law, Measure.map_fst_prod, measure_univ, one_smul] at h + exact h.symm + +/-! ### (d) Transport of an integral along `s` to the marginal law -/ + +/-- For a measurable `f : ℝ → ℝ`, `∫ f (s ω) dP = ∫ f t d(gaussianReal 0 (5/3))` +(`integral_map` along the a.e.-measurable `s`, then `tm_law_s`). -/ +theorem tm_integral_comp_s (I : ConeMomentInterface) (f : ℝ → ℝ) (hf : Measurable f) : + ∫ ω, f (I.s ω) ∂I.P = ∫ t, f t ∂(gaussianReal 0 (5 / 3 : NNReal)) := by + let := I.mΩ + have := I.isProb + rw [← tm_law_s I] + exact (integral_map (tm_aemeasurable_s I) hf.aestronglyMeasurable).symm + +/-! ## `rl_`: the law of `R^2` (`RayleighSquareLaw`) -/ + +/-! ### (a) Elementary integrals -/ + +/-- The right-hand side of `RayleighSquareLaw`: `∫_0^a exp (-z/2)/2 dz = 1 - exp (-a/2)`, by +the fundamental theorem of calculus with antiderivative `z ↦ -exp (-z/2)`. -/ +theorem rl_rhs_integral (a : ℝ) : + ∫ z in (0 : ℝ)..a, Real.exp (-z / 2) / 2 = 1 - Real.exp (-a / 2) := by + have hderiv : ∀ z ∈ Set.uIcc (0 : ℝ) a, + HasDerivAt (fun z : ℝ => -Real.exp (-z / 2)) (Real.exp (-z / 2) / 2) z := by + intro z _ + have h1 : HasDerivAt (fun z : ℝ => -z / 2) (-1 / 2) z := + ((hasDerivAt_id' z).neg).div_const 2 + have h2 := h1.exp.neg + convert h2 using 1 + ring + have hint : IntervalIntegrable (fun z : ℝ => Real.exp (-z / 2) / 2) volume 0 a := + (by fun_prop : Continuous fun z : ℝ => Real.exp (-z / 2) / 2).intervalIntegrable 0 a + rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hderiv hint] + simp + ring + +/-- The radial integral of the planar Gaussian density over the disk of radius `sqrt a`: +`∫_0^{sqrt a} r (2π)⁻¹ exp (-r^2/2) dr = (2π)⁻¹ (1 - exp (-a/2))`, by the fundamental +theorem of calculus with antiderivative `r ↦ -(2π)⁻¹ exp (-r^2/2)`. -/ +theorem rl_radial_integral (a : ℝ) (ha : 0 ≤ a) : + ∫ r in (0 : ℝ)..Real.sqrt a, r * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2)) = + (2 * Real.pi)⁻¹ * (1 - Real.exp (-a / 2)) := by + have hderiv : ∀ r ∈ Set.uIcc (0 : ℝ) (Real.sqrt a), + HasDerivAt (fun r : ℝ => -((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2))) + (r * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2))) r := by + intro r _ + have h1 : HasDerivAt (fun r : ℝ => -(r ^ 2) / 2) + (-(((2 : ℕ) : ℝ) * r ^ (2 - 1) * 1) / 2) r := + (((hasDerivAt_id' r).fun_pow 2).neg).div_const 2 + have h2 := (h1.exp.const_mul ((2 * Real.pi)⁻¹)).neg + convert h2 using 1 + push_cast + ring + have hcont : Continuous fun r : ℝ => r * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2)) := by + fun_prop + have hint : IntervalIntegrable (fun r : ℝ => r * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2))) + volume 0 (Real.sqrt a) := + hcont.intervalIntegrable 0 (Real.sqrt a) + rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hderiv hint, Real.sq_sqrt ha] + simp + ring + +/-- The angular integral: `∫_{Ioo (-π) π} 1 = 2π`. -/ +theorem rl_angular_integral : ∫ _θ in Set.Ioo (-Real.pi) Real.pi, (1 : ℝ) = 2 * Real.pi := by + rw [setIntegral_const, smul_eq_mul, mul_one, measureReal_def, Real.volume_Ioo, + ENNReal.toReal_ofReal (by linarith [Real.pi_pos])] + ring + +/-! ### (b) The planar standard Gaussian density -/ + +/-- The product of two standard Gaussian densities is the planar density +`(2π)⁻¹ exp (-(x^2 + y^2)/2)`. -/ +theorem rl_pdf_mul (x y : ℝ) : + gaussianPDFReal 0 1 x * gaussianPDFReal 0 1 y = + (2 * Real.pi)⁻¹ * Real.exp (-(x ^ 2 + y ^ 2) / 2) := by + simp only [gaussianPDFReal_def, NNReal.coe_one, mul_one, sub_zero] + have hs : (Real.sqrt (2 * Real.pi))⁻¹ * (Real.sqrt (2 * Real.pi))⁻¹ = + (2 * Real.pi)⁻¹ := by + rw [← mul_inv, Real.mul_self_sqrt (by positivity)] + calc (Real.sqrt (2 * Real.pi))⁻¹ * Real.exp (-x ^ 2 / 2) * + ((Real.sqrt (2 * Real.pi))⁻¹ * Real.exp (-y ^ 2 / 2)) + = ((Real.sqrt (2 * Real.pi))⁻¹ * (Real.sqrt (2 * Real.pi))⁻¹) * + (Real.exp (-x ^ 2 / 2) * Real.exp (-y ^ 2 / 2)) := by ring + _ = (2 * Real.pi)⁻¹ * Real.exp (-(x ^ 2 + y ^ 2) / 2) := by + rw [hs, ← Real.exp_add] + congr 2 + ring + +/-- The closed disk `{p | p.1^2 + p.2^2 ≤ a}` is measurable. -/ +theorem rl_measurableSet_disk (a : ℝ) : + MeasurableSet {p : ℝ × ℝ | p.1 ^ 2 + p.2 ^ 2 ≤ a} := + measurableSet_le (by fun_prop) measurable_const + +/-- The set `{r | r^2 ≤ a}` is measurable. -/ +theorem rl_measurableSet_radial (a : ℝ) : MeasurableSet {r : ℝ | r ^ 2 ≤ a} := + measurableSet_le (by fun_prop) measurable_const + +/-- In polar coordinates the disk integrand `p.1 • 1_D (density ∘ polarCoord.symm) p` is the +radial function `1_{r^2 ≤ a} (r (2π)⁻¹ exp (-r^2/2))` of `p.1` times the constant `1` of +`p.2` (the form `setIntegral_prod_mul` separates). -/ +theorem rl_polar_integrand (a : ℝ) (p : ℝ × ℝ) : + p.1 • Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ a} + (fun q => gaussianPDFReal 0 1 q.1 * gaussianPDFReal 0 1 q.2) (polarCoord.symm p) = + Set.indicator {r : ℝ | r ^ 2 ≤ a} + (fun r => r * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2))) p.1 * + (fun _ : ℝ => (1 : ℝ)) p.2 := by + have hsq : (p.1 * Real.cos p.2) ^ 2 + (p.1 * Real.sin p.2) ^ 2 = p.1 ^ 2 := by + rw [mul_pow, mul_pow, ← mul_add, Real.cos_sq_add_sin_sq, mul_one] + simp only [polarCoord_symm_apply, Set.indicator, Set.mem_ofPred_eq, rl_pdf_mul, hsq, + smul_eq_mul, mul_one] + split_ifs <;> simp + +/-- For `a ≥ 0`, `Ioi 0 ∩ {r | r^2 ≤ a} = Ioc 0 (sqrt a)`. -/ +theorem rl_Ioi_inter_eq (a : ℝ) (ha : 0 ≤ a) : + Set.Ioi (0 : ℝ) ∩ {r : ℝ | r ^ 2 ≤ a} = Set.Ioc 0 (Real.sqrt a) := by + ext r + simp only [Set.mem_inter_iff, Set.mem_Ioi, Set.mem_ofPred_eq, Set.mem_Ioc] + constructor + · rintro ⟨h0, hr⟩ + exact ⟨h0, (Real.le_sqrt h0.le ha).2 hr⟩ + · rintro ⟨h0, hr⟩ + exact ⟨h0, (Real.le_sqrt h0.le ha).1 hr⟩ + +/-- The Bochner integral of the planar standard Gaussian density over the closed disk of +radius `sqrt a` is `1 - exp (-a/2)`: polar coordinates (`integral_comp_polarCoord_symm`), +separation of variables (`setIntegral_prod_mul`), the angular integral `2π` and the radial +integral `(2π)⁻¹ (1 - exp (-a/2))`. -/ +theorem rl_disk_integral (a : ℝ) (ha : 0 ≤ a) : + ∫ p in {p : ℝ × ℝ | p.1 ^ 2 + p.2 ^ 2 ≤ a}, + gaussianPDFReal 0 1 p.1 * gaussianPDFReal 0 1 p.2 = 1 - Real.exp (-a / 2) := by + have h1 : ∫ p in {p : ℝ × ℝ | p.1 ^ 2 + p.2 ^ 2 ≤ a}, + gaussianPDFReal 0 1 p.1 * gaussianPDFReal 0 1 p.2 = + ∫ p, Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ a} + (fun q => gaussianPDFReal 0 1 q.1 * gaussianPDFReal 0 1 q.2) p := + (integral_indicator (rl_measurableSet_disk a)).symm + have h2 : ∫ p, Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ a} + (fun q => gaussianPDFReal 0 1 q.1 * gaussianPDFReal 0 1 q.2) p = + ∫ p in polarCoord.target, p.1 • Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ a} + (fun q => gaussianPDFReal 0 1 q.1 * gaussianPDFReal 0 1 q.2) (polarCoord.symm p) := + (integral_comp_polarCoord_symm _).symm + have h3 : ∫ p in polarCoord.target, + p.1 • Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ a} + (fun q => gaussianPDFReal 0 1 q.1 * gaussianPDFReal 0 1 q.2) (polarCoord.symm p) = + ∫ p in Set.Ioi (0 : ℝ) ×ˢ Set.Ioo (-Real.pi) Real.pi, + Set.indicator {r : ℝ | r ^ 2 ≤ a} + (fun r => r * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2))) p.1 * + (fun _ : ℝ => (1 : ℝ)) p.2 := by + rw [polarCoord_target] + exact integral_congr_ae (Filter.Eventually.of_forall (fun p => rl_polar_integrand a p)) + have h4 := setIntegral_prod_mul (μ := (volume : Measure ℝ)) (ν := (volume : Measure ℝ)) + (Set.indicator {r : ℝ | r ^ 2 ≤ a} + (fun r => r * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2)))) + (fun _ : ℝ => (1 : ℝ)) (Set.Ioi (0 : ℝ)) (Set.Ioo (-Real.pi) Real.pi) + have h5 : ∫ r in Set.Ioi (0 : ℝ), Set.indicator {r : ℝ | r ^ 2 ≤ a} + (fun r => r * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2))) r = + (2 * Real.pi)⁻¹ * (1 - Real.exp (-a / 2)) := by + rw [setIntegral_indicator (rl_measurableSet_radial a), rl_Ioi_inter_eq a ha, + ← intervalIntegral.integral_of_le (Real.sqrt_nonneg a), rl_radial_integral a ha] + rw [h1, h2, h3] + refine h4.trans ?_ + rw [h5, rl_angular_integral] + field_simp + +/-- The measure of the closed disk of radius `sqrt a` under the standard planar Gaussian +`gaussianReal 0 1 ⊗ gaussianReal 0 1` is `1 - exp (-a/2)`: the product of the two +`withDensity` measures is `withDensity` of the product density (`prod_withDensity`), whose +value on the (measurable) disk is the `lintegral` of the density there (`withDensity_apply`), +which is `ENNReal.ofReal` of the Bochner integral `rl_disk_integral` +(`ofReal_integral_eq_lintegral_ofReal`). -/ +theorem rl_disk_measure (a : ℝ) (ha : 0 ≤ a) : + ((gaussianReal 0 1).prod (gaussianReal 0 1)) {p : ℝ × ℝ | p.1 ^ 2 + p.2 ^ 2 ≤ a} = + ENNReal.ofReal (1 - Real.exp (-a / 2)) := by + have hD := rl_measurableSet_disk a + rw [gaussianReal_of_var_ne_zero 0 one_ne_zero, + prod_withDensity (measurable_gaussianPDF 0 1) (measurable_gaussianPDF 0 1), + ← Measure.volume_eq_prod ℝ ℝ, withDensity_apply _ hD, ← rl_disk_integral a ha] + have hint : Integrable (fun p : ℝ × ℝ => gaussianPDFReal 0 1 p.1 * gaussianPDFReal 0 1 p.2) + (volume : Measure (ℝ × ℝ)) := + (integrable_gaussianPDFReal 0 1).mul_prod (integrable_gaussianPDFReal 0 1) + have hnn : 0 ≤ᵐ[(volume : Measure (ℝ × ℝ)).restrict + {p : ℝ × ℝ | p.1 ^ 2 + p.2 ^ 2 ≤ a}] + (fun p : ℝ × ℝ => gaussianPDFReal 0 1 p.1 * gaussianPDFReal 0 1 p.2) := + Filter.Eventually.of_forall + (fun p => mul_nonneg (gaussianPDFReal_nonneg _ _ _) (gaussianPDFReal_nonneg _ _ _)) + rw [ofReal_integral_eq_lintegral_ofReal hint.integrableOn hnn] + refine setLIntegral_congr_fun hD (fun p _ => ?_) + simp only [gaussianPDF] + rw [ENNReal.ofReal_mul (gaussianPDFReal_nonneg _ _ _)] + +/-! ### (c) The interface: measurability and the law of `(x, y)` + +A.e.-measurability of the triple `(s, x, y)` is `tm_aemeasurable_triple` above (the `rl_` copy +was removed at integration). -/ + +/-- The pair `(x, y)` is a.e.-measurable, as the second component of the triple. -/ +theorem rl_aemeasurable_xy (I : ConeMomentInterface) : + @AEMeasurable _ _ _ I.mΩ (fun ω => (I.x ω, I.y ω)) I.P := + (tm_aemeasurable_triple I).snd + +/-- The law of `(x, y)` is `gaussianReal 0 1 ⊗ gaussianReal 0 1`: the second marginal of the +product law `sxy_law` (`AEMeasurable.map_map_of_aemeasurable`, `Measure.map_snd_prod`; the +composite `Prod.snd ∘ (s, x, y)` is `(x, y)`). -/ +theorem rl_law_xy (I : ConeMomentInterface) : + @Measure.map _ _ I.mΩ _ (fun ω => (I.x ω, I.y ω)) I.P = + (gaussianReal 0 1).prod (gaussianReal 0 1) := by + let := I.mΩ + have := I.isProb + have hT := tm_aemeasurable_triple I + have h := AEMeasurable.map_map_of_aemeasurable (measurable_snd.aemeasurable) hT + rw [I.sxy_law, Measure.map_snd_prod, measure_univ, one_smul] at h + exact h.symm + +/-- The probability of `{x^2 + y^2 ≤ a}` under `P` is the Gaussian measure of the closed disk: +the event is measurable, so `Measure.map_apply_of_aemeasurable` applies to the law `rl_law_xy`. -/ +theorem rl_prob_eq_disk_measure (I : ConeMomentInterface) (a : ℝ) : + I.P {ω | I.x ω ^ 2 + I.y ω ^ 2 ≤ a} = + ((gaussianReal 0 1).prod (gaussianReal 0 1)) {p : ℝ × ℝ | p.1 ^ 2 + p.2 ^ 2 ≤ a} := by + let := I.mΩ + have := I.isProb + rw [← rl_law_xy I, + Measure.map_apply_of_aemeasurable (rl_aemeasurable_xy I) (rl_measurableSet_disk a)] + rfl + +end Aux + +/-! ## The `m = 1` cone moment identity for every inhabitant of the interface -/ + +/-- Proves: `UniversalLaw.Spec.Side24.ConeMomentD1Identity` (claim `side24-coefficient`). +Method: transport along the law field `I.A_law` (`Aux.c1_integral_indicator_eq`, applied to +`I.Ω`, `I.mΩ`, `I.P`, `I.isProb`, `I.A`; it uses `integral_map` after a.e.-measurability of +`I.A` from `Aux.c1_aemeasurable_of_map_eq_gaussian`, the Dirac-versus-`NullSingletonClass` +argument), then the negative half of the second moment of +`gaussianReal 0 (8/3)` is `(8/3)/2` (`Aux.c1_setIntegral_sq_Iio`, from `integral_add_compl`, +`integral_Ici_eq_integral_Ioi`, the symmetry `gaussianReal_map_neg` and the variance +`variance_fun_id_gaussianReal`), and `push_cast`/`norm_num` give `4/3 = coneMomentD1`. +Does not establish: the informal identity `D_1 = E[A^2 1{A<0}] = 4/3` of the source as the +source's object (the alignment of `ConeMomentInterface`, its field `A_law` and `coneMomentD1` +with the source's `A ~ N(0, 8/3)` and `D_1` is PENDING_INDEPENDENT_REVIEW); that `D_1` is the +parent's cone moment `D_u` of (15.1) for the reference law; that `ConeMomentInterface` has an +inhabitant, or that its laws are the conditional law `A | V = 0` of the reference field; any +enclosure of the coefficient; `ConeMomentD2Identity`, `TraceMoments` or `RayleighSquareLaw`; +the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the +register, not derived here; nonauthor review or acceptance of +SIDE24-COEFFICIENT-D23-20260924-v1. -/ +theorem coneMomentD1Identity (I : ConeMomentInterface) : + UniversalLaw.Spec.Side24.ConeMomentD1Identity I := by + unfold UniversalLaw.Spec.Side24.ConeMomentD1Identity coneMomentD1 + rw [@Aux.c1_integral_indicator_eq I.Ω I.mΩ I.P I.isProb I.A I.A_law, + Aux.c1_setIntegral_sq_Iio (8 / 3 : NNReal) (by norm_num)] + push_cast + norm_num + +/-! ## The Section 1 trace moments for every inhabitant of the interface -/ + +/-- Proves: `UniversalLaw.Spec.Side24.TraceMoments` (claim `side24-coefficient`). +Method: the law of `I.s` is `gaussianReal 0 (5/3)` (`Aux.tm_law_s`: the first marginal of the +interface's product law `sxy_law`, after `Aux.tm_aemeasurable_triple` derives a.e.-measurability +of the triple from the law by the Dirac-versus-`NullSingletonClass` argument); each of the three +integrals is transported to that law by `integral_map` (`Aux.tm_integral_comp_s`) and computed +there: `∫ t^2 = 5/3` (`Aux.c1_second_moment_gaussianReal`, shared with `ConeMomentD1Identity`, +from `variance_fun_id_gaussianReal`), +`∫ t^4 = 3 (5/3)^2 = 25/3` (`Aux.tm_fourth_moment`, from `iteratedDeriv_mgf_zero` and the +explicit fourth derivative `Aux.tm_iteratedDeriv_four`), and `∫ exp (-t^2/2) = sqrt (3/8)` +(`Aux.tm_exp_moment`, from `integral_gaussianReal_eq_integral_smul` and `integral_gaussian`). +Does not establish: the informal Section 1 display of the source as the source's object (the +alignment of the interface fields `s`, `P`, `sxy_law` and of `gaussianReal` with the source's +`s ~ N(0, 5/3)` is PENDING_INDEPENDENT_REVIEW); that any inhabitant of `ConeMomentInterface` +exists, or that its laws are the conditional law `A | V = 0` of the reference field (the laws are +hypothesis fields of the given `I`, not derived here); the main-side constants +`UniversalLaw.Side24.fourth_moment_of_trace` and `exponential_moment_of_trace`, which +kernel-check `3 (5/3)^2 = 25/3` and `1 + 5/3 = 8/3` on main and are neither imported nor +re-stated here (the same arithmetic is discharged inline by `ring`); `ConeMomentD2Identity`, +`RayleighSquareLaw` or any cone moment; any enclosure of the coefficient; the statement module's +own `Does not claim:` text (the statement packet is not edited here); the disposition of +`side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; +nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1. -/ +theorem traceMoments (I : ConeMomentInterface) : UniversalLaw.Spec.Side24.TraceMoments I := by + unfold UniversalLaw.Spec.Side24.TraceMoments + refine ⟨?_, ?_, ?_⟩ + · have e : ∫ ω, I.s ω ^ 2 ∂I.P = ∫ t, t ^ 2 ∂(gaussianReal 0 (5 / 3 : NNReal)) := + Aux.tm_integral_comp_s I (fun t => t ^ 2) (by fun_prop) + rw [e, Aux.c1_second_moment_gaussianReal] + push_cast + ring + · have e : ∫ ω, I.s ω ^ 4 ∂I.P = ∫ t, t ^ 4 ∂(gaussianReal 0 (5 / 3 : NNReal)) := + Aux.tm_integral_comp_s I (fun t => t ^ 4) (by fun_prop) + rw [e, Aux.tm_fourth_moment] + push_cast + ring + · have e : ∫ ω, Real.exp (-(I.s ω ^ 2) / 2) ∂I.P = + ∫ t, Real.exp (-(t ^ 2) / 2) ∂(gaussianReal 0 (5 / 3 : NNReal)) := + Aux.tm_integral_comp_s I (fun t => Real.exp (-(t ^ 2) / 2)) (by fun_prop) + rw [e, Aux.tm_exp_moment] + +/-! ## The law of `R^2` for every inhabitant of the interface -/ + +/-- Proves: `UniversalLaw.Spec.Side24.RayleighSquareLaw` (claim `side24-coefficient`). +Method: for `a ≥ 0`, `P {x^2 + y^2 ≤ a}` is the measure of the closed disk of radius `sqrt a` +under `gaussianReal 0 1 ⊗ gaussianReal 0 1` (`Aux.rl_prob_eq_disk_measure`, from the law +`Aux.rl_law_xy` of `(x, y)` obtained as the second marginal of the interface's product law +`sxy_law`, after `Aux.tm_aemeasurable_triple`, shared with `TraceMoments`, derives +a.e.-measurability of the triple from the +law by the Dirac-versus-`NullSingletonClass` argument); that disk measure is +`ENNReal.ofReal (1 - exp (-a/2))` (`Aux.rl_disk_measure`: `prod_withDensity`, +`withDensity_apply`, `ofReal_integral_eq_lintegral_ofReal`, then polar coordinates +`integral_comp_polarCoord_symm` and `setIntegral_prod_mul` in `Aux.rl_disk_integral`, with the +angular integral `2π` and the radial integral by the fundamental theorem of calculus); the +right-hand side `∫_0^a exp (-z/2)/2 dz` is the same number `1 - exp (-a/2)` +(`Aux.rl_rhs_integral`), and `ENNReal.toReal_ofReal` closes. +Does not establish: the informal Section 1 display "The variable R^2 has density e^(-z/2)/2" of +the source as the source's object, nor its review and acceptance (the alignment of the interface +fields `x`, `y`, `P`, `sxy_law` and of `gaussianReal` with the source's `x, y ~ N(0, 1)` is +PENDING_INDEPENDENT_REVIEW); from the Prop's own `Does not claim:` line, "the law (statement +only)": the law is proved here only for the formal objects of the Prop, as the distribution +function `P(R^2 ≤ a)` for `a ≥ 0`, not as a density and not for the source's variable; and +"measurability of the event (if the event is not measurable, `P` is its outer measure)": +measurability of the formal event `{x^2 + y^2 ≤ a}` is proved here (`Aux.rl_measurableSet_disk`, +`Measure.map_apply_of_aemeasurable`), so the outer-measure reading does not arise for this +event, but nothing is established about any other event; that any inhabitant of +`ConeMomentInterface` exists, or that its laws are the conditional law `A | V = 0` of the +reference field (the laws are hypothesis fields of the given `I`, not derived here); +`ConeMomentD2Identity` or any cone moment; any enclosure of the coefficient; the statement +module's own text (the statement packet is not edited here); the disposition of +`side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; +nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1. -/ +theorem rayleighSquareLaw (I : ConeMomentInterface) : UniversalLaw.Spec.Side24.RayleighSquareLaw I := by + unfold UniversalLaw.Spec.Side24.RayleighSquareLaw + intro a ha + rw [Aux.rl_prob_eq_disk_measure I a, Aux.rl_disk_measure a ha, Aux.rl_rhs_integral, + ENNReal.toReal_ofReal] + have : Real.exp (-a / 2) ≤ 1 := by + rw [Real.exp_le_one_iff] + linarith + linarith + +end UniversalLaw.Proofs.Side24Interfaces From 48bfcbae7834c2d6f4d3123d85c69d0c5afedfa6 Mon Sep 17 00:00:00 2001 From: Claude Date: Sun, 11 Oct 2026 22:28:40 +0000 Subject: [PATCH 2/3] Statement register and packet docs: record the 13 round-4 proof-packet theorems and the triage of the Props without a theorem (no hosted receipt yet) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Register (specifications/lean_statements_20261010/REGISTER.json): 13 `proved` statement entries for the theorems of SpecificationProofs/{Side24Analysis, Side24Interfaces, RNCluster}.lean committed at 80e80243 (12 under side24-coefficient, 1 under OA-C6-RARE-CLUSTER-20260929-v1), each with the Prop's own anchor and the docstring's "Does not establish" field; the 10 proof-packet kernel_evidence rows regenerated with four receipt tiers (runs 38116277597, 38121713840, 38149888699 / 38149524759, each named with its PR head and actual tested merge commit, corroborated on PR head 12e072d6 by run 38150363063 / merge 86a6e8c8…), and the 13 round-4 theorems marked as having NO hosted receipt yet (the first lean-specification-proofs.yml run on this branch is their first receipt); not_established / next_step of side24-coefficient, math.side24-coefficient and OA-C6-RARE-CLUSTER regenerated; one round-4 triage sentence appended to next_step of the other 27 triaged rows (per-Prop verdicts PROVABLE_WITH_WORK / BLOCKED_MISSING_MATHLIB with the named Mathlib gap / BLOCKED_INTERFACE with its refuting inhabitant / PREDICATE_VOCABULARY). Claim-level labels unchanged on all 171 rows (138 none, 33 specified); statements 339 -> 352, proved entries 142 -> 155, kernel_evidence rows 27 -> 28. Statement packet README.md / SCOPE.md: Declarations and Claims sections re-rendered from the register (72 of 141 Props now have a theorem; the 13 round-4 theorems carry the dagger "no hosted receipt yet" in every table and header). Proof packet README.md / SCOPE.md: inventory 73 theorems over 72 Props; the triage's corrections of the previous informal reasons recorded row by row as corrections of reasons, not statement defects; triage verdicts are scheduling labels with no authority over register labels. Generated by scripts kept outside the repository (update_register8.py, render_docs8.py), after this lane's own adversarial verification of the first rendering (same account, credit 0) found and fixed: BLOCKED_MISSING_MATHLIB items that named present lemmas instead of the absent Kac-Rice / Rice formula; an over-general "superseded" sentence now scoped to the twelve Side24 Props proved in round 4; the omitted antecedent 0 < a of Theorem O; "proved Props" vocabulary; singular grammar; a receipt-free row that inherited other rows' receipt text. Checks at this tree: statement check.py --allow-dirty SOURCE_CHECK_PASS (352 statements, 150 local declarations); test_check.py 97 tests OK in normal and -O modes; proof check.py --allow-dirty SOURCE_CHECK_PASS (73 theorems). No Lean byte changed. Scientific effect NONE; alignment PENDING_INDEPENDENT_REVIEW; organizational-independence credit 0; a proof-packet theorem is evidence about its registered Prop for the formal objects only, never about the informal claim. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01Cz7WZybv8znP64SpPj6sWY --- .../lean_statement_proofs_20261011/README.md | 37 ++- .../lean_statement_proofs_20261011/SCOPE.md | 6 +- .../lean_statements_20261010/README.md | 54 +++-- .../lean_statements_20261010/REGISTER.json | 229 ++++++++++++++---- .../lean_statements_20261010/SCOPE.md | 110 ++++----- 5 files changed, 290 insertions(+), 146 deletions(-) diff --git a/specifications/lean_statement_proofs_20261011/README.md b/specifications/lean_statement_proofs_20261011/README.md index 2ed4c371c..5e5eac856 100644 --- a/specifications/lean_statement_proofs_20261011/README.md +++ b/specifications/lean_statement_proofs_20261011/README.md @@ -43,20 +43,35 @@ register gives it. ## Registered Props without a theorem here, and why -Inventory at this commit: 60 theorems, 59 of the 141 registered `def … : Prop` declarations (`TriangleNeedsThree` -is proved twice). The other 82 fall into the groups below. None is weakened, restated or relabelled: the register +Inventory at this commit: 73 theorems, 72 of the 141 registered `def … : Prop` declarations (`TriangleNeedsThree` +is proved twice). The other 69 fall into the groups below. None is weakened, restated or relabelled: the register keeps every one at `specified`, and a group's reason is a statement about what Mathlib and this packet can reach -today, not about the truth of the Prop. +today, not about the truth of the Prop. The round-4 triage (workflow `wf_a3531fd8-891`, same account, +organizational-independence credit 0) tested each reason this table used to give against the pinned Mathlib; +several were found not to be load-bearing and are corrected below, as corrections of previous informal reasons, +not as statement defects. Its verdicts (PROVABLE_NOW, PROVABLE_WITH_WORK, BLOCKED_MISSING_MATHLIB, +BLOCKED_INTERFACE, PREDICATE_VOCABULARY) are work-scheduling labels and carry no authority over any register +label; a probe is scratch work, never cited as evidence, and where a row reports a probe-level finding it says +so. The 13 theorems of the three round-4 modules (`Side24Analysis`, `Side24Interfaces`, `RNCluster`) have **no +hosted receipt at this commit**: `check.py --allow-dirty` passes in source mode and the round-4 commit records a +clean local `lake build` with every theorem of the new modules on `propext`, `Classical.choice`, `Quot.sound` +only, but a local run is not evidence about any commit; the first `lean-specification-proofs.yml` run on this +branch is their first receipt. | group | Props (names without `UniversalLaw.Spec..`) | why no theorem here | |---|---|---| | P15 parametrised predicates | `IsRealizedFamily`, `IsGood`, `IsKDecomposable`, `IsGeneratorCover`, `IsPaletteFeasible`, `IsMinimalForbidden`, `IsTransversal`, `IsProperDownset` | predicates with free arguments, not closed statements: a theorem of their type would assert them for every argument, which is false (`IsGood F Finset.univ` fails for every realized family). They are the vocabulary of the other Props, not targets. | | P15 full-price note, formerly listed here | — | the hazard interpolation `CoordinatewiseConcavity`, `HazardProductBound` (F6), `HazardPhiBound` (F7), `FullGroundPriceBound` (F8), `FullGroundBudgetIff`, the full budget `LocalPriceBoundFull` (F12), `FullTransformedPriceBudget` (F2) and the enclosure `RhoStarEnclosure` (F3) have theorems in `P15Interpolation` since round 3: the concavity from the one-coordinate split of `μ_p(A)` (affine in `e^{-t_i}`, coefficients `0 ≤ S_in ≤ S_out` for a downset) and `concaveOn_of_deriv2_nonpos`, not from the undisplayed derivative formulas (F4)-(F5), which stay unspecified; (F6)-(F8), (F12) and (F2) as their consequences through `CapacityWorstCase`, `GlobalAssemblyHazard` and `FullBlockCoverIff`; (F3) by `Real.exp_bound` at degrees 26 and 16 compared by `norm_num` against the 20-digit rationals, a different certificate from the source's `full_price.py` procedure, which stays unspecified. Every theorem-shaped Prop of claim `p15-full-price` now has a theorem; the claim stays `specified`. | -| Side24 derivative bounds | `IsUnitTuple` (predicate), `DerivativePairingBound`, `DerivativeBoundAtLeastOne`, `DerivativeBoundAtZero` | explicit `iteratedFDeriv ℝ q (fun x => exp (-‖x‖² / 2))` in arbitrary unit directions: Mathlib has no closed form for the iterated Fréchet derivatives of a Gaussian (the Hermite expansion behind the pairing sums); the arithmetic of the pairing sums is kernel-proved on main (`UniversalLaw.Side24.pairing_sum_six`, …). | -| Side24 lattice sums | `ThetaSummable`, `LatticeImageSumBound`, `ImageBound` | summability of `n ↦ exp (-‖z + 24 n‖² / 2)` over `Fin d → ℤ` and the explicit tail `≤ 1458 e^{-288}`: Gaussian-decay summability over `ℤ^d` and a shell count; Mathlib's Jacobi-theta results are one-dimensional. | -| Side24 covariance and density | `NegDefCone` (predicate), `ReferenceOddAndPinBlocks`, `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound`, `DensityComparison`, `IntegrandRatioBounds`, `ReferencePinDensityProduct` | built on the derivative contractions above (`jointCov`, `gradCov`, `pinCov`, `derivContraction`); the matrix arithmetic is kernel-proved on main, the evaluation of the Fréchet derivatives is not available. | -| Side24 interface statements | `ConeMomentD1Identity`, `ConeMomentD2Identity`, `TraceMoments`, `RayleighSquareLaw` (over `ConeMomentInterface`); `CoefficientEnclosure`, `ReferenceComparison` (over `PeriodicCoefficientInterface`) | quantify over an interface whose fields are given, not constructed: the Gaussian moment identities need the conditional law `A_u \| V_u = 0` built in Mathlib's Gaussian measure theory; the enclosure needs the numerical pipeline. `ConeMomentD2Algebra`, `ElementaryConeIntegral` and `ReferenceCoefficientIdentity` (proved) are the parts that need neither. | -| RN beyond (N1)-(N5), Lifetime, Field | the 22 RN Props other than `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5`; all 21 Lifetime and 10 Field Props | statements about the Gaussian field through interface structures (`PinnedLawInterface`, `GaussianFieldInterface`, …): the D5 window bounds, the dimension lift, the Palm and Fourier routes, the remote-collision and annulus-bridge bounds, the parent Theorems A/B/C, the planar elder gap, the remainder (R1), and the critical-point predicates. These are the programme's open research claims: a proof would be the research result itself, not a formalisation of an existing argument, and the interfaces do not carry the Kac–Rice data the informal arguments use. | +| Side24 derivative bounds | `IsUnitTuple` (predicate) | `IsUnitTuple d q u` (`∀ i, ‖u i‖ = 1`) has free arguments and is consumed only as a hypothesis of the derivative Props; a theorem of its type would assert it of every tuple, which is false (`u = 0`). Vocabulary, not a target. | +| Side24 derivative bounds, formerly listed here | — | `DerivativePairingBound`, `DerivativeBoundAtLeastOne` and `DerivativeBoundAtZero` have theorems in `Side24Analysis` since round 4. The previous reason ("Mathlib has no closed form for the iterated Fréchet derivatives of a Gaussian") was not load-bearing and is corrected here, as a correction of an informal reason, not a statement defect: no closed form is needed. The operator norms `b_q = ‖iteratedFDeriv ℝ q (Kinf d) x‖` satisfy the Hermite-type recursion `b_{q+2} ≤ ‖x‖ b_{q+1} + (q+1) b_q`, from `norm_iteratedFDeriv_fderiv`, the explicit derivative `fderiv ℝ (Kinf d) y = Kinf d y • (-(innerSL ℝ y))` and the Leibniz bound `norm_iteratedFDeriv_smul_le` (the linear factor's derivatives of order `≥ 2` vanish); with `b_0 = Kinf d x`, `b_1 = Kinf d x ‖x‖` the orders `q ≤ 6` match the unfolded `pairingSum q ‖x‖`, and the contraction at a unit tuple is dominated by the operator norm (`ContinuousMultilinearMap.le_opNorm`). The two corollaries follow from `pairingSum q r ≤ 76 r⁶` for `1 ≤ r` and `pairingSum q 0 ≤ 15` (values `1, 0, 1, 0, 3, 0, 15`). The pairing-sum arithmetic kernel-proved on main (`UniversalLaw.Side24.pairing_sum_six`, …) is neither cited nor relied on; nothing is stated for `q > 6`. | +| Side24 lattice sums | `ImageBound` | BLOCKED_MISSING_MATHLIB (weeks); not a statement defect (`K24` is `C^∞`, so Lean's total `iteratedFDeriv` is the genuine derivative, and `e^{-288} < 10^{-125}` leaves the strict inequality room). The exact missing piece is the exchange of `iteratedFDeriv` with the `tsum` over `Fin d → ℤ` near `0`: Mathlib `d13f23b7` has only the global `iteratedFDeriv_tsum`, `iteratedFDeriv_tsum_apply` and `contDiff_tsum_of_eventually` (`Analysis/Calculus/SmoothSeries.lean`), which need a summable bound `‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i` for every `x`, which the translates `x ↦ Kinf d (x + 24 n)` do not have (their supremum over `x` does not depend on `n`), and its only local (open-set) result, `hasFDerivAt_tsum_of_isPreconnected`, is first order. A cutoff workaround (`ContDiffBump` times each term, transfer back by `Filter.EventuallyEq.iteratedFDeriv`, the normalisation `1 / thetaSum24 d 0` by `iteratedFDeriv_const_smul_apply`) is a multi-week development on top of the derivative and lattice theorems now in `Side24Analysis`. | +| Side24 lattice sums, formerly listed here | — | `ThetaSummable` (for every `d`) and `LatticeImageSumBound` have theorems in `Side24Analysis` since round 4. The previous reason ("Gaussian-decay summability over `ℤ^d` and a shell count; Mathlib's Jacobi-theta results are one-dimensional") was not load-bearing and is corrected here: the one-dimensional `HurwitzKernelBounds.summable_f_int` lifts to `Fin d → ℤ` through a `Fintype.piFinset` product bound (`Finset.prod_univ_sum`, `summable_of_sum_le`), with the dominating family `exp(‖z‖²/2) ∏ i, exp(-144 n_i²)` for `ThetaSummable`; `LatticeImageSumBound` by the same lift after the power-mean bound `(∑ n_i²)³ ≤ 9 ∑ n_i⁶` for `d ≤ 3`, the one-dimensional sums bounded by `2` and `4 e^{-288}` through a geometric tail from the registered `SuccessiveTermRatio` (`432 e^{-288} ≤ 1458 e^{-288}`), a different route from the source's shell count `729 ∑ j⁹ e^{-288 j²}`, which stays unspecified. | +| Side24 covariance and density | `NegDefCone` (predicate), `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound` | `NegDefCone q` (`(-(transverseMatrix …)).PosDef`) has a free argument and is the set-builder predicate inside `coneIntegrand`; a theorem of its type is false at `q = (0, 0, 0)`. Vocabulary, not a target. `CovarianceOrdering` and `EntrywiseCovarianceBound`: BLOCKED_MISSING_MATHLIB (weeks), both true as stated. Two of the four pieces the triage listed as missing are theorems since round 4 (`latticeImageSumBound` for the `ℤ^d` tail `∑_{n ≠ 0} \|n\|⁶ e^{-288 \|n\|²} ≤ 1458 e^{-288}`; `derivativePairingBound` and its corollaries for the constants `76` and `15` that the entrywise constant `2 imageConstE` needs). The exact pieces still missing from Mathlib `d13f23b7` are (i) local term-wise iterated differentiation of `thetaSum24` at `0` with bounds summable only near `0` (the same gap as `ImageBound`: `iteratedFDeriv_tsum` needs bounds uniform over all `x`, the only local result `hasFDerivAt_tsum_of_isPreconnected` is first order, and the hypothesis `ThetaSummable d` gives pointwise summability only) and, for the `IsHermitian` half of `PosSemidef` in `CovarianceOrdering`, (ii) symmetry of `iteratedFDeriv` under permutation of the two direction blocks beyond order two (Mathlib has `isSymmSndFDerivAt_iff_iteratedFDeriv` only). The display-(3) ordering would even survive the crude bound `norm_iteratedFDeriv_comp_le` (`60 E' < epsilon` with a factor of about 8 to spare). `ReferenceCovarianceLowerBound`: PROVABLE_WITH_WORK (weeks), no missing theory: the entries of `Cref` from the contraction values already in `Side24Analysis` (orders 2, 4, 6) and the vanishing odd orders at `0` (order 5 needs one more general-direction level of the peeling), then an explicit sum-of-squares certificate for the block matrix (odd block `[[1, -3], [-3, 15]]`, `I_{d-1}`, Hessian block with eigenvalues `4, 2, 2` resp. `5, 2, …, 2`) minus `I/3` through `Matrix.posSemidef_iff_dotProduct_mulVec`. | +| Side24 covariance and density, formerly listed here | — | `ReferenceOddAndPinBlocks`, `ReferencePinDensityProduct`, `IntegrandRatioBounds` and `DensityComparison` have theorems in `Side24Analysis` since round 4. The previous reason ("built on the derivative contractions …; the evaluation of the Fréchet derivatives is not available") was not load-bearing and is corrected here: the values of `iteratedFDeriv ℝ n (Kinf d) 0` in frame directions through order six are computable with existing API, peeling one direction at a time (`iteratedFDeriv_succ_apply_right` + `iteratedFDeriv_clm_apply_const_apply`) with explicit `HasFDerivAt` closed forms of the intermediate functions, giving `-⟪a, b⟫`, the three pairings and `-15 ⟪u, u⟫³`, hence the five blocks (`Cov(G) = I`, `Var(t) = 15`, `Cov(t, G) = (-3, 0, …)`, `pinCov = diagonal (3, 1, …, 1)`, `τ² = 15 - 9 = 6`) in an orthonormal frame and the pin density product `(2π)^{-d}/√3`; `IntegrandRatioBounds` was misgrouped: it is pure real arithmetic in `epsilon`, `exponentA`, `exponentB` (Bernoulli's inequality, exponent monotonicity and two exact rational inequalities decided by `norm_num`) with no derivative input; `DensityComparison` is matrix analysis for positive-definite real matrices (Loewner determinant monotonicity and inverse antitonicity through `CFC.sqrt`, developed in the module's `Aux`), with no derivative input either. The matrix arithmetic kernel-proved on main (`UniversalLaw.Side24.conditional_third_derivative_variance`, …) is neither cited nor relied on. | +| Side24 interface statements | `ConeMomentD2Identity` (over `ConeMomentInterface`); `CoefficientEnclosure`, `ReferenceComparison` (over `PeriodicCoefficientInterface`) | `ConeMomentD2Identity`: PROVABLE_WITH_WORK (weeks), no missing theory: transport along `sxy_law` as for the three proved interface Props, the cone identification `NegDefCone (s, x, y) ↔ s < 0 ∧ x² + y² < s²` (both directions elaborate in a probe), Fubini with the integrable bound `(s² + x² + y²)²`, the inner truncated polar integral reducing to the proved `elementaryConeIntegral`, and the outer `gaussianReal 0 (5/3)` moments already in `Side24Interfaces`; several hundred lines. `CoefficientEnclosure`, `ReferenceComparison`: BLOCKED_INTERFACE; they are predicates on the interface object, as their own `Interface:` lines say: `PeriodicCoefficientInterface` takes `coneMoment` as a GIVEN field, tied to `c24` only through `eq_15_2`, so doubling `coneMoment` and `c24` turns any inhabitant into another, and no inhabitant can satisfy both `d2Lower < c24 2 < d2Upper` and its doubled version (`2 d2Lower > d2Upper`), nor both `\|q - 1\| < 10⁻¹⁰⁶` and `\|2q - 1\| < 10⁻¹⁰⁶`. Hence `(∀ I, CoefficientEnclosure I) → IsEmpty PeriodicCoefficientInterface`, and likewise for `ReferenceComparison`: a theorem of either type would exist only if the interface is empty, which nobody claims. Probe-level finding (elaborated without `sorry` in scratch work), not a receipt and not evidence. What would make the Props determinate is outside the statement packet and outside Mathlib: the conditional Gaussian law `A_u \| V_u = 0` of the `K24` field pinning `coneMoment` (closest Mathlib: `multivariateGaussian`, `condDistrib`), plus the numerical enclosure pipeline. The register's `specified` label is the right one; nothing in this packet encloses the coefficient `c_{d,24}`. | +| Side24 interface statements, formerly listed here | — | `ConeMomentD1Identity`, `TraceMoments` and `RayleighSquareLaw` have theorems in `Side24Interfaces` since round 4, for a GIVEN `I : ConeMomentInterface`. The previous reason ("the Gaussian moment identities need the conditional law `A_u \| V_u = 0` built in Mathlib's Gaussian measure theory") was not load-bearing for these three and is corrected here: they follow from the interface's law fields alone (`A_law : Measure.map A P = gaussianReal 0 (8/3)`, `sxy_law` as a product of `gaussianReal`s), the laws being hypothesis fields of `I`, not constructed. Route: a.e.-measurability of `A` and of the triple `(s, x, y)` from the law fields by the Dirac-versus-`NullSingletonClass` argument (`Measure.map_of_not_aemeasurable_of_ne_zero` against `nullSingletonClass_gaussianReal`), `integral_map` to the Gaussian law, then: for `D_1` the negative half of the second moment by `gaussianReal_map_neg` symmetry and `variance_fun_id_gaussianReal` (`(8/3)/2 = 4/3`); for the trace moments `∫ t² = 5/3`, `∫ t⁴ = 25/3` (`iteratedDeriv_mgf_zero` with the explicit fourth derivative of the mgf) and `∫ exp(-t²/2) = √(3/8)` (`integral_gaussianReal_eq_integral_smul`, `integral_gaussian`); for the Rayleigh law the disk measure under `gaussianReal 0 1 ⊗ gaussianReal 0 1` by `prod_withDensity`, polar coordinates (`integral_comp_polarCoord_symm`) and the fundamental theorem of calculus, equal to `∫_0^a exp(-z/2)/2 dz = 1 - exp(-a/2)`, measurability of the event proved so the outer-measure reading does not arise. Not established: that `ConeMomentInterface` has an inhabitant, or that its laws are the conditional law `A \| V = 0` of the reference field; `ConeMomentD2Identity`; any cone moment of the `K24` field. `ConeMomentD2Algebra`, `ElementaryConeIntegral` and `ReferenceCoefficientIdentity` (proved earlier) need no interface. | +| RN beyond (N1)-(N5) and (O), Lifetime, Field | the 21 RN Props other than `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5`, `RareClusterPoissonObstructionO`; all 21 Lifetime and 10 Field Props | 52 Props in three classes, per the round-4 triage. (a) 22 BLOCKED_INTERFACE: the 19 RN Props over `PinnedLawInterface` (the D5 window bounds D5-a to D5-f, the annulus bridge, the four dimension-lift theorems, Theorem A(2)/A(3) of the fixed remote window, the fixed-annulus candidate, the Fourier cutoff (F4), the Palm route (Q, Theta), Corollary D and the OPEN witness-collision target) and the 3 Lifetime Props over `PairLawInterface` (`ParentUniformSelectionA`, `PlanarElderLowerE2`, `PlanarTwoSidedRateE4`). The previous reason here ("a proof would be the research result itself") was not the sharp one and is corrected: the interfaces constrain the sample paths and the law only at the two pins (`Q_pinned`, `W_typed`, `Z_pos`, `QW_eq`) or only through `p_nonneg`/`p_le_one`, and admit concrete falsifying inhabitants: a Dirac (or two-point) law at a smooth periodic pinned path with one or two extra nondegenerate critical points placed where the Prop counts them (a null `E`, the midpoint, a shell, a remote point), the free kernel field `Λ := 0` for Theorem A(2), or the constant laws `p ≡ 0` / `p ≡ 1`, so no theorem of the universal closure can exist for any Mathlib. This is under-determination of the interface (the Kac–Rice representation and the Gaussian regression law are not transcribed, as the interface docstrings say), not a statement defect; the reduction steps and the `p ≡ 0`/`p ≡ 1` refutations elaborate in probes (probe-level, not a receipt; the path constructions are not formalised). (b) 4 Lifetime Props BLOCKED_MISSING_MATHLIB (weeks): `ParentCompactWindowDensityB`, `ParentUnrestrictedLeadingDensityC`, `RemainderBoundedR1`, `PlanarDensityGapE11`, whose truth value the field interface's `gaussian` field does determine, but whose conclusions need density versions (`IsLifetimeDensityVersion`: a Kac–Rice representation of the expected pair count with lifetime in `A`, absolute continuity in `ℓ`), a.s. Morse genericity and the source's asymptotics, none of which Mathlib `d13f23b7` has; in `PlanarDensityGapE11` the hypothesis `ParentUniformSelectionA … PL …` is inert (it constrains only `PL.p`, which the conclusion does not mention, and `p ≡ 1` satisfies it), a shape observation for the register. (c) 26 predicates with free arguments (`IsCriticalPoint`, `HessianHasInertia` in RN; the 14 of Lifetime and the 10 of Field: `HessianHasInertia` and the `Is…` family in each): vocabulary of the other Props, not targets. Across the whole table the triage's counts for the 69 Props without a theorem are 28 predicates (these 26 plus `IsUnitTuple`, `NegDefCone`), 24 BLOCKED_INTERFACE (these 22 plus `CoefficientEnclosure`, `ReferenceComparison`), 7 BLOCKED_MISSING_MATHLIB (these 4 plus `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`), 2 PROVABLE_WITH_WORK (`ReferenceCovarianceLowerBound`, `ConeMomentD2Identity`) and the 8 P15 predicates it did not re-examine. | +| RN rare cluster (Theorem O), formerly listed here | — | `RareClusterPoissonObstructionO` has a theorem in `RNCluster` since round 4. The previous reason ("a proof would be the research result itself, and the interfaces do not carry the Kac–Rice data") was not load-bearing for this Prop and is corrected here: the Prop is the abstract implication (H1)+(H2) ⇒ (3.1) for a GIVEN family of pmfs `μ r : ℕ → ℝ`, exactly what the source names as its only analytic inputs, and it ties `μ` to no field. Route: `2a ≤ A` is derived from the antecedents at `r0` (as the source remarks; never a hypothesis); the upper half with the witness `lam = 0`, where `pmfTV (μ r) (poissonPMF 0) = 1 - μ r 0 ≤ ∑ n μ r n ≤ A r³`; the lower half by testing the tail `{n ≥ 2}` when `lam ≤ r²` (Poisson tail `≤ lam²` from `Real.add_one_le_exp`) and the tail `{n ≥ 1}` when `lam > r²` (`x/2 ≤ 1 - exp(-x)` on `[0, 1/2]` from `Real.abs_exp_sub_one_sub_id_le`), with the Poisson total mass from `ProbabilityTheory.hasSum_one_poissonMeasure`, at `rStar = min r0 (min (a/2) (min (1/(a + 2A)) (1/2)))`. Not established: that (H1) or (H2) hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta; neither is revalidated here), the mean-matched statement `lambda = mu`, Theorems C and S, the source's infimum form verbatim (the existential with witness `lam = 0` is what is proved). The claim `OA-C6-RARE-CLUSTER-20260929-v1` keeps its register labels. | The register (`../lean_statements_20261010/REGISTER.json`) says the same per claim row in its `not_established` and `next_step` fields; this table is the packet-side summary and decides nothing. @@ -68,8 +83,10 @@ README.md SCOPE.md check.py test_check.py replay.sh lakefile.toml lean-toolchain lake-manifest.json SpecificationProofs.lean SpecificationProofs/.lean namespace UniversalLaw.Proofs.; P15 is split by theme into P15Elementary, P15Structure, P15Hazard, P15Covers, P15Budget, - P15Interpolation (in import order), Side24Elementary and RNProbability - are one each + P15Interpolation; Side24 into Side24Elementary, Side24Analysis, + Side24Interfaces; RN into RNProbability, RNCluster (each list in + import order; the root imports the eight modules of rounds 1-3 and + then Side24Analysis, Side24Interfaces, RNCluster) ``` Rules for every module (enforced by `check.py`, see its module docstring): diff --git a/specifications/lean_statement_proofs_20261011/SCOPE.md b/specifications/lean_statement_proofs_20261011/SCOPE.md index 523a53d27..fbe2f3c17 100644 --- a/specifications/lean_statement_proofs_20261011/SCOPE.md +++ b/specifications/lean_statement_proofs_20261011/SCOPE.md @@ -31,9 +31,9 @@ statement proved here: **PENDING_INDEPENDENT_REVIEW**. Organizational-independen | family | the theorems establish | not established | |---|---|---| | P15 (`Specifications.P15`, modules P15Elementary, P15Structure, P15Hazard, P15Covers, P15Budget, P15Interpolation) | exact finite arithmetic and finite combinatorics of the registered Props: generator prices, the demand-one refutation data, `ρ* < 6/7` and its certificate, the 816-benchmark counts, palette identities, cover (816 and not 815) and union bound (P11); Proposition P1; both halves of Theorem P2 as the Props state them (the cover criterion (P7) and the hazard bound (P8) for prices `0 ≤ c_v ≤ p_v`) with the local statements (P2)-(P5); the restricted transformed-price budget (T1) with (T4), (T5) and the six-block transformed range; (F9), (F13), the sharpness clause of `ρ*` for the triple block; the hazard interpolation (F5)-(F8) (separate coordinatewise concavity of `t ↦ -log μ_{p(t)}(A)` for a proper downset, the product bound, the `φ` bound, the full-ground price bound and its exactness `iff`), the local bound (F12), Theorem F as `FullTransformedPriceBudget` states it ((F2): for every `p ∈ [0,1]^X`, every `0 ≤ c_v ≤ φ(p_v)` and `K ≥ K_H(d)`, `coverCost ≤ cappedHazard ρ* μ`) and the 20-digit enclosure (F3) of `ρ*`; all for the formal `RealizedData` record, downset, `coverCost` (an `sInf`) and `mu` | the informal P15 claims (P15-PRICE-BOUNDARY, P15-REALIZED-COVERS, FULL-PRICE, PRICE-BUDGET), their dispositions, Theorem F itself: the hazard interpolation (F4)-(F8), the block bound (F12) and the budget (F2), the 20-digit `ρ*` enclosure (F3); attainment of the `sInf`; alignment of `RealizedData`/`IsRealizedFamily`/`downset`/`coverCost`/`mu` with the informal objects (PENDING_INDEPENDENT_REVIEW), any prize statement | -| Side24 (`Specifications.Side24`) | elementary exponential bounds, a term-ratio bound, the D2 cone-moment algebra, the reference-coefficient identity, the elementary cone integral, as the Props state them | the coefficient enclosure `c_{d,24}` (REVIEWED_SCOPED is transcribed, not derived), the Gaussian-moment and derivative-bound statements, the lattice image sum over a `tsum`, the interface-level statements | -| RN (`Specifications.RN`) | the elementary probability identities and bounds N1–N5 as the Props state them, for the interface objects they quantify over | the Radon–Nikodym uniformity lemma `D3-LEMMA-RN-UNIF` (OPEN), the five validity premises of Theorem D1 v2.2(2) (OPEN), any identification of the interface with the actual field | -| Lifetime, Field | nothing is expected to be provable (interface-level statements); no theorem is inventoried | everything | +| Side24 (`Specifications.Side24`, modules Side24Elementary, Side24Analysis, Side24Interfaces) | elementary exponential bounds, a term-ratio bound, the D2 cone-moment algebra, the reference-coefficient identity, the elementary cone integral; the summability of the theta family `n ↦ exp(-‖z + 24 n‖²/2)` over `Fin d → ℤ` for every `d` and `z`; the Section 2 derivative bounds for Mathlib's `iteratedFDeriv ℝ q (Kinf d)` at unit tuples and `q ≤ 6` (the pairing bound `exp(-‖x‖²/2) pairingSum q ‖x‖`, `76 ‖x‖⁶ exp(-‖x‖²/2)` at `‖x‖ ≥ 1`, `15` at `0`); the lattice image sum (summable and `≤ 1458 e^{-288}` for `d ≤ 3`); the Section 1 reference blocks in an orthonormal frame (`Cov(G) = I`, `Var(t) = 15`, `Cov(t, G) = (-3, 0, …)`, `pinCov = diagonal (3, 1, …, 1)`, `τ² = 6`) and the pin density product `(2π)^{-d}/√3` for the formal `gaussDensity`; the Section 4 integrand ratio bounds (arithmetic in `epsilon`, `exponentA`, `exponentB`) and the density comparison for positive-definite matrices with `(1 - epsilon) • C0 ≤ C ≤ (1 + epsilon) • C0` in the Loewner order; and, for a GIVEN `I : ConeMomentInterface`, `D_1 = 4/3`, the trace moments `5/3`, `25/3`, `√(3/8)` and the Rayleigh law of `x² + y²` as a distribution function, from the interface's law fields alone; all as the Props state them, for the formal objects (`E d`, `Kinf`, `latticeVec`, `derivContraction`, `gaussDensity`, the interface structures) | the coefficient enclosure `c_{d,24}` (REVIEWED_SCOPED is transcribed, not derived): nothing here encloses the coefficient, and `CoefficientEnclosure` / `ReferenceComparison` are predicates on `PeriodicCoefficientInterface`, which takes the cone moment as a GIVEN field; `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound` (no local term-wise iterated differentiation of the lattice sum at Mathlib `d13f23b7`), `ReferenceCovarianceLowerBound`, `ConeMomentD2Identity`; that `ConeMomentInterface` or `PeriodicCoefficientInterface` has an inhabitant, or that their law fields are the conditional law `A_u \| V_u = 0` of any field (no field is constructed); that any `gradCov`/`pinCov` matrix is the covariance of a random object; the source's shell count and closed-form derivative displays, which stay unspecified; alignment of the formal objects with the source's (PENDING_INDEPENDENT_REVIEW) | +| RN (`Specifications.RN`, modules RNProbability, RNCluster) | the elementary probability identities and bounds N1–N5 as the Props state them, for the interface objects they quantify over; Theorem O's obstruction as `RareClusterPoissonObstructionO` states it, for a GIVEN family of pmfs `μ r : ℕ → ℝ` satisfying (H1) `∑ n μ r n ≤ A r³` and (H2) `a r³ ≤ 1 - μ r 0 - μ r 1` on `(0, r0]`: for all small `r`, every Poisson law is at `pmfTV` distance at least `(a/2) r³` from `μ r`, and one (`lam = 0`) is within `A r³`, with `2a ≤ A` derived, not assumed | the Radon–Nikodym uniformity lemma `D3-LEMMA-RN-UNIF` (OPEN), the five validity premises of Theorem D1 v2.2(2) (OPEN), any identification of the interface with the actual field; that (H1) or (H2) hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta; neither is revalidated here, and the Prop ties `μ` to no field), the mean-matched statement `lambda = mu`, Theorems C and S, the source's infimum form verbatim (the existential with witness `lam = 0` is what is proved); the 19 remaining RN Props over `PinnedLawInterface` (the D5 window bounds, the dimension lift, Theorem A(2)/A(3), the Palm and Fourier routes, Corollary D, the OPEN witness-collision target), which the interface under-determines (README table) | +| Lifetime, Field | no theorem is inventoried; the round-4 triage (README table) classifies the 31 Props as 24 predicates (vocabulary), 3 interface-blocked (falsified by the constant laws `p ≡ 0` / `p ≡ 1` of `PairLawInterface`) and 4 Lifetime Props blocked on Mathlib developments (density versions, Kac–Rice), not on the interface | everything | The three lines of each theorem's docstring (`Proves:`, `Method:`, `Does not establish:`) are the per-theorem version of this table and are required by the gate. diff --git a/specifications/lean_statements_20261010/README.md b/specifications/lean_statements_20261010/README.md index 6e9a1e3b0..bb5f487de 100644 --- a/specifications/lean_statements_20261010/README.md +++ b/specifications/lean_statements_20261010/README.md @@ -4,10 +4,11 @@ Scientific effect: **NONE**. Organizational-independence credit of this packet: Claude, delegated AI work). Alignment status of every statement: `PENDING_INDEPENDENT_REVIEW`. Formal progress of every declaration here: `specified`. The sibling proof packet `specifications/lean_statement_proofs_20261011` holds theorems whose type is exactly one of these -Props (59 of the 141 Props, 60 theorems); the register lists them under the claim rows as +Props (72 of the 141 Props, 73 theorems); the register lists them under the claim rows as statement entries of kind `theorem` with label `proved` (52 of them kernel-checked by the hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…, same tree as the head; the 8 of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194…), by run 38149888699: -PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96…, the head of PR #422, same tree as the head), +PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96…, the head of PR #422, same tree as the head; the 13 added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c8…), +have no hosted receipt yet and are marked † in the tables), which changes no label of this packet and no claim-level label. No claim, premise, obligation or disposition changes because this packet exists, builds, or is green in CI, or because a Prop of it is proved. @@ -51,11 +52,11 @@ ladder; it is `specified`. ## Declarations -150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified` in this package; the 123 auxiliary definitions and abbreviations of the modules are inventory only and are not register statements. Names are given without the module namespace `UniversalLaw.Spec..`; the source column is the pinned source named by the docstring's `Source:` line. Anchors and "Not established" cells are copied verbatim from the Lean docstrings (`Anchor:` and `Does not claim:` lines) through REGISTER.json. The last column names the theorems of the sibling proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly the declaration (namespace `UniversalLaw.Proofs..`): 59 of the 141 Props have one (60 theorems; `TriangleNeedsThree` is proved twice), each listed in the register under the claim rows as a statement entry of kind `theorem` with label `proved` (the label of a committed proof package; `kernel-checked` appears only in a run receipt: the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, so each receipt records as checked_commit that merge and not the head: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; each merge has the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads). That label belongs to the theorem, not to the declaration here, which stays `specified`, and not to any claim row; a kernel-checked theorem is evidence about its Prop only, and alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. +150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified` in this package; the 123 auxiliary definitions and abbreviations of the modules are inventory only and are not register statements. Names are given without the module namespace `UniversalLaw.Spec..`; the source column is the pinned source named by the docstring's `Source:` line. Anchors and "Not established" cells are copied verbatim from the Lean docstrings (`Anchor:` and `Does not claim:` lines) through REGISTER.json. The last column names the theorems of the sibling proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly the declaration (namespace `UniversalLaw.Proofs..`): 72 of the 141 Props have one (73 theorems; `TriangleNeedsThree` is proved twice), each listed in the register under the claim rows as a statement entry of kind `theorem` with label `proved` (the label of a committed proof package; `kernel-checked` appears only in a run receipt: the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, so each receipt records as checked_commit that merge and not the head: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; each merge has the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems marked † (`Side24Analysis`, `Side24Interfaces` and `RNCluster`) were added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), and have no hosted receipt yet: the first hosted `lean-specification-proofs.yml` run on the round-4 branch is their first receipt). That label belongs to the theorem, not to the declaration here, which stays `specified`, and not to any claim row; a kernel-checked theorem is evidence about its Prop only, and alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. ### `Specifications/Field.lean` (12 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | | `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | @@ -72,7 +73,7 @@ ladder; it is `specified`. ### `Specifications/Lifetime.lean` (24 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | | `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | @@ -101,32 +102,32 @@ ladder; it is `specified`. ### `Specifications/Side24.lean` (28 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---| -| `ThetaSummable` | def_prop | `side24-proof` | `side24-coefficient` | — | +| `ThetaSummable` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.thetaSummable†` | | `IsUnitTuple` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `DerivativePairingBound` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `DerivativeBoundAtZero` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `LatticeImageSumBound` | def_prop | `side24-proof` | `side24-coefficient` | — | +| `DerivativePairingBound` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.derivativePairingBound†` | +| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.derivativeBoundAtLeastOne†` | +| `DerivativeBoundAtZero` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.derivativeBoundAtZero†` | +| `LatticeImageSumBound` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.latticeImageSumBound†` | | `SuccessiveTermRatio` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.successiveTermRatio` | | `ExponentialBounds` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.exponentialBounds` | | `ImageBound` | def_prop | `side24-proof` | `side24-coefficient` | — | | `CovarianceOrdering` | def_prop | `side24-proof` | `side24-coefficient` | — | | `EntrywiseCovarianceBound` | def_prop | `side24-proof` | `side24-coefficient` | — | | `ReferenceCovarianceLowerBound` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `DensityComparison` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `IntegrandRatioBounds` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `ReferencePinDensityProduct` | def_prop | `side24-proof` | `side24-coefficient` | — | +| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.referenceOddAndPinBlocks†` | +| `DensityComparison` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.densityComparison†` | +| `IntegrandRatioBounds` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.integrandRatioBounds†` | +| `ReferencePinDensityProduct` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.referencePinDensityProduct†` | | `ReferenceCoefficientIdentity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.referenceCoefficientIdentity` | | `ConeMomentInterface` | structure | `side24-proof` | `side24-coefficient` | — | | `NegDefCone` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `ConeMomentD1Identity` | def_prop | `side24-proof` | `side24-coefficient` | — | +| `ConeMomentD1Identity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.coneMomentD1Identity†` | | `ConeMomentD2Identity` | def_prop | `side24-proof` | `side24-coefficient` | — | | `ConeMomentD2Algebra` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.coneMomentD2Algebra` | -| `TraceMoments` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `RayleighSquareLaw` | def_prop | `side24-proof` | `side24-coefficient` | — | +| `TraceMoments` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.traceMoments†` | +| `RayleighSquareLaw` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.rayleighSquareLaw†` | | `ElementaryConeIntegral` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.elementaryConeIntegral` | | `PeriodicCoefficientInterface` | structure | `side24-proof` | `side24-coefficient`, `math.side24-coefficient` | — | | `CoefficientEnclosure` | def_prop | `side24-proof` | `side24-coefficient`, `math.side24-coefficient` | — | @@ -134,7 +135,7 @@ ladder; it is `specified`. ### `Specifications/RN.lean` (28 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---| | `IsCriticalPoint` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | — | | `HessianHasInertia` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | — | @@ -161,13 +162,13 @@ ladder; it is `specified`. | `FourierCutoffPlanarSecondFactorialF4` | def_prop | `c6-fourier-cutoff` | `OA-C6-FOURIER-20260929-v1` | — | | `PalmRouteFactorialMomentsQ` | def_prop | `c6-palm-route` | `CL-C6-PALM-20260929-v1` | — | | `PalmRouteSecondFactorialTheta` | def_prop | `c6-palm-route` | `CL-C6-PALM-20260929-v1` | — | -| `RareClusterPoissonObstructionO` | def_prop | `c6-rare-cluster` | `OA-C6-RARE-CLUSTER-20260929-v1` | — | +| `RareClusterPoissonObstructionO` | def_prop | `c6-rare-cluster` | `OA-C6-RARE-CLUSTER-20260929-v1` | `RNCluster.rareClusterPoissonObstructionO†` | | `RemoteCollisionSecondFactorialD` | def_prop | `remote-collision` | `math.rn-region.witness-collision`, `CL-D5-REMOTE-COLLISION-20260928-v1` | — | | `WitnessCollisionNearPinPairsTarget` | def_prop | `d5-reconciliation` | `math.rn-region.witness-collision` | — | ### `Specifications/P15.lean` (58 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---| | `RealizedData` | structure | `p15-realized-covers` | `p15-realized-covers` | — | | `IsRealizedFamily` | def_prop | `p15-realized-covers` | `p15-realized-covers` | — | @@ -230,11 +231,11 @@ ladder; it is `specified`. ## Claims -171 claim rows: 10 `landing_claim`, 33 `graph_node`, 2 `open_obligation`, 95 `proof_index_result`, 7 `existing_lean_package`, 12 `main_experiment_proof`, 12 `exact_certificate`. Claim-level formal progress: 33 `specified` (a `def … : Prop` stating the claim itself is registered), 138 `none`; no research claim is `proved` or `kernel-checked`. 197 statement entries point at the 150 declarations of this package (all `specified`). 142 statement entries carry the label `proved` and are theorems of other packages, listed as the arithmetic skeleton or the Prop-level proof of the claim they sit under and never as the claim: 31 are the 31 Side24 arithmetic targets of main's `formal/` at origin/main `f6deeba7`; 41 are the 28 P15 arithmetic targets of main's `formal/` at `91cffbb5` (main PR #358, not yet on main; kernel evidence the hosted workspace-landing run 38096827443), each listed on every P15 row whose layer 0 source it anchors to; 70 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (60 theorems, 59 distinct Props, each listed on every row that registers its Prop; the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 as kernel-checked and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, which the receipt records as checked_commit: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; same trees as the heads; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads). Exactly one Prop registered as stating a claim itself has such a theorem: `DemandOneRefutation` on rows `p15-price-boundary` and `math.p15-price-boundary`; their claim-level label stays `specified` (the register admits only none/specified at claim level, alignment of the Prop with the source paragraph is PENDING_INDEPENDENT_REVIEW, and a receipt is about the Prop only), as the rows' `not established` cells record. `disposition` is the verbatim LANDING_CLAIMS value or `—` (null). Nothing in these tables is a scientific status, a review verdict or an acceptance; alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. +171 claim rows: 10 `landing_claim`, 33 `graph_node`, 2 `open_obligation`, 95 `proof_index_result`, 7 `existing_lean_package`, 12 `main_experiment_proof`, 12 `exact_certificate`. Claim-level formal progress: 33 `specified` (a `def … : Prop` stating the claim itself is registered), 138 `none`; no research claim is `proved` or `kernel-checked`. 197 statement entries point at the 150 declarations of this package (all `specified`). 155 statement entries carry the label `proved` and are theorems of other packages, listed as the arithmetic skeleton or the Prop-level proof of the claim they sit under and never as the claim: 31 are the 31 Side24 arithmetic targets of main's `formal/` at origin/main `f6deeba7`; 41 are the 28 P15 arithmetic targets of main's `formal/` at `91cffbb5` (main PR #358, not yet on main; kernel evidence the hosted workspace-landing run 38096827443), each listed on every P15 row whose layer 0 source it anchors to; 83 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (73 theorems, 72 distinct Props, each listed on every row that registers its Prop; the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 as kernel-checked and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, which the receipt records as checked_commit: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; same trees as the heads; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems marked † (`Side24Analysis`, `Side24Interfaces` and `RNCluster`) were added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), and have no hosted receipt yet: the first hosted `lean-specification-proofs.yml` run on the round-4 branch is their first receipt). Exactly one Prop registered as stating a claim itself has such a theorem: `DemandOneRefutation` on rows `p15-price-boundary` and `math.p15-price-boundary`; their claim-level label stays `specified` (the register admits only none/specified at claim level, alignment of the Prop with the source paragraph is PENDING_INDEPENDENT_REVIEW, and a receipt is about the Prop only), as the rows' `not established` cells record. `disposition` is the verbatim LANDING_CLAIMS value or `—` (null). Nothing in these tables is a scientific status, a review verdict or an acceptance; alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. | claim_id | kind | disposition (LANDING_CLAIMS, verbatim) | formal_progress | layer 0 source | statements | |---|---|---|---|---|---| -| `side24-coefficient` | landing_claim | REVIEWED_SCOPED | `specified` | `side24-proof` | 28 specified: `ThetaSummable`, `IsUnitTuple`, `DerivativePairingBound`, `DerivativeBoundAtLeastOne`, `DerivativeBoundAtZero`, `LatticeImageSumBound`, `SuccessiveTermRatio`, `ExponentialBounds`, `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound`, `ReferenceOddAndPinBlocks`, `DensityComparison`, `IntegrandRatioBounds`, `ReferencePinDensityProduct`, `ReferenceCoefficientIdentity`, `ConeMomentInterface`, `NegDefCone`, `ConeMomentD1Identity`, `ConeMomentD2Identity`, `ConeMomentD2Algebra`, `TraceMoments`, `RayleighSquareLaw`, `ElementaryConeIntegral`, `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison`; 27 proved (main, origin/main f6deeba7): `pairing_sum_six`, `pairing_sum_le_six`, `pairing_terms_six`, `sixth_moment_double_factorial`, `lattice_shell_constant`, `geometric_ratio_constants`, `period_exponent`, `image_constant_value`, `exp_taylor_partial_sum_gt_ten`, `covariance_relative_bound`, `odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`, `exponent_bounds`, `exponent_values`, `ratio_constant_ordering`, `density_comparison_below_reported`, `conditional_third_derivative_variance`, `transverse_covariance_entries_thirds`, `cone_moment_m1_thirds`, `trace_and_traceless_variances`, `fourth_moment_of_trace`, `exponential_moment_of_trace`, `cone_moment_m2_algebra`, `cube_root_simplification`, `pin_determinant_and_joint_dimension`, `hessian_block_eigenvalues`; 5 proved (proof packet): `Side24Elementary.exponentialBounds`, `Side24Elementary.successiveTermRatio`, `Side24Elementary.coneMomentD2Algebra`, `Side24Elementary.referenceCoefficientIdentity`, `Side24Elementary.elementaryConeIntegral` | +| `side24-coefficient` | landing_claim | REVIEWED_SCOPED | `specified` | `side24-proof` | 28 specified: `ThetaSummable`, `IsUnitTuple`, `DerivativePairingBound`, `DerivativeBoundAtLeastOne`, `DerivativeBoundAtZero`, `LatticeImageSumBound`, `SuccessiveTermRatio`, `ExponentialBounds`, `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound`, `ReferenceOddAndPinBlocks`, `DensityComparison`, `IntegrandRatioBounds`, `ReferencePinDensityProduct`, `ReferenceCoefficientIdentity`, `ConeMomentInterface`, `NegDefCone`, `ConeMomentD1Identity`, `ConeMomentD2Identity`, `ConeMomentD2Algebra`, `TraceMoments`, `RayleighSquareLaw`, `ElementaryConeIntegral`, `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison`; 27 proved (main, origin/main f6deeba7): `pairing_sum_six`, `pairing_sum_le_six`, `pairing_terms_six`, `sixth_moment_double_factorial`, `lattice_shell_constant`, `geometric_ratio_constants`, `period_exponent`, `image_constant_value`, `exp_taylor_partial_sum_gt_ten`, `covariance_relative_bound`, `odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`, `exponent_bounds`, `exponent_values`, `ratio_constant_ordering`, `density_comparison_below_reported`, `conditional_third_derivative_variance`, `transverse_covariance_entries_thirds`, `cone_moment_m1_thirds`, `trace_and_traceless_variances`, `fourth_moment_of_trace`, `exponential_moment_of_trace`, `cone_moment_m2_algebra`, `cube_root_simplification`, `pin_determinant_and_joint_dimension`, `hessian_block_eigenvalues`; 17 proved (proof packet): `Side24Elementary.exponentialBounds`, `Side24Elementary.successiveTermRatio`, `Side24Elementary.coneMomentD2Algebra`, `Side24Elementary.referenceCoefficientIdentity`, `Side24Elementary.elementaryConeIntegral`, `Side24Analysis.thetaSummable†`, `Side24Analysis.derivativePairingBound†`, `Side24Analysis.derivativeBoundAtLeastOne†`, `Side24Analysis.derivativeBoundAtZero†`, `Side24Analysis.referenceOddAndPinBlocks†`, `Side24Analysis.referencePinDensityProduct†`, `Side24Analysis.integrandRatioBounds†`, `Side24Analysis.latticeImageSumBound†`, `Side24Analysis.densityComparison†`, `Side24Interfaces.coneMomentD1Identity†`, `Side24Interfaces.traceMoments†`, `Side24Interfaces.rayleighSquareLaw†` | | `lifetime-remainder` | landing_claim | REVIEWED_SCOPED | `specified` | `lifetime-remainder` | 4 specified: `RemainderBoundedR1`, `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `IsLifetimeDensityVersion` | | `rn-count-interface` | landing_claim | HOLD_WITH_DOMAIN | `none` | `rn-count-interface` | 5 specified: `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5`; 5 proved (proof packet): `RNProbability.conditionalMeanIdentityN3`, `RNProbability.holderCountBoundN1`, `RNProbability.holderConsequenceN2`, `RNProbability.uniformCounterexampleN4`, `RNProbability.exponentialTailLayerCakeN5` | | `rn-fixed-remote-window` | landing_claim | HOLD_WITH_DOMAIN | `specified` | `rn-fixed-remote-window` | 5 specified: `PinnedLawInterface`, `IsCriticalPoint`, `HessianHasInertia`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | @@ -288,7 +289,7 @@ ladder; it is `specified`. | `CL-D5-DIMENSION-LIFT-20260929-v1` | proof_index_result | — | `specified` | `d5-dimension-lift` | 4 specified: `DimensionLiftPuncturedPinBallP`, `DimensionLiftCompactCollarC`, `DimensionLiftIntermediateShellsI`, `DimensionLiftGlobalWindowG` | | `OA-C6-FOURIER-20260929-v1` | proof_index_result | — | `none` | `c6-fourier-cutoff` | 1 specified: `FourierCutoffPlanarSecondFactorialF4` | | `CL-C6-PALM-20260929-v1` | proof_index_result | — | `specified` | `c6-palm-route` | 2 specified: `PalmRouteFactorialMomentsQ`, `PalmRouteSecondFactorialTheta` | -| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | `c6-rare-cluster` | 1 specified: `RareClusterPoissonObstructionO` | +| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | `c6-rare-cluster` | 1 specified: `RareClusterPoissonObstructionO`; 1 proved (proof packet): `RNCluster.rareClusterPoissonObstructionO†` | | `CL-D5-REMOTE-COLLISION-20260928-v1` | proof_index_result | — | `none` | `remote-collision` | 1 specified: `RemoteCollisionSecondFactorialD` | | `cumulative-transfer-correction` | proof_index_result | — | `none` | `cumulative-transfer-correction` | — | | `hist.D0-RN-UNIF-historical-predicates` | open_obligation | — | `none` | `downstream-graph` | — | @@ -354,7 +355,7 @@ Packets listed by PROOF_INDEX by id only (82 rows, kind `proof_index_result`, fo * Not a scientific status. Dispositions in the register are transcribed verbatim from `claims/LANDING_CLAIMS.json` at the pinned commit; the checker refuses any other value. * Not changed by the proof packet. `specifications/lean_statement_proofs_20261011` depends on this - packet by path and holds 60 theorems whose type is exactly one of 59 registered Props (P15, Side24 + packet by path and holds 73 theorems whose type is exactly one of 72 registered Props (P15, Side24 and RN families; see the last column of the declaration tables). The register lists those theorems under the claim rows with label `proved`, the label of a committed proof package; `kernel-checked` appears only in that packet's hosted run receipts (`lean-specification-proofs.yml`; run 38121713840 receipted the 52 @@ -362,7 +363,8 @@ Packets listed by PROOF_INDEX by id only (82 rows, kind `proof_index_result`, fo base that a pull_request run checks out, not the head itself: run 38121713840 PR head 910809d6, actual tested merge 5b198699…; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8…; both with base b2d70d77… and each with the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194…), are receipted by run 38149888699: PR head 0dcb0016, actual tested - merge 4363b692… with base 79731b96…, the head of PR #422, and first by run 38149524759: PR head 88c21058, actual tested merge 3306233b…, same base). + merge 4363b692… with base 79731b96…, the head of PR #422, and first by run 38149524759: PR head 88c21058, actual tested merge 3306233b…, same base; the 13 theorems + added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…, receipt theorem_count 60) have no hosted receipt yet, marked †). The 150 declarations here remain `specified`; a kernel-checked theorem is evidence about its Prop only, never about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; no claim-level label is `proved`, including the one row (`p15-price-boundary`) whose diff --git a/specifications/lean_statements_20261010/REGISTER.json b/specifications/lean_statements_20261010/REGISTER.json index 4bcb02696..6e3f670b5 100644 --- a/specifications/lean_statements_20261010/REGISTER.json +++ b/specifications/lean_statements_20261010/REGISTER.json @@ -778,6 +778,114 @@ "formal_progress": "proved", "anchor": "integral_0^a (a-z)^2 e^(-z/2) dz/2 = a^2-4a+8-8e^(-a/2).", "does_not_claim": "the Rayleigh law of `R^2` (`RayleighSquareLaw`), the cone-moment identity `ConeMomentD2Identity`, any enclosure of the coefficient, nonauthor review or acceptance." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Analysis.lean", + "declaration": "UniversalLaw.Proofs.Side24Analysis.thetaSummable", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "K_24(z) = sum_{n in Z^d} exp(-|z+24n|^2/2)", + "does_not_claim": "the informal convergence statement of the source as the source's object (the alignment of `E d`, `latticeVec` and this family with the source's `R^d`, `24 Z^d` and `K_24` is PENDING_INDEPENDENT_REVIEW); any value, positivity, smoothness or Poisson-summation property of `thetaSum24` or `K24`; `ImageBound`, `LatticeImageSumBound` or the derivative bounds of Section 2; any enclosure of the coefficient; the statement module's own `Does not claim:` text (the statement packet is not edited here); the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Analysis.lean", + "declaration": "UniversalLaw.Proofs.Side24Analysis.derivativePairingBound", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "pairings bound the absolute derivative by", + "does_not_claim": "the informal Section 2 display or its review and acceptance; the alignment of the formal objects (`Kinf`, `IsUnitTuple`, `pairingSum`, Mathlib's `iteratedFDeriv ℝ q`) with the source's (PENDING_INDEPENDENT_REVIEW); validity for `q > 6` (the Prop restricts to `q ≤ 6`, as the source does, and nothing beyond it is stated or proved here); any enclosure of the coefficient `c_{d,24}`; the disposition of `side24-coefficient`, which is REVIEWED_SCOPED as transcribed in the register, not derived here." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Analysis.lean", + "declaration": "UniversalLaw.Proofs.Side24Analysis.derivativeBoundAtLeastOne", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "At |x|>=1 this is at most76 |x|^6 e^(-|x|^2/2).", + "does_not_claim": "the informal Section 2 display or its review and acceptance; the alignment of the formal objects (`Kinf`, `IsUnitTuple`, `pairingSum`, Mathlib's `iteratedFDeriv ℝ q`) with the source's (PENDING_INDEPENDENT_REVIEW); any identification with the main-branch targets `UniversalLaw.Side24.pairing_sum_six` and `pairing_sum_le_six`, which kernel-check the coefficient arithmetic `1+15+45+15=76` and `76` bounding the smaller orders there, not here; validity for `q > 6` or for `‖x‖ < 1` (the Prop restricts to `q ≤ 6` and `1 ≤ ‖x‖`, as the source does); any enclosure of the coefficient `c_{d,24}`; the disposition of `side24-coefficient`, which is REVIEWED_SCOPED as transcribed in the register, not derived here." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Analysis.lean", + "declaration": "UniversalLaw.Proofs.Side24Analysis.derivativeBoundAtZero", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "contraction has absolute value at most15.", + "does_not_claim": "the informal Section 2 display or its review and acceptance; the alignment of the formal objects (`Kinf`, `IsUnitTuple`, `pairingSum`, Mathlib's `iteratedFDeriv ℝ q` at `0`) with the source's (PENDING_INDEPENDENT_REVIEW); the double-factorial identity `5!! = 15` as such, which is kernel-checked on main (`UniversalLaw.Side24.sixth_moment_double_factorial`) and is neither re-proved nor cited here (only `pairingSum 6 0 = 15` is computed); validity for `q > 6` (the Prop restricts to `q ≤ 6`, as the source does); any value of a contraction (only an upper bound on its absolute value); any enclosure of the coefficient `c_{d,24}`; the disposition of `side24-coefficient`, which is REVIEWED_SCOPED as transcribed in the register, not derived here." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Analysis.lean", + "declaration": "UniversalLaw.Proofs.Side24Analysis.referenceOddAndPinBlocks", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "Var(t)=15, Cov(t,G)=(-3,0,...,0),", + "does_not_claim": "the informal Section 1 block values of the source as the source's objects (the alignment of `E d`, `Kinf`, `derivContraction`, `gradCov`, `pinCov`, `thirdGradCov`, `thirdVar` and `condThirdVariance` with the source's `R^d`, `K_infty`, covariances and conditional variance is PENDING_INDEPENDENT_REVIEW); that any of these matrices is the covariance of a random vector or field (none is constructed in the statement module); the statement module's own `Does not claim:` text (\"the identities (statement only); `15 - 9 = 6` is kernel-checked on main (`UniversalLaw.Side24.conditional_third_derivative_variance`), not here\"), which describes the statement packet, is not edited here, and main's theorem is neither cited nor relied on; `ReferencePinDensityProduct`, `CovarianceOrdering`, `EntrywiseCovarianceBound` or any other Prop of the module; any enclosure of the coefficient; the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Analysis.lean", + "declaration": "UniversalLaw.Proofs.Side24Analysis.referencePinDensityProduct", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3).", + "does_not_claim": "the informal Section 1 display or its review and acceptance; the alignment of the formal objects (`gaussDensity`, `gradCov`, `pinCov`, `derivContraction` through Mathlib's `iteratedFDeriv ℝ` at `0`, `Orthonormal ℝ R`) with the source's (PENDING_INDEPENDENT_REVIEW); the determinant identity `det diag(3,1,…,1) = 3` as the main-branch target `UniversalLaw.Side24.pin_determinant_and_joint_dimension`, which is kernel-checked on main and is neither cited nor re-used here (the determinant is recomputed by `Matrix.det_diagonal` in `Aux.rp_det_pinDiagonal`); that `gaussDensity` is the density of any random vector or that any field is constructed; the remaining conjuncts of `ReferenceOddAndPinBlocks` (`Var(t) = 15`, `Cov(t, G) = (-3, 0, …, 0)`, `tau^2 = 6`), which this theorem does not use; the value for `d ∉ {2, 3}` (the Prop restricts to `d = 2 ∨ d = 3`, as the source does); any enclosure of the coefficient `c_{d,24}`; the disposition of `side24-coefficient`, which is REVIEWED_SCOPED as transcribed in the register, not derived here." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Analysis.lean", + "declaration": "UniversalLaw.Proofs.Side24Analysis.integrandRatioBounds", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "ratio between1-13epsilon and", + "does_not_claim": "the informal Section 4 display of the source as the source's object (the alignment of `epsilon`, `exponentA`, `exponentB` and these power quotients with the source's `epsilon`, `a`, `b` and integrand ratio is PENDING_INDEPENDENT_REVIEW); that the integrand ratio of the source equals these power quotients (not specified in the statement packet); the arithmetic `13 ≤ 32`, `28 ≤ 32` and `32*10^(-108) < 10^(-106)` (kernel-checked on main in `UniversalLaw.Side24.ratio_constant_ordering` and `density_comparison_below_reported`, not here); `DensityComparison` or any covariance ordering; any enclosure of the coefficient; the statement module's own `Does not claim:` text (the statement packet is not edited here); the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Analysis.lean", + "declaration": "UniversalLaw.Proofs.Side24Analysis.latticeImageSumBound", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "<=1458 e^(-288).", + "does_not_claim": "the informal Section 2 lattice display of the source as the source's object (the alignment of `intNormSq`, `imageTerm` and this `tsum` over `Fin d → ℤ` with the source's `sum_{n ≠ 0} |n|^6 e^(-288|n|^2)` is PENDING_INDEPENDENT_REVIEW); the intermediate shell count `729 ∑_j j^9 e^(-288 j^2)` of the source, which the Prop does not specify and this proof does not reproduce; the arithmetic `27·27 = 729`, `729·2 = 1458` kernel-checked on main, not here; `ImageBound`, the derivative bounds of Section 2 or any enclosure of the coefficient; the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Analysis.lean", + "declaration": "UniversalLaw.Proofs.Side24Analysis.densityComparison", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "[(1-epsilon)/(1+epsilon)]^(n/2) phi_((1-epsilon)C0)", + "does_not_claim": "the informal Section 4 display of the source as the source's object (the alignment of `gaussDensity`, `symCoordDim`, `epsilon` and the positive-semidefinite hypotheses with the source's densities, `n = m(m+1)/2`, `epsilon` and ordering is PENDING_INDEPENDENT_REVIEW); the cone-moment consequence `(1 ± epsilon)^m D0` and the scaling `a^m` of `h(A) = det(A)^2 1{A<0}` (not specified in the statement packet); that `(1 ± epsilon) • C0` or `C` are covariances of any random object; `CovarianceOrdering` or any ordering of `C24` against `Cref`; any enclosure of the coefficient; the statement module's own `Does not claim:` text (the statement packet is not edited here); the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Interfaces.lean", + "declaration": "UniversalLaw.Proofs.Side24Interfaces.coneMomentD1Identity", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "D_1=E[A^2 1{A<0}]=4/3.", + "does_not_claim": "the informal identity `D_1 = E[A^2 1{A<0}] = 4/3` of the source as the source's object (the alignment of `ConeMomentInterface`, its field `A_law` and `coneMomentD1` with the source's `A ~ N(0, 8/3)` and `D_1` is PENDING_INDEPENDENT_REVIEW); that `D_1` is the parent's cone moment `D_u` of (15.1) for the reference law; that `ConeMomentInterface` has an inhabitant, or that its laws are the conditional law `A | V = 0` of the reference field; any enclosure of the coefficient; `ConeMomentD2Identity`, `TraceMoments` or `RayleighSquareLaw`; the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Interfaces.lean", + "declaration": "UniversalLaw.Proofs.Side24Interfaces.traceMoments", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "E s^2=5/3, E s^4=25/3 and", + "does_not_claim": "the informal Section 1 display of the source as the source's object (the alignment of the interface fields `s`, `P`, `sxy_law` and of `gaussianReal` with the source's `s ~ N(0, 5/3)` is PENDING_INDEPENDENT_REVIEW); that any inhabitant of `ConeMomentInterface` exists, or that its laws are the conditional law `A | V = 0` of the reference field (the laws are hypothesis fields of the given `I`, not derived here); the main-side constants `UniversalLaw.Side24.fourth_moment_of_trace` and `exponential_moment_of_trace`, which kernel-check `3 (5/3)^2 = 25/3` and `1 + 5/3 = 8/3` on main and are neither imported nor re-stated here (the same arithmetic is discharged inline by `ring`); `ConeMomentD2Identity`, `RayleighSquareLaw` or any cone moment; any enclosure of the coefficient; the statement module's own `Does not claim:` text (the statement packet is not edited here); the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Interfaces.lean", + "declaration": "UniversalLaw.Proofs.Side24Interfaces.rayleighSquareLaw", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "The variable R^2 has density e^(-z/2)/2.", + "does_not_claim": "the informal Section 1 display \"The variable R^2 has density e^(-z/2)/2\" of the source as the source's object, nor its review and acceptance (the alignment of the interface fields `x`, `y`, `P`, `sxy_law` and of `gaussianReal` with the source's `x, y ~ N(0, 1)` is PENDING_INDEPENDENT_REVIEW); from the Prop's own `Does not claim:` line, \"the law (statement only)\": the law is proved here only for the formal objects of the Prop, as the distribution function `P(R^2 ≤ a)` for `a ≥ 0`, not as a density and not for the source's variable; and \"measurability of the event (if the event is not measurable, `P` is its outer measure)\": measurability of the formal event `{x^2 + y^2 ≤ a}` is proved here (`Aux.rl_measurableSet_disk`, `Measure.map_apply_of_aemeasurable`), so the outer-measure reading does not arise for this event, but nothing is established about any other event; that any inhabitant of `ConeMomentInterface` exists, or that its laws are the conditional law `A | V = 0` of the reference field (the laws are hypothesis fields of the given `I`, not derived here); `ConeMomentD2Identity` or any cone moment; any enclosure of the coefficient; the statement module's own text (the statement packet is not edited here); the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." } ], "kernel_evidence": [ @@ -788,12 +896,12 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3)", - "note": "5 theorem(s) of this row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 60 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation}.lean, 59 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); theorems added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60) have no hosted receipt yet: the next hosted lean-specification-proofs.yml run is their first receipt", + "note": "17 theorems of this row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral, Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). 12 (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them. `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], - "not_established": "Nothing here encloses the coefficient. CoefficientEnclosure and ReferenceComparison are statements for any object satisfying PeriodicCoefficientInterface; the interface takes the cone moment D_u of (15.1) as a GIVEN field, so c_{d,24} is not uniquely determined in Lean and no inhabitant is constructed. The 27 main theorems listed as `proved` are the integer/rational arithmetic skeleton of PROOF.md sections 1-4 (coefficient sums, shell constants, Taylor partial sum, block minors, cone-moment algebra), not the Gaussian conditioning, the Rayleigh cone integration, the Stirling log-gamma step (section 5, not specified), the interval arithmetic of coefficient.py, or the enclosure itself. ExponentialBounds, SuccessiveTermRatio, ConeMomentD2Algebra, ReferenceCoefficientIdentity and ElementaryConeIntegral have theorems of exactly their type in the proof packet specifications/lean_statement_proofs_20261011 (5 entries below, label `proved`; the hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked them (receipt formalization_status kernel-checked, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound})): elementary real inequalities, the term ratio, the D2 cone-moment algebra, the reference-coefficient identity as stated (its left side is not identified with the parent formula (15.2) at the reference law: rotational invariance of K_infty is not specified) and the elementary cone integral; none of them is the Gaussian conditioning, ConeMomentD1Identity / ConeMomentD2Identity, RayleighSquareLaw, LatticeImageSumBound or any enclosure of the coefficient, and the claim-level label stays `specified`. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed.", - "next_step": "Formal lane (Math- Mathlib package, by the lane's admission rules): construct the conditional law A_u | V_u = 0 and prove ConeMomentD1Identity / ConeMomentD2Identity for the reference law under ConeMomentInterface, RayleighSquareLaw and LatticeImageSumBound (ReferenceCoefficientIdentity is proved as stated; the identification of its left side with (15.2) at the reference law is not specified); the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked the 5 proved entries; the enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md). Independent reviewer: alignment verdicts on the 28 Side24 statements." + "not_established": "Nothing here encloses the coefficient. CoefficientEnclosure and ReferenceComparison are statements for any object satisfying PeriodicCoefficientInterface; the interface takes the cone moment D_u of (15.1) as a GIVEN field, so c_{d,24} is not uniquely determined in Lean and no inhabitant is constructed. The 27 main theorems listed as `proved` are the integer/rational arithmetic skeleton of PROOF.md sections 1-4 (coefficient sums, shell constants, Taylor partial sum, block minors, cone-moment algebra), not the Gaussian conditioning, the Rayleigh cone integration, the Stirling log-gamma step (section 5, not specified), the interval arithmetic of coefficient.py, or the enclosure itself. Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, LatticeImageSumBound, SuccessiveTermRatio, ExponentialBounds, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (17 entries below, label `proved`; 5 of them (Side24Elementary.successiveTermRatio, Side24Elementary.exponentialBounds, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.coneMomentD2Algebra, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.latticeImageSumBound, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.densityComparison, Side24Analysis.integrandRatioBounds, Side24Analysis.referencePinDensityProduct, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them). What those theorems prove, for the formal objects of the statement module only: elementary real inequalities (ExponentialBounds), the term ratio (SuccessiveTermRatio), the D2 cone-moment algebra (ConeMomentD2Algebra), the reference-coefficient identity as stated (ReferenceCoefficientIdentity; its left side is not identified with the parent formula (15.2) at the reference law: rotational invariance of K_infty is not specified) and the elementary cone integral (ElementaryConeIntegral); summability of the formal theta family n |-> exp(-|z + 24 n|^2/2) over Fin d -> Z for every d (ThetaSummable: the statement entry's does_not_claim \"that the family is summable (true, but not shown here)\" describes the statement packet, which is not edited; the summability is now shown in the proof packet, for the formal family, by comparison with a product of one-dimensional Gaussian sums); the Section 2 derivative bounds for q <= 6 as the Props state them (DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero: a Hermite-type recursion on the operator norms of iteratedFDeriv, no closed form of the Gaussian derivatives, the contraction at a unit tuple bounded by the operator norm; main's pairing_sum_six, pairing_sum_le_six and sixth_moment_double_factorial are neither cited nor relied on (the 1 + 15 + 45 + 15 = 76 arithmetic is recomputed inline by norm_num); nothing for q > 6); the lattice image sum (LatticeImageSumBound: summability and the bound 1458 e^(-288), via 432 e^(-288); the source's shell count 729 sum_j j^9 e^(-288 j^2) is not reproduced and not specified); the five reference block identities and the pin density product for the formal matrices (ReferenceOddAndPinBlocks, ReferencePinDensityProduct: derivative contractions of Kinf at 0 in an orthonormal frame; none of these matrices is shown to be the covariance of any random vector or field); pure real arithmetic in epsilon and the exponents a, b for d = 2, 3 (IntegrandRatioBounds; that the source's integrand ratio equals these power quotients is not specified); Loewner determinant and inverse monotonicity for positive-definite real matrices (DensityComparison; the cone-moment consequence (1 +/- epsilon)^m D0 and the scaling a^m are not specified); and, over ConeMomentInterface, the m = 1 cone moment 4/3, the trace moments 5/3, 25/3, sqrt(3/8) and the distribution function P(R^2 <= a) = integral_0^a e^(-z/2)/2 dz for a >= 0 (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw: proved for every inhabitant from the law fields A_law, sxy_law and isProb alone; no inhabitant is constructed, the laws are not derived as the conditional law A | V = 0 of the reference field, and the Rayleigh law is proved as a distribution function, not as a density). None of them is the Gaussian conditioning, ConeMomentD2Identity, ImageBound, the covariance ordering or any enclosure of the coefficient, and the claim-level label stays `specified` (claim-level vocabulary none/specified; a theorem of a Prop is evidence about that Prop, for the formal objects, not acceptance of the informal theorem; alignment PENDING_INDEPENDENT_REVIEW). Theorem-shaped Props of this row still `specified` with no theorem: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison; the parametrised predicates IsUnitTuple, NegDefCone are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited.", + "next_step": "Math- Mathlib lane: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); the 12 theorems of this row added after PR head 12e072d6 (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) await the next hosted lean-specification-proofs.yml run, their first receipt; the other theorems of this row have a hosted receipt: the 5 of PR head 00d2ac6e (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); each of those receipted theorems corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, ReferenceOddAndPinBlocks, IntegrandRatioBounds, ReferencePinDensityProduct, ConeMomentD1Identity, TraceMoments PROVABLE_NOW and LatticeImageSumBound, DensityComparison, RayleighSquareLaw PROVABLE_WITH_WORK: proved in round 4 (see entries); ReferenceCovarianceLowerBound PROVABLE_WITH_WORK (weeks); ConeMomentD2Identity PROVABLE_WITH_WORK (weeks); ImageBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum; the global version is inapplicable to lattice translates; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), iteratedFDeriv_tsum_apply (SmoothSeries.lean)); CovarianceOrdering BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: local term-wise iterated differentiation of the lattice sum, the d-dimensional Gaussian tail bound and higher-order symmetry of iteratedFDeriv; display (3) needs only the crude norm_iteratedFDeriv_comp_le, with a factor about 8 to spare; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), hasFDerivAt_tsum_of_isPreconnected (SmoothSeries.lean)); EntrywiseCovarianceBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum for the term-wise iterated differentiation of the lattice sum (the same blocker as CovarianceOrdering; the finding records no further shape observation); present but insufficient: norm_iteratedFDeriv_comp_le (Bounds.lean), iteratedFDeriv_tsum (SmoothSeries.lean)); CoefficientEnclosure BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); ReferenceComparison BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); IsUnitTuple, NegDefCone PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. The enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md) and an interface that determines c_{d,24} (the cone moment is a given field; ConeMomentD2Identity for the reference law under ConeMomentInterface is the open Rayleigh-integration step). Proof packet README: for the Props proved in round 4 (ConeMomentD1Identity, DensityComparison, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, DerivativePairingBound, IntegrandRatioBounds, LatticeImageSumBound, RayleighSquareLaw, ReferenceOddAndPinBlocks, ReferencePinDensityProduct, ThetaSummable, TraceMoments) the informal reasons the README gave were found not load-bearing by the triage and are superseded by their theorems; the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure and ReferenceComparison stand (for the last two the triage confirms the README's reason as the precise missing ingredient) (the proof-packet maintainers' regeneration, not this register's). Independent reviewer: alignment verdicts on the 28 Side24 statements, in particular of the 17 Props of this row that have a theorem of exactly their type, with their anchors (ThetaSummable with the source's R^d, 24 Z^d and K_24; the derivative Props with Section 2; the interface Props with the source's A ~ N(0, 8/3), s ~ N(0, 5/3), x, y ~ N(0, 1))." }, { "claim_id": "lifetime-remainder", @@ -845,7 +953,7 @@ ], "kernel_evidence": [], "not_established": "RemainderBoundedR1 states the shape of (R1) for versions of the densities under the interface structures; the result is not proved. Not stated: (R19)-(R20), the observation transform and determinant lemmas R3.1/R3.2, the cap-loss bound, the parent interfaces D1-A-E it imports, the marked-cylinder-cap support. The coefficient c enters only through IsUnrestrictedLeadingCoefficient (c > 0 and the leading-term role), not through the representation (15.2). No numerical C or ell_*, no second coefficient, no convergence of the remainder, no compact-window O(ell^(2/3)) difference without cutoffs. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "State (R19) (cumulative form) as a further Prop; transcribe the parent interfaces consumed (section 8 comparison, section 9 as replaced by v1.1, section 10, (14.1), (15.2)) as explicit hypotheses; independent reviewer: alignment verdict on RemainderBoundedR1 against (R1), including the \"versions\" clause and the exclusion of the essential global maximum." + "next_step": "State (R19) (cumulative form) as a further Prop; transcribe the parent interfaces consumed (section 8 comparison, section 9 as replaced by v1.1, section 10, (14.1), (15.2)) as explicit hypotheses; independent reviewer: alignment verdict on RemainderBoundedR1 against (R1), including the \"versions\" clause and the exclusion of the essential global maximum. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemainderBoundedR1 BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean), ncard_le_ncard (Card.lean)); IsLifetimeDensityVersion, IsUnrestrictedLeadingCoefficient, IsUnrestrictedPair PREDICATE_VOCABULARY: vocabulary of the other Props, not a target." }, { "claim_id": "rn-count-interface", @@ -952,8 +1060,8 @@ "kernel_evidence": [ { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3)", - "note": "5 theorem(s) of this row (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 60 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation}.lean, 59 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", + "note": "5 theorems of this row (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "No Prop states Theorem N as a whole, so the claim-level label stays none: (N1)-(N5) (Hoelder bound, its power-law consequence, the conditional-mean identity, the uniform counterexample, the exponential-tail layer cake) are stated as abstract probability-space Props and each has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (5 entries below, label `proved`; the hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked them (receipt formalization_status kernel-checked, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound})); (N6) (the full-pin weighted Kac-Rice expression for E_Q^W N_B) and (N7) (the enclosure-to-rate implication under Z_r >= z_* r^2) are NOT stated, and they are the Gaussian numerator interface that the disposition reason calls \"a target rather than a discharged global RN theorem\". The five theorems prove the Props for the abstract probability spaces and counts they quantify over: nothing about the Gaussian model is stated or proved; no RN rate, no 24-jet enclosure, no ENV-RESCOV discharge; D3-LEMMA-RN-UNIF stays not closed. Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -1018,7 +1126,7 @@ ], "kernel_evidence": [], "not_established": "FixedRemoteMeanMeasureA2 and FixedRemoteCountBoundA3 state the shapes of displays (2) and (3) for any object satisfying PinnedLawInterface (the regression law Q_r, the weight W_r, the normalizer and the contact kernel Lambda_j are interface fields, gated on 0 < r < L); no inhabitant of the interface is constructed and the kernel is not identified with formula (13). Not stated: the observation transform U_r and target v_r, the coupling (5), the weight expansion (10)-(11), the Kac-Rice formula (12), the kernel formula (13), the unnormalized numerator bound (14), and the multi-witness Section 6 displays (15)-(16). Nothing is proved; no numerical C or r_*; no shrinking exclusion rho -> 0, no collisions, no global RN complement (the disposition reason). Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "State Section 6 ((15)-(16), separated m-tuples) as Props; record an inhabitant construction of PinnedLawInterface as a proof-lane target (Dirac mass at one admissible smooth path); independent reviewer: alignment of A2/A3 with displays (2)-(3), in particular the uniformity in b, k, R, j, E and the fixed rho." + "next_step": "State Section 6 ((15)-(16), separated m-tuples) as Props; record an inhabitant construction of PinnedLawInterface as a proof-lane target (Dirac mass at one admissible smooth path); independent reviewer: alignment of A2/A3 with displays (2)-(3), in particular the uniformity in b, k, R, j, E and the fixed rho. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point); HessianHasInertia, IsCriticalPoint PREDICATE_VOCABULARY: vocabulary of the other Props, not a target." }, { "claim_id": "rn-fixed-annulus-window", @@ -1043,7 +1151,7 @@ ], "kernel_evidence": [], "not_established": "FixedAnnulusWindowCandidateA2 states the shape of (A2) (d = 2, fixed L, fixed scaled annulus, compact positive gaps, between-pin height window) over PinnedLawInterface; the result is not proved. Not stated: the six-pin regression (A3)-(A5), normalizer (A8), contact vector (A10)-(A12), cutoff (A13)-(A14), density (A17), Hessian bounds (A18)-(A20), intensities (A21)-(A26), the TWO_SCALE_ADDENDUM S6-S21 support (not pinned here), the eight reviewed interfaces as separate statements. No all-height assertion, no positive leading coefficient, no probability lower bound, no uniformity as k -> 0, B0 -> infinity, in L or for d >= 3; no global D5/RN/JETMOD transfer. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "Pin TWO_SCALE_ADDENDUM.md and state (S3)-(S5) and (S20)-(S21); independent reviewer: alignment of FixedAnnulusWindowCandidateA2 with (A2), checking that the height window is strict and the constants depend only on L, A0, B0 and the mark compacts." + "next_step": "Pin TWO_SCALE_ADDENDUM.md and state (S3)-(S5) and (S20)-(S21); independent reviewer: alignment of FixedAnnulusWindowCandidateA2 with (A2), checking that the height window is strict and the constants depend only on L, A0, B0 and the mark compacts. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedAnnulusWindowCandidateA2 BLOCKED_INTERFACE (a Dirac inhabitant with an index-j window critical point on a null E0 (a singleton in the annulus); no field of PinnedLawInterface supplies the a.s. absence that the weighted Kac-Rice formula (A21) gives)." }, { "claim_id": "p15-realized-covers", @@ -1596,8 +1704,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3)", - "note": "20 theorem(s) of this row (P15Elementary.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Price, P15Structure.elementaryPaletteFormula, P15Structure.triangleNeedsThree, P15Structure.benchmark816Palette, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.benchmark816WholeGround, P15Structure.minimalForbiddenCharacterisation, P15Structure.benchmark816MinimalForbiddenCounts, P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Covers.benchmark816Cover, P15Covers.localRestriction, P15Covers.paletteNumberAttained, P15Covers.benchmark816UnionBound, P15Budget.coverCostHazardBound) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 60 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation}.lean, 59 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 13 of them (P15Elementary.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Price, P15Structure.elementaryPaletteFormula, P15Structure.triangleNeedsThree, P15Structure.benchmark816Palette, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.benchmark816WholeGround, P15Structure.minimalForbiddenCharacterisation, P15Structure.benchmark816MinimalForbiddenCounts) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 7 (P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Covers.benchmark816Cover, P15Covers.localRestriction, P15Covers.paletteNumberAttained, P15Covers.benchmark816UnionBound, P15Budget.coverCostHazardBound) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", + "note": "20 theorems of this row (P15Elementary.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Price, P15Structure.elementaryPaletteFormula, P15Structure.triangleNeedsThree, P15Structure.benchmark816Palette, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.benchmark816WholeGround, P15Structure.minimalForbiddenCharacterisation, P15Structure.benchmark816MinimalForbiddenCounts, P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Covers.benchmark816Cover, P15Covers.localRestriction, P15Covers.paletteNumberAttained, P15Covers.benchmark816UnionBound, P15Budget.coverCostHazardBound) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 13 of them (P15Elementary.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Price, P15Structure.elementaryPaletteFormula, P15Structure.triangleNeedsThree, P15Structure.benchmark816Palette, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.benchmark816WholeGround, P15Structure.minimalForbiddenCharacterisation, P15Structure.benchmark816MinimalForbiddenCounts) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 7 (P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Covers.benchmark816Cover, P15Covers.localRestriction, P15Covers.paletteNumberAttained, P15Covers.benchmark816UnionBound, P15Budget.coverCostHazardBound) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: PaletteNumberAttained, MinimalForbiddenCharacterisation, LocalRestriction, LocalDecomposability, LocalObstructionIsFullBlock, LocalPriceChain, IndependentBlockHazard, FullBlockCoverIff, CoverCostHazardBound, WholeGroundChromaticNumber, ElementaryPaletteFormula, TriangleNeedsThree, Benchmark816IsRealized, Benchmark816Palette, Benchmark816Cover, Benchmark816WholeGround, Benchmark816MinimalForbiddenCounts, Benchmark816Price, Benchmark816UnionBound (20 entries below, label `proved`; 13 of them (P15Structure.minimalForbiddenCharacterisation, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.elementaryPaletteFormula, P15Elementary.triangleNeedsThree, P15Structure.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Palette, P15Structure.benchmark816WholeGround, P15Structure.benchmark816MinimalForbiddenCounts, P15Structure.benchmark816Price) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 7 of them (P15Covers.paletteNumberAttained, P15Covers.localRestriction, P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Budget.coverCostHazardBound, P15Covers.benchmark816Cover, P15Covers.benchmark816UnionBound) added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN)); among them Proposition P1 (MinimalForbiddenCharacterisation), both halves of Theorem P2 (FullBlockCoverIff, display (P7), and CoverCostHazardBound, display (P8)), the local statements (P2)-(P5) and the benchmark Props including the union bound (P11). Theorem-shaped Props of this row still `specified` with no theorem: none; the parametrised predicates IsRealizedFamily, IsGood, IsKDecomposable, IsGeneratorCover, IsPaletteFeasible, IsMinimalForbidden, IsTransversal are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. Theorem P2 is proved as the two Props state it (for the formal RealizedData record and the formal downset, prices 0 <= c_v <= p_v), which is a statement about those Props, not acceptance of the informal theorem: the claim-level label stays `specified` (this register admits only none/specified at claim level), alignment of RealizedData / IsRealizedFamily / downset / coverCost / mu with the informal objects is PENDING_INDEPENDENT_REVIEW, and the price range p_v < c_v <= phi(p_v) is a different claim row. The 12 main P15 arithmetic theorems listed as `proved` (finite arithmetic of Section 6: 409, 2454, 15, 500616, 419743994415, 816, 815*3 < 2448, 818, 409p = 1/100, the (P11) rational bound, positivity of 6/40900^409, the elementary instances of (P9)) are at commit 91cffbb5 of main PR #358, not on main; they do not count the minimal forbidders, show the palette is 816 or evaluate 6/40900^409, and the proof-packet theorems that do (Benchmark816MinimalForbiddenCounts, Benchmark816Palette, Benchmark816Price) do so for the formal RealizedData record, whose identification with the informal six-block construction is an alignment question. Nothing is asserted for prices p_v < c_v <= phi(p_v) (row p15-price-budget-restricted), for arbitrary P15 inputs or for the prize (original_prize_closed stays false). Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -1731,8 +1839,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3)", - "note": "3 theorem(s) of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.logOnePlusBelowIdentity, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 60 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation}.lean, 59 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 3 of them (P15Elementary.twoCoordGeneratorPrices, P15Elementary.logOnePlusBelowIdentity, P15Elementary.demandOneRefutation) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", + "note": "3 theorems of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.logOnePlusBelowIdentity, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 3 of them (P15Elementary.twoCoordGeneratorPrices, P15Elementary.logOnePlusBelowIdentity, P15Elementary.demandOneRefutation) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "DemandOneRefutation states the refutation (the two-coordinate data form a realized family with K_H(d) = 1, obstruction {X}, admissible prices 2/3, mu_p(D) = 3/4, cover cost 4/9 > 1/3 > log(4/3)). DemandOneRefutation, TwoCoordGeneratorPrices and LogOnePlusBelowIdentity each have a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011 (listed below as `proved`, the label of a committed proof package; the hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked them (receipt formalization_status kernel-checked, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound})); the two logarithm inequalities phi(1/2) = log 2 > 2/3 and log(4/3) < 1/3 are conjuncts of DemandOneRefutation and are proved inside its theorem from Mathlib's Real.log bounds. The claim-level label stays `specified`, not `proved`: this register admits only none/specified at claim level (no research claim of this programme is proved), whether DemandOneRefutation says what the source paragraph says is the independent reviewer's alignment question (PENDING_INDEPENDENT_REVIEW), the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked the Prop and nothing more, and a kernel-checked Prop is evidence about that Prop only. This is the one row of the register where a theorem proves the Prop registered as stating the claim itself; it is recorded here for the operator and not acted on. The 6 main P15 arithmetic theorems listed as `proved` (good sets, obstruction, 3/4 over denominator 4, prices 9,6,6,4 over 9, 3*4 > 9) are at commit 91cffbb5 of main PR #358, not on main; they are finite arithmetic, not the refutation. The counterexample does not refute d_i >= 2 successors or original P15-B (the disposition reason; the theorem's docstring says the same). Disposition EXACT_COUNTEREXAMPLE is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -1929,8 +2037,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3)", - "note": "8 theorem(s) of this row (P15Elementary.logRatioBound, P15Elementary.capTestAtTwoThirds, P15Elementary.ratioExponentBound, P15Structure.benchmark816RationalPrice, P15Budget.restrictedLocalBudget, P15Budget.rhoTestSufficient, P15Budget.restrictedTransformedPriceBudget, P15Budget.benchmark816TransformedRange) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 60 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation}.lean, 59 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 4 of them (P15Elementary.logRatioBound, P15Elementary.capTestAtTwoThirds, P15Elementary.ratioExponentBound, P15Structure.benchmark816RationalPrice) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 4 (P15Budget.restrictedLocalBudget, P15Budget.rhoTestSufficient, P15Budget.restrictedTransformedPriceBudget, P15Budget.benchmark816TransformedRange) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", + "note": "8 theorems of this row (P15Elementary.logRatioBound, P15Elementary.capTestAtTwoThirds, P15Elementary.ratioExponentBound, P15Structure.benchmark816RationalPrice, P15Budget.restrictedLocalBudget, P15Budget.rhoTestSufficient, P15Budget.restrictedTransformedPriceBudget, P15Budget.benchmark816TransformedRange) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 4 of them (P15Elementary.logRatioBound, P15Elementary.capTestAtTwoThirds, P15Elementary.ratioExponentBound, P15Structure.benchmark816RationalPrice) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 4 (P15Budget.restrictedLocalBudget, P15Budget.rhoTestSufficient, P15Budget.restrictedTransformedPriceBudget, P15Budget.benchmark816TransformedRange) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "RestrictedTransformedPriceBudget states (T1) for every realized family with a_i >= 1, d_i >= 2, p_v <= 1/4, c_v <= phi(p_v) and is proved as stated in the proof packet; that is evidence about the Prop (for the formal RealizedData record, downset, coverCost and mu), not acceptance of the informal theorem: the claim-level label stays `specified` (claim-level vocabulary none/specified; alignment PENDING_INDEPENDENT_REVIEW). Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: LogRatioBound, RatioExponentBound, RestrictedLocalBudget, RestrictedTransformedPriceBudget, RhoTestSufficient, CapTestAtTwoThirds, Benchmark816TransformedRange, Benchmark816RationalPrice (8 entries below, label `proved`; 4 of them (P15Elementary.logRatioBound, P15Elementary.ratioExponentBound, P15Elementary.capTestAtTwoThirds, P15Structure.benchmark816RationalPrice) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 4 of them (P15Budget.restrictedLocalBudget, P15Budget.restrictedTransformedPriceBudget, P15Budget.rhoTestSufficient, P15Budget.benchmark816TransformedRange) added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN)). Theorem-shaped Props of this row still `specified` with no theorem: none. The intermediate ratio steps of (T3) are proved inside the theorems that use them; the integral representation behind (T2), the admissibility integral and the Section 5 checker program are not specified. The 3 main P15 arithmetic theorems listed as `proved` (the a = 1, n = 3 instance 16/27, finite instances of the (T3) bound and the rational identity 81801/3345620000) are at commit 91cffbb5 of main PR #358, not on main; they are not (T1). The dependency on claim p15-realized-covers is discharged only to the extent that row records. Disposition HOLD_WITH_DOMAIN (\"independent theorem review not yet source-bound in manifest\") is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -2370,8 +2478,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); the theorems added after PR head 79731b96 (module P15Interpolation) are receipted by run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 60, worktree CLEAN) and were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422; same trees as the heads", - "note": "19 theorem(s) of this row (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.sixSeventhsCertificate, P15Elementary.hStarExceedsOne, P15Elementary.phiAtPStar, P15Elementary.hazardStarPositive, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.demandOneBoundary, P15Elementary.capacityHazardIsBinomial, P15Elementary.sharpnessOfRhoStar, P15Hazard.globalAssemblyHazard, P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull, P15Interpolation.fullTransformedPriceBudget) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 60 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation}.lean, 59 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 8 of them (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.sixSeventhsCertificate, P15Elementary.hStarExceedsOne, P15Elementary.phiAtPStar, P15Elementary.hazardStarPositive, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.demandOneBoundary) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 3 (P15Elementary.capacityHazardIsBinomial, P15Elementary.sharpnessOfRhoStar, P15Hazard.globalAssemblyHazard) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). 8 (P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull, P15Interpolation.fullTransformedPriceBudget) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head. `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); the theorems added after PR head 79731b96 (module P15Interpolation) are receipted by run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 60, worktree CLEAN) and were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422; same trees as the heads; corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", + "note": "19 theorems of this row (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.sixSeventhsCertificate, P15Elementary.hStarExceedsOne, P15Elementary.phiAtPStar, P15Elementary.hazardStarPositive, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.demandOneBoundary, P15Elementary.capacityHazardIsBinomial, P15Elementary.sharpnessOfRhoStar, P15Hazard.globalAssemblyHazard, P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull, P15Interpolation.fullTransformedPriceBudget) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 8 of them (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.sixSeventhsCertificate, P15Elementary.hStarExceedsOne, P15Elementary.phiAtPStar, P15Elementary.hazardStarPositive, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.demandOneBoundary) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 3 (P15Elementary.capacityHazardIsBinomial, P15Elementary.sharpnessOfRhoStar, P15Hazard.globalAssemblyHazard) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). 8 (P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull, P15Interpolation.fullTransformedPriceBudget) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head. The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "FullTransformedPriceBudget states (F2) and is proved as stated; RhoStarEnclosure states (F3) and is proved as stated; SharpnessOfRhoStar states the attainment clause and is proved as stated. The claim-level label stays `specified` (claim-level vocabulary none/specified; a theorem of a Prop is evidence about that Prop, for the formal objects, not acceptance of Theorem F; alignment PENDING_INDEPENDENT_REVIEW). Theorem F is proved as FullTransformedPriceBudget states it: for the formal RealizedData record with a_i >= 1, d_i >= 2, every p in [0,1]^X, every 0 <= c_v <= phi(p_v) and the same K >= K_H(d), coverCost <= cappedHazard rho_star mu, through the coordinatewise concavity (CoordinatewiseConcavity, a theorem of exactly that Prop), (F6)-(F8), (F12) and (F13); that is evidence about those Props, not acceptance of the informal theorem. RhoStarEnclosure is proved by a rational certificate (Real.exp_bound series enclosures for e and for exp with explicit remainders, compared by norm_num), not by the source's full_price.py Fraction procedure, which remains unspecified. Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: FullTransformedPriceBudget, RhoStarEnclosure, RhoStarBelowSixSevenths, HazardStarPositive, CoordinatewiseConcavity, HazardProductBound, HazardPhiBound, FullGroundPriceBound, PhiAtPStar, FullGroundBudgetIff, CapacityHazardIsBinomial, OddMajorityIdentity, CapacityWorstCase, HStarExceedsOne, LocalPriceBoundFull, GlobalAssemblyHazard, SharpnessOfRhoStar, DemandOneBoundary, SixSeventhsCertificate (19 entries below, label `proved`; 8 of them (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.hazardStarPositive, P15Elementary.phiAtPStar, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.hStarExceedsOne, P15Elementary.demandOneBoundary, P15Elementary.sixSeventhsCertificate) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1); 3 of them (P15Elementary.capacityHazardIsBinomial, P15Hazard.globalAssemblyHazard, P15Elementary.sharpnessOfRhoStar) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN); 8 of them (P15Interpolation.fullTransformedPriceBudget, P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head). Theorem-shaped Props of this row still `specified` with no theorem: none; the parametrised predicates IsProperDownset are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. The derivative formulas (F4)-(F5) are not specified; CoordinatewiseConcavity, their consequence, is proved from the one-coordinate split of mu_p(A) (display (F4): affine in exp(-t_i) with coefficients 0 <= S_in <= S_out for a downset) and the second-derivative criterion for s |-> -log(a + b exp(-s)). SixSeventhsCertificate uses Mathlib's decimal bound on e in place of the source's series argument. The 7 main P15 arithmetic theorems listed as `proved` (finite arithmetic of the Section 6 certificate chain: adjacent 20-digit endpoints as Nat, 16/27 < 6/7, n >= 2a+1 instances, the rational orderings and the degree-5 Taylor partial sum) are at commit 91cffbb5 of main PR #358, not on main; they do not place rho_star = 1/(3 - log(3e - 2)) between the endpoints, do not prove e < 87/32 or 197/32 < exp(11/6) as real-number facts (the SixSeventhsCertificate theorem does, as that Prop states them), and are not Theorem F. The full_price.py Fraction procedure is not a kernel proof and is not specified. No arbitrary-downset or unrestricted P15 prize statement (original_prize_closed stays false). Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -2694,7 +2802,7 @@ ], "kernel_evidence": [], "not_established": "ParentUniformSelectionA, ParentCompactWindowDensityB and ParentUnrestrictedLeadingDensityC state the shapes of (1.1), (1.2), (1.3) for every object satisfying the interfaces; nothing is proved. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. The reading rule that the GRAPH binds to this node (congruence erratum E1, Section 9 replacement v1.1, marked-cylinder cap sections 1-5, reconciliation record with W1 and the embedding radius r < L/(4 sqrt 2)) is NOT transcribed into any hypothesis; the Props are stated from the parent bytes alone. Not stated: the regression law Q_{r,b,k,R}, the typed weight W_r, the full normalizer Z_r and Q^W (p_r is a primitive interface field), the marked Kac-Rice representation (Sections 9-10, radial ledger (10.2)), the Gaussian/angular representation (15.2) of c_{d,L} and (11.3) of c_{B,K}, (12.1)-(12.3), (14.1), (5.5), Sections 2-8. No numerical r_*, C, z_*, c_{B,K}, c_{d,L}; no lower bound for d >= 3. The GRAPH classification PROVED_REVIEWED and the review disposition ACCEPT_AT_EXISTENTIAL_SCOPE are source identity, not re-derived; the five validity premises of the hardening-branch claim graph are not touched. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p or the field: the parent-shape Props are predicates on the interface objects, not closed statements; the universal closure of ParentUniformSelectionA over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed.", - "next_step": "Pin ERRATUM_CONGRUENCE.md, REPAIR.md (Section 9 v1.1) and the reconciliation record, and add the reading rule as explicit hypotheses or separate Props; state (15.2) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of the three parent Props against (1.1)-(1.3) with the honesty checklist (quantifier order, existential constants outside the universal quantifiers, \"versions\", compact marks, all frames)." + "next_step": "Pin ERRATUM_CONGRUENCE.md, REPAIR.md (Section 9 v1.1) and the reconciliation record, and add the reading rule as explicit hypotheses or separate Props; state (15.2) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of the three parent Props against (1.1)-(1.3) with the honesty checklist (quantifier order, existential constants outside the universal quantifiers, \"versions\", compact marks, all frames). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ParentCompactWindowDensityB BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, expected-number-of-zeros or critical points; present but insufficient: gaussianReal (Real.lean), Measure.map (Map.lean)); ParentUnrestrictedLeadingDensityC BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: IsEquivalent (AsymptoticEquivalent.lean), ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean)); ParentUniformSelectionA BLOCKED_INTERFACE (p == 0 (docstring inhabitant, probe Lifetime-B_1)); HessianHasInertia, IsCandidatePair, IsCriticalPoint, IsFiniteBar, IsIndexSaddle, IsLPeriodic, IsLifetimeDensityVersion, IsMorseDistinct, IsNondegMax, IsSmooth, IsUnrestrictedLeadingCoefficient, IsUnrestrictedPair, IsWindowElderPair, IsWindowPair (Lifetime and Field copies) PREDICATE_VOCABULARY: vocabulary of the other Props, not a target." }, { "claim_id": "math.d1-component.congruence-erratum", @@ -2785,7 +2893,7 @@ ], "kernel_evidence": [], "not_established": "See row lifetime-remainder: (R1) is stated, not proved; the imports of parent interfaces D1-A-E are not transcribed; Theorem A's cubic bound (1.1) is not consumed by R and is not a hypothesis here. The GRAPH classification and review disposition are source identity only. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "As for row lifetime-remainder." + "next_step": "As for row lifetime-remainder. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemainderBoundedR1 BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean), ncard_le_ncard (Card.lean))." }, { "claim_id": "math.side24-coefficient", @@ -2833,8 +2941,8 @@ "note": "The 31 Side24 targets at origin/main f6deeba7 are decide-proved integer or cross-multiplied rational facts (manifest label `proved`; `kernel-checked` only in a run receipt). Arithmetic skeleton only: formal/SCOPE.md says none of them is the coefficient enclosure." } ], - "not_established": "See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the \"evaluates_eq_15_2\" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. The 27 arithmetic theorems of main and the 5 proof-packet theorems (ExponentialBounds, SuccessiveTermRatio, ConeMomentD2Algebra, ReferenceCoefficientIdentity, ElementaryConeIntegral; none of them the enclosure; kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)) are listed on the landing row. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed.", - "next_step": "As for row side24-coefficient." + "not_established": "See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the \"evaluates_eq_15_2\" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. Of the three statements of this node, none has a theorem; stated, not proved: CoefficientEnclosure, ReferenceComparison. The 27 arithmetic theorems of main and the 17 proof-packet theorems of the landing row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral, Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw; none of them the enclosure; 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them) are listed on the landing row, never here and never as the node; the interface theorems among them (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw) hold for every inhabitant of ConeMomentInterface from its law fields alone and construct nothing. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited.", + "next_step": "As for row side24-coefficient. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): CoefficientEnclosure BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface elaborates, by doubling coneMoment and c24, so no theorem of its type can exist unless the interface is empty); ReferenceComparison BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, ReferenceComparison I) -> IsEmpty PeriodicCoefficientInterface elaborates likewise)." }, { "claim_id": "math.rn-count-interface", @@ -2941,8 +3049,8 @@ "kernel_evidence": [ { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3)", - "note": "5 theorem(s) of this row (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 60 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation}.lean, 59 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", + "note": "5 theorems of this row (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "See row rn-count-interface: (N1)-(N5) stated and each proved as a theorem of the proof packet (5 entries below, `proved`; kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)), (N6)-(N7) not stated, so no Prop states the interface as a whole and the node stays none; nothing about the Gaussian model is proved. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -2989,7 +3097,7 @@ ], "kernel_evidence": [], "not_established": "See row rn-fixed-remote-window: displays (2)-(3) stated over PinnedLawInterface, not proved; Section 6 not stated; no inhabitant of the interface. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "As for row rn-fixed-remote-window." + "next_step": "As for row rn-fixed-remote-window. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point)." }, { "claim_id": "math.rn-region.fixed-remote", @@ -3023,7 +3131,7 @@ ], "kernel_evidence": [], "not_established": "The region's mathematical content is the fixed-remote Theorem A at exactly this scope, stated by A2/A3 and not proved. The coverage classification itself (COVERED_BY_CANDIDATE, coverage_source) is graph bookkeeping and is not a Lean statement. No all-height bound on the remote region, no rho -> 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "As for row rn-fixed-remote-window; nothing region-specific to add." + "next_step": "As for row rn-fixed-remote-window; nothing region-specific to add. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point)." }, { "claim_id": "math.rn-region.mesoscopic-scaled-annulus", @@ -3057,7 +3165,7 @@ ], "kernel_evidence": [], "not_established": "D5dSummedPinNeighbourhoodC2 states the summed pin-neighbourhood bound (all heights, scaled ball of radius R minus the pins) and AnnulusBridgeAllHeight2 the all-height fixed-annulus bridge; neither is proved, and the \"covers the reviewed fixed annulus\" reading (R >= B) is not a Lean statement. No d > 2, no numerical constants, no elder selection, no JETMOD or witness-collision closure. The eight reading-rule component nodes are source identity, not transcribed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "Independent reviewer: alignment of D5dSummedPinNeighbourhoodC2 with row D5-d; proof lane: none yet (the composition D5-a + D5-c at eta = 1/4 is a proof, not a statement)." + "next_step": "Independent reviewer: alignment of D5dSummedPinNeighbourhoodC2 with row D5-d; proof lane: none yet (the composition D5-a + D5-c at eta = 1/4 is a proof, not a statement). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): AnnulusBridgeAllHeight2 BLOCKED_INTERFACE (a Dirac-tilted law at a pinned smooth periodic path with one extra nondegenerate critical point in the annulus violates display (2) for a null Borel E); D5dSummedPinNeighbourhoodC2 BLOCKED_INTERFACE (a Dirac-law inhabitant with one extra nondegenerate critical point at the midpoint gives count >= 1 > C r^3 as r -> 0)." }, { "claim_id": "math.rn-region.pin-collision", @@ -3100,7 +3208,7 @@ ], "kernel_evidence": [], "not_established": "The three row shapes are stated for any object satisfying PinnedLawInterface and are not proved; the P2 source (PUNCTURED_PIN_PROOF.md) is not pinned here, so the Props are anchored to the reconciliation's transcription of the rows, not to the proof's own bytes. Including the deterministic pin atoms would invalidate the vanishing bounds (the reconciliation's caveat); the Props puncture the disks accordingly. No d > 2, no numerical constants, no uniformity in kappa. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "Pin PUNCTURED_PIN_PROOF.md and COLLAR_PROOF.md (reviews/d5_local_collar_20260928) and re-anchor D5a/D5b/D5c to their own displays; independent reviewer: alignment of the three Props with rows D5-a, D5-b, D5-d." + "next_step": "Pin PUNCTURED_PIN_PROOF.md and COLLAR_PROOF.md (reviews/d5_local_collar_20260928) and re-anchor D5a/D5b/D5c to their own displays; independent reviewer: alignment of the three Props with rows D5-a, D5-b, D5-d. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): BLOCKED_INTERFACE (interface under-determination, not statement defects; each refuted by a concrete Dirac-law inhabitant): D5aPuncturedPinDisks (extra critical point in the punctured disk, null E), D5bMicrodisk (extra critical point at distance r^2 from M), D5dSummedPinNeighbourhoodC2 (extra critical point at the midpoint, count >= 1 > C r^3)." }, { "claim_id": "math.rn-region.intermediate-r-to-rho", @@ -3125,7 +3233,7 @@ ], "kernel_evidence": [], "not_established": "D5eIntermediateShellsI3I4 states the window-only bound E N <= C r^3 (A_0^(-2) + rho^2) on {A_0 r <= |X| <= rho} with side conditions taken from dimension-lift (1.6); not proved. The I3/I4 source (frontiers/intermediate_window_20260928) is not pinned here; no all-height intermediate count exists or is claimed; no numerical constants. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "Pin intermediate_window_20260928/PROOF.md and anchor the single-shell display there; independent reviewer: alignment of D5eIntermediateShellsI3I4 with row D5-e and with dimension-lift (1.6)." + "next_step": "Pin intermediate_window_20260928/PROOF.md and anchor the single-shell display there; independent reviewer: alignment of D5eIntermediateShellsI3I4 with row D5-e and with dimension-lift (1.6). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): D5eIntermediateShellsI3I4 BLOCKED_INTERFACE (a Dirac-law inhabitant with one extra window-height critical point at distance 8r from the midpoint)." }, { "claim_id": "math.rn-region.witness-collision", @@ -3159,7 +3267,7 @@ ], "kernel_evidence": [], "not_established": "WitnessCollisionNearPinPairsTarget states the OPEN obligation as a target Prop, not as a result: nothing asserts that the target holds, is attainable, or follows from any reviewed source. RemoteCollisionSecondFactorialD states only the fixed-remote, fixed-eta part (Corollary D of remote_collision), which the GRAPH records as reviewed evidence and NOT as a proof of this node. The Palm route's Theorem Q (q = 2) would imply the target but is an author-side candidate stated separately (PalmRouteFactorialMomentsQ) and not consumed here; the c6_witness_collision reconciliation's proposed discharge is declarative with execution pending. The node stays OPEN_ACTIVE; this register cannot and does not change that. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "None for Lean beyond the target statement: the obligation needs a source-bound proof and nonauthor review in Math-; if Theorem Q is folded, re-anchor the target to the fold record." + "next_step": "None for Lean beyond the target statement: the obligation needs a source-bound proof and nonauthor review in Math-; if Theorem Q is folded, re-anchor the target to the fold record. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemoteCollisionSecondFactorialD BLOCKED_INTERFACE (a Dirac inhabitant with two prescribed remote critical points falsifies both displays; no field gives a.s. absence of index-j window pairs on null remote sets); WitnessCollisionNearPinPairsTarget BLOCKED_INTERFACE (a Dirac inhabitant with two window critical points within eta of a pin gives expectation 2 for all small r; consistent with the node's OPEN status)." }, { "claim_id": "math.rn-mesoscopic-reduction", @@ -3313,11 +3421,11 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); the theorems added after PR head 79731b96 (module P15Interpolation) are receipted by run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 60, worktree CLEAN) and were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422; same trees as the heads", - "note": "3 theorem(s) of this row (P15Elementary.sharpnessOfRhoStar, P15Interpolation.rhoStarEnclosure, P15Interpolation.fullTransformedPriceBudget) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 60 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation}.lean, 59 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 1 (P15Elementary.sharpnessOfRhoStar) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). 2 (P15Interpolation.rhoStarEnclosure, P15Interpolation.fullTransformedPriceBudget) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head. `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); the theorems added after PR head 79731b96 (module P15Interpolation) are receipted by run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 60, worktree CLEAN) and were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422; same trees as the heads; corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", + "note": "3 theorems of this row (P15Elementary.sharpnessOfRhoStar, P15Interpolation.rhoStarEnclosure, P15Interpolation.fullTransformedPriceBudget) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 1 (P15Elementary.sharpnessOfRhoStar) was added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). 2 (P15Interpolation.rhoStarEnclosure, P15Interpolation.fullTransformedPriceBudget) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head. The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], - "not_established": "See row p15-full-price. Of the three Props of this node, with a theorem of exactly their type in the proof packet: FullTransformedPriceBudget, RhoStarEnclosure, SharpnessOfRhoStar (3 entries below, label `proved`; 1 of them (P15Elementary.sharpnessOfRhoStar) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN); 2 of them (P15Interpolation.fullTransformedPriceBudget, P15Interpolation.rhoStarEnclosure) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head); none of the three is left unproved. A theorem of SharpnessOfRhoStar is about the formal triple block and the formal cover cost; a theorem of FullTransformedPriceBudget is about the formal RealizedData record, downset, coverCost and mu; none of this is Theorem F, and the node-level label stays `specified`. The 7 main P15 arithmetic theorems of source p15-full-price (at commit 91cffbb5 of main PR #358, not on main) are listed here as the arithmetic skeleton, never as the node. The price-boundary refutation is a scope boundary (GRAPH edge boundary_counterexample_not_required_premise), not a hypothesis. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", + "not_established": "See row p15-full-price. Of the three Props of this node, with a theorem of exactly their type in the proof packet: FullTransformedPriceBudget, RhoStarEnclosure, SharpnessOfRhoStar (3 entries below, label `proved`; 1 of them (P15Elementary.sharpnessOfRhoStar) was added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN); 2 of them (P15Interpolation.fullTransformedPriceBudget, P15Interpolation.rhoStarEnclosure) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head); none of the three is left unproved. A theorem of SharpnessOfRhoStar is about the formal triple block and the formal cover cost; a theorem of FullTransformedPriceBudget is about the formal RealizedData record, downset, coverCost and mu; none of this is Theorem F, and the node-level label stays `specified`. The 7 main P15 arithmetic theorems of source p15-full-price (at commit 91cffbb5 of main PR #358, not on main) are listed here as the arithmetic skeleton, never as the node. The price-boundary refutation is a scope boundary (GRAPH edge boundary_counterexample_not_required_premise), not a hypothesis. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", "next_step": "As for row p15-full-price." }, { @@ -3430,8 +3538,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3)", - "note": "2 theorem(s) of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 60 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation}.lean, 59 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 2 of them (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", + "note": "2 theorems of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 2 of them (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "See row p15-price-boundary: TwoCoordGeneratorPrices and DemandOneRefutation (the two Props of this node) have theorems of exactly their type in the proof packet (listed as `proved`; kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)), the two logarithm inequalities included; the claim-level label stays `specified` for the reasons given there (claim-level vocabulary none/specified, alignment PENDING_INDEPENDENT_REVIEW, a Prop-level proof is evidence about the Prop only, whatever the hosted run 38116277597 receipted). The 6 main P15 arithmetic theorems of source p15-price-boundary (at commit 91cffbb5 of main PR #358, not on main) are listed as the arithmetic skeleton, never as the node. REFUTED is the graph classification of the extension, transcribed as source identity. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -3475,7 +3583,7 @@ ], "kernel_evidence": [], "not_established": "See row rn-fixed-annulus-window: (A2) stated, not proved; no global D5/RN/JETMOD, k -> 0, d >= 3, pin-neighbourhood, intermediate-scale, collision or elder-selection closure. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "As for row rn-fixed-annulus-window." + "next_step": "As for row rn-fixed-annulus-window. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedAnnulusWindowCandidateA2 BLOCKED_INTERFACE (a Dirac inhabitant with an index-j window critical point on a null E0 (a singleton in the annulus); no field of PinnedLawInterface supplies the a.s. absence that the weighted Kac-Rice formula (A21) gives)." }, { "claim_id": "math.rn-region.fixed-annulus-window", @@ -3500,7 +3608,7 @@ ], "kernel_evidence": [], "not_established": "The region's content is (A2) at its exact scope, stated and not proved; the coverage classification is graph bookkeeping, not a Lean statement; the superseded high-jet route (regional.fixed-annulus.high-jet-route, SUPERSEDED_NONBLOCKING) is not represented. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "As for row rn-fixed-annulus-window." + "next_step": "As for row rn-fixed-annulus-window. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedAnnulusWindowCandidateA2 BLOCKED_INTERFACE (a Dirac inhabitant with an index-j window critical point on a null E0 (a singleton in the annulus); no field of PinnedLawInterface supplies the a.s. absence that the weighted Kac-Rice formula (A21) gives)." }, { "claim_id": "math.d5-pin-neighborhood-first-moment", @@ -3570,7 +3678,7 @@ ], "kernel_evidence": [], "not_established": "The six row shapes are stated for any object satisfying PinnedLawInterface (sample space Omega, gated pin regime 0 < r < L) and are not proved. The consumption graph (D5-d from D5-a + D5-c, D5-f from D5-d + I4 + remote Theorem A) is a proof structure and is not stated; the component sources PUNCTURED_PIN_PROOF.md, COLLAR_PROOF.md and intermediate_window PROOF.md are not pinned here, so D5a/b/c/e/f are anchored to the reconciliation's row text. The lower bound c r^3 <= P(A_r) of D5-f has its source in remote_collision Corollary F-, not pinned. No d > 2, no numerical constants, no uniformity in T, R, eta, k -> 0; no elder pairing; no RN/24-jet/JETMOD; the witness-collision region stays OPEN_ACTIVE. The STATUS.md note that D5 GRAPH transitions were not executed by that summary is a recorded discrepancy, not resolved here. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "Pin the three component proofs and re-anchor; independent reviewer: alignment of the six Props with the section 2 table (all-height versus window I_r per row, exclusion of M and S)." + "next_step": "Pin the three component proofs and re-anchor; independent reviewer: alignment of the six Props with the section 2 table (all-height versus window I_r per row, exclusion of M and S). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): BLOCKED_INTERFACE (interface under-determination, not statement defects; each refuted by a concrete Dirac-law inhabitant): D5aPuncturedPinDisks (extra critical point in the punctured disk, null E), D5bMicrodisk (extra critical point at distance r^2 from M), D5cCollarC1 (extra critical point at the midpoint, E = {0}), D5dSummedPinNeighbourhoodC2 (extra critical point at the midpoint, count >= 1 > C r^3), D5eIntermediateShellsI3I4 (extra window-height critical point at distance 8r from the midpoint), D5fGlobalWindowFirstMomentI5 (one or no extra window critical point refutes either direction; the lower bound is Set.ncard-sensitive)." }, { "claim_id": "math.d5-component.punctured-pin-proof", @@ -3604,7 +3712,7 @@ ], "kernel_evidence": [], "not_established": "The Props state rows D5-a and D5-b as the reconciliation transcribes P2, not P2 from its own bytes (not pinned); the slices P4, P9, P17, P20 and the continuum argument reviewed are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "Pin PUNCTURED_PIN_PROOF.md (SHA-256 f972e50b… (GRAPH.json fingerprint)) and anchor D5a/D5b to its own displays." + "next_step": "Pin PUNCTURED_PIN_PROOF.md (SHA-256 f972e50b… (GRAPH.json fingerprint)) and anchor D5a/D5b to its own displays. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): D5aPuncturedPinDisks BLOCKED_INTERFACE (a Dirac-law inhabitant with one extra critical point in the punctured disk and a null E); D5bMicrodisk BLOCKED_INTERFACE (the same Dirac-law inhabitant with the extra critical point at distance r^2 from M (kappa = 1))." }, { "claim_id": "math.d5-component.punctured-pin-review-111", @@ -3749,7 +3857,7 @@ ], "kernel_evidence": [], "not_established": "D5cCollarC1 states row D5-c as the reconciliation transcribes C1, not C1 from its own bytes (not pinned); the normalizer (P14) it imports and the corollary C2 are not stated separately; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "Pin COLLAR_PROOF.md (SHA-256 794babe0… (GRAPH.json fingerprint)) and anchor D5cCollarC1 to display (C1)." + "next_step": "Pin COLLAR_PROOF.md (SHA-256 794babe0… (GRAPH.json fingerprint)) and anchor D5cCollarC1 to display (C1). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): D5cCollarC1 BLOCKED_INTERFACE (a Dirac-law inhabitant with an extra critical point at the midpoint and E = {0} in the collar)." }, { "claim_id": "math.d5-component.annulus-proof", @@ -3774,7 +3882,7 @@ ], "kernel_evidence": [], "not_established": "AnnulusBridgeAllHeight2 states display (2) (d = 2, fixed L, compact birth, k_- > 0, 1 < A < B, all heights) over PinnedLawInterface; not proved. Not stated: displays (4)-(20), the R1-R7 review interfaces as separate statements, the deterministic three-Hessian lemma, the crossover factor min(1,(r/|v|)^6). No matching lower bound, no numerical constant or radius, no pin neighbourhoods, no k -> 0 uniformity, no d >= 3, no fixed-remote all-height bound, no all-scales RN theorem. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "State the R1-R7 interfaces as Props; independent reviewer: alignment of AnnulusBridgeAllHeight2 with display (2), checking \"all heights\" and the uniformity over E and j." + "next_step": "State the R1-R7 interfaces as Props; independent reviewer: alignment of AnnulusBridgeAllHeight2 with display (2), checking \"all heights\" and the uniformity over E and j. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): AnnulusBridgeAllHeight2 BLOCKED_INTERFACE (a Dirac-tilted law at a pinned smooth periodic path with one extra nondegenerate critical point in the annulus violates display (2) for a null Borel E)." }, { "claim_id": "math.d5-component.intermediate-window-proof", @@ -3808,7 +3916,7 @@ ], "kernel_evidence": [], "not_established": "The Props state rows D5-e and D5-f as the reconciliation transcribes I3/I4/I5, not from the proof's own bytes (not pinned); the tiling N1 of the Claude review and the planar F+ lower bound source are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "Pin intermediate_window_20260928/PROOF.md (SHA-256 b3eb9456… (GRAPH.json fingerprint)) and anchor the shell displays there." + "next_step": "Pin intermediate_window_20260928/PROOF.md (SHA-256 b3eb9456… (GRAPH.json fingerprint)) and anchor the shell displays there. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): D5eIntermediateShellsI3I4 BLOCKED_INTERFACE (a Dirac-law inhabitant with one extra window-height critical point at distance 8r from the midpoint); D5fGlobalWindowFirstMomentI5 BLOCKED_INTERFACE (Dirac-law inhabitants refute both directions (one extra window critical point: probability 1 > C r^3; none: probability 0 < c r^3), and the lower bound is sensitive to the Set.ncard convention (infinitely many window critical points count as 0))." }, { "claim_id": "math.d5-component.remote-window-proof", @@ -3842,7 +3950,7 @@ ], "kernel_evidence": [], "not_established": "See rows rn-fixed-remote-window and math.rn-fixed-remote-window; the D5 consumption at rho = s_0 is a proof step, not stated. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "As for row rn-fixed-remote-window." + "next_step": "As for row rn-fixed-remote-window. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point)." }, { "claim_id": "D1-THM-A", @@ -3876,7 +3984,7 @@ ], "kernel_evidence": [], "not_established": "ParentUniformSelectionA states the shape of (1.1): existential r_* > 0, C < infinity outside the universal quantifiers over 0 < r <= r_*, b in [bLo,bHi], k in [kLo,kHi] and all frames, for the interface field p of PairLawInterface. Not stated or proved: the full-normalizer floor Z_r >= z_* r^2 (5.5), the transverse-matrix boundary layer, the regression law and typed weight, the reading-rule amendments (E1, Section 9 v1.1, W1, r < L/(4 sqrt 2)), any numerical r_*, C, z_*. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. No globally uniform constant over unbounded b, k; no lower bound. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: ParentUniformSelectionA is a predicate on the interface objects, not a closed statement; its universal closure over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed.", - "next_step": "Independent reviewer: alignment of ParentUniformSelectionA with (1.1) under the honesty checklist; formal lane: none until the regression law is constructed." + "next_step": "Independent reviewer: alignment of ParentUniformSelectionA with (1.1) under the honesty checklist; formal lane: none until the regression law is constructed. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ParentUniformSelectionA BLOCKED_INTERFACE (the falsifying inhabitant p == 0 named in the docstring, confirmed by elaboration (probe Lifetime-B_1))." }, { "claim_id": "D1-THM-B", @@ -3928,7 +4036,7 @@ ], "kernel_evidence": [], "not_established": "ParentCompactWindowDensityB states the shape of (1.2) (asymptotics of versions of nu_cand and nu_eld, c_{B,K} > 0, the O(ell^(2/3)) difference) for compact B, K of positive length; not proved. Not stated: the exact coefficient formula (11.3), the radial ledger (10.2), the pushforward r = (ell/k)^(1/3); the distance cutoff is quantified existentially and not identified with the r_* of (1.1). The difference is NOT asserted for unrestricted densities; no numerical c_{B,K}, C_{B,K}; the historical H3 finite-band floor is not re-certified. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "State (11.3) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of ParentCompactWindowDensityB with (1.2), including the positive-length requirement." + "next_step": "State (11.3) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of ParentCompactWindowDensityB with (1.2), including the positive-length requirement. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ParentCompactWindowDensityB BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, expected-number-of-zeros or critical points; present but insufficient: gaussianReal (Real.lean), Measure.map (Map.lean)); IsLifetimeDensityVersion, IsWindowElderPair, IsWindowPair PREDICATE_VOCABULARY: vocabulary of the other Props, not a target." }, { "claim_id": "D1-THM-C", @@ -3971,7 +4079,7 @@ ], "kernel_evidence": [], "not_established": "ParentUnrestrictedLeadingDensityC states the shape of (1.3) (leading term only) with the coefficient as an abstract positive constant c in IsUnrestrictedLeadingCoefficient; not proved. The Gaussian/angular representation (15.2), the target-growth majorant (13.3)-(13.6), the integrability (13.5) and the off-diagonal bound (14.1) are not stated; the SIDE24 enclosure of c_{d,24} is a separate row. No unrestricted remainder or difference rate (that is Theorem R), no elementary closed form. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "State (15.2) and (14.1) as Props; independent reviewer: alignment of ParentUnrestrictedLeadingDensityC with (1.3), in particular the exclusion of the essential global-maximum class." + "next_step": "State (15.2) and (14.1) as Props; independent reviewer: alignment of ParentUnrestrictedLeadingDensityC with (1.3), in particular the exclusion of the essential global-maximum class. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ParentUnrestrictedLeadingDensityC BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: IsEquivalent (AsymptoticEquivalent.lean), ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean)); IsUnrestrictedLeadingCoefficient, IsUnrestrictedPair PREDICATE_VOCABULARY: vocabulary of the other Props, not a target." }, { "claim_id": "OA-ELDER-LOWER-DENSITY-GAP-20260928-v1", @@ -4014,7 +4122,7 @@ ], "kernel_evidence": [], "not_established": "PlanarElderLowerE2 states (E2) for d = 2 over PairLawInterface; PlanarTwoSidedRateE4 states (E4) CONDITIONALLY on the parent (1.1) as the source's Corollary E2 does; PlanarDensityGapE11 states (E11) under the same (1.1) hypothesis, which the source needs only for the upper half (the lower half is thereby stated under a hypothesis the source does not need, disclosed in the docstring). None is proved. Not stated: (E3), (E7)-(E10), the polygonal-path and tilted jet-box argument. No lower bound for d >= 3 (separate packets CL-ELDER-LOWER-ALL-D / OA-ELDER-DIMENSION-LIFT, not extracted here); no numerical c or r_*. The review verdict of reviews/d1_elder_lower_claude_20260928 is not transcribed here; the pinned bytes carry their own disposition line. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: PlanarElderLowerE2, PlanarTwoSidedRateE4 and PlanarDensityGapE11 are predicates on the interface objects, not closed statements; the universal closure of E2 and E4 over all inhabitants is false whenever the interfaces are inhabited (take p identically 1), and `specified` refers to the statement shape at the source's object, which is not constructed.", - "next_step": "Independent reviewer: record whether (E11)'s lower half should be stated without the (1.1) hypothesis (a narrower-scope verdict is the reviewer's to write); transcribe the review verdict once pinned." + "next_step": "Independent reviewer: record whether (E11)'s lower half should be stated without the (1.1) hypothesis (a narrower-scope verdict is the reviewer's to write); transcribe the review verdict once pinned. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): PlanarDensityGapE11 BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: basisFun (PiL2.lean), ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean); its hypothesis ParentUniformSelectionA is inert (no PairLawInterface field ties p to G or I; p == 1 satisfies it), so a theorem of E11 would be a theorem of its conclusion alone); PlanarElderLowerE2 BLOCKED_INTERFACE (the falsifying inhabitant p == 1 named in the docstring, confirmed by elaboration (probe Lifetime-B_1)); PlanarTwoSidedRateE4 BLOCKED_INTERFACE (p == 1 satisfies the hypothesis and the lower half fails (probe Lifetime-B_1))." }, { "claim_id": "two-scale-s6-s21", @@ -4111,7 +4219,7 @@ ], "kernel_evidence": [], "not_established": "The four theorem shapes (1.2), (1.3), (1.5)-(1.6), (1.7) are stated over PinnedLawInterface for fixed d >= 2 and not proved. Not stated: display (1.4), Corollaries F_d (1.8) and M_d (1.9), the five devices D1-D5, the consumed sources ([LP] sections 2-5, (5.5), the planar templates, [RM], [RC], [EDL], [LOCAL]). No all-height intermediate or global O(r^3) count; no torus-wide second factorial moment; no elder selection; no uniformity in k, L, d; no numerical C, c, r_*, s_0; no recovery of historical RN/JETMOD/24-jet carriers. The reviews (Math-#141 OpenAI, source-exposed) are source identity; the 2026-10-05 erratum may amend these statements and was not read. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "Read reviews/d5_dimension_lift_erratum_20261005 and amend the four Props if the displays changed; state Corollaries F_d and M_d; independent reviewer: alignment of the four Props with (1.2)-(1.7)." + "next_step": "Read reviews/d5_dimension_lift_erratum_20261005 and amend the four Props if the displays changed; state Corollaries F_d and M_d; independent reviewer: alignment of the four Props with (1.2)-(1.7). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): BLOCKED_INTERFACE (interface under-determination, not statement defects; each refuted by a concrete Dirac-law inhabitant): DimensionLiftCompactCollarC (extra critical point at the midpoint, E = {0}, every d >= 2), DimensionLiftGlobalWindowG (extra window-height critical point at the midpoint), DimensionLiftIntermediateShellsI (extra window-height critical point at distance 8r from the midpoint), DimensionLiftPuncturedPinBallP (extra critical point in the punctured pin ball, null E)." }, { "claim_id": "OA-C6-FOURIER-20260929-v1", @@ -4136,7 +4244,7 @@ ], "kernel_evidence": [], "not_established": "No Prop states Theorem F (the cap functional Psi_F and its stretched-exponential tail (F1)-(F2)), Theorem W in general d and p (F3), or Proposition B; only the planar consequence (F4) is stated, GIVEN the hypothesis G_2 (E N <= C r^3), so the claim as a whole is not specified. Nothing is proved; the logarithm is neither shown sharp nor removable here; no numerical cutoff or parameter-uniform constants; the abstract barrier is not a Gaussian counterexample. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "State (F1)-(F2) with Psi_F as an interface field (a measurable functional dominating the total critical count) and (F3) for general d, p; then the claim would be `specified`." + "next_step": "State (F1)-(F2) with Psi_F as an interface field (a measurable functional dominating the total critical count) and (F3) for general d, p; then the claim would be `specified`. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FourierCutoffPlanarSecondFactorialF4 BLOCKED_INTERFACE (the antecedent G_2 plus the interface fields admit a rare large cluster (two-point law, cluster size >= 1/r with probability r^3/n_r) with E N(N-1) of order r^2 while E N = r^3; Theorem W's Gaussian input is not transcribed)." }, { "claim_id": "CL-C6-PALM-20260929-v1", @@ -4170,7 +4278,7 @@ ], "kernel_evidence": [], "not_established": "PalmRouteFactorialMomentsQ states (1.1) (with r_* allowed to depend on q) and PalmRouteSecondFactorialTheta states (1.2) over PinnedLawInterface; not proved. Corollary P (size-biased law) and Sections 3-7 (the cap Psi, the parametric Rouche tail, the marked Kac-Rice formula with Hoelder insertion, regimes R1-R4) are not stated; the dependence on [C6L] and [DL] at their bytes is not transcribed; the lower bounds' source [EDL] (A4) is not stated. No lower bound for q >= 3; no distribution of N beyond factorial moments; no elder selection; no numerical constants; for d >= 3 conditional on [DL]. The GRAPH records that this global result is not a proof of the regional witness-collision node. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "State Corollary P; independent reviewer: alignment of the two Props with (1.1)-(1.2), including whether r_* may depend on q." + "next_step": "State Corollary P; independent reviewer: alignment of the two Props with (1.1)-(1.2), including whether r_* may depend on q. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): PalmRouteFactorialMomentsQ BLOCKED_INTERFACE (a Dirac inhabitant whose single path has two window critical points off the pins gives E (N)_2 = 2 for all small r); PalmRouteSecondFactorialTheta BLOCKED_INTERFACE (the lower bounds c r^3 fail for an inhabitant with no window critical points off the pins and for one with infinitely many (windowCountAll uses Set.ncard, 0 on infinite sets))." }, { "claim_id": "OA-C6-RARE-CLUSTER-20260929-v1", @@ -4191,11 +4299,26 @@ "formal_progress": "specified", "anchor": "(a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3.", "does_not_claim": "the result (Theorem O) or its proof; that `2a ≤ A` (derived in the source, not a hypothesis here); that H1 or H2 hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment matching or the independent-replica limit; the conditional-law liminf remark; that the upper half is the source's infimum bound verbatim: `∃ lam ≥ 0, dTV(Law N_r, Pois(lam)) ≤ A r^3` implies `inf_{lam ≥ 0} dTV ≤ A r^3` and is stronger unless the infimum is attained, and the existential form is used because the source's proof exhibits the witness `lam = 0` (the source: The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p); nonauthor review or acceptance." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/RNCluster.lean", + "declaration": "UniversalLaw.Proofs.RNCluster.rareClusterPoissonObstructionO", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "(a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3.", + "does_not_claim": "the informal Theorem O (3.1) of the source as the source's statement (the alignment of `pmfTV`, `poissonPMF` and the family `μ` with the source's `dTV`, `Pois(lambda)` and `Law N_r` is PENDING_INDEPENDENT_REVIEW); the result (Theorem O) or its proof as a reviewed fact; `2a ≤ A` as a registered statement (it is derived internally from the antecedents at `r0`, as in the source, and is neither a hypothesis nor a conclusion of the Prop); that (H1) or (H2) hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment matching or the independent-replica limit; the conditional-law liminf remark; that the upper half is the source's infimum bound verbatim (the Prop's existential form with witness `lam = 0` is what is proved); the statement module's own `Does not claim:` text (the statement packet is not edited here); the disposition of `OA-C6-RARE-CLUSTER-20260929-v1`, for which the register row transcribes none (`disposition_transcribed: null`; title: PROOF_INDEX reviewed scoped result, conditional packet) and which is in any case transcribed there, not derived here; nonauthor review or acceptance of OA-C6-RARE-CLUSTER-20260929-v1." } ], - "kernel_evidence": [], - "not_established": "Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified. Nothing is proved; H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count; for A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); Theorem O is a candidate kernel target (finite total-variation arithmetic)." + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "receipt_or_workflow": "Hosted receipt: none yet. The proof-packet theorem of this row (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60, which does not name it); the next hosted lean-specification-proofs.yml run is the first receipt.", + "note": "1 theorem of this row (RNCluster.rareClusterPoissonObstructionO) is inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 1 (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them. `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + } + ], + "not_established": "Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified and the claim-level label stays `none` (the register admits only none/specified at claim level, and this row's claim is the packet, not Theorem O alone). RareClusterPoissonObstructionO has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (1 entry below, label `proved`; 1 of them (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them): it proves Theorem O's inequality as the Prop states it, for every family mu and constants a, A satisfying the Prop's own antecedents (0 < a, nonnegativity, summability and total mass one of each mu r, summable mean, (H1), (H2) on (0, r0]) and no others: the upper half with the witness lam = 0 (the Prop's existential form, not the source's infimum verbatim), the lower half through tail events {n >= 2} and {n >= 1}; 2a <= A is derived internally from the antecedents at r0, as in the source, and is neither a hypothesis nor a conclusion of the Prop. That theorem is evidence about the Prop, for the formal objects pmfTV, poissonPMF and the family mu : R -> N -> R, not acceptance of Theorem O; alignment of those objects with the source's dTV, Pois(lambda) and Law N_r is PENDING_INDEPENDENT_REVIEW. H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count (the Prop does not tie mu to any field; the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement lambda = mu, Theorems C and S, the factorial-moment matching, the independent-replica limit (7.1) and the conditional-law liminf remark are neither stated nor proved. For A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics; the hypotheses imply 2a <= A, so A > 0 is derivable and the Prop is not vacuous in general (the triage's example mu r = (1 - r^3) delta_0 + r^3 delta_2 with a = 1, A = 2 satisfies the antecedents for r <= 1). Disposition: none transcribed (disposition_transcribed null; the title's \"PROOF_INDEX reviewed scoped result\" is the source's wording, not a status decided here). Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", + "next_step": "State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); the 1 theorem of this row added after PR head 12e072d6 (RNCluster.rareClusterPoissonObstructionO) await the next hosted lean-specification-proofs.yml run, their first receipt. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RareClusterPoissonObstructionO PROVABLE_NOW: proved in round 4 (see entries). Independent reviewer: alignment of RareClusterPoissonObstructionO with Theorem O (3.1), in particular the existential form of the upper half and the (H1), (H2) antecedents on (0, r0]." }, { "claim_id": "CL-D5-REMOTE-COLLISION-20260928-v1", @@ -4220,7 +4343,7 @@ ], "kernel_evidence": [], "not_established": "Only Corollary D (second factorial moment <= C r^5 |E| in the fixed remote region) is stated; Theorem C (ordered near-pair count), Corollaries E and F (event asymptotics, torus-wide lower bound) are not, so the packet is not specified as a whole. Nothing proved; pairs with a witness near the pins are excluded; no torus-wide second factorial moment; this is explicitly NOT the open witness-collision mechanism. Author Anthropic/Claude: same provider as this packet, independence credit 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", - "next_step": "State Theorem C and Corollaries E, F; independent (non-Anthropic) reviewer for both the source and the Props." + "next_step": "State Theorem C and Corollaries E, F; independent (non-Anthropic) reviewer for both the source and the Props. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemoteCollisionSecondFactorialD BLOCKED_INTERFACE (a Dirac inhabitant with two prescribed remote critical points falsifies both displays; no field gives a.s. absence of index-j window pairs on null remote sets)." }, { "claim_id": "cumulative-transfer-correction", diff --git a/specifications/lean_statements_20261010/SCOPE.md b/specifications/lean_statements_20261010/SCOPE.md index 5e809d269..828e88406 100644 --- a/specifications/lean_statements_20261010/SCOPE.md +++ b/specifications/lean_statements_20261010/SCOPE.md @@ -6,8 +6,9 @@ Scientific effect: **NONE**. Formal progress: `specified` for every declaration. ## Boundary This package states; it does not prove. Proofs of some of its Props live in the sibling proof packet -`specifications/lean_statement_proofs_20261011` (60 theorems typed exactly as 59 of the 141 Props; -the last column of the declaration tables and the `proved (proof packet …)` entries of the claims table); +`specifications/lean_statement_proofs_20261011` (73 theorems typed exactly as 72 of the 141 Props; +the last column of the declaration tables and the `proved (proof packet …)` entries of the claims table, where † marks +a theorem added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…) with no hosted receipt yet); that packet depends on this one by path, never edits it, and its theorems change no label here and no claim-level label. The five modules under `Specifications/` declare, as `def … : Prop` and interface `structure`s in the namespaces `UniversalLaw.Spec.`, the @@ -59,11 +60,11 @@ a theorem between string literals, an indented Prop definition, a proof written ## Declarations -150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified` in this package; the 123 auxiliary definitions and abbreviations of the modules are inventory only and are not register statements. Names are given without the module namespace `UniversalLaw.Spec..`; the source column is the pinned source named by the docstring's `Source:` line. Anchors and "Not established" cells are copied verbatim from the Lean docstrings (`Anchor:` and `Does not claim:` lines) through REGISTER.json. The last column names the theorems of the sibling proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly the declaration (namespace `UniversalLaw.Proofs..`): 59 of the 141 Props have one (60 theorems; `TriangleNeedsThree` is proved twice), each listed in the register under the claim rows as a statement entry of kind `theorem` with label `proved` (the label of a committed proof package; `kernel-checked` appears only in a run receipt: the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, so each receipt records as checked_commit that merge and not the head: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; each merge has the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads). That label belongs to the theorem, not to the declaration here, which stays `specified`, and not to any claim row; a kernel-checked theorem is evidence about its Prop only, and alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. +150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified` in this package; the 123 auxiliary definitions and abbreviations of the modules are inventory only and are not register statements. Names are given without the module namespace `UniversalLaw.Spec..`; the source column is the pinned source named by the docstring's `Source:` line. Anchors and "Not established" cells are copied verbatim from the Lean docstrings (`Anchor:` and `Does not claim:` lines) through REGISTER.json. The last column names the theorems of the sibling proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly the declaration (namespace `UniversalLaw.Proofs..`): 72 of the 141 Props have one (73 theorems; `TriangleNeedsThree` is proved twice), each listed in the register under the claim rows as a statement entry of kind `theorem` with label `proved` (the label of a committed proof package; `kernel-checked` appears only in a run receipt: the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, so each receipt records as checked_commit that merge and not the head: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; each merge has the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems marked † (`Side24Analysis`, `Side24Interfaces` and `RNCluster`) were added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), and have no hosted receipt yet: the first hosted `lean-specification-proofs.yml` run on the round-4 branch is their first receipt). That label belongs to the theorem, not to the declaration here, which stays `specified`, and not to any claim row; a kernel-checked theorem is evidence about its Prop only, and alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. ### `Specifications/Field.lean` (12 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance | anything about the field. | — | | `IsSmooth` | def_prop | `lifetime-parent` | In particular the field has a smooth version and all finite moments of its global derivative suprema exist. | existence of a smooth version, or moment bounds on derivative suprema. | — | @@ -80,7 +81,7 @@ a theorem between string literals, an indented Prop definition, a proof written ### `Specifications/Lifetime.lean` (24 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance | anything about the field. | — | | `IsSmooth` | def_prop | `lifetime-parent` | In particular the field has a smooth version and all finite moments of its global derivative suprema exist. | existence of a smooth version, or moment bounds on derivative suprema. | — | @@ -109,32 +110,32 @@ a theorem between string literals, an indented Prop definition, a proof written ### `Specifications/Side24.lean` (28 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---|---| -| `ThetaSummable` | def_prop | `side24-proof` | K_24(z) = sum_{n in Z^d} exp(-\|z+24n\|^2/2) | that the family is summable (true, but not shown here). | — | +| `ThetaSummable` | def_prop | `side24-proof` | K_24(z) = sum_{n in Z^d} exp(-\|z+24n\|^2/2) | that the family is summable (true, but not shown here). | `Side24Analysis.thetaSummable†` | | `IsUnitTuple` | def_prop | `side24-proof` | For q<=6 unit directional differentiations of phi(x)=exp(-\|x\|^2/2), product-rule | anything. | — | -| `DerivativePairingBound` | def_prop | `side24-proof` | pairings bound the absolute derivative by | the bound (statement only); validity for `q > 6` (the source restricts to `q ≤ 6`); nothing here encloses the coefficient. | — | -| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | At \|x\|>=1 this is at most76 \|x\|^6 e^(-\|x\|^2/2). | the bound (statement only); the coefficient arithmetic `1+15+45+15=76` and `76` bounding the smaller orders are kernel-checked on main (`UniversalLaw.Side24.pairing_sum_six`, `pairing_sum_le_six`), not here; nothing here encloses the coefficient. | — | -| `DerivativeBoundAtZero` | def_prop | `side24-proof` | contraction has absolute value at most15. | the bound (statement only); `5!! = 15` is kernel-checked on main (`UniversalLaw.Side24.sixth_moment_double_factorial`), not here. | — | -| `LatticeImageSumBound` | def_prop | `side24-proof` | <=1458 e^(-288). | the bound (statement only); the intermediate `729 sum_j j^9 e^(-288 j^2)` shell count is not specified; `27*27 = 729`, `729*2 = 1458` are kernel-checked on main (`UniversalLaw.Side24.lattice_shell_constant`), not here. | — | +| `DerivativePairingBound` | def_prop | `side24-proof` | pairings bound the absolute derivative by | the bound (statement only); validity for `q > 6` (the source restricts to `q ≤ 6`); nothing here encloses the coefficient. | `Side24Analysis.derivativePairingBound†` | +| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | At \|x\|>=1 this is at most76 \|x\|^6 e^(-\|x\|^2/2). | the bound (statement only); the coefficient arithmetic `1+15+45+15=76` and `76` bounding the smaller orders are kernel-checked on main (`UniversalLaw.Side24.pairing_sum_six`, `pairing_sum_le_six`), not here; nothing here encloses the coefficient. | `Side24Analysis.derivativeBoundAtLeastOne†` | +| `DerivativeBoundAtZero` | def_prop | `side24-proof` | contraction has absolute value at most15. | the bound (statement only); `5!! = 15` is kernel-checked on main (`UniversalLaw.Side24.sixth_moment_double_factorial`), not here. | `Side24Analysis.derivativeBoundAtZero†` | +| `LatticeImageSumBound` | def_prop | `side24-proof` | <=1458 e^(-288). | the bound (statement only); the intermediate `729 sum_j j^9 e^(-288 j^2)` shell count is not specified; `27*27 = 729`, `729*2 = 1458` are kernel-checked on main (`UniversalLaw.Side24.lattice_shell_constant`), not here. | `Side24Analysis.latticeImageSumBound†` | | `SuccessiveTermRatio` | def_prop | `side24-proof` | successive terms have ratio at most512e^(-864)<1/2. | the bounds (statement only); `2^9 = 512`, `288*3 = 864` are kernel-checked on main (`UniversalLaw.Side24.geometric_ratio_constants`), not here. | `Side24Elementary.successiveTermRatio` | | `ExponentialBounds` | def_prop | `side24-proof` | hence e^(-288)<10^(-125) and the geometric-ratio inequality follows too. | the inequalities (statement only); the rational Taylor sum through degree 20 exceeding `10` is kernel-checked on main (the `UniversalLaw.Side24` Taylor-sum target in `Ledger.lean`), not here. | `Side24Elementary.exponentialBounds` | | `ImageBound` | def_prop | `side24-proof` | \|D^q K_24(0)-D^q K_infty(0)\| | the bound (statement only); differentiability of `K_24` (if `K_24` is not `C^q` the derivative is `0` in Lean); nothing here encloses the coefficient; nonauthor review or acceptance. | — | | `CovarianceOrdering` | def_prop | `side24-proof` | The inequalities are in the positive-semidefinite ordering and hold in every | the ordering (statement only); the passage to marginal and conditional covariances (not specified); `60E < epsilon` is kernel-checked on main (`UniversalLaw.Side24.covariance_relative_bound`), not here; nothing here encloses the coefficient; nonauthor review or acceptance. | — | | `EntrywiseCovarianceBound` | def_prop | `side24-proof` | Each covariance entry differs from the reference by at most2E, by (2). | the bound (statement only); the spectral-norm bound `20E` (not specified); nonauthor review or acceptance. | — | | `ReferenceCovarianceLowerBound` | def_prop | `side24-proof` | Thus C_ref >= I/3 and | the bound (statement only); the odd-block eigenvalues `8 ± sqrt 58` and the minors `2/3`, `7/9` are kernel-checked on main (`UniversalLaw.Side24.odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`), not here. | — | -| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | Var(t)=15, Cov(t,G)=(-3,0,...,0), | the identities (statement only); `15 - 9 = 6` is kernel-checked on main (`UniversalLaw.Side24.conditional_third_derivative_variance`), not here. | — | -| `DensityComparison` | def_prop | `side24-proof` | [(1-epsilon)/(1+epsilon)]^(n/2) phi_((1-epsilon)C0) | the comparison (statement only); the cone-moment consequence `(1 ± epsilon)^m D0` and the scaling `a^m` of `h(A) = det(A)^2 1{A<0}` (not specified); nothing here encloses the coefficient; nonauthor review or acceptance. | — | -| `IntegrandRatioBounds` | def_prop | `side24-proof` | ratio between1-13epsilon and | the bounds (statement only); that the integrand ratio of the source equals these power quotients (not specified); `13 ≤ 32`, `28 ≤ 32` and `32*10^(-108) < 10^(-106)` are kernel-checked on main (`UniversalLaw.Side24.ratio_constant_ordering`, `density_comparison_below_reported`), not here. | — | -| `ReferencePinDensityProduct` | def_prop | `side24-proof` | p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3). | the identity (statement only); `det diag(3,1,…,1) = 3` is kernel-checked on main (`UniversalLaw.Side24.pin_determinant_and_joint_dimension`), not here. | — | +| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | Var(t)=15, Cov(t,G)=(-3,0,...,0), | the identities (statement only); `15 - 9 = 6` is kernel-checked on main (`UniversalLaw.Side24.conditional_third_derivative_variance`), not here. | `Side24Analysis.referenceOddAndPinBlocks†` | +| `DensityComparison` | def_prop | `side24-proof` | [(1-epsilon)/(1+epsilon)]^(n/2) phi_((1-epsilon)C0) | the comparison (statement only); the cone-moment consequence `(1 ± epsilon)^m D0` and the scaling `a^m` of `h(A) = det(A)^2 1{A<0}` (not specified); nothing here encloses the coefficient; nonauthor review or acceptance. | `Side24Analysis.densityComparison†` | +| `IntegrandRatioBounds` | def_prop | `side24-proof` | ratio between1-13epsilon and | the bounds (statement only); that the integrand ratio of the source equals these power quotients (not specified); `13 ≤ 32`, `28 ≤ 32` and `32*10^(-108) < 10^(-106)` are kernel-checked on main (`UniversalLaw.Side24.ratio_constant_ordering`, `density_comparison_below_reported`), not here. | `Side24Analysis.integrandRatioBounds†` | +| `ReferencePinDensityProduct` | def_prop | `side24-proof` | p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3). | the identity (statement only); `det diag(3,1,…,1) = 3` is kernel-checked on main (`UniversalLaw.Side24.pin_determinant_and_joint_dimension`), not here. | `Side24Analysis.referencePinDensityProduct†` | | `ReferenceCoefficientIdentity` | def_prop | `side24-proof` | This eliminates BOTH the angular and cone integrals for the reference expression. | the identity (statement only); that the left side is the parent formula (15.2) evaluated at the reference law (the angular integrand is constant by rotational invariance of `K_infty`, a fact not specified here); `6^2*2 = 3*24` is kernel-checked on main (`UniversalLaw.Side24.cube_root_simplification`), not here; nonauthor review or acceptance. | `Side24Elementary.referenceCoefficientIdentity` | | `ConeMomentInterface` | structure | `side24-proof` | Var(s)=5/3 and Var(x)=Var(y)=1. The negative-definite cone is s<-R, | that these laws are the conditional law `A \| V = 0` of the reference field (the source derives them; here they are hypotheses); the derivation of `Var(s) = 5/3` from the `(2/3) delta delta + …` formula (kernel-checked arithmetic on main, `UniversalLaw.Side24.trace_and_traceless_variances`). | — | | `NegDefCone` | def_prop | `side24-proof` | R=sqrt(x^2+y^2), and det A=s^2-R^2. | the source's reduction of the cone to `s < -R` (not specified; the cone is taken by its definition). | — | -| `ConeMomentD1Identity` | def_prop | `side24-proof` | D_1=E[A^2 1{A<0}]=4/3. | the identity (statement only); that `D_1` is the parent's cone moment `D_u` of (15.1) for the reference law; nothing here encloses the coefficient. | — | +| `ConeMomentD1Identity` | def_prop | `side24-proof` | D_1=E[A^2 1{A<0}]=4/3. | the identity (statement only); that `D_1` is the parent's cone moment `D_u` of (15.1) for the reference law; nothing here encloses the coefficient. | `Side24Interfaces.coneMomentD1Identity†` | | `ConeMomentD2Identity` | def_prop | `side24-proof` | D_2 = (1/2)[25/3-20/3+8-8sqrt(3/8)] | the identity (statement only); the Rayleigh integration route; that `D_2` is the parent's `D_u` for the reference law; nothing here encloses the coefficient; nonauthor review or acceptance. | — | | `ConeMomentD2Algebra` | def_prop | `side24-proof` | = 29/6-sqrt(6). | the identity (statement only); its rational part and `4 sqrt(3/8) = sqrt 6` are kernel-checked on main (`UniversalLaw.Side24.cone_moment_m2_algebra`), not here. | `Side24Elementary.coneMomentD2Algebra` | -| `TraceMoments` | def_prop | `side24-proof` | E s^2=5/3, E s^4=25/3 and | the identities (statement only); `3 (5/3)^2 = 25/3` and `1 + 5/3 = 8/3` are kernel-checked on main (`UniversalLaw.Side24.fourth_moment_of_trace`, `exponential_moment_of_trace`), not here. | — | -| `RayleighSquareLaw` | def_prop | `side24-proof` | The variable R^2 has density e^(-z/2)/2. | the law (statement only); measurability of the event (if the event is not measurable, `P` is its outer measure). | — | +| `TraceMoments` | def_prop | `side24-proof` | E s^2=5/3, E s^4=25/3 and | the identities (statement only); `3 (5/3)^2 = 25/3` and `1 + 5/3 = 8/3` are kernel-checked on main (`UniversalLaw.Side24.fourth_moment_of_trace`, `exponential_moment_of_trace`), not here. | `Side24Interfaces.traceMoments†` | +| `RayleighSquareLaw` | def_prop | `side24-proof` | The variable R^2 has density e^(-z/2)/2. | the law (statement only); measurability of the event (if the event is not measurable, `P` is its outer measure). | `Side24Interfaces.rayleighSquareLaw†` | | `ElementaryConeIntegral` | def_prop | `side24-proof` | integral_0^a (a-z)^2 e^(-z/2) dz/2 = a^2-4a+8-8e^(-a/2). | the identity (statement only). | `Side24Elementary.elementaryConeIntegral` | | `PeriodicCoefficientInterface` | structure | `side24-proof` | Let c_d,24 be precisely its equation (15.2), using ordinary sphere area, ordered | existence of an object satisfying this interface; that `c24` is the finite-bar asymptotic constant (that reading is conditional on the parent's results, which are not touched); that the parent's "ordered maximum/saddle roles and the full height/gradient pin Jacobian" are encoded beyond the prefactor of (15.2). | — | | `CoefficientEnclosure` | def_prop | `side24-proof` | 0.07340691930603427103 < c_2,24 < 0.07340691930603427104, | the enclosures (statement only: nothing here encloses the coefficient); that an object satisfying the interface exists or is the coefficient of the source; the parent's results or the persistence interpretation; any finite-radius error estimate, unrestricted remainder or RN/24-jet statement; the arithmetic skeleton (endpoint adjacency, `d3 < d2`, endpoints in `(0, 1/10)`) is kernel-checked on main (31 `UniversalLaw.Side24` targets at origin/main f6deeba7), not here; nonauthor review or acceptance. | — | @@ -142,7 +143,7 @@ a theorem between string literals, an indented Prop definition, a proof written ### `Specifications/RN.lean` (28 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---|---| | `IsCriticalPoint` | def_prop | `lifetime-parent` | grad f(M)=grad f(S)=0. | differentiability of `g` at `x` (if `g` is not differentiable, `fderiv` is `0`). | — | | `HessianHasInertia` | def_prop | `lifetime-parent` | The index counts negative eigenvalues. | equivalence with an eigenvalue count for a non-symmetric form. | — | @@ -169,13 +170,13 @@ a theorem between string literals, an indented Prop definition, a proof written | `FourierCutoffPlanarSecondFactorialF4` | def_prop | `c6-fourier-cutoff` | E_Qr^W N(N-1) <= C r^3 log(1/r), | Theorem F (the cap `Psi_F` and its tail (F1)-(F2)); Theorem W for general `d` and `p` (F3); that `G_2` holds (the source takes it from the reviewed planar I5 composition); Proposition B; that the logarithm is sharp or removable; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. | — | | `PalmRouteFactorialMomentsQ` | def_prop | `c6-palm-route` | E_(Q_r^W)[ (N)_q ] <= C_q r^3. | the result or its proof (the cap `Psi` in `d` dimensions, the parametric Rouché tail, the marked Kac-Rice formula with Hölder insertion, the regimes R1-R4); the dependence on [C6L] and [DL] at their bytes; a lower bound for `q ≥ 3`; Corollary P; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. | — | | `PalmRouteSecondFactorialTheta` | def_prop | `c6-palm-route` | c r^3 <= E_(Q_r^W)[ N(N-1) ] <= C r^3, | the lower bounds' source ([EDL] (A4)) or the upper bounds' sources (Theorem Q, [DL] Corollary M_d); that the logarithm of the Fourier cutoff is removed as a reviewed fact; any statement about the positions of the pairs; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. | — | -| `RareClusterPoissonObstructionO` | def_prop | `c6-rare-cluster` | (a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3. | the result (Theorem O) or its proof; that `2a ≤ A` (derived in the source, not a hypothesis here); that H1 or H2 hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment matching or the independent-replica limit; the conditional-law liminf remark; that the upper half is the source's infimum bound verbatim: `∃ lam ≥ 0, dTV(Law N_r, Pois(lam)) ≤ A r^3` implies `inf_{lam ≥ 0} dTV ≤ A r^3` and is stronger unless the infimum is attained, and the existential form is used because the source's proof exhibits the witness `lam = 0` (the source: The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p); nonauthor review or acceptance. | — | +| `RareClusterPoissonObstructionO` | def_prop | `c6-rare-cluster` | (a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3. | the result (Theorem O) or its proof; that `2a ≤ A` (derived in the source, not a hypothesis here); that H1 or H2 hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment matching or the independent-replica limit; the conditional-law liminf remark; that the upper half is the source's infimum bound verbatim: `∃ lam ≥ 0, dTV(Law N_r, Pois(lam)) ≤ A r^3` implies `inf_{lam ≥ 0} dTV ≤ A r^3` and is stronger unless the infimum is attained, and the existential form is used because the source's proof exhibits the witness `lam = 0` (the source: The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p); nonauthor review or acceptance. | `RNCluster.rareClusterPoissonObstructionO†` | | `RemoteCollisionSecondFactorialD` | def_prop | `remote-collision` | E_(Q_r^W) N(E)(N(E)-1) <= C r^5 \|E\|, | Theorem C (the ordered near-pair count) or its proof (Lemmas 1-5); Corollaries E and F; a torus-wide second factorial moment or `math.rn-region.witness-collision` in its full sense; pairs with a witness near the pins; equality of `N(E)` with the all-critical-point count `windowCountAll` (they agree only off the degenerate locus); any numerical constant; nonauthor review or acceptance. | — | | `WitnessCollisionNearPinPairsTarget` | def_prop | `d5-reconciliation` | A torus-wide second factorial moment `E_{Q_r^W} N(N−1)` for the window count, with pairs where at least | that this target holds, is attainable, or is implied by any reviewed source (the Palm route's Theorem Q is an author-side candidate whose `q = 2` case would imply it; that candidate is stated separately as `PalmRouteFactorialMomentsQ` and is not consumed here); that the lower obstruction `E N(N−1) ≥ 2c r³` is stated here; the optimal order; any status transition of the node. | — | ### `Specifications/P15.lean` (58 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | |---|---|---|---|---|---| | `RealizedData` | structure | `p15-realized-covers` | Choose positive integers a_i,d_i and disjoint original-coordinate blocks X_i with | that the data satisfy the sources' side conditions (that is `IsRealizedFamily`). | — | | `IsRealizedFamily` | def_prop | `p15-realized-covers` | n_i=\|X_i\|=a_i d_i+1. | that any particular data record satisfies it. | — | @@ -238,42 +239,42 @@ a theorem between string literals, an indented Prop definition, a proof written ## Claims and what is not established -171 claim rows: 10 `landing_claim`, 33 `graph_node`, 2 `open_obligation`, 95 `proof_index_result`, 7 `existing_lean_package`, 12 `main_experiment_proof`, 12 `exact_certificate`. Claim-level formal progress: 33 `specified` (a `def … : Prop` stating the claim itself is registered), 138 `none`; no research claim is `proved` or `kernel-checked`. 197 statement entries point at the 150 declarations of this package (all `specified`). 142 statement entries carry the label `proved` and are theorems of other packages, listed as the arithmetic skeleton or the Prop-level proof of the claim they sit under and never as the claim: 31 are the 31 Side24 arithmetic targets of main's `formal/` at origin/main `f6deeba7`; 41 are the 28 P15 arithmetic targets of main's `formal/` at `91cffbb5` (main PR #358, not yet on main; kernel evidence the hosted workspace-landing run 38096827443), each listed on every P15 row whose layer 0 source it anchors to; 70 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (60 theorems, 59 distinct Props, each listed on every row that registers its Prop; the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 as kernel-checked and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, which the receipt records as checked_commit: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; same trees as the heads; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads). Exactly one Prop registered as stating a claim itself has such a theorem: `DemandOneRefutation` on rows `p15-price-boundary` and `math.p15-price-boundary`; their claim-level label stays `specified` (the register admits only none/specified at claim level, alignment of the Prop with the source paragraph is PENDING_INDEPENDENT_REVIEW, and a receipt is about the Prop only), as the rows' `not established` cells record. `disposition` is the verbatim LANDING_CLAIMS value or `—` (null). Nothing in these tables is a scientific status, a review verdict or an acceptance; alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. +171 claim rows: 10 `landing_claim`, 33 `graph_node`, 2 `open_obligation`, 95 `proof_index_result`, 7 `existing_lean_package`, 12 `main_experiment_proof`, 12 `exact_certificate`. Claim-level formal progress: 33 `specified` (a `def … : Prop` stating the claim itself is registered), 138 `none`; no research claim is `proved` or `kernel-checked`. 197 statement entries point at the 150 declarations of this package (all `specified`). 155 statement entries carry the label `proved` and are theorems of other packages, listed as the arithmetic skeleton or the Prop-level proof of the claim they sit under and never as the claim: 31 are the 31 Side24 arithmetic targets of main's `formal/` at origin/main `f6deeba7`; 41 are the 28 P15 arithmetic targets of main's `formal/` at `91cffbb5` (main PR #358, not yet on main; kernel evidence the hosted workspace-landing run 38096827443), each listed on every P15 row whose layer 0 source it anchors to; 83 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (73 theorems, 72 distinct Props, each listed on every row that registers its Prop; the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 as kernel-checked and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, which the receipt records as checked_commit: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; same trees as the heads; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems marked † (`Side24Analysis`, `Side24Interfaces` and `RNCluster`) were added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), and have no hosted receipt yet: the first hosted `lean-specification-proofs.yml` run on the round-4 branch is their first receipt). Exactly one Prop registered as stating a claim itself has such a theorem: `DemandOneRefutation` on rows `p15-price-boundary` and `math.p15-price-boundary`; their claim-level label stays `specified` (the register admits only none/specified at claim level, alignment of the Prop with the source paragraph is PENDING_INDEPENDENT_REVIEW, and a receipt is about the Prop only), as the rows' `not established` cells record. `disposition` is the verbatim LANDING_CLAIMS value or `—` (null). Nothing in these tables is a scientific status, a review verdict or an acceptance; alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. | claim_id | kind | disposition | formal_progress | statements | not established | next step | |---|---|---|---|---|---|---| -| `side24-coefficient` | landing_claim | REVIEWED_SCOPED | `specified` | 28 specified: `ThetaSummable`, `IsUnitTuple`, `DerivativePairingBound`, `DerivativeBoundAtLeastOne`, `DerivativeBoundAtZero`, `LatticeImageSumBound`, `SuccessiveTermRatio`, `ExponentialBounds`, `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound`, `ReferenceOddAndPinBlocks`, `DensityComparison`, `IntegrandRatioBounds`, `ReferencePinDensityProduct`, `ReferenceCoefficientIdentity`, `ConeMomentInterface`, `NegDefCone`, `ConeMomentD1Identity`, `ConeMomentD2Identity`, `ConeMomentD2Algebra`, `TraceMoments`, `RayleighSquareLaw`, `ElementaryConeIntegral`, `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison`; 27 proved (main, origin/main f6deeba7): `pairing_sum_six`, `pairing_sum_le_six`, `pairing_terms_six`, `sixth_moment_double_factorial`, `lattice_shell_constant`, `geometric_ratio_constants`, `period_exponent`, `image_constant_value`, `exp_taylor_partial_sum_gt_ten`, `covariance_relative_bound`, `odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`, `exponent_bounds`, `exponent_values`, `ratio_constant_ordering`, `density_comparison_below_reported`, `conditional_third_derivative_variance`, `transverse_covariance_entries_thirds`, `cone_moment_m1_thirds`, `trace_and_traceless_variances`, `fourth_moment_of_trace`, `exponential_moment_of_trace`, `cone_moment_m2_algebra`, `cube_root_simplification`, `pin_determinant_and_joint_dimension`, `hessian_block_eigenvalues`; 5 proved (proof packet): `Side24Elementary.exponentialBounds`, `Side24Elementary.successiveTermRatio`, `Side24Elementary.coneMomentD2Algebra`, `Side24Elementary.referenceCoefficientIdentity`, `Side24Elementary.elementaryConeIntegral` | Nothing here encloses the coefficient. CoefficientEnclosure and ReferenceComparison are statements for any object satisfying PeriodicCoefficientInterface; the interface takes the cone moment D_u of (15.1) as a GIVEN field, so c_{d,24} is not uniquely determined in Lean and no inhabitant is constructed. The 27 main theorems listed as `proved` are the integer/rational arithmetic skeleton of PROOF.md sections 1-4 (coefficient sums, shell constants, Taylor partial sum, block minors, cone-moment algebra), not the Gaussian conditioning, the Rayleigh cone integration, the Stirling log-gamma step (section 5, not specified), the interval arithmetic of coefficient.py, or the enclosure itself. ExponentialBounds, SuccessiveTermRatio, ConeMomentD2Algebra, ReferenceCoefficientIdentity and ElementaryConeIntegral have theorems of exactly their type in the proof packet specifications/lean_statement_proofs_20261011 (5 entries below, label `proved`; the hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked them (receipt formalization_status kernel-checked, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound})): elementary real inequalities, the term ratio, the D2 cone-moment algebra, the reference-coefficient identity as stated (its left side is not identified with the parent formula (15.2) at the reference law: rotational invariance of K_infty is not specified) and the elementary cone integral; none of them is the Gaussian conditioning, ConeMomentD1Identity / ConeMomentD2Identity, RayleighSquareLaw, LatticeImageSumBound or any enclosure of the coefficient, and the claim-level label stays `specified`. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. | Formal lane (Math- Mathlib package, by the lane's admission rules): construct the conditional law A_u \| V_u = 0 and prove ConeMomentD1Identity / ConeMomentD2Identity for the reference law under ConeMomentInterface, RayleighSquareLaw and LatticeImageSumBound (ReferenceCoefficientIdentity is proved as stated; the identification of its left side with (15.2) at the reference law is not specified); the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked the 5 proved entries; the enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md). Independent reviewer: alignment verdicts on the 28 Side24 statements. | -| `lifetime-remainder` | landing_claim | REVIEWED_SCOPED | `specified` | 4 specified: `RemainderBoundedR1`, `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `IsLifetimeDensityVersion` | RemainderBoundedR1 states the shape of (R1) for versions of the densities under the interface structures; the result is not proved. Not stated: (R19)-(R20), the observation transform and determinant lemmas R3.1/R3.2, the cap-loss bound, the parent interfaces D1-A-E it imports, the marked-cylinder-cap support. The coefficient c enters only through IsUnrestrictedLeadingCoefficient (c > 0 and the leading-term role), not through the representation (15.2). No numerical C or ell_*, no second coefficient, no convergence of the remainder, no compact-window O(ell^(2/3)) difference without cutoffs. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (R19) (cumulative form) as a further Prop; transcribe the parent interfaces consumed (section 8 comparison, section 9 as replaced by v1.1, section 10, (14.1), (15.2)) as explicit hypotheses; independent reviewer: alignment verdict on RemainderBoundedR1 against (R1), including the "versions" clause and the exclusion of the essential global maximum. | +| `side24-coefficient` | landing_claim | REVIEWED_SCOPED | `specified` | 28 specified: `ThetaSummable`, `IsUnitTuple`, `DerivativePairingBound`, `DerivativeBoundAtLeastOne`, `DerivativeBoundAtZero`, `LatticeImageSumBound`, `SuccessiveTermRatio`, `ExponentialBounds`, `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound`, `ReferenceOddAndPinBlocks`, `DensityComparison`, `IntegrandRatioBounds`, `ReferencePinDensityProduct`, `ReferenceCoefficientIdentity`, `ConeMomentInterface`, `NegDefCone`, `ConeMomentD1Identity`, `ConeMomentD2Identity`, `ConeMomentD2Algebra`, `TraceMoments`, `RayleighSquareLaw`, `ElementaryConeIntegral`, `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison`; 27 proved (main, origin/main f6deeba7): `pairing_sum_six`, `pairing_sum_le_six`, `pairing_terms_six`, `sixth_moment_double_factorial`, `lattice_shell_constant`, `geometric_ratio_constants`, `period_exponent`, `image_constant_value`, `exp_taylor_partial_sum_gt_ten`, `covariance_relative_bound`, `odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`, `exponent_bounds`, `exponent_values`, `ratio_constant_ordering`, `density_comparison_below_reported`, `conditional_third_derivative_variance`, `transverse_covariance_entries_thirds`, `cone_moment_m1_thirds`, `trace_and_traceless_variances`, `fourth_moment_of_trace`, `exponential_moment_of_trace`, `cone_moment_m2_algebra`, `cube_root_simplification`, `pin_determinant_and_joint_dimension`, `hessian_block_eigenvalues`; 17 proved (proof packet): `Side24Elementary.exponentialBounds`, `Side24Elementary.successiveTermRatio`, `Side24Elementary.coneMomentD2Algebra`, `Side24Elementary.referenceCoefficientIdentity`, `Side24Elementary.elementaryConeIntegral`, `Side24Analysis.thetaSummable†`, `Side24Analysis.derivativePairingBound†`, `Side24Analysis.derivativeBoundAtLeastOne†`, `Side24Analysis.derivativeBoundAtZero†`, `Side24Analysis.referenceOddAndPinBlocks†`, `Side24Analysis.referencePinDensityProduct†`, `Side24Analysis.integrandRatioBounds†`, `Side24Analysis.latticeImageSumBound†`, `Side24Analysis.densityComparison†`, `Side24Interfaces.coneMomentD1Identity†`, `Side24Interfaces.traceMoments†`, `Side24Interfaces.rayleighSquareLaw†` | Nothing here encloses the coefficient. CoefficientEnclosure and ReferenceComparison are statements for any object satisfying PeriodicCoefficientInterface; the interface takes the cone moment D_u of (15.1) as a GIVEN field, so c_{d,24} is not uniquely determined in Lean and no inhabitant is constructed. The 27 main theorems listed as `proved` are the integer/rational arithmetic skeleton of PROOF.md sections 1-4 (coefficient sums, shell constants, Taylor partial sum, block minors, cone-moment algebra), not the Gaussian conditioning, the Rayleigh cone integration, the Stirling log-gamma step (section 5, not specified), the interval arithmetic of coefficient.py, or the enclosure itself. Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, LatticeImageSumBound, SuccessiveTermRatio, ExponentialBounds, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (17 entries below, label `proved`; 5 of them (Side24Elementary.successiveTermRatio, Side24Elementary.exponentialBounds, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.coneMomentD2Algebra, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.latticeImageSumBound, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.densityComparison, Side24Analysis.integrandRatioBounds, Side24Analysis.referencePinDensityProduct, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them). What those theorems prove, for the formal objects of the statement module only: elementary real inequalities (ExponentialBounds), the term ratio (SuccessiveTermRatio), the D2 cone-moment algebra (ConeMomentD2Algebra), the reference-coefficient identity as stated (ReferenceCoefficientIdentity; its left side is not identified with the parent formula (15.2) at the reference law: rotational invariance of K_infty is not specified) and the elementary cone integral (ElementaryConeIntegral); summability of the formal theta family n \|-> exp(-\|z + 24 n\|^2/2) over Fin d -> Z for every d (ThetaSummable: the statement entry's does_not_claim "that the family is summable (true, but not shown here)" describes the statement packet, which is not edited; the summability is now shown in the proof packet, for the formal family, by comparison with a product of one-dimensional Gaussian sums); the Section 2 derivative bounds for q <= 6 as the Props state them (DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero: a Hermite-type recursion on the operator norms of iteratedFDeriv, no closed form of the Gaussian derivatives, the contraction at a unit tuple bounded by the operator norm; main's pairing_sum_six, pairing_sum_le_six and sixth_moment_double_factorial are neither cited nor relied on (the 1 + 15 + 45 + 15 = 76 arithmetic is recomputed inline by norm_num); nothing for q > 6); the lattice image sum (LatticeImageSumBound: summability and the bound 1458 e^(-288), via 432 e^(-288); the source's shell count 729 sum_j j^9 e^(-288 j^2) is not reproduced and not specified); the five reference block identities and the pin density product for the formal matrices (ReferenceOddAndPinBlocks, ReferencePinDensityProduct: derivative contractions of Kinf at 0 in an orthonormal frame; none of these matrices is shown to be the covariance of any random vector or field); pure real arithmetic in epsilon and the exponents a, b for d = 2, 3 (IntegrandRatioBounds; that the source's integrand ratio equals these power quotients is not specified); Loewner determinant and inverse monotonicity for positive-definite real matrices (DensityComparison; the cone-moment consequence (1 +/- epsilon)^m D0 and the scaling a^m are not specified); and, over ConeMomentInterface, the m = 1 cone moment 4/3, the trace moments 5/3, 25/3, sqrt(3/8) and the distribution function P(R^2 <= a) = integral_0^a e^(-z/2)/2 dz for a >= 0 (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw: proved for every inhabitant from the law fields A_law, sxy_law and isProb alone; no inhabitant is constructed, the laws are not derived as the conditional law A \| V = 0 of the reference field, and the Rayleigh law is proved as a distribution function, not as a density). None of them is the Gaussian conditioning, ConeMomentD2Identity, ImageBound, the covariance ordering or any enclosure of the coefficient, and the claim-level label stays `specified` (claim-level vocabulary none/specified; a theorem of a Prop is evidence about that Prop, for the formal objects, not acceptance of the informal theorem; alignment PENDING_INDEPENDENT_REVIEW). Theorem-shaped Props of this row still `specified` with no theorem: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison; the parametrised predicates IsUnitTuple, NegDefCone are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited. | Math- Mathlib lane: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); the 12 theorems of this row added after PR head 12e072d6 (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) await the next hosted lean-specification-proofs.yml run, their first receipt; the other theorems of this row have a hosted receipt: the 5 of PR head 00d2ac6e (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); each of those receipted theorems corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, ReferenceOddAndPinBlocks, IntegrandRatioBounds, ReferencePinDensityProduct, ConeMomentD1Identity, TraceMoments PROVABLE_NOW and LatticeImageSumBound, DensityComparison, RayleighSquareLaw PROVABLE_WITH_WORK: proved in round 4 (see entries); ReferenceCovarianceLowerBound PROVABLE_WITH_WORK (weeks); ConeMomentD2Identity PROVABLE_WITH_WORK (weeks); ImageBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum; the global version is inapplicable to lattice translates; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), iteratedFDeriv_tsum_apply (SmoothSeries.lean)); CovarianceOrdering BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: local term-wise iterated differentiation of the lattice sum, the d-dimensional Gaussian tail bound and higher-order symmetry of iteratedFDeriv; display (3) needs only the crude norm_iteratedFDeriv_comp_le, with a factor about 8 to spare; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), hasFDerivAt_tsum_of_isPreconnected (SmoothSeries.lean)); EntrywiseCovarianceBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum for the term-wise iterated differentiation of the lattice sum (the same blocker as CovarianceOrdering; the finding records no further shape observation); present but insufficient: norm_iteratedFDeriv_comp_le (Bounds.lean), iteratedFDeriv_tsum (SmoothSeries.lean)); CoefficientEnclosure BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); ReferenceComparison BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); IsUnitTuple, NegDefCone PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. The enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md) and an interface that determines c_{d,24} (the cone moment is a given field; ConeMomentD2Identity for the reference law under ConeMomentInterface is the open Rayleigh-integration step). Proof packet README: for the Props proved in round 4 (ConeMomentD1Identity, DensityComparison, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, DerivativePairingBound, IntegrandRatioBounds, LatticeImageSumBound, RayleighSquareLaw, ReferenceOddAndPinBlocks, ReferencePinDensityProduct, ThetaSummable, TraceMoments) the informal reasons the README gave were found not load-bearing by the triage and are superseded by their theorems; the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure and ReferenceComparison stand (for the last two the triage confirms the README's reason as the precise missing ingredient) (the proof-packet maintainers' regeneration, not this register's). Independent reviewer: alignment verdicts on the 28 Side24 statements, in particular of the 17 Props of this row that have a theorem of exactly their type, with their anchors (ThetaSummable with the source's R^d, 24 Z^d and K_24; the derivative Props with Section 2; the interface Props with the source's A ~ N(0, 8/3), s ~ N(0, 5/3), x, y ~ N(0, 1)). | +| `lifetime-remainder` | landing_claim | REVIEWED_SCOPED | `specified` | 4 specified: `RemainderBoundedR1`, `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `IsLifetimeDensityVersion` | RemainderBoundedR1 states the shape of (R1) for versions of the densities under the interface structures; the result is not proved. Not stated: (R19)-(R20), the observation transform and determinant lemmas R3.1/R3.2, the cap-loss bound, the parent interfaces D1-A-E it imports, the marked-cylinder-cap support. The coefficient c enters only through IsUnrestrictedLeadingCoefficient (c > 0 and the leading-term role), not through the representation (15.2). No numerical C or ell_*, no second coefficient, no convergence of the remainder, no compact-window O(ell^(2/3)) difference without cutoffs. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (R19) (cumulative form) as a further Prop; transcribe the parent interfaces consumed (section 8 comparison, section 9 as replaced by v1.1, section 10, (14.1), (15.2)) as explicit hypotheses; independent reviewer: alignment verdict on RemainderBoundedR1 against (R1), including the "versions" clause and the exclusion of the essential global maximum. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemainderBoundedR1 BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean), ncard_le_ncard (Card.lean)); IsLifetimeDensityVersion, IsUnrestrictedLeadingCoefficient, IsUnrestrictedPair PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. | | `rn-count-interface` | landing_claim | HOLD_WITH_DOMAIN | `none` | 5 specified: `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5`; 5 proved (proof packet): `RNProbability.conditionalMeanIdentityN3`, `RNProbability.holderCountBoundN1`, `RNProbability.holderConsequenceN2`, `RNProbability.uniformCounterexampleN4`, `RNProbability.exponentialTailLayerCakeN5` | No Prop states Theorem N as a whole, so the claim-level label stays none: (N1)-(N5) (Hoelder bound, its power-law consequence, the conditional-mean identity, the uniform counterexample, the exponential-tail layer cake) are stated as abstract probability-space Props and each has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (5 entries below, label `proved`; the hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked them (receipt formalization_status kernel-checked, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound})); (N6) (the full-pin weighted Kac-Rice expression for E_Q^W N_B) and (N7) (the enclosure-to-rate implication under Z_r >= z_* r^2) are NOT stated, and they are the Gaussian numerator interface that the disposition reason calls "a target rather than a discharged global RN theorem". The five theorems prove the Props for the abstract probability spaces and counts they quantify over: nothing about the Gaussian model is stated or proved; no RN rate, no 24-jet enclosure, no ENV-RESCOV discharge; D3-LEMMA-RN-UNIF stays not closed. Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | State (N7) as an implication with the Kac-Rice numerator as an interface field (the integral of (N6) as a given quantity) and Z_r >= z_* r^2 as a hypothesis; then the claim would be `specified`. The hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked the five proved entries. Independent reviewer: alignment of the five Props with (N1)-(N5). | -| `rn-fixed-remote-window` | landing_claim | HOLD_WITH_DOMAIN | `specified` | 5 specified: `PinnedLawInterface`, `IsCriticalPoint`, `HessianHasInertia`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | FixedRemoteMeanMeasureA2 and FixedRemoteCountBoundA3 state the shapes of displays (2) and (3) for any object satisfying PinnedLawInterface (the regression law Q_r, the weight W_r, the normalizer and the contact kernel Lambda_j are interface fields, gated on 0 < r < L); no inhabitant of the interface is constructed and the kernel is not identified with formula (13). Not stated: the observation transform U_r and target v_r, the coupling (5), the weight expansion (10)-(11), the Kac-Rice formula (12), the kernel formula (13), the unnormalized numerator bound (14), and the multi-witness Section 6 displays (15)-(16). Nothing is proved; no numerical C or r_*; no shrinking exclusion rho -> 0, no collisions, no global RN complement (the disposition reason). Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Section 6 ((15)-(16), separated m-tuples) as Props; record an inhabitant construction of PinnedLawInterface as a proof-lane target (Dirac mass at one admissible smooth path); independent reviewer: alignment of A2/A3 with displays (2)-(3), in particular the uniformity in b, k, R, j, E and the fixed rho. | -| `rn-fixed-annulus-window` | landing_claim | REVIEWED_SCOPED | `specified` | 1 specified: `FixedAnnulusWindowCandidateA2` | FixedAnnulusWindowCandidateA2 states the shape of (A2) (d = 2, fixed L, fixed scaled annulus, compact positive gaps, between-pin height window) over PinnedLawInterface; the result is not proved. Not stated: the six-pin regression (A3)-(A5), normalizer (A8), contact vector (A10)-(A12), cutoff (A13)-(A14), density (A17), Hessian bounds (A18)-(A20), intensities (A21)-(A26), the TWO_SCALE_ADDENDUM S6-S21 support (not pinned here), the eight reviewed interfaces as separate statements. No all-height assertion, no positive leading coefficient, no probability lower bound, no uniformity as k -> 0, B0 -> infinity, in L or for d >= 3; no global D5/RN/JETMOD transfer. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin TWO_SCALE_ADDENDUM.md and state (S3)-(S5) and (S20)-(S21); independent reviewer: alignment of FixedAnnulusWindowCandidateA2 with (A2), checking that the height window is strict and the constants depend only on L, A0, B0 and the mark compacts. | +| `rn-fixed-remote-window` | landing_claim | HOLD_WITH_DOMAIN | `specified` | 5 specified: `PinnedLawInterface`, `IsCriticalPoint`, `HessianHasInertia`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | FixedRemoteMeanMeasureA2 and FixedRemoteCountBoundA3 state the shapes of displays (2) and (3) for any object satisfying PinnedLawInterface (the regression law Q_r, the weight W_r, the normalizer and the contact kernel Lambda_j are interface fields, gated on 0 < r < L); no inhabitant of the interface is constructed and the kernel is not identified with formula (13). Not stated: the observation transform U_r and target v_r, the coupling (5), the weight expansion (10)-(11), the Kac-Rice formula (12), the kernel formula (13), the unnormalized numerator bound (14), and the multi-witness Section 6 displays (15)-(16). Nothing is proved; no numerical C or r_*; no shrinking exclusion rho -> 0, no collisions, no global RN complement (the disposition reason). Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Section 6 ((15)-(16), separated m-tuples) as Props; record an inhabitant construction of PinnedLawInterface as a proof-lane target (Dirac mass at one admissible smooth path); independent reviewer: alignment of A2/A3 with displays (2)-(3), in particular the uniformity in b, k, R, j, E and the fixed rho. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point); HessianHasInertia, IsCriticalPoint PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. | +| `rn-fixed-annulus-window` | landing_claim | REVIEWED_SCOPED | `specified` | 1 specified: `FixedAnnulusWindowCandidateA2` | FixedAnnulusWindowCandidateA2 states the shape of (A2) (d = 2, fixed L, fixed scaled annulus, compact positive gaps, between-pin height window) over PinnedLawInterface; the result is not proved. Not stated: the six-pin regression (A3)-(A5), normalizer (A8), contact vector (A10)-(A12), cutoff (A13)-(A14), density (A17), Hessian bounds (A18)-(A20), intensities (A21)-(A26), the TWO_SCALE_ADDENDUM S6-S21 support (not pinned here), the eight reviewed interfaces as separate statements. No all-height assertion, no positive leading coefficient, no probability lower bound, no uniformity as k -> 0, B0 -> infinity, in L or for d >= 3; no global D5/RN/JETMOD transfer. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin TWO_SCALE_ADDENDUM.md and state (S3)-(S5) and (S20)-(S21); independent reviewer: alignment of FixedAnnulusWindowCandidateA2 with (A2), checking that the height window is strict and the constants depend only on L, A0, B0 and the mark compacts. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedAnnulusWindowCandidateA2 BLOCKED_INTERFACE (a Dirac inhabitant with an index-j window critical point on a null E0 (a singleton in the annulus); no field of PinnedLawInterface supplies the a.s. absence that the weighted Kac-Rice formula (A21) gives). | | `p15-realized-covers` | landing_claim | HOLD_WITH_DOMAIN | `specified` | 27 specified: `RealizedData`, `IsRealizedFamily`, `IsGood`, `IsKDecomposable`, `IsGeneratorCover`, `IsPaletteFeasible`, `PaletteNumberAttained`, `IsMinimalForbidden`, `IsTransversal`, `MinimalForbiddenCharacterisation`, `LocalRestriction`, `LocalDecomposability`, `LocalObstructionIsFullBlock`, `LocalPriceChain`, `IndependentBlockHazard`, `FullBlockCoverIff`, `CoverCostHazardBound`, `WholeGroundChromaticNumber`, `ElementaryPaletteFormula`, `TriangleNeedsThree`, `Benchmark816IsRealized`, `Benchmark816Palette`, `Benchmark816Cover`, `Benchmark816WholeGround`, `Benchmark816MinimalForbiddenCounts`, `Benchmark816Price`, `Benchmark816UnionBound`; 12 proved (main 91cffbb5, PR #358, not on main): `benchmark_block_and_ground_sizes`, `four_block_supports_fifteen`, `minimal_forbidder_counts`, `palette_816_from_two_label_classes`, `palette_815_fails_by_counting`, `whole_ground_chromatic_818`, `elementary_palette_formula_instances`, `macro_triangle_three_colors`, `ceil_block_over_capacity_instances`, `block_occupancy_one_percent`, `low_failure_union_bound_arithmetic`, `cover_price_numerator_and_denominator_positive`; 20 proved (proof packet): `P15Elementary.triangleNeedsThree`, `P15Structure.benchmark816IsRealized`, `P15Structure.benchmark816Price`, `P15Structure.elementaryPaletteFormula`, `P15Structure.triangleNeedsThree`, `P15Structure.benchmark816Palette`, `P15Structure.localDecomposability`, `P15Structure.localObstructionIsFullBlock`, `P15Structure.fullBlockCoverIff`, `P15Structure.wholeGroundChromaticNumber`, `P15Structure.benchmark816WholeGround`, `P15Structure.minimalForbiddenCharacterisation`, `P15Structure.benchmark816MinimalForbiddenCounts`, `P15Hazard.localPriceChain`, `P15Hazard.independentBlockHazard`, `P15Covers.benchmark816Cover`, `P15Covers.localRestriction`, `P15Covers.paletteNumberAttained`, `P15Covers.benchmark816UnionBound`, `P15Budget.coverCostHazardBound` | Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: PaletteNumberAttained, MinimalForbiddenCharacterisation, LocalRestriction, LocalDecomposability, LocalObstructionIsFullBlock, LocalPriceChain, IndependentBlockHazard, FullBlockCoverIff, CoverCostHazardBound, WholeGroundChromaticNumber, ElementaryPaletteFormula, TriangleNeedsThree, Benchmark816IsRealized, Benchmark816Palette, Benchmark816Cover, Benchmark816WholeGround, Benchmark816MinimalForbiddenCounts, Benchmark816Price, Benchmark816UnionBound (20 entries below, label `proved`; 13 of them (P15Structure.minimalForbiddenCharacterisation, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.elementaryPaletteFormula, P15Elementary.triangleNeedsThree, P15Structure.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Palette, P15Structure.benchmark816WholeGround, P15Structure.benchmark816MinimalForbiddenCounts, P15Structure.benchmark816Price) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 7 of them (P15Covers.paletteNumberAttained, P15Covers.localRestriction, P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Budget.coverCostHazardBound, P15Covers.benchmark816Cover, P15Covers.benchmark816UnionBound) added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN)); among them Proposition P1 (MinimalForbiddenCharacterisation), both halves of Theorem P2 (FullBlockCoverIff, display (P7), and CoverCostHazardBound, display (P8)), the local statements (P2)-(P5) and the benchmark Props including the union bound (P11). Theorem-shaped Props of this row still `specified` with no theorem: none; the parametrised predicates IsRealizedFamily, IsGood, IsKDecomposable, IsGeneratorCover, IsPaletteFeasible, IsMinimalForbidden, IsTransversal are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. Theorem P2 is proved as the two Props state it (for the formal RealizedData record and the formal downset, prices 0 <= c_v <= p_v), which is a statement about those Props, not acceptance of the informal theorem: the claim-level label stays `specified` (this register admits only none/specified at claim level), alignment of RealizedData / IsRealizedFamily / downset / coverCost / mu with the informal objects is PENDING_INDEPENDENT_REVIEW, and the price range p_v < c_v <= phi(p_v) is a different claim row. The 12 main P15 arithmetic theorems listed as `proved` (finite arithmetic of Section 6: 409, 2454, 15, 500616, 419743994415, 816, 815*3 < 2448, 818, 409p = 1/100, the (P11) rational bound, positivity of 6/40900^409, the elementary instances of (P9)) are at commit 91cffbb5 of main PR #358, not on main; they do not count the minimal forbidders, show the palette is 816 or evaluate 6/40900^409, and the proof-packet theorems that do (Benchmark816MinimalForbiddenCounts, Benchmark816Palette, Benchmark816Price) do so for the formal RealizedData record, whose identification with the informal six-block construction is an alignment question. Nothing is asserted for prices p_v < c_v <= phi(p_v) (row p15-price-budget-restricted), for arbitrary P15 inputs or for the prize (original_prize_closed stays false). Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | Proof lane: every theorem-shaped Prop of this row has a theorem (the parametrised predicates IsRealizedFamily, IsGood, IsKDecomposable, IsGeneratorCover, IsPaletteFeasible, IsMinimalForbidden, IsTransversal are vocabulary, not targets); every theorem of this row has a hosted receipt (run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)). Independent reviewer: alignment of MinimalForbiddenCharacterisation with Proposition P1 (completeness clause), of FullBlockCoverIff with (P7), of CoverCostHazardBound with (P8) (the sInf cover cost, the cappedHazard endpoint convention) and of the Benchmark816 data record with the Section 6 construction. main: PR #358 merge (operator) so the 12 arithmetic theorems are on main. | | `p15-price-boundary` | landing_claim | EXACT_COUNTEREXAMPLE | `specified` | 3 specified: `TwoCoordGeneratorPrices`, `DemandOneRefutation`, `LogOnePlusBelowIdentity`; 6 proved (main 91cffbb5, PR #358, not on main): `smallest_member_block_size`, `demand_one_good_sets`, `demand_one_obstruction`, `good_measure_three_quarters`, `generator_prices_ninths`, `minimum_generator_price_exceeds_third`; 3 proved (proof packet): `P15Elementary.twoCoordGeneratorPrices`, `P15Elementary.logOnePlusBelowIdentity`, `P15Elementary.demandOneRefutation` | DemandOneRefutation states the refutation (the two-coordinate data form a realized family with K_H(d) = 1, obstruction {X}, admissible prices 2/3, mu_p(D) = 3/4, cover cost 4/9 > 1/3 > log(4/3)). DemandOneRefutation, TwoCoordGeneratorPrices and LogOnePlusBelowIdentity each have a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011 (listed below as `proved`, the label of a committed proof package; the hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked them (receipt formalization_status kernel-checked, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound})); the two logarithm inequalities phi(1/2) = log 2 > 2/3 and log(4/3) < 1/3 are conjuncts of DemandOneRefutation and are proved inside its theorem from Mathlib's Real.log bounds. The claim-level label stays `specified`, not `proved`: this register admits only none/specified at claim level (no research claim of this programme is proved), whether DemandOneRefutation says what the source paragraph says is the independent reviewer's alignment question (PENDING_INDEPENDENT_REVIEW), the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked the Prop and nothing more, and a kernel-checked Prop is evidence about that Prop only. This is the one row of the register where a theorem proves the Prop registered as stating the claim itself; it is recorded here for the operator and not acted on. The 6 main P15 arithmetic theorems listed as `proved` (good sets, obstruction, 3/4 over denominator 4, prices 9,6,6,4 over 9, 3*4 > 9) are at commit 91cffbb5 of main PR #358, not on main; they are finite arithmetic, not the refutation. The counterexample does not refute d_i >= 2 successors or original P15-B (the disposition reason; the theorem's docstring says the same). Disposition EXACT_COUNTEREXAMPLE is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | The hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) produced the first receipt (`kernel-checked` appears only there); later commits need their own run. Independent (non-Anthropic) reviewer: alignment of DemandOneRefutation with the source paragraph (same two-coordinate family, prices, cover cost and the "does not contradict original P15-B" scope); any claim-level label beyond none/specified is a register-schema decision for the operator after that review, which no receipt produces. main: PR #358 merge (operator) so the 6 arithmetic theorems are on main. | | `p15-price-budget-restricted` | landing_claim | HOLD_WITH_DOMAIN | `specified` | 8 specified: `LogRatioBound`, `RatioExponentBound`, `RestrictedLocalBudget`, `RestrictedTransformedPriceBudget`, `RhoTestSufficient`, `CapTestAtTwoThirds`, `Benchmark816TransformedRange`, `Benchmark816RationalPrice`; 3 proved (main 91cffbb5, PR #358, not on main): `t3_ratio_at_minimal_block`, `t3_bound_finite_instances`, `explicit_rational_price_above_p`; 8 proved (proof packet): `P15Elementary.logRatioBound`, `P15Elementary.capTestAtTwoThirds`, `P15Elementary.ratioExponentBound`, `P15Structure.benchmark816RationalPrice`, `P15Budget.restrictedLocalBudget`, `P15Budget.rhoTestSufficient`, `P15Budget.restrictedTransformedPriceBudget`, `P15Budget.benchmark816TransformedRange` | RestrictedTransformedPriceBudget states (T1) for every realized family with a_i >= 1, d_i >= 2, p_v <= 1/4, c_v <= phi(p_v) and is proved as stated in the proof packet; that is evidence about the Prop (for the formal RealizedData record, downset, coverCost and mu), not acceptance of the informal theorem: the claim-level label stays `specified` (claim-level vocabulary none/specified; alignment PENDING_INDEPENDENT_REVIEW). Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: LogRatioBound, RatioExponentBound, RestrictedLocalBudget, RestrictedTransformedPriceBudget, RhoTestSufficient, CapTestAtTwoThirds, Benchmark816TransformedRange, Benchmark816RationalPrice (8 entries below, label `proved`; 4 of them (P15Elementary.logRatioBound, P15Elementary.ratioExponentBound, P15Elementary.capTestAtTwoThirds, P15Structure.benchmark816RationalPrice) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 4 of them (P15Budget.restrictedLocalBudget, P15Budget.restrictedTransformedPriceBudget, P15Budget.rhoTestSufficient, P15Budget.benchmark816TransformedRange) added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN)). Theorem-shaped Props of this row still `specified` with no theorem: none. The intermediate ratio steps of (T3) are proved inside the theorems that use them; the integral representation behind (T2), the admissibility integral and the Section 5 checker program are not specified. The 3 main P15 arithmetic theorems listed as `proved` (the a = 1, n = 3 instance 16/27, finite instances of the (T3) bound and the rational identity 81801/3345620000) are at commit 91cffbb5 of main PR #358, not on main; they are not (T1). The dependency on claim p15-realized-covers is discharged only to the extent that row records. Disposition HOLD_WITH_DOMAIN ("independent theorem review not yet source-bound in manifest") is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | Math- Mathlib lane: every theorem-shaped Prop of this row has a theorem; every theorem of this row has a hosted receipt (run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)). Independent reviewer: alignment of RestrictedTransformedPriceBudget with (T1) including the per-coordinate price hypothesis, the K >= K_H(d) palette and the cappedHazard endpoint convention. main: PR #358 merge (operator) so the 3 arithmetic theorems are on main. | | `p15-full-price` | landing_claim | REVIEWED_SCOPED | `specified` | 20 specified: `FullTransformedPriceBudget`, `RhoStarEnclosure`, `RhoStarBelowSixSevenths`, `IsProperDownset`, `HazardStarPositive`, `CoordinatewiseConcavity`, `HazardProductBound`, `HazardPhiBound`, `FullGroundPriceBound`, `PhiAtPStar`, `FullGroundBudgetIff`, `CapacityHazardIsBinomial`, `OddMajorityIdentity`, `CapacityWorstCase`, `HStarExceedsOne`, `LocalPriceBoundFull`, `GlobalAssemblyHazard`, `SharpnessOfRhoStar`, `DemandOneBoundary`, `SixSeventhsCertificate`; 7 proved (main 91cffbb5, PR #358, not on main): `rho_star_endpoints_adjacent`, `rho_star_upper_endpoint_below_six_sevenths`, `sixteen_twentysevenths_below_six_sevenths`, `minimal_block_size_instances`, `e_rational_bounds_ordering`, `exp_eleven_sixths_taylor_certificate`, `h_star_rational_certificate_constants`; 19 proved (proof packet): `P15Elementary.rhoStarBelowSixSevenths`, `P15Elementary.sixSeventhsCertificate`, `P15Elementary.hStarExceedsOne`, `P15Elementary.phiAtPStar`, `P15Elementary.hazardStarPositive`, `P15Elementary.oddMajorityIdentity`, `P15Elementary.capacityWorstCase`, `P15Elementary.demandOneBoundary`, `P15Elementary.capacityHazardIsBinomial`, `P15Elementary.sharpnessOfRhoStar`, `P15Hazard.globalAssemblyHazard`, `P15Interpolation.rhoStarEnclosure`, `P15Interpolation.coordinatewiseConcavity`, `P15Interpolation.hazardProductBound`, `P15Interpolation.hazardPhiBound`, `P15Interpolation.fullGroundPriceBound`, `P15Interpolation.fullGroundBudgetIff`, `P15Interpolation.localPriceBoundFull`, `P15Interpolation.fullTransformedPriceBudget` | FullTransformedPriceBudget states (F2) and is proved as stated; RhoStarEnclosure states (F3) and is proved as stated; SharpnessOfRhoStar states the attainment clause and is proved as stated. The claim-level label stays `specified` (claim-level vocabulary none/specified; a theorem of a Prop is evidence about that Prop, for the formal objects, not acceptance of Theorem F; alignment PENDING_INDEPENDENT_REVIEW). Theorem F is proved as FullTransformedPriceBudget states it: for the formal RealizedData record with a_i >= 1, d_i >= 2, every p in [0,1]^X, every 0 <= c_v <= phi(p_v) and the same K >= K_H(d), coverCost <= cappedHazard rho_star mu, through the coordinatewise concavity (CoordinatewiseConcavity, a theorem of exactly that Prop), (F6)-(F8), (F12) and (F13); that is evidence about those Props, not acceptance of the informal theorem. RhoStarEnclosure is proved by a rational certificate (Real.exp_bound series enclosures for e and for exp with explicit remainders, compared by norm_num), not by the source's full_price.py Fraction procedure, which remains unspecified. Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: FullTransformedPriceBudget, RhoStarEnclosure, RhoStarBelowSixSevenths, HazardStarPositive, CoordinatewiseConcavity, HazardProductBound, HazardPhiBound, FullGroundPriceBound, PhiAtPStar, FullGroundBudgetIff, CapacityHazardIsBinomial, OddMajorityIdentity, CapacityWorstCase, HStarExceedsOne, LocalPriceBoundFull, GlobalAssemblyHazard, SharpnessOfRhoStar, DemandOneBoundary, SixSeventhsCertificate (19 entries below, label `proved`; 8 of them (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.hazardStarPositive, P15Elementary.phiAtPStar, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.hStarExceedsOne, P15Elementary.demandOneBoundary, P15Elementary.sixSeventhsCertificate) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1); 3 of them (P15Elementary.capacityHazardIsBinomial, P15Hazard.globalAssemblyHazard, P15Elementary.sharpnessOfRhoStar) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN); 8 of them (P15Interpolation.fullTransformedPriceBudget, P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head). Theorem-shaped Props of this row still `specified` with no theorem: none; the parametrised predicates IsProperDownset are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. The derivative formulas (F4)-(F5) are not specified; CoordinatewiseConcavity, their consequence, is proved from the one-coordinate split of mu_p(A) (display (F4): affine in exp(-t_i) with coefficients 0 <= S_in <= S_out for a downset) and the second-derivative criterion for s \|-> -log(a + b exp(-s)). SixSeventhsCertificate uses Mathlib's decimal bound on e in place of the source's series argument. The 7 main P15 arithmetic theorems listed as `proved` (finite arithmetic of the Section 6 certificate chain: adjacent 20-digit endpoints as Nat, 16/27 < 6/7, n >= 2a+1 instances, the rational orderings and the degree-5 Taylor partial sum) are at commit 91cffbb5 of main PR #358, not on main; they do not place rho_star = 1/(3 - log(3e - 2)) between the endpoints, do not prove e < 87/32 or 197/32 < exp(11/6) as real-number facts (the SixSeventhsCertificate theorem does, as that Prop states them), and are not Theorem F. The full_price.py Fraction procedure is not a kernel proof and is not specified. No arbitrary-downset or unrestricted P15 prize statement (original_prize_closed stays false). Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | Math- Mathlib lane: every theorem-shaped Prop of this row has a theorem (the parametrised predicates IsProperDownset are vocabulary, not targets); every theorem of this row has a hosted receipt: the 8 added after PR head 79731b96 (P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull, P15Interpolation.fullTransformedPriceBudget) by run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) and first by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3), the others by run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2). Independent reviewer: alignment of FullTransformedPriceBudget with (F2), in particular the quantification over ALL p in [0,1]^X and the SAME palette K >= K_H(d), and of the proved interpolation chain (CoordinatewiseConcavity through LocalPriceBoundFull) with Sections 2-3 of the source. main: PR #358 merge (operator) so the 7 arithmetic theorems are on main. | | `downstream-hard-gate` | landing_claim | ENGINEERING_HOLD | `none` | — | An engineering control, not a mathematical claim: there is no theorem to state in Lean and none is stated. The gate's terminal classifications and non-discharge tokens are transcribed in the graph_node rows of this register only as source identity. Disposition ENGINEERING_HOLD ("repository-wide external controlling-source binding remains unresolved") is transcribed, not re-derived. | No Lean target. The control's semantics live in Math- frontiers/downstream_gate_20260925 and its workflow; the register only records that every GRAPH math.* node has a row here. | -| `math.uniform-matrix-cap-lifetime` | graph_node | — | `specified` | 32 specified: `IsLPeriodic`, `IsSmooth`, `GaussianFieldInterface`, `IsCriticalPoint`, `HessianHasInertia`, `IsNondegMax`, `IsIndexSaddle`, `IsCandidatePair`, `IsMorseDistinct`, `SuperlevelBarInterface`, `IsFiniteBar`, `IsLifetimeDensityVersion`, `IsLPeriodic`, `IsSmooth`, `GaussianFieldInterface`, `IsCriticalPoint`, `HessianHasInertia`, `IsNondegMax`, `IsIndexSaddle`, `IsCandidatePair`, `IsMorseDistinct`, `SuperlevelBarInterface`, `IsFiniteBar`, `IsLifetimeDensityVersion`, `IsWindowPair`, `IsWindowElderPair`, `IsUnrestrictedPair`, `PairLawInterface`, `ParentUniformSelectionA`, `ParentCompactWindowDensityB`, `IsUnrestrictedLeadingCoefficient`, `ParentUnrestrictedLeadingDensityC` | ParentUniformSelectionA, ParentCompactWindowDensityB and ParentUnrestrictedLeadingDensityC state the shapes of (1.1), (1.2), (1.3) for every object satisfying the interfaces; nothing is proved. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. The reading rule that the GRAPH binds to this node (congruence erratum E1, Section 9 replacement v1.1, marked-cylinder cap sections 1-5, reconciliation record with W1 and the embedding radius r < L/(4 sqrt 2)) is NOT transcribed into any hypothesis; the Props are stated from the parent bytes alone. Not stated: the regression law Q_{r,b,k,R}, the typed weight W_r, the full normalizer Z_r and Q^W (p_r is a primitive interface field), the marked Kac-Rice representation (Sections 9-10, radial ledger (10.2)), the Gaussian/angular representation (15.2) of c_{d,L} and (11.3) of c_{B,K}, (12.1)-(12.3), (14.1), (5.5), Sections 2-8. No numerical r_*, C, z_*, c_{B,K}, c_{d,L}; no lower bound for d >= 3. The GRAPH classification PROVED_REVIEWED and the review disposition ACCEPT_AT_EXISTENTIAL_SCOPE are source identity, not re-derived; the five validity premises of the hardening-branch claim graph are not touched. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p or the field: the parent-shape Props are predicates on the interface objects, not closed statements; the universal closure of ParentUniformSelectionA over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed. | Pin ERRATUM_CONGRUENCE.md, REPAIR.md (Section 9 v1.1) and the reconciliation record, and add the reading rule as explicit hypotheses or separate Props; state (15.2) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of the three parent Props against (1.1)-(1.3) with the honesty checklist (quantifier order, existential constants outside the universal quantifiers, "versions", compact marks, all frames). | +| `math.uniform-matrix-cap-lifetime` | graph_node | — | `specified` | 32 specified: `IsLPeriodic`, `IsSmooth`, `GaussianFieldInterface`, `IsCriticalPoint`, `HessianHasInertia`, `IsNondegMax`, `IsIndexSaddle`, `IsCandidatePair`, `IsMorseDistinct`, `SuperlevelBarInterface`, `IsFiniteBar`, `IsLifetimeDensityVersion`, `IsLPeriodic`, `IsSmooth`, `GaussianFieldInterface`, `IsCriticalPoint`, `HessianHasInertia`, `IsNondegMax`, `IsIndexSaddle`, `IsCandidatePair`, `IsMorseDistinct`, `SuperlevelBarInterface`, `IsFiniteBar`, `IsLifetimeDensityVersion`, `IsWindowPair`, `IsWindowElderPair`, `IsUnrestrictedPair`, `PairLawInterface`, `ParentUniformSelectionA`, `ParentCompactWindowDensityB`, `IsUnrestrictedLeadingCoefficient`, `ParentUnrestrictedLeadingDensityC` | ParentUniformSelectionA, ParentCompactWindowDensityB and ParentUnrestrictedLeadingDensityC state the shapes of (1.1), (1.2), (1.3) for every object satisfying the interfaces; nothing is proved. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. The reading rule that the GRAPH binds to this node (congruence erratum E1, Section 9 replacement v1.1, marked-cylinder cap sections 1-5, reconciliation record with W1 and the embedding radius r < L/(4 sqrt 2)) is NOT transcribed into any hypothesis; the Props are stated from the parent bytes alone. Not stated: the regression law Q_{r,b,k,R}, the typed weight W_r, the full normalizer Z_r and Q^W (p_r is a primitive interface field), the marked Kac-Rice representation (Sections 9-10, radial ledger (10.2)), the Gaussian/angular representation (15.2) of c_{d,L} and (11.3) of c_{B,K}, (12.1)-(12.3), (14.1), (5.5), Sections 2-8. No numerical r_*, C, z_*, c_{B,K}, c_{d,L}; no lower bound for d >= 3. The GRAPH classification PROVED_REVIEWED and the review disposition ACCEPT_AT_EXISTENTIAL_SCOPE are source identity, not re-derived; the five validity premises of the hardening-branch claim graph are not touched. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p or the field: the parent-shape Props are predicates on the interface objects, not closed statements; the universal closure of ParentUniformSelectionA over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed. | Pin ERRATUM_CONGRUENCE.md, REPAIR.md (Section 9 v1.1) and the reconciliation record, and add the reading rule as explicit hypotheses or separate Props; state (15.2) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of the three parent Props against (1.1)-(1.3) with the honesty checklist (quantifier order, existential constants outside the universal quantifiers, "versions", compact marks, all frames). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ParentCompactWindowDensityB BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, expected-number-of-zeros or critical points; present but insufficient: gaussianReal (Real.lean), Measure.map (Map.lean)); ParentUnrestrictedLeadingDensityC BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: IsEquivalent (AsymptoticEquivalent.lean), ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean)); ParentUniformSelectionA BLOCKED_INTERFACE (p == 0 (docstring inhabitant, probe Lifetime-B_1)); HessianHasInertia, IsCandidatePair, IsCriticalPoint, IsFiniteBar, IsIndexSaddle, IsLPeriodic, IsLifetimeDensityVersion, IsMorseDistinct, IsNondegMax, IsSmooth, IsUnrestrictedLeadingCoefficient, IsUnrestrictedPair, IsWindowElderPair, IsWindowPair (Lifetime and Field copies) PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. | | `math.d1-component.congruence-erratum` | graph_node | — | `none` | — | No Lean statement: the erratum's source bytes are not pinned in this packet and no Prop states the corrected congruence D_r H_i D_r or det(D_r H_i D_r) = det(H_i)/r. The parent Props of this register do not transcribe section 5 at all, so the amendment has nothing to attach to yet. | Pin ERRATUM_CONGRUENCE.md (blob 213594d6, 1782 B); state the corrected congruence as a matrix identity over Matrix (Fin d) (Fin d) R (a candidate kernel-proof target: block-matrix algebra). | | `math.d1-component.section9-replacement-v1_1` | graph_node | — | `none` | — | No Lean statement: the repair's bytes are not pinned and no Prop states the weighted pair Kac-Rice identity (9.1) or its monotone-class extension to bounded Borel marks. In this register the marked Kac-Rice representation is deliberately NOT transcribed (densities are pinned only through IsLifetimeDensityVersion), so the Borel elder mark enters only as the ElderPartner field of SuperlevelBarInterface, whose measurability is disclaimed there. | Pin REPAIR.md (blob fe9b9ce4, 9062 B); state (9.1)-(9.4) as Props over MeasureTheory.Measure on the pair domain with the Armentano-Azais-Leon identity as an interface hypothesis. | | `math.d1-component.marked-cylinder-cap` | graph_node | — | `none` | — | No Prop of this package states the deterministic cap theorem (unique transverse ridge, closed cap through S separating M from every older point, elder death partner of M is S under conditions (1)). The pinned bytes exist (sources/marked-cylinder-cap). Kernel evidence exists outside this package: Math- frontiers/cap_first_exit_lean_20261002 proves the topological global step and the analytic chain L1-L13, L16-L20 (torus_capHyp_blocks, torus_elder_death_level_blocks), but its 152 declarations are not registered here with anchors, its alignment review is NOT_CLAIMED, and it does not formalize that the field on G_r satisfies SP section 2, L14/L15, L23 or the persistence-pairing identification. The probabilistic corollary (2)-(3) and sections 6-8 are not consumed by D1 and not stated. | Register the CapFirstExit declarations here with their ALIGNMENT.md anchors against the pinned CAP bytes (statements of kind theorem, label proved, kernel_evidence cap-first-exit-lean.yml), or state the deterministic theorem as a def : Prop in Lifetime.lean so that the Math- lane's delta review has a target; independent (non-Claude) reviewer for the v1.5-v2.6 delta. | | `math.d1-component.reconciliation-record` | graph_node | — | `none` | — | A review/reading-rule record, not a mathematical statement: nothing to state in Lean beyond what its components say. The embedding-radius condition r < L/(4 sqrt 2) and the W1 wording are not transcribed into any hypothesis of the parent Props. Its ACCEPT verdicts are not re-derived; the OpenAI author-provider check it records is non-discharge by its own terms. | Add the embedding radius as an explicit hypothesis to ParentUniformSelectionA / PairLawInterface once the reading rule is pinned; no proof target. | -| `math.lifetime-remainder` | graph_node | — | `specified` | 1 specified: `RemainderBoundedR1` | See row lifetime-remainder: (R1) is stated, not proved; the imports of parent interfaces D1-A-E are not transcribed; Theorem A's cubic bound (1.1) is not consumed by R and is not a hypothesis here. The GRAPH classification and review disposition are source identity only. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row lifetime-remainder. | -| `math.side24-coefficient` | graph_node | — | `specified` | 3 specified: `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison` | See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the "evaluates_eq_15_2" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. The 27 arithmetic theorems of main and the 5 proof-packet theorems (ExponentialBounds, SuccessiveTermRatio, ConeMomentD2Algebra, ReferenceCoefficientIdentity, ElementaryConeIntegral; none of them the enclosure; kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)) are listed on the landing row. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. | As for row side24-coefficient. | +| `math.lifetime-remainder` | graph_node | — | `specified` | 1 specified: `RemainderBoundedR1` | See row lifetime-remainder: (R1) is stated, not proved; the imports of parent interfaces D1-A-E are not transcribed; Theorem A's cubic bound (1.1) is not consumed by R and is not a hypothesis here. The GRAPH classification and review disposition are source identity only. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row lifetime-remainder. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemainderBoundedR1 BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean), ncard_le_ncard (Card.lean)). | +| `math.side24-coefficient` | graph_node | — | `specified` | 3 specified: `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison` | See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the "evaluates_eq_15_2" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. Of the three statements of this node, none has a theorem; stated, not proved: CoefficientEnclosure, ReferenceComparison. The 27 arithmetic theorems of main and the 17 proof-packet theorems of the landing row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral, Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw; none of them the enclosure; 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them) are listed on the landing row, never here and never as the node; the interface theorems among them (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw) hold for every inhabitant of ConeMomentInterface from its law fields alone and construct nothing. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited. | As for row side24-coefficient. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): CoefficientEnclosure BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface elaborates, by doubling coneMoment and c24, so no theorem of its type can exist unless the interface is empty); ReferenceComparison BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, ReferenceComparison I) -> IsEmpty PeriodicCoefficientInterface elaborates likewise). | | `math.rn-count-interface` | graph_node | — | `none` | 5 specified: `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5`; 5 proved (proof packet): `RNProbability.conditionalMeanIdentityN3`, `RNProbability.holderCountBoundN1`, `RNProbability.holderConsequenceN2`, `RNProbability.uniformCounterexampleN4`, `RNProbability.exponentialTailLayerCakeN5` | See row rn-count-interface: (N1)-(N5) stated and each proved as a theorem of the proof packet (5 entries below, `proved`; kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)), (N6)-(N7) not stated, so no Prop states the interface as a whole and the node stays none; nothing about the Gaussian model is proved. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | As for row rn-count-interface. | -| `math.rn-fixed-remote-window` | graph_node | — | `specified` | 3 specified: `PinnedLawInterface`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | See row rn-fixed-remote-window: displays (2)-(3) stated over PinnedLawInterface, not proved; Section 6 not stated; no inhabitant of the interface. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window. | -| `math.rn-region.fixed-remote` | graph_node | — | `specified` | 2 specified: `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | The region's mathematical content is the fixed-remote Theorem A at exactly this scope, stated by A2/A3 and not proved. The coverage classification itself (COVERED_BY_CANDIDATE, coverage_source) is graph bookkeeping and is not a Lean statement. No all-height bound on the remote region, no rho -> 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window; nothing region-specific to add. | -| `math.rn-region.mesoscopic-scaled-annulus` | graph_node | — | `specified` | 2 specified: `D5dSummedPinNeighbourhoodC2`, `AnnulusBridgeAllHeight2` | D5dSummedPinNeighbourhoodC2 states the summed pin-neighbourhood bound (all heights, scaled ball of radius R minus the pins) and AnnulusBridgeAllHeight2 the all-height fixed-annulus bridge; neither is proved, and the "covers the reviewed fixed annulus" reading (R >= B) is not a Lean statement. No d > 2, no numerical constants, no elder selection, no JETMOD or witness-collision closure. The eight reading-rule component nodes are source identity, not transcribed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Independent reviewer: alignment of D5dSummedPinNeighbourhoodC2 with row D5-d; proof lane: none yet (the composition D5-a + D5-c at eta = 1/4 is a proof, not a statement). | -| `math.rn-region.pin-collision` | graph_node | — | `specified` | 3 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk`, `D5dSummedPinNeighbourhoodC2` | The three row shapes are stated for any object satisfying PinnedLawInterface and are not proved; the P2 source (PUNCTURED_PIN_PROOF.md) is not pinned here, so the Props are anchored to the reconciliation's transcription of the rows, not to the proof's own bytes. Including the deterministic pin atoms would invalidate the vanishing bounds (the reconciliation's caveat); the Props puncture the disks accordingly. No d > 2, no numerical constants, no uniformity in kappa. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin PUNCTURED_PIN_PROOF.md and COLLAR_PROOF.md (reviews/d5_local_collar_20260928) and re-anchor D5a/D5b/D5c to their own displays; independent reviewer: alignment of the three Props with rows D5-a, D5-b, D5-d. | -| `math.rn-region.intermediate-r-to-rho` | graph_node | — | `specified` | 1 specified: `D5eIntermediateShellsI3I4` | D5eIntermediateShellsI3I4 states the window-only bound E N <= C r^3 (A_0^(-2) + rho^2) on {A_0 r <= \|X\| <= rho} with side conditions taken from dimension-lift (1.6); not proved. The I3/I4 source (frontiers/intermediate_window_20260928) is not pinned here; no all-height intermediate count exists or is claimed; no numerical constants. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin intermediate_window_20260928/PROOF.md and anchor the single-shell display there; independent reviewer: alignment of D5eIntermediateShellsI3I4 with row D5-e and with dimension-lift (1.6). | -| `math.rn-region.witness-collision` | open_obligation | — | `specified` | 2 specified: `WitnessCollisionNearPinPairsTarget`, `RemoteCollisionSecondFactorialD` | WitnessCollisionNearPinPairsTarget states the OPEN obligation as a target Prop, not as a result: nothing asserts that the target holds, is attainable, or follows from any reviewed source. RemoteCollisionSecondFactorialD states only the fixed-remote, fixed-eta part (Corollary D of remote_collision), which the GRAPH records as reviewed evidence and NOT as a proof of this node. The Palm route's Theorem Q (q = 2) would imply the target but is an author-side candidate stated separately (PalmRouteFactorialMomentsQ) and not consumed here; the c6_witness_collision reconciliation's proposed discharge is declarative with execution pending. The node stays OPEN_ACTIVE; this register cannot and does not change that. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | None for Lean beyond the target statement: the obligation needs a source-bound proof and nonauthor review in Math-; if Theorem Q is folded, re-anchor the target to the fold record. | +| `math.rn-fixed-remote-window` | graph_node | — | `specified` | 3 specified: `PinnedLawInterface`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | See row rn-fixed-remote-window: displays (2)-(3) stated over PinnedLawInterface, not proved; Section 6 not stated; no inhabitant of the interface. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point). | +| `math.rn-region.fixed-remote` | graph_node | — | `specified` | 2 specified: `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | The region's mathematical content is the fixed-remote Theorem A at exactly this scope, stated by A2/A3 and not proved. The coverage classification itself (COVERED_BY_CANDIDATE, coverage_source) is graph bookkeeping and is not a Lean statement. No all-height bound on the remote region, no rho -> 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window; nothing region-specific to add. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point). | +| `math.rn-region.mesoscopic-scaled-annulus` | graph_node | — | `specified` | 2 specified: `D5dSummedPinNeighbourhoodC2`, `AnnulusBridgeAllHeight2` | D5dSummedPinNeighbourhoodC2 states the summed pin-neighbourhood bound (all heights, scaled ball of radius R minus the pins) and AnnulusBridgeAllHeight2 the all-height fixed-annulus bridge; neither is proved, and the "covers the reviewed fixed annulus" reading (R >= B) is not a Lean statement. No d > 2, no numerical constants, no elder selection, no JETMOD or witness-collision closure. The eight reading-rule component nodes are source identity, not transcribed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Independent reviewer: alignment of D5dSummedPinNeighbourhoodC2 with row D5-d; proof lane: none yet (the composition D5-a + D5-c at eta = 1/4 is a proof, not a statement). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): AnnulusBridgeAllHeight2 BLOCKED_INTERFACE (a Dirac-tilted law at a pinned smooth periodic path with one extra nondegenerate critical point in the annulus violates display (2) for a null Borel E); D5dSummedPinNeighbourhoodC2 BLOCKED_INTERFACE (a Dirac-law inhabitant with one extra nondegenerate critical point at the midpoint gives count >= 1 > C r^3 as r -> 0). | +| `math.rn-region.pin-collision` | graph_node | — | `specified` | 3 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk`, `D5dSummedPinNeighbourhoodC2` | The three row shapes are stated for any object satisfying PinnedLawInterface and are not proved; the P2 source (PUNCTURED_PIN_PROOF.md) is not pinned here, so the Props are anchored to the reconciliation's transcription of the rows, not to the proof's own bytes. Including the deterministic pin atoms would invalidate the vanishing bounds (the reconciliation's caveat); the Props puncture the disks accordingly. No d > 2, no numerical constants, no uniformity in kappa. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin PUNCTURED_PIN_PROOF.md and COLLAR_PROOF.md (reviews/d5_local_collar_20260928) and re-anchor D5a/D5b/D5c to their own displays; independent reviewer: alignment of the three Props with rows D5-a, D5-b, D5-d. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): BLOCKED_INTERFACE (interface under-determination, not statement defects; each refuted by a concrete Dirac-law inhabitant): D5aPuncturedPinDisks (extra critical point in the punctured disk, null E), D5bMicrodisk (extra critical point at distance r^2 from M), D5dSummedPinNeighbourhoodC2 (extra critical point at the midpoint, count >= 1 > C r^3). | +| `math.rn-region.intermediate-r-to-rho` | graph_node | — | `specified` | 1 specified: `D5eIntermediateShellsI3I4` | D5eIntermediateShellsI3I4 states the window-only bound E N <= C r^3 (A_0^(-2) + rho^2) on {A_0 r <= \|X\| <= rho} with side conditions taken from dimension-lift (1.6); not proved. The I3/I4 source (frontiers/intermediate_window_20260928) is not pinned here; no all-height intermediate count exists or is claimed; no numerical constants. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin intermediate_window_20260928/PROOF.md and anchor the single-shell display there; independent reviewer: alignment of D5eIntermediateShellsI3I4 with row D5-e and with dimension-lift (1.6). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): D5eIntermediateShellsI3I4 BLOCKED_INTERFACE (a Dirac-law inhabitant with one extra window-height critical point at distance 8r from the midpoint). | +| `math.rn-region.witness-collision` | open_obligation | — | `specified` | 2 specified: `WitnessCollisionNearPinPairsTarget`, `RemoteCollisionSecondFactorialD` | WitnessCollisionNearPinPairsTarget states the OPEN obligation as a target Prop, not as a result: nothing asserts that the target holds, is attainable, or follows from any reviewed source. RemoteCollisionSecondFactorialD states only the fixed-remote, fixed-eta part (Corollary D of remote_collision), which the GRAPH records as reviewed evidence and NOT as a proof of this node. The Palm route's Theorem Q (q = 2) would imply the target but is an author-side candidate stated separately (PalmRouteFactorialMomentsQ) and not consumed here; the c6_witness_collision reconciliation's proposed discharge is declarative with execution pending. The node stays OPEN_ACTIVE; this register cannot and does not change that. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | None for Lean beyond the target statement: the obligation needs a source-bound proof and nonauthor review in Math-; if Theorem Q is folded, re-anchor the target to the fold record. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemoteCollisionSecondFactorialD BLOCKED_INTERFACE (a Dirac inhabitant with two prescribed remote critical points falsifies both displays; no field gives a.s. absence of index-j window pairs on null remote sets); WitnessCollisionNearPinPairsTarget BLOCKED_INTERFACE (a Dirac inhabitant with two window critical points within eta of a pin gives expectation 2 for all small r; consistent with the node's OPEN status). | | `math.rn-mesoscopic-reduction` | graph_node | — | `none` | — | No Lean statement: the PR7 route's bytes (frontiers/rn_mesoscopic_20260925/PROOF.md) are not pinned here and its Lemma A (positive-definite covariance of the rows of (U_0, J_0(y)) on K_AB) is not stated. The supersession is a graph classification, not mathematics. | Pin rn_mesoscopic_20260925/PROOF.md and state Lemma A (Matrix.PosDef lower bound on a compact annulus) if the route is ever revived; otherwise none. | -| `math.p15-full-price` | graph_node | — | `specified` | 3 specified: `FullTransformedPriceBudget`, `RhoStarEnclosure`, `SharpnessOfRhoStar`; 7 proved (main 91cffbb5, PR #358, not on main): `rho_star_endpoints_adjacent`, `rho_star_upper_endpoint_below_six_sevenths`, `sixteen_twentysevenths_below_six_sevenths`, `minimal_block_size_instances`, `e_rational_bounds_ordering`, `exp_eleven_sixths_taylor_certificate`, `h_star_rational_certificate_constants`; 3 proved (proof packet): `P15Elementary.sharpnessOfRhoStar`, `P15Interpolation.rhoStarEnclosure`, `P15Interpolation.fullTransformedPriceBudget` | See row p15-full-price. Of the three Props of this node, with a theorem of exactly their type in the proof packet: FullTransformedPriceBudget, RhoStarEnclosure, SharpnessOfRhoStar (3 entries below, label `proved`; 1 of them (P15Elementary.sharpnessOfRhoStar) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN); 2 of them (P15Interpolation.fullTransformedPriceBudget, P15Interpolation.rhoStarEnclosure) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head); none of the three is left unproved. A theorem of SharpnessOfRhoStar is about the formal triple block and the formal cover cost; a theorem of FullTransformedPriceBudget is about the formal RealizedData record, downset, coverCost and mu; none of this is Theorem F, and the node-level label stays `specified`. The 7 main P15 arithmetic theorems of source p15-full-price (at commit 91cffbb5 of main PR #358, not on main) are listed here as the arithmetic skeleton, never as the node. The price-boundary refutation is a scope boundary (GRAPH edge boundary_counterexample_not_required_premise), not a hypothesis. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | As for row p15-full-price. | +| `math.p15-full-price` | graph_node | — | `specified` | 3 specified: `FullTransformedPriceBudget`, `RhoStarEnclosure`, `SharpnessOfRhoStar`; 7 proved (main 91cffbb5, PR #358, not on main): `rho_star_endpoints_adjacent`, `rho_star_upper_endpoint_below_six_sevenths`, `sixteen_twentysevenths_below_six_sevenths`, `minimal_block_size_instances`, `e_rational_bounds_ordering`, `exp_eleven_sixths_taylor_certificate`, `h_star_rational_certificate_constants`; 3 proved (proof packet): `P15Elementary.sharpnessOfRhoStar`, `P15Interpolation.rhoStarEnclosure`, `P15Interpolation.fullTransformedPriceBudget` | See row p15-full-price. Of the three Props of this node, with a theorem of exactly their type in the proof packet: FullTransformedPriceBudget, RhoStarEnclosure, SharpnessOfRhoStar (3 entries below, label `proved`; 1 of them (P15Elementary.sharpnessOfRhoStar) was added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN); 2 of them (P15Interpolation.fullTransformedPriceBudget, P15Interpolation.rhoStarEnclosure) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head); none of the three is left unproved. A theorem of SharpnessOfRhoStar is about the formal triple block and the formal cover cost; a theorem of FullTransformedPriceBudget is about the formal RealizedData record, downset, coverCost and mu; none of this is Theorem F, and the node-level label stays `specified`. The 7 main P15 arithmetic theorems of source p15-full-price (at commit 91cffbb5 of main PR #358, not on main) are listed here as the arithmetic skeleton, never as the node. The price-boundary refutation is a scope boundary (GRAPH edge boundary_counterexample_not_required_premise), not a hypothesis. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | As for row p15-full-price. | | `math.p15-price-boundary` | graph_node | — | `specified` | 2 specified: `TwoCoordGeneratorPrices`, `DemandOneRefutation`; 6 proved (main 91cffbb5, PR #358, not on main): `smallest_member_block_size`, `demand_one_good_sets`, `demand_one_obstruction`, `good_measure_three_quarters`, `generator_prices_ninths`, `minimum_generator_price_exceeds_third`; 2 proved (proof packet): `P15Elementary.twoCoordGeneratorPrices`, `P15Elementary.demandOneRefutation` | See row p15-price-boundary: TwoCoordGeneratorPrices and DemandOneRefutation (the two Props of this node) have theorems of exactly their type in the proof packet (listed as `proved`; kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)), the two logarithm inequalities included; the claim-level label stays `specified` for the reasons given there (claim-level vocabulary none/specified, alignment PENDING_INDEPENDENT_REVIEW, a Prop-level proof is evidence about the Prop only, whatever the hosted run 38116277597 receipted). The 6 main P15 arithmetic theorems of source p15-price-boundary (at commit 91cffbb5 of main PR #358, not on main) are listed as the arithmetic skeleton, never as the node. REFUTED is the graph classification of the extension, transcribed as source identity. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | As for row p15-price-boundary. | | `math.rn-selector-region-crosswalk` | graph_node | — | `none` | — | An engineering inventory, not a mathematical claim; nothing to state in Lean and nothing stated. | No Lean target. | -| `math.rn-fixed-annulus-window` | graph_node | — | `specified` | 1 specified: `FixedAnnulusWindowCandidateA2` | See row rn-fixed-annulus-window: (A2) stated, not proved; no global D5/RN/JETMOD, k -> 0, d >= 3, pin-neighbourhood, intermediate-scale, collision or elder-selection closure. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-annulus-window. | -| `math.rn-region.fixed-annulus-window` | graph_node | — | `specified` | 1 specified: `FixedAnnulusWindowCandidateA2` | The region's content is (A2) at its exact scope, stated and not proved; the coverage classification is graph bookkeeping, not a Lean statement; the superseded high-jet route (regional.fixed-annulus.high-jet-route, SUPERSEDED_NONBLOCKING) is not represented. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-annulus-window. | -| `math.d5-pin-neighborhood-first-moment` | graph_node | — | `specified` | 6 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk`, `D5cCollarC1`, `D5dSummedPinNeighbourhoodC2`, `D5eIntermediateShellsI3I4`, `D5fGlobalWindowFirstMomentI5` | The six row shapes are stated for any object satisfying PinnedLawInterface (sample space Omega, gated pin regime 0 < r < L) and are not proved. The consumption graph (D5-d from D5-a + D5-c, D5-f from D5-d + I4 + remote Theorem A) is a proof structure and is not stated; the component sources PUNCTURED_PIN_PROOF.md, COLLAR_PROOF.md and intermediate_window PROOF.md are not pinned here, so D5a/b/c/e/f are anchored to the reconciliation's row text. The lower bound c r^3 <= P(A_r) of D5-f has its source in remote_collision Corollary F-, not pinned. No d > 2, no numerical constants, no uniformity in T, R, eta, k -> 0; no elder pairing; no RN/24-jet/JETMOD; the witness-collision region stays OPEN_ACTIVE. The STATUS.md note that D5 GRAPH transitions were not executed by that summary is a recorded discrepancy, not resolved here. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin the three component proofs and re-anchor; independent reviewer: alignment of the six Props with the section 2 table (all-height versus window I_r per row, exclusion of M and S). | -| `math.d5-component.punctured-pin-proof` | graph_node | — | `specified` | 2 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk` | The Props state rows D5-a and D5-b as the reconciliation transcribes P2, not P2 from its own bytes (not pinned); the slices P4, P9, P17, P20 and the continuum argument reviewed are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin PUNCTURED_PIN_PROOF.md (SHA-256 f972e50b… (GRAPH.json fingerprint)) and anchor D5a/D5b to its own displays. | +| `math.rn-fixed-annulus-window` | graph_node | — | `specified` | 1 specified: `FixedAnnulusWindowCandidateA2` | See row rn-fixed-annulus-window: (A2) stated, not proved; no global D5/RN/JETMOD, k -> 0, d >= 3, pin-neighbourhood, intermediate-scale, collision or elder-selection closure. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-annulus-window. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedAnnulusWindowCandidateA2 BLOCKED_INTERFACE (a Dirac inhabitant with an index-j window critical point on a null E0 (a singleton in the annulus); no field of PinnedLawInterface supplies the a.s. absence that the weighted Kac-Rice formula (A21) gives). | +| `math.rn-region.fixed-annulus-window` | graph_node | — | `specified` | 1 specified: `FixedAnnulusWindowCandidateA2` | The region's content is (A2) at its exact scope, stated and not proved; the coverage classification is graph bookkeeping, not a Lean statement; the superseded high-jet route (regional.fixed-annulus.high-jet-route, SUPERSEDED_NONBLOCKING) is not represented. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-annulus-window. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedAnnulusWindowCandidateA2 BLOCKED_INTERFACE (a Dirac inhabitant with an index-j window critical point on a null E0 (a singleton in the annulus); no field of PinnedLawInterface supplies the a.s. absence that the weighted Kac-Rice formula (A21) gives). | +| `math.d5-pin-neighborhood-first-moment` | graph_node | — | `specified` | 6 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk`, `D5cCollarC1`, `D5dSummedPinNeighbourhoodC2`, `D5eIntermediateShellsI3I4`, `D5fGlobalWindowFirstMomentI5` | The six row shapes are stated for any object satisfying PinnedLawInterface (sample space Omega, gated pin regime 0 < r < L) and are not proved. The consumption graph (D5-d from D5-a + D5-c, D5-f from D5-d + I4 + remote Theorem A) is a proof structure and is not stated; the component sources PUNCTURED_PIN_PROOF.md, COLLAR_PROOF.md and intermediate_window PROOF.md are not pinned here, so D5a/b/c/e/f are anchored to the reconciliation's row text. The lower bound c r^3 <= P(A_r) of D5-f has its source in remote_collision Corollary F-, not pinned. No d > 2, no numerical constants, no uniformity in T, R, eta, k -> 0; no elder pairing; no RN/24-jet/JETMOD; the witness-collision region stays OPEN_ACTIVE. The STATUS.md note that D5 GRAPH transitions were not executed by that summary is a recorded discrepancy, not resolved here. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin the three component proofs and re-anchor; independent reviewer: alignment of the six Props with the section 2 table (all-height versus window I_r per row, exclusion of M and S). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): BLOCKED_INTERFACE (interface under-determination, not statement defects; each refuted by a concrete Dirac-law inhabitant): D5aPuncturedPinDisks (extra critical point in the punctured disk, null E), D5bMicrodisk (extra critical point at distance r^2 from M), D5cCollarC1 (extra critical point at the midpoint, E = {0}), D5dSummedPinNeighbourhoodC2 (extra critical point at the midpoint, count >= 1 > C r^3), D5eIntermediateShellsI3I4 (extra window-height critical point at distance 8r from the midpoint), D5fGlobalWindowFirstMomentI5 (one or no extra window critical point refutes either direction; the lower bound is Set.ncard-sensitive). | +| `math.d5-component.punctured-pin-proof` | graph_node | — | `specified` | 2 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk` | The Props state rows D5-a and D5-b as the reconciliation transcribes P2, not P2 from its own bytes (not pinned); the slices P4, P9, P17, P20 and the continuum argument reviewed are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin PUNCTURED_PIN_PROOF.md (SHA-256 f972e50b… (GRAPH.json fingerprint)) and anchor D5a/D5b to its own displays. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): D5aPuncturedPinDisks BLOCKED_INTERFACE (a Dirac-law inhabitant with one extra critical point in the punctured disk and a null E); D5bMicrodisk BLOCKED_INTERFACE (the same Dirac-law inhabitant with the extra critical point at distance r^2 from M (kappa = 1)). | | `math.d5-component.punctured-pin-review-111` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | | `math.d5-component.punctured-pin-continuum-review` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | | `math.d5-component.punctured-pin-continuum-check` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | @@ -282,22 +283,22 @@ a theorem between string literals, an indented Prop definition, a proof written | `math.d5-component.i5-review-xai` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | | `math.d5-component.i5-review-claude` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | | `math.d5-component.offpin-second-moment-review` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | -| `math.d5-component.collar-proof` | graph_node | — | `specified` | 1 specified: `D5cCollarC1` | D5cCollarC1 states row D5-c as the reconciliation transcribes C1, not C1 from its own bytes (not pinned); the normalizer (P14) it imports and the corollary C2 are not stated separately; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin COLLAR_PROOF.md (SHA-256 794babe0… (GRAPH.json fingerprint)) and anchor D5cCollarC1 to display (C1). | -| `math.d5-component.annulus-proof` | graph_node | — | `specified` | 1 specified: `AnnulusBridgeAllHeight2` | AnnulusBridgeAllHeight2 states display (2) (d = 2, fixed L, compact birth, k_- > 0, 1 < A < B, all heights) over PinnedLawInterface; not proved. Not stated: displays (4)-(20), the R1-R7 review interfaces as separate statements, the deterministic three-Hessian lemma, the crossover factor min(1,(r/\|v\|)^6). No matching lower bound, no numerical constant or radius, no pin neighbourhoods, no k -> 0 uniformity, no d >= 3, no fixed-remote all-height bound, no all-scales RN theorem. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State the R1-R7 interfaces as Props; independent reviewer: alignment of AnnulusBridgeAllHeight2 with display (2), checking "all heights" and the uniformity over E and j. | -| `math.d5-component.intermediate-window-proof` | graph_node | — | `specified` | 2 specified: `D5eIntermediateShellsI3I4`, `D5fGlobalWindowFirstMomentI5` | The Props state rows D5-e and D5-f as the reconciliation transcribes I3/I4/I5, not from the proof's own bytes (not pinned); the tiling N1 of the Claude review and the planar F+ lower bound source are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin intermediate_window_20260928/PROOF.md (SHA-256 b3eb9456… (GRAPH.json fingerprint)) and anchor the shell displays there. | -| `math.d5-component.remote-window-proof` | graph_node | — | `specified` | 2 specified: `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | See rows rn-fixed-remote-window and math.rn-fixed-remote-window; the D5 consumption at rho = s_0 is a proof step, not stated. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window. | -| `D1-THM-A` | proof_index_result | — | `specified` | 2 specified: `PairLawInterface`, `ParentUniformSelectionA` | ParentUniformSelectionA states the shape of (1.1): existential r_* > 0, C < infinity outside the universal quantifiers over 0 < r <= r_*, b in [bLo,bHi], k in [kLo,kHi] and all frames, for the interface field p of PairLawInterface. Not stated or proved: the full-normalizer floor Z_r >= z_* r^2 (5.5), the transverse-matrix boundary layer, the regression law and typed weight, the reading-rule amendments (E1, Section 9 v1.1, W1, r < L/(4 sqrt 2)), any numerical r_*, C, z_*. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. No globally uniform constant over unbounded b, k; no lower bound. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: ParentUniformSelectionA is a predicate on the interface objects, not a closed statement; its universal closure over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed. | Independent reviewer: alignment of ParentUniformSelectionA with (1.1) under the honesty checklist; formal lane: none until the regression law is constructed. | -| `D1-THM-B` | proof_index_result | — | `specified` | 4 specified: `IsWindowPair`, `IsWindowElderPair`, `IsLifetimeDensityVersion`, `ParentCompactWindowDensityB` | ParentCompactWindowDensityB states the shape of (1.2) (asymptotics of versions of nu_cand and nu_eld, c_{B,K} > 0, the O(ell^(2/3)) difference) for compact B, K of positive length; not proved. Not stated: the exact coefficient formula (11.3), the radial ledger (10.2), the pushforward r = (ell/k)^(1/3); the distance cutoff is quantified existentially and not identified with the r_* of (1.1). The difference is NOT asserted for unrestricted densities; no numerical c_{B,K}, C_{B,K}; the historical H3 finite-band floor is not re-certified. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (11.3) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of ParentCompactWindowDensityB with (1.2), including the positive-length requirement. | -| `D1-THM-C` | proof_index_result | — | `specified` | 3 specified: `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `ParentUnrestrictedLeadingDensityC` | ParentUnrestrictedLeadingDensityC states the shape of (1.3) (leading term only) with the coefficient as an abstract positive constant c in IsUnrestrictedLeadingCoefficient; not proved. The Gaussian/angular representation (15.2), the target-growth majorant (13.3)-(13.6), the integrability (13.5) and the off-diagonal bound (14.1) are not stated; the SIDE24 enclosure of c_{d,24} is a separate row. No unrestricted remainder or difference rate (that is Theorem R), no elementary closed form. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (15.2) and (14.1) as Props; independent reviewer: alignment of ParentUnrestrictedLeadingDensityC with (1.3), in particular the exclusion of the essential global-maximum class. | -| `OA-ELDER-LOWER-DENSITY-GAP-20260928-v1` | proof_index_result | — | `specified` | 3 specified: `PlanarElderLowerE2`, `PlanarTwoSidedRateE4`, `PlanarDensityGapE11` | PlanarElderLowerE2 states (E2) for d = 2 over PairLawInterface; PlanarTwoSidedRateE4 states (E4) CONDITIONALLY on the parent (1.1) as the source's Corollary E2 does; PlanarDensityGapE11 states (E11) under the same (1.1) hypothesis, which the source needs only for the upper half (the lower half is thereby stated under a hypothesis the source does not need, disclosed in the docstring). None is proved. Not stated: (E3), (E7)-(E10), the polygonal-path and tilted jet-box argument. No lower bound for d >= 3 (separate packets CL-ELDER-LOWER-ALL-D / OA-ELDER-DIMENSION-LIFT, not extracted here); no numerical c or r_*. The review verdict of reviews/d1_elder_lower_claude_20260928 is not transcribed here; the pinned bytes carry their own disposition line. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: PlanarElderLowerE2, PlanarTwoSidedRateE4 and PlanarDensityGapE11 are predicates on the interface objects, not closed statements; the universal closure of E2 and E4 over all inhabitants is false whenever the interfaces are inhabited (take p identically 1), and `specified` refers to the statement shape at the source's object, which is not constructed. | Independent reviewer: record whether (E11)'s lower half should be stated without the (1.1) hypothesis (a narrower-scope verdict is the reviewer's to write); transcribe the review verdict once pinned. | +| `math.d5-component.collar-proof` | graph_node | — | `specified` | 1 specified: `D5cCollarC1` | D5cCollarC1 states row D5-c as the reconciliation transcribes C1, not C1 from its own bytes (not pinned); the normalizer (P14) it imports and the corollary C2 are not stated separately; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin COLLAR_PROOF.md (SHA-256 794babe0… (GRAPH.json fingerprint)) and anchor D5cCollarC1 to display (C1). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): D5cCollarC1 BLOCKED_INTERFACE (a Dirac-law inhabitant with an extra critical point at the midpoint and E = {0} in the collar). | +| `math.d5-component.annulus-proof` | graph_node | — | `specified` | 1 specified: `AnnulusBridgeAllHeight2` | AnnulusBridgeAllHeight2 states display (2) (d = 2, fixed L, compact birth, k_- > 0, 1 < A < B, all heights) over PinnedLawInterface; not proved. Not stated: displays (4)-(20), the R1-R7 review interfaces as separate statements, the deterministic three-Hessian lemma, the crossover factor min(1,(r/\|v\|)^6). No matching lower bound, no numerical constant or radius, no pin neighbourhoods, no k -> 0 uniformity, no d >= 3, no fixed-remote all-height bound, no all-scales RN theorem. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State the R1-R7 interfaces as Props; independent reviewer: alignment of AnnulusBridgeAllHeight2 with display (2), checking "all heights" and the uniformity over E and j. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): AnnulusBridgeAllHeight2 BLOCKED_INTERFACE (a Dirac-tilted law at a pinned smooth periodic path with one extra nondegenerate critical point in the annulus violates display (2) for a null Borel E). | +| `math.d5-component.intermediate-window-proof` | graph_node | — | `specified` | 2 specified: `D5eIntermediateShellsI3I4`, `D5fGlobalWindowFirstMomentI5` | The Props state rows D5-e and D5-f as the reconciliation transcribes I3/I4/I5, not from the proof's own bytes (not pinned); the tiling N1 of the Claude review and the planar F+ lower bound source are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin intermediate_window_20260928/PROOF.md (SHA-256 b3eb9456… (GRAPH.json fingerprint)) and anchor the shell displays there. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): D5eIntermediateShellsI3I4 BLOCKED_INTERFACE (a Dirac-law inhabitant with one extra window-height critical point at distance 8r from the midpoint); D5fGlobalWindowFirstMomentI5 BLOCKED_INTERFACE (Dirac-law inhabitants refute both directions (one extra window critical point: probability 1 > C r^3; none: probability 0 < c r^3), and the lower bound is sensitive to the Set.ncard convention (infinitely many window critical points count as 0)). | +| `math.d5-component.remote-window-proof` | graph_node | — | `specified` | 2 specified: `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | See rows rn-fixed-remote-window and math.rn-fixed-remote-window; the D5 consumption at rho = s_0 is a proof step, not stated. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point). | +| `D1-THM-A` | proof_index_result | — | `specified` | 2 specified: `PairLawInterface`, `ParentUniformSelectionA` | ParentUniformSelectionA states the shape of (1.1): existential r_* > 0, C < infinity outside the universal quantifiers over 0 < r <= r_*, b in [bLo,bHi], k in [kLo,kHi] and all frames, for the interface field p of PairLawInterface. Not stated or proved: the full-normalizer floor Z_r >= z_* r^2 (5.5), the transverse-matrix boundary layer, the regression law and typed weight, the reading-rule amendments (E1, Section 9 v1.1, W1, r < L/(4 sqrt 2)), any numerical r_*, C, z_*. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. No globally uniform constant over unbounded b, k; no lower bound. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: ParentUniformSelectionA is a predicate on the interface objects, not a closed statement; its universal closure over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed. | Independent reviewer: alignment of ParentUniformSelectionA with (1.1) under the honesty checklist; formal lane: none until the regression law is constructed. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ParentUniformSelectionA BLOCKED_INTERFACE (the falsifying inhabitant p == 0 named in the docstring, confirmed by elaboration (probe Lifetime-B_1)). | +| `D1-THM-B` | proof_index_result | — | `specified` | 4 specified: `IsWindowPair`, `IsWindowElderPair`, `IsLifetimeDensityVersion`, `ParentCompactWindowDensityB` | ParentCompactWindowDensityB states the shape of (1.2) (asymptotics of versions of nu_cand and nu_eld, c_{B,K} > 0, the O(ell^(2/3)) difference) for compact B, K of positive length; not proved. Not stated: the exact coefficient formula (11.3), the radial ledger (10.2), the pushforward r = (ell/k)^(1/3); the distance cutoff is quantified existentially and not identified with the r_* of (1.1). The difference is NOT asserted for unrestricted densities; no numerical c_{B,K}, C_{B,K}; the historical H3 finite-band floor is not re-certified. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (11.3) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of ParentCompactWindowDensityB with (1.2), including the positive-length requirement. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ParentCompactWindowDensityB BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, expected-number-of-zeros or critical points; present but insufficient: gaussianReal (Real.lean), Measure.map (Map.lean)); IsLifetimeDensityVersion, IsWindowElderPair, IsWindowPair PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. | +| `D1-THM-C` | proof_index_result | — | `specified` | 3 specified: `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `ParentUnrestrictedLeadingDensityC` | ParentUnrestrictedLeadingDensityC states the shape of (1.3) (leading term only) with the coefficient as an abstract positive constant c in IsUnrestrictedLeadingCoefficient; not proved. The Gaussian/angular representation (15.2), the target-growth majorant (13.3)-(13.6), the integrability (13.5) and the off-diagonal bound (14.1) are not stated; the SIDE24 enclosure of c_{d,24} is a separate row. No unrestricted remainder or difference rate (that is Theorem R), no elementary closed form. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (15.2) and (14.1) as Props; independent reviewer: alignment of ParentUnrestrictedLeadingDensityC with (1.3), in particular the exclusion of the essential global-maximum class. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ParentUnrestrictedLeadingDensityC BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: IsEquivalent (AsymptoticEquivalent.lean), ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean)); IsUnrestrictedLeadingCoefficient, IsUnrestrictedPair PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. | +| `OA-ELDER-LOWER-DENSITY-GAP-20260928-v1` | proof_index_result | — | `specified` | 3 specified: `PlanarElderLowerE2`, `PlanarTwoSidedRateE4`, `PlanarDensityGapE11` | PlanarElderLowerE2 states (E2) for d = 2 over PairLawInterface; PlanarTwoSidedRateE4 states (E4) CONDITIONALLY on the parent (1.1) as the source's Corollary E2 does; PlanarDensityGapE11 states (E11) under the same (1.1) hypothesis, which the source needs only for the upper half (the lower half is thereby stated under a hypothesis the source does not need, disclosed in the docstring). None is proved. Not stated: (E3), (E7)-(E10), the polygonal-path and tilted jet-box argument. No lower bound for d >= 3 (separate packets CL-ELDER-LOWER-ALL-D / OA-ELDER-DIMENSION-LIFT, not extracted here); no numerical c or r_*. The review verdict of reviews/d1_elder_lower_claude_20260928 is not transcribed here; the pinned bytes carry their own disposition line. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: PlanarElderLowerE2, PlanarTwoSidedRateE4 and PlanarDensityGapE11 are predicates on the interface objects, not closed statements; the universal closure of E2 and E4 over all inhabitants is false whenever the interfaces are inhabited (take p identically 1), and `specified` refers to the statement shape at the source's object, which is not constructed. | Independent reviewer: record whether (E11)'s lower half should be stated without the (1.1) hypothesis (a narrower-scope verdict is the reviewer's to write); transcribe the review verdict once pinned. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): PlanarDensityGapE11 BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: basisFun (PiL2.lean), ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean); its hypothesis ParentUniformSelectionA is inert (no PairLawInterface field ties p to G or I; p == 1 satisfies it), so a theorem of E11 would be a theorem of its conclusion alone); PlanarElderLowerE2 BLOCKED_INTERFACE (the falsifying inhabitant p == 1 named in the docstring, confirmed by elaboration (probe Lifetime-B_1)); PlanarTwoSidedRateE4 BLOCKED_INTERFACE (p == 1 satisfies the hypothesis and the lower half fails (probe Lifetime-B_1)). | | `two-scale-s6-s21` | proof_index_result | — | `none` | — | No Lean statement: the addendum is not pinned here and none of (S3)-(S5), (S20)-(S21) is stated; it enters the register only as the named support of FixedAnnulusWindowCandidateA2. Does not establish uniform O(r^3) on a fixed scaled-width annulus, anything at pin sites, near u = 0, r << \|x\| << rho, witness collisions, d >= 3, k_- = 0, uniform L. | Pin TWO_SCALE_ADDENDUM.md (blob 89cae3a9, 15902 B) and state (S3)-(S5) over PinnedLawInterface with the Stecconi weighted Kac-Rice input as an interface field. | | `axial-density` | proof_index_result | — | `none` | — | No Lean statement: not pinned, not stated. The bound contains no pin-density factor, spatial Jacobian or Palm weight; nothing about the height-conditioned density, Hessian moments, the endpoint-weighted count, the full normalizer, the remaining thin belt, collisions, d = 3 or numerical constants. | Pin axial_density_20260925/PROOF.md (blob a72418d7) and state (1) as a density bound over PinnedLawInterface with the conditional gradient density as an interface field. | | `fixed-transverse-chart` | proof_index_result | — | `none` | — | No Lean statement: not pinned, not stated. Constants are qualitative and not uniform as eta -> 0, B -> infinity or the gap interval approaches zero; no pin collisions, no intermediate distances, no d = 3, no positive leading coefficient, no probability lower bound. | Pin TRANSVERSE_BOUND_CANDIDATE.md (blob 024d9277) and state the chart bound as a Prop parallel to FixedAnnulusWindowCandidateA2 with the extra \|v\| >= eta restriction. | -| `CL-D5-DIMENSION-LIFT-20260929-v1` | proof_index_result | — | `specified` | 4 specified: `DimensionLiftPuncturedPinBallP`, `DimensionLiftCompactCollarC`, `DimensionLiftIntermediateShellsI`, `DimensionLiftGlobalWindowG` | The four theorem shapes (1.2), (1.3), (1.5)-(1.6), (1.7) are stated over PinnedLawInterface for fixed d >= 2 and not proved. Not stated: display (1.4), Corollaries F_d (1.8) and M_d (1.9), the five devices D1-D5, the consumed sources ([LP] sections 2-5, (5.5), the planar templates, [RM], [RC], [EDL], [LOCAL]). No all-height intermediate or global O(r^3) count; no torus-wide second factorial moment; no elder selection; no uniformity in k, L, d; no numerical C, c, r_*, s_0; no recovery of historical RN/JETMOD/24-jet carriers. The reviews (Math-#141 OpenAI, source-exposed) are source identity; the 2026-10-05 erratum may amend these statements and was not read. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Read reviews/d5_dimension_lift_erratum_20261005 and amend the four Props if the displays changed; state Corollaries F_d and M_d; independent reviewer: alignment of the four Props with (1.2)-(1.7). | -| `OA-C6-FOURIER-20260929-v1` | proof_index_result | — | `none` | 1 specified: `FourierCutoffPlanarSecondFactorialF4` | No Prop states Theorem F (the cap functional Psi_F and its stretched-exponential tail (F1)-(F2)), Theorem W in general d and p (F3), or Proposition B; only the planar consequence (F4) is stated, GIVEN the hypothesis G_2 (E N <= C r^3), so the claim as a whole is not specified. Nothing is proved; the logarithm is neither shown sharp nor removable here; no numerical cutoff or parameter-uniform constants; the abstract barrier is not a Gaussian counterexample. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (F1)-(F2) with Psi_F as an interface field (a measurable functional dominating the total critical count) and (F3) for general d, p; then the claim would be `specified`. | -| `CL-C6-PALM-20260929-v1` | proof_index_result | — | `specified` | 2 specified: `PalmRouteFactorialMomentsQ`, `PalmRouteSecondFactorialTheta` | PalmRouteFactorialMomentsQ states (1.1) (with r_* allowed to depend on q) and PalmRouteSecondFactorialTheta states (1.2) over PinnedLawInterface; not proved. Corollary P (size-biased law) and Sections 3-7 (the cap Psi, the parametric Rouche tail, the marked Kac-Rice formula with Hoelder insertion, regimes R1-R4) are not stated; the dependence on [C6L] and [DL] at their bytes is not transcribed; the lower bounds' source [EDL] (A4) is not stated. No lower bound for q >= 3; no distribution of N beyond factorial moments; no elder selection; no numerical constants; for d >= 3 conditional on [DL]. The GRAPH records that this global result is not a proof of the regional witness-collision node. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Corollary P; independent reviewer: alignment of the two Props with (1.1)-(1.2), including whether r_* may depend on q. | -| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | 1 specified: `RareClusterPoissonObstructionO` | Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified. Nothing is proved; H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count; for A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); Theorem O is a candidate kernel target (finite total-variation arithmetic). | -| `CL-D5-REMOTE-COLLISION-20260928-v1` | proof_index_result | — | `none` | 1 specified: `RemoteCollisionSecondFactorialD` | Only Corollary D (second factorial moment <= C r^5 \|E\| in the fixed remote region) is stated; Theorem C (ordered near-pair count), Corollaries E and F (event asymptotics, torus-wide lower bound) are not, so the packet is not specified as a whole. Nothing proved; pairs with a witness near the pins are excluded; no torus-wide second factorial moment; this is explicitly NOT the open witness-collision mechanism. Author Anthropic/Claude: same provider as this packet, independence credit 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Theorem C and Corollaries E, F; independent (non-Anthropic) reviewer for both the source and the Props. | +| `CL-D5-DIMENSION-LIFT-20260929-v1` | proof_index_result | — | `specified` | 4 specified: `DimensionLiftPuncturedPinBallP`, `DimensionLiftCompactCollarC`, `DimensionLiftIntermediateShellsI`, `DimensionLiftGlobalWindowG` | The four theorem shapes (1.2), (1.3), (1.5)-(1.6), (1.7) are stated over PinnedLawInterface for fixed d >= 2 and not proved. Not stated: display (1.4), Corollaries F_d (1.8) and M_d (1.9), the five devices D1-D5, the consumed sources ([LP] sections 2-5, (5.5), the planar templates, [RM], [RC], [EDL], [LOCAL]). No all-height intermediate or global O(r^3) count; no torus-wide second factorial moment; no elder selection; no uniformity in k, L, d; no numerical C, c, r_*, s_0; no recovery of historical RN/JETMOD/24-jet carriers. The reviews (Math-#141 OpenAI, source-exposed) are source identity; the 2026-10-05 erratum may amend these statements and was not read. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Read reviews/d5_dimension_lift_erratum_20261005 and amend the four Props if the displays changed; state Corollaries F_d and M_d; independent reviewer: alignment of the four Props with (1.2)-(1.7). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): BLOCKED_INTERFACE (interface under-determination, not statement defects; each refuted by a concrete Dirac-law inhabitant): DimensionLiftCompactCollarC (extra critical point at the midpoint, E = {0}, every d >= 2), DimensionLiftGlobalWindowG (extra window-height critical point at the midpoint), DimensionLiftIntermediateShellsI (extra window-height critical point at distance 8r from the midpoint), DimensionLiftPuncturedPinBallP (extra critical point in the punctured pin ball, null E). | +| `OA-C6-FOURIER-20260929-v1` | proof_index_result | — | `none` | 1 specified: `FourierCutoffPlanarSecondFactorialF4` | No Prop states Theorem F (the cap functional Psi_F and its stretched-exponential tail (F1)-(F2)), Theorem W in general d and p (F3), or Proposition B; only the planar consequence (F4) is stated, GIVEN the hypothesis G_2 (E N <= C r^3), so the claim as a whole is not specified. Nothing is proved; the logarithm is neither shown sharp nor removable here; no numerical cutoff or parameter-uniform constants; the abstract barrier is not a Gaussian counterexample. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (F1)-(F2) with Psi_F as an interface field (a measurable functional dominating the total critical count) and (F3) for general d, p; then the claim would be `specified`. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FourierCutoffPlanarSecondFactorialF4 BLOCKED_INTERFACE (the antecedent G_2 plus the interface fields admit a rare large cluster (two-point law, cluster size >= 1/r with probability r^3/n_r) with E N(N-1) of order r^2 while E N = r^3; Theorem W's Gaussian input is not transcribed). | +| `CL-C6-PALM-20260929-v1` | proof_index_result | — | `specified` | 2 specified: `PalmRouteFactorialMomentsQ`, `PalmRouteSecondFactorialTheta` | PalmRouteFactorialMomentsQ states (1.1) (with r_* allowed to depend on q) and PalmRouteSecondFactorialTheta states (1.2) over PinnedLawInterface; not proved. Corollary P (size-biased law) and Sections 3-7 (the cap Psi, the parametric Rouche tail, the marked Kac-Rice formula with Hoelder insertion, regimes R1-R4) are not stated; the dependence on [C6L] and [DL] at their bytes is not transcribed; the lower bounds' source [EDL] (A4) is not stated. No lower bound for q >= 3; no distribution of N beyond factorial moments; no elder selection; no numerical constants; for d >= 3 conditional on [DL]. The GRAPH records that this global result is not a proof of the regional witness-collision node. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Corollary P; independent reviewer: alignment of the two Props with (1.1)-(1.2), including whether r_* may depend on q. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): PalmRouteFactorialMomentsQ BLOCKED_INTERFACE (a Dirac inhabitant whose single path has two window critical points off the pins gives E (N)_2 = 2 for all small r); PalmRouteSecondFactorialTheta BLOCKED_INTERFACE (the lower bounds c r^3 fail for an inhabitant with no window critical points off the pins and for one with infinitely many (windowCountAll uses Set.ncard, 0 on infinite sets)). | +| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | 1 specified: `RareClusterPoissonObstructionO`; 1 proved (proof packet): `RNCluster.rareClusterPoissonObstructionO†` | Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified and the claim-level label stays `none` (the register admits only none/specified at claim level, and this row's claim is the packet, not Theorem O alone). RareClusterPoissonObstructionO has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (1 entry below, label `proved`; 1 of them (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them): it proves Theorem O's inequality as the Prop states it, for every family mu and constants a, A satisfying the Prop's own antecedents (0 < a, nonnegativity, summability and total mass one of each mu r, summable mean, (H1), (H2) on (0, r0]) and no others: the upper half with the witness lam = 0 (the Prop's existential form, not the source's infimum verbatim), the lower half through tail events {n >= 2} and {n >= 1}; 2a <= A is derived internally from the antecedents at r0, as in the source, and is neither a hypothesis nor a conclusion of the Prop. That theorem is evidence about the Prop, for the formal objects pmfTV, poissonPMF and the family mu : R -> N -> R, not acceptance of Theorem O; alignment of those objects with the source's dTV, Pois(lambda) and Law N_r is PENDING_INDEPENDENT_REVIEW. H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count (the Prop does not tie mu to any field; the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement lambda = mu, Theorems C and S, the factorial-moment matching, the independent-replica limit (7.1) and the conditional-law liminf remark are neither stated nor proved. For A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics; the hypotheses imply 2a <= A, so A > 0 is derivable and the Prop is not vacuous in general (the triage's example mu r = (1 - r^3) delta_0 + r^3 delta_2 with a = 1, A = 2 satisfies the antecedents for r <= 1). Disposition: none transcribed (disposition_transcribed null; the title's "PROOF_INDEX reviewed scoped result" is the source's wording, not a status decided here). Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); the 1 theorem of this row added after PR head 12e072d6 (RNCluster.rareClusterPoissonObstructionO) await the next hosted lean-specification-proofs.yml run, their first receipt. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RareClusterPoissonObstructionO PROVABLE_NOW: proved in round 4 (see entries). Independent reviewer: alignment of RareClusterPoissonObstructionO with Theorem O (3.1), in particular the existential form of the upper half and the (H1), (H2) antecedents on (0, r0]. | +| `CL-D5-REMOTE-COLLISION-20260928-v1` | proof_index_result | — | `none` | 1 specified: `RemoteCollisionSecondFactorialD` | Only Corollary D (second factorial moment <= C r^5 \|E\| in the fixed remote region) is stated; Theorem C (ordered near-pair count), Corollaries E and F (event asymptotics, torus-wide lower bound) are not, so the packet is not specified as a whole. Nothing proved; pairs with a witness near the pins are excluded; no torus-wide second factorial moment; this is explicitly NOT the open witness-collision mechanism. Author Anthropic/Claude: same provider as this packet, independence credit 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Theorem C and Corollaries E, F; independent (non-Anthropic) reviewer for both the source and the Props. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemoteCollisionSecondFactorialD BLOCKED_INTERFACE (a Dirac inhabitant with two prescribed remote critical points falsifies both displays; no field gives a.s. absence of index-j window pairs on null remote sets). | | `cumulative-transfer-correction` | proof_index_result | — | `none` | — | No Prop states the corrected cumulative theorem (sigma-finite (Z, lambda), alpha > -1, m > 0, ratio hypothesis h/(kappa r^m) -> 1, limit N(ell)/ell^beta -> (1/(alpha+1)) int B0 kappa^(-beta)), although its bytes are pinned. Nothing is proved; the correction does not license inferring a density asymptotic from a cumulative one; Section A of NOTE.md is outside both acceptances. | State the corrected theorem as a def : Prop over MeasureTheory.Measure Z (an abstract measure-space statement with no Gaussian input; a natural Mathlib-lane proof target afterwards). | | `hist.D0-RN-UNIF-historical-predicates` | open_obligation | — | `none` | — | No Lean statement: the historical numerical RN all-cell / 24-jet certification predicates have no in-tree source text to anchor (their carriers are ABSENT by fingerprint) and no Prop states any of them. The GRAPH records no complete proof where not explicitly superseded, and hist.lemma_closed is FALSE; this register changes none of that. The regional bypass (regional.fixed-annulus.high-jet-route, SUPERSEDED_NONBLOCKING) does not reopen or close these predicates in their original replay scopes; no numerical certificate exists in tree. | None for Lean until a source text exists; recovery of the absent carriers is a custody question (reviews/d0_custody_audit_20260926), not a formalization one. | | `lean:main-formal-UniversalLaw` | existing_lean_package | — | `none` | 4 proved (main, origin/main f6deeba7): `d2_endpoints_adjacent`, `d3_endpoints_adjacent`, `d3_below_d2`, `endpoints_in_unit_tenth` | The package states no research claim as a Prop, so the claim-level label is none; its 31 theorems at origin/main f6deeba7 are finite integer/cross-multiplied rational facts proved by decide (label `proved` in the manifest, `kernel-checked` only in a run receipt). The four Endpoints theorems listed here show the published 20-digit endpoints are adjacent, ordered and below 1/10 as Nat; they do not place c_{2,24} or c_{3,24} between them and say nothing about the true coefficients (the module's own note). The 28 P15 arithmetic targets are at commit 91cffbb5 of main PR #358 (not on main), are listed as `proved` statements on the rows p15-price-boundary, math.p15-price-boundary, p15-realized-covers, p15-price-budget-restricted, p15-full-price and math.p15-full-price, and have the hosted workspace-landing run 38096827443 (job formal / formal-evidence, artifact formal-verification-38096827443) as kernel evidence; they are finite arithmetic and none is Theorem F (F2), Theorem (T1), Theorem P2, Proposition P1, the enclosure (F3) of rho_star or any inequality involving log, exp or e (main formal/SCOPE.md). main has NO mechanism for `specified`: its gate refuses any manifest label but `proved`, so statement-only work cannot live in this package. Alignment review by a non-Anthropic lane is an open ask (formal/reviews holds only the template). | PR #358 merge is the operator's decision; until then the 28 P15 targets are on a branch, not on main. Obtain a non-Anthropic alignment review of the 59 targets; keep real-number statements in the Math- Mathlib lane (the proof packet specifications/lean_statement_proofs_20261011 now holds the real-number P15 theorems, kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)). | @@ -441,4 +442,5 @@ about the registered Prop it names (and, for main's `formal/` targets, about a f identity), never about the claim row it sits under; `kernel-checked` is a run-receipt word (the proof packet's hosted run 38121713840 receipted the 52 theorems of PR head 910809d6: actual tested merge 5b198699… with base b2d70d77…, the synthetic merge a pull_request run checks out, same tree as the head; the 8 of `P15Interpolation` by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base -79731b96…, the head of PR #422). +79731b96…, the head of PR #422; theorems added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8… with base 79731b96…) await the first hosted +`lean-specification-proofs.yml` run on the round-4 branch, their first receipt). From be01479435f98f34b802258ee9ec3c3578f37169 Mon Sep 17 00:00:00 2001 From: Claude Date: Sun, 11 Oct 2026 23:14:48 +0000 Subject: [PATCH 3/3] Proof packet round 4: bind the 13 theorems to their first hosted receipt (run 38180705952) and apply review amendments R4-PROSE-01/02 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit One forward commit after the nonauthor review of head 48bfcbae (Math- #434 review 5485371911: the 13-theorem-to-registered-Prop comparison passes; two prose amendments requested). Receipt binding (REGISTER.json kernel_evidence of the 10 proof-packet rows, not_established / next_step of side24-coefficient, math.side24-coefficient and OA-C6-RARE-CLUSTER-20260929-v1; statement README.md / SCOPE.md and proof README.md re-rendered): the 13 theorems of Side24Analysis, Side24Interfaces and RNCluster are now recorded as kernel-checked by hosted run 38180705952 (job proof-evidence 114592458899) on PR head 48bfcbae, actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d6 (tree equal to the head's), receipt theorem_count 73, leanchecker PASS, worktree CLEAN, artifact lean-specification-proofs-38180705952-1 sha256 87309efa... (GitHub digest); the "no hosted receipt yet" clauses and dagger marks for them are removed; the 60 earlier theorems keep their tiers and gain one corroboration clause naming the same run, head and merge. A receipt is a record of a computation about the registered Props, never evidence about an informal claim. R4-PROSE-01 (thread 4243584161): the Rayleigh docstring of rayleighSquareLaw, the helper docstring of Aux.rl_prob_eq_disk_measure, the register's does_not_claim (still equal to the normalized docstring) and the proof README now say what the proof shows: the event is the preimage of a measurable disk under the a.e.-measurable coordinate map, so Measure.map_apply_of_aemeasurable transports the measure; the sample-space preimage is only null-measurable (I.mΩ is not assumed complete) and MeasurableSet of it is not asserted. Docstring text only; no declaration, statement, proof term or premise changed. R4-PROSE-02 (thread 4243584162): the CovarianceOrdering triage clause in REGISTER.json and its SCOPE.md copy name the remaining gaps (local term-wise iterated differentiation of the lattice sum, higher-order symmetry of iteratedFDeriv) and record that the 2026-10-11 triage table, kept as the historical record, also listed the d-dimensional Gaussian tail bound (since discharged by latticeImageSumBound) and the derivative constants (since supplied by the derivative theorems). No verdict changes. Checks at this tree: Side24Interfaces.lean elaborates with -DwarningAsError; statement check.py --allow-dirty SOURCE_CHECK_PASS (352 statements) and test_check.py in both Python modes; proof check.py --allow-dirty SOURCE_CHECK_PASS (73 theorems) and test_check.py in both modes. Claim-level labels unchanged on all 171 rows. Scientific effect NONE; alignment PENDING_INDEPENDENT_REVIEW; organizational-independence credit 0. Before the push, this lane's own adversarial verification of the first form of this commit (three verifiers, same account, credit 0) refuted one sentence: the OA-C6-RARE-CLUSTER receipt field said the theorem 'was first receipted by that run' with run 38150363063 (which does not name it) as the nearest antecedent; it now names run 38180705952 with PR head and tested merge. Folded into the same commit: 'the merge has the same tree as its head' in place of a dangling 'same tree as the head' after a sentence about the base; the declaration-table headers move the receipt facts out of the parenthesis that describes the base; the earlier-tier corroboration wording is made explicit ('each of those 5 theorems', so it cannot be read as covering the 12 added later); the table headers say the merge has the same tree as PR head 48bfcbae; null-measurability and completeness are stated relative to I.P (the measure), and the helper docstring says the preimage is only shown to be null-measurable; the triage-table clause names where the table is kept (Math- reading branch claude/round4-triage-table-20261011, commit 43a975ec); 'every control classified as intended' replaces 'every negative control refused' (the receipt also lists accepted positive controls). Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01Cz7WZybv8znP64SpPj6sWY --- .../lean_statement_proofs_20261011/README.md | 16 +++--- .../SpecificationProofs/Side24Interfaces.lean | 17 ++++-- .../lean_statements_20261010/README.md | 51 +++++++++--------- .../lean_statements_20261010/REGISTER.json | 52 +++++++++--------- .../lean_statements_20261010/SCOPE.md | 53 +++++++++---------- 5 files changed, 100 insertions(+), 89 deletions(-) diff --git a/specifications/lean_statement_proofs_20261011/README.md b/specifications/lean_statement_proofs_20261011/README.md index 5e5eac856..4c8163d92 100644 --- a/specifications/lean_statement_proofs_20261011/README.md +++ b/specifications/lean_statement_proofs_20261011/README.md @@ -52,11 +52,15 @@ several were found not to be load-bearing and are corrected below, as correction not as statement defects. Its verdicts (PROVABLE_NOW, PROVABLE_WITH_WORK, BLOCKED_MISSING_MATHLIB, BLOCKED_INTERFACE, PREDICATE_VOCABULARY) are work-scheduling labels and carry no authority over any register label; a probe is scratch work, never cited as evidence, and where a row reports a probe-level finding it says -so. The 13 theorems of the three round-4 modules (`Side24Analysis`, `Side24Interfaces`, `RNCluster`) have **no -hosted receipt at this commit**: `check.py --allow-dirty` passes in source mode and the round-4 commit records a -clean local `lake build` with every theorem of the new modules on `propext`, `Classical.choice`, `Quot.sound` -only, but a local run is not evidence about any commit; the first `lean-specification-proofs.yml` run on this -branch is their first receipt. +so. The 13 theorems of the three round-4 modules (`Side24Analysis`, `Side24Interfaces`, `RNCluster`) are **receipted +by the hosted `lean-specification-proofs.yml` run 38180705952** (event pull_request on PR #434, head `48bfcbae`; the actually +tested checkout, the receipt's `checked_commit`, is the synthetic merge `9833d6d9…` of that head into base `12e072d6…`, the +head of PR #423, with the same tree as the head; job proof-evidence: fresh build, `leanchecker` PASS, receipt +`formalization_status: "kernel-checked"`, theorem_count 73, covered_props 72, every control classified as +intended (negative controls refused, positive controls accepted), worktree CLEAN; artifact `lean-specification-proofs-38180705952-1`, sha256 `87309efa…` as GitHub +reports it), their first receipt; the same run receipts all 73 theorems of that head. The round-4 commit's clean local +`lake build` with every theorem of the new modules on `propext`, `Classical.choice`, `Quot.sound` only, like any local +run, is evidence about no commit; later commits need their own run. | group | Props (names without `UniversalLaw.Spec..`) | why no theorem here | |---|---|---| @@ -69,7 +73,7 @@ branch is their first receipt. | Side24 covariance and density | `NegDefCone` (predicate), `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound` | `NegDefCone q` (`(-(transverseMatrix …)).PosDef`) has a free argument and is the set-builder predicate inside `coneIntegrand`; a theorem of its type is false at `q = (0, 0, 0)`. Vocabulary, not a target. `CovarianceOrdering` and `EntrywiseCovarianceBound`: BLOCKED_MISSING_MATHLIB (weeks), both true as stated. Two of the four pieces the triage listed as missing are theorems since round 4 (`latticeImageSumBound` for the `ℤ^d` tail `∑_{n ≠ 0} \|n\|⁶ e^{-288 \|n\|²} ≤ 1458 e^{-288}`; `derivativePairingBound` and its corollaries for the constants `76` and `15` that the entrywise constant `2 imageConstE` needs). The exact pieces still missing from Mathlib `d13f23b7` are (i) local term-wise iterated differentiation of `thetaSum24` at `0` with bounds summable only near `0` (the same gap as `ImageBound`: `iteratedFDeriv_tsum` needs bounds uniform over all `x`, the only local result `hasFDerivAt_tsum_of_isPreconnected` is first order, and the hypothesis `ThetaSummable d` gives pointwise summability only) and, for the `IsHermitian` half of `PosSemidef` in `CovarianceOrdering`, (ii) symmetry of `iteratedFDeriv` under permutation of the two direction blocks beyond order two (Mathlib has `isSymmSndFDerivAt_iff_iteratedFDeriv` only). The display-(3) ordering would even survive the crude bound `norm_iteratedFDeriv_comp_le` (`60 E' < epsilon` with a factor of about 8 to spare). `ReferenceCovarianceLowerBound`: PROVABLE_WITH_WORK (weeks), no missing theory: the entries of `Cref` from the contraction values already in `Side24Analysis` (orders 2, 4, 6) and the vanishing odd orders at `0` (order 5 needs one more general-direction level of the peeling), then an explicit sum-of-squares certificate for the block matrix (odd block `[[1, -3], [-3, 15]]`, `I_{d-1}`, Hessian block with eigenvalues `4, 2, 2` resp. `5, 2, …, 2`) minus `I/3` through `Matrix.posSemidef_iff_dotProduct_mulVec`. | | Side24 covariance and density, formerly listed here | — | `ReferenceOddAndPinBlocks`, `ReferencePinDensityProduct`, `IntegrandRatioBounds` and `DensityComparison` have theorems in `Side24Analysis` since round 4. The previous reason ("built on the derivative contractions …; the evaluation of the Fréchet derivatives is not available") was not load-bearing and is corrected here: the values of `iteratedFDeriv ℝ n (Kinf d) 0` in frame directions through order six are computable with existing API, peeling one direction at a time (`iteratedFDeriv_succ_apply_right` + `iteratedFDeriv_clm_apply_const_apply`) with explicit `HasFDerivAt` closed forms of the intermediate functions, giving `-⟪a, b⟫`, the three pairings and `-15 ⟪u, u⟫³`, hence the five blocks (`Cov(G) = I`, `Var(t) = 15`, `Cov(t, G) = (-3, 0, …)`, `pinCov = diagonal (3, 1, …, 1)`, `τ² = 15 - 9 = 6`) in an orthonormal frame and the pin density product `(2π)^{-d}/√3`; `IntegrandRatioBounds` was misgrouped: it is pure real arithmetic in `epsilon`, `exponentA`, `exponentB` (Bernoulli's inequality, exponent monotonicity and two exact rational inequalities decided by `norm_num`) with no derivative input; `DensityComparison` is matrix analysis for positive-definite real matrices (Loewner determinant monotonicity and inverse antitonicity through `CFC.sqrt`, developed in the module's `Aux`), with no derivative input either. The matrix arithmetic kernel-proved on main (`UniversalLaw.Side24.conditional_third_derivative_variance`, …) is neither cited nor relied on. | | Side24 interface statements | `ConeMomentD2Identity` (over `ConeMomentInterface`); `CoefficientEnclosure`, `ReferenceComparison` (over `PeriodicCoefficientInterface`) | `ConeMomentD2Identity`: PROVABLE_WITH_WORK (weeks), no missing theory: transport along `sxy_law` as for the three proved interface Props, the cone identification `NegDefCone (s, x, y) ↔ s < 0 ∧ x² + y² < s²` (both directions elaborate in a probe), Fubini with the integrable bound `(s² + x² + y²)²`, the inner truncated polar integral reducing to the proved `elementaryConeIntegral`, and the outer `gaussianReal 0 (5/3)` moments already in `Side24Interfaces`; several hundred lines. `CoefficientEnclosure`, `ReferenceComparison`: BLOCKED_INTERFACE; they are predicates on the interface object, as their own `Interface:` lines say: `PeriodicCoefficientInterface` takes `coneMoment` as a GIVEN field, tied to `c24` only through `eq_15_2`, so doubling `coneMoment` and `c24` turns any inhabitant into another, and no inhabitant can satisfy both `d2Lower < c24 2 < d2Upper` and its doubled version (`2 d2Lower > d2Upper`), nor both `\|q - 1\| < 10⁻¹⁰⁶` and `\|2q - 1\| < 10⁻¹⁰⁶`. Hence `(∀ I, CoefficientEnclosure I) → IsEmpty PeriodicCoefficientInterface`, and likewise for `ReferenceComparison`: a theorem of either type would exist only if the interface is empty, which nobody claims. Probe-level finding (elaborated without `sorry` in scratch work), not a receipt and not evidence. What would make the Props determinate is outside the statement packet and outside Mathlib: the conditional Gaussian law `A_u \| V_u = 0` of the `K24` field pinning `coneMoment` (closest Mathlib: `multivariateGaussian`, `condDistrib`), plus the numerical enclosure pipeline. The register's `specified` label is the right one; nothing in this packet encloses the coefficient `c_{d,24}`. | -| Side24 interface statements, formerly listed here | — | `ConeMomentD1Identity`, `TraceMoments` and `RayleighSquareLaw` have theorems in `Side24Interfaces` since round 4, for a GIVEN `I : ConeMomentInterface`. The previous reason ("the Gaussian moment identities need the conditional law `A_u \| V_u = 0` built in Mathlib's Gaussian measure theory") was not load-bearing for these three and is corrected here: they follow from the interface's law fields alone (`A_law : Measure.map A P = gaussianReal 0 (8/3)`, `sxy_law` as a product of `gaussianReal`s), the laws being hypothesis fields of `I`, not constructed. Route: a.e.-measurability of `A` and of the triple `(s, x, y)` from the law fields by the Dirac-versus-`NullSingletonClass` argument (`Measure.map_of_not_aemeasurable_of_ne_zero` against `nullSingletonClass_gaussianReal`), `integral_map` to the Gaussian law, then: for `D_1` the negative half of the second moment by `gaussianReal_map_neg` symmetry and `variance_fun_id_gaussianReal` (`(8/3)/2 = 4/3`); for the trace moments `∫ t² = 5/3`, `∫ t⁴ = 25/3` (`iteratedDeriv_mgf_zero` with the explicit fourth derivative of the mgf) and `∫ exp(-t²/2) = √(3/8)` (`integral_gaussianReal_eq_integral_smul`, `integral_gaussian`); for the Rayleigh law the disk measure under `gaussianReal 0 1 ⊗ gaussianReal 0 1` by `prod_withDensity`, polar coordinates (`integral_comp_polarCoord_symm`) and the fundamental theorem of calculus, equal to `∫_0^a exp(-z/2)/2 dz = 1 - exp(-a/2)`, measurability of the event proved so the outer-measure reading does not arise. Not established: that `ConeMomentInterface` has an inhabitant, or that its laws are the conditional law `A \| V = 0` of the reference field; `ConeMomentD2Identity`; any cone moment of the `K24` field. `ConeMomentD2Algebra`, `ElementaryConeIntegral` and `ReferenceCoefficientIdentity` (proved earlier) need no interface. | +| Side24 interface statements, formerly listed here | — | `ConeMomentD1Identity`, `TraceMoments` and `RayleighSquareLaw` have theorems in `Side24Interfaces` since round 4, for a GIVEN `I : ConeMomentInterface`. The previous reason ("the Gaussian moment identities need the conditional law `A_u \| V_u = 0` built in Mathlib's Gaussian measure theory") was not load-bearing for these three and is corrected here: they follow from the interface's law fields alone (`A_law : Measure.map A P = gaussianReal 0 (8/3)`, `sxy_law` as a product of `gaussianReal`s), the laws being hypothesis fields of `I`, not constructed. Route: a.e.-measurability of `A` and of the triple `(s, x, y)` from the law fields by the Dirac-versus-`NullSingletonClass` argument (`Measure.map_of_not_aemeasurable_of_ne_zero` against `nullSingletonClass_gaussianReal`), `integral_map` to the Gaussian law, then: for `D_1` the negative half of the second moment by `gaussianReal_map_neg` symmetry and `variance_fun_id_gaussianReal` (`(8/3)/2 = 4/3`); for the trace moments `∫ t² = 5/3`, `∫ t⁴ = 25/3` (`iteratedDeriv_mgf_zero` with the explicit fourth derivative of the mgf) and `∫ exp(-t²/2) = √(3/8)` (`integral_gaussianReal_eq_integral_smul`, `integral_gaussian`); for the Rayleigh law the disk measure under `gaussianReal 0 1 ⊗ gaussianReal 0 1` by `prod_withDensity`, polar coordinates (`integral_comp_polarCoord_symm`) and the fundamental theorem of calculus, equal to `∫_0^a exp(-z/2)/2 dz = 1 - exp(-a/2)`; the event is the preimage of a measurable disk under the a.e.-measurable coordinate map, so `P` of it is transported by `Measure.map_apply_of_aemeasurable` (the sample-space preimage is only shown to be null-measurable for `I.P`, which is not assumed complete; `MeasurableSet` of it is not asserted). Not established: that `ConeMomentInterface` has an inhabitant, or that its laws are the conditional law `A \| V = 0` of the reference field; `ConeMomentD2Identity`; any cone moment of the `K24` field. `ConeMomentD2Algebra`, `ElementaryConeIntegral` and `ReferenceCoefficientIdentity` (proved earlier) need no interface. | | RN beyond (N1)-(N5) and (O), Lifetime, Field | the 21 RN Props other than `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5`, `RareClusterPoissonObstructionO`; all 21 Lifetime and 10 Field Props | 52 Props in three classes, per the round-4 triage. (a) 22 BLOCKED_INTERFACE: the 19 RN Props over `PinnedLawInterface` (the D5 window bounds D5-a to D5-f, the annulus bridge, the four dimension-lift theorems, Theorem A(2)/A(3) of the fixed remote window, the fixed-annulus candidate, the Fourier cutoff (F4), the Palm route (Q, Theta), Corollary D and the OPEN witness-collision target) and the 3 Lifetime Props over `PairLawInterface` (`ParentUniformSelectionA`, `PlanarElderLowerE2`, `PlanarTwoSidedRateE4`). The previous reason here ("a proof would be the research result itself") was not the sharp one and is corrected: the interfaces constrain the sample paths and the law only at the two pins (`Q_pinned`, `W_typed`, `Z_pos`, `QW_eq`) or only through `p_nonneg`/`p_le_one`, and admit concrete falsifying inhabitants: a Dirac (or two-point) law at a smooth periodic pinned path with one or two extra nondegenerate critical points placed where the Prop counts them (a null `E`, the midpoint, a shell, a remote point), the free kernel field `Λ := 0` for Theorem A(2), or the constant laws `p ≡ 0` / `p ≡ 1`, so no theorem of the universal closure can exist for any Mathlib. This is under-determination of the interface (the Kac–Rice representation and the Gaussian regression law are not transcribed, as the interface docstrings say), not a statement defect; the reduction steps and the `p ≡ 0`/`p ≡ 1` refutations elaborate in probes (probe-level, not a receipt; the path constructions are not formalised). (b) 4 Lifetime Props BLOCKED_MISSING_MATHLIB (weeks): `ParentCompactWindowDensityB`, `ParentUnrestrictedLeadingDensityC`, `RemainderBoundedR1`, `PlanarDensityGapE11`, whose truth value the field interface's `gaussian` field does determine, but whose conclusions need density versions (`IsLifetimeDensityVersion`: a Kac–Rice representation of the expected pair count with lifetime in `A`, absolute continuity in `ℓ`), a.s. Morse genericity and the source's asymptotics, none of which Mathlib `d13f23b7` has; in `PlanarDensityGapE11` the hypothesis `ParentUniformSelectionA … PL …` is inert (it constrains only `PL.p`, which the conclusion does not mention, and `p ≡ 1` satisfies it), a shape observation for the register. (c) 26 predicates with free arguments (`IsCriticalPoint`, `HessianHasInertia` in RN; the 14 of Lifetime and the 10 of Field: `HessianHasInertia` and the `Is…` family in each): vocabulary of the other Props, not targets. Across the whole table the triage's counts for the 69 Props without a theorem are 28 predicates (these 26 plus `IsUnitTuple`, `NegDefCone`), 24 BLOCKED_INTERFACE (these 22 plus `CoefficientEnclosure`, `ReferenceComparison`), 7 BLOCKED_MISSING_MATHLIB (these 4 plus `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`), 2 PROVABLE_WITH_WORK (`ReferenceCovarianceLowerBound`, `ConeMomentD2Identity`) and the 8 P15 predicates it did not re-examine. | | RN rare cluster (Theorem O), formerly listed here | — | `RareClusterPoissonObstructionO` has a theorem in `RNCluster` since round 4. The previous reason ("a proof would be the research result itself, and the interfaces do not carry the Kac–Rice data") was not load-bearing for this Prop and is corrected here: the Prop is the abstract implication (H1)+(H2) ⇒ (3.1) for a GIVEN family of pmfs `μ r : ℕ → ℝ`, exactly what the source names as its only analytic inputs, and it ties `μ` to no field. Route: `2a ≤ A` is derived from the antecedents at `r0` (as the source remarks; never a hypothesis); the upper half with the witness `lam = 0`, where `pmfTV (μ r) (poissonPMF 0) = 1 - μ r 0 ≤ ∑ n μ r n ≤ A r³`; the lower half by testing the tail `{n ≥ 2}` when `lam ≤ r²` (Poisson tail `≤ lam²` from `Real.add_one_le_exp`) and the tail `{n ≥ 1}` when `lam > r²` (`x/2 ≤ 1 - exp(-x)` on `[0, 1/2]` from `Real.abs_exp_sub_one_sub_id_le`), with the Poisson total mass from `ProbabilityTheory.hasSum_one_poissonMeasure`, at `rStar = min r0 (min (a/2) (min (1/(a + 2A)) (1/2)))`. Not established: that (H1) or (H2) hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta; neither is revalidated here), the mean-matched statement `lambda = mu`, Theorems C and S, the source's infimum form verbatim (the existential with witness `lam = 0` is what is proved). The claim `OA-C6-RARE-CLUSTER-20260929-v1` keeps its register labels. | diff --git a/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Interfaces.lean b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Interfaces.lean index 16085e2e3..4b2dd98c9 100644 --- a/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Interfaces.lean +++ b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Interfaces.lean @@ -499,8 +499,10 @@ theorem rl_law_xy (I : ConeMomentInterface) : rw [I.sxy_law, Measure.map_snd_prod, measure_univ, one_smul] at h exact h.symm -/-- The probability of `{x^2 + y^2 ≤ a}` under `P` is the Gaussian measure of the closed disk: -the event is measurable, so `Measure.map_apply_of_aemeasurable` applies to the law `rl_law_xy`. -/ +/-- The probability of `{x^2 + y^2 ≤ a}` under `P` is the Gaussian measure of the closed disk: the +disk is a measurable set and `(x, y)` is a.e.-measurable, so `Measure.map_apply_of_aemeasurable` +transports the law `rl_law_xy` (a.e.-measurable transport; the sample-space preimage is only +shown to be null-measurable for `I.P`, which is not assumed complete). -/ theorem rl_prob_eq_disk_measure (I : ConeMomentInterface) (a : ℝ) : I.P {ω | I.x ω ^ 2 + I.y ω ^ 2 ≤ a} = ((gaussianReal 0 1).prod (gaussianReal 0 1)) {p : ℝ × ℝ | p.1 ^ 2 + p.2 ^ 2 ≤ a} := by @@ -603,9 +605,14 @@ PENDING_INDEPENDENT_REVIEW); from the Prop's own `Does not claim:` line, "the la only)": the law is proved here only for the formal objects of the Prop, as the distribution function `P(R^2 ≤ a)` for `a ≥ 0`, not as a density and not for the source's variable; and "measurability of the event (if the event is not measurable, `P` is its outer measure)": -measurability of the formal event `{x^2 + y^2 ≤ a}` is proved here (`Aux.rl_measurableSet_disk`, -`Measure.map_apply_of_aemeasurable`), so the outer-measure reading does not arise for this -event, but nothing is established about any other event; that any inhabitant of +the formal event `{x^2 + y^2 ≤ a}` is the preimage of the closed disk, a measurable set +(`Aux.rl_measurableSet_disk`), under the a.e.-measurable coordinate map `(x, y)` +(`Aux.rl_aemeasurable_xy`); `Measure.map_apply_of_aemeasurable` transports the measure along that +map, so `P` of the event equals the disk measure under the law. The preimage is thereby +null-measurable for `I.P` (`AEMeasurable.nullMeasurableSet_preimage`); `MeasurableSet` of the +sample-space preimage is not asserted (`ConeMomentInterface` assumes neither that `I.P` is +complete nor that the coordinate fields are measurable), and nothing is established about any +other event; that any inhabitant of `ConeMomentInterface` exists, or that its laws are the conditional law `A | V = 0` of the reference field (the laws are hypothesis fields of the given `I`, not derived here); `ConeMomentD2Identity` or any cone moment; any enclosure of the coefficient; the statement diff --git a/specifications/lean_statements_20261010/README.md b/specifications/lean_statements_20261010/README.md index bb5f487de..e421fa7f4 100644 --- a/specifications/lean_statements_20261010/README.md +++ b/specifications/lean_statements_20261010/README.md @@ -7,8 +7,8 @@ Formal progress of every declaration here: `specified`. The sibling proof packet Props (72 of the 141 Props, 73 theorems); the register lists them under the claim rows as statement entries of kind `theorem` with label `proved` (52 of them kernel-checked by the hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…, same tree as the head; the 8 of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194…), by run 38149888699: -PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96…, the head of PR #422, same tree as the head; the 13 added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c8…), -have no hosted receipt yet and are marked † in the tables), +PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96…, the head of PR #422, same tree as the head; the 13 of `Side24Analysis`, `Side24Interfaces` and `RNCluster`, added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), by run 38180705952: +PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6…, the head of PR #423, same tree as the head), which changes no label of this packet and no claim-level label. No claim, premise, obligation or disposition changes because this packet exists, builds, or is green in CI, or because a Prop of it is proved. @@ -52,11 +52,11 @@ ladder; it is `specified`. ## Declarations -150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified` in this package; the 123 auxiliary definitions and abbreviations of the modules are inventory only and are not register statements. Names are given without the module namespace `UniversalLaw.Spec..`; the source column is the pinned source named by the docstring's `Source:` line. Anchors and "Not established" cells are copied verbatim from the Lean docstrings (`Anchor:` and `Does not claim:` lines) through REGISTER.json. The last column names the theorems of the sibling proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly the declaration (namespace `UniversalLaw.Proofs..`): 72 of the 141 Props have one (73 theorems; `TriangleNeedsThree` is proved twice), each listed in the register under the claim rows as a statement entry of kind `theorem` with label `proved` (the label of a committed proof package; `kernel-checked` appears only in a run receipt: the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, so each receipt records as checked_commit that merge and not the head: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; each merge has the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems marked † (`Side24Analysis`, `Side24Interfaces` and `RNCluster`) were added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), and have no hosted receipt yet: the first hosted `lean-specification-proofs.yml` run on the round-4 branch is their first receipt). That label belongs to the theorem, not to the declaration here, which stays `specified`, and not to any claim row; a kernel-checked theorem is evidence about its Prop only, and alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. +150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified` in this package; the 123 auxiliary definitions and abbreviations of the modules are inventory only and are not register statements. Names are given without the module namespace `UniversalLaw.Spec..`; the source column is the pinned source named by the docstring's `Source:` line. Anchors and "Not established" cells are copied verbatim from the Lean docstrings (`Anchor:` and `Does not claim:` lines) through REGISTER.json. The last column names the theorems of the sibling proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly the declaration (namespace `UniversalLaw.Proofs..`): 72 of the 141 Props have one (73 theorems; `TriangleNeedsThree` is proved twice), each listed in the register under the claim rows as a statement entry of kind `theorem` with label `proved` (the label of a committed proof package; `kernel-checked` appears only in a run receipt: the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, so each receipt records as checked_commit that merge and not the head: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; each merge has the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems of `Side24Analysis`, `Side24Interfaces` and `RNCluster`, added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), are kernel-checked by the hosted run 38180705952 on PR head 48bfcbae (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, the head of PR #423; receipt theorem_count 73, leanchecker PASS; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd) and were first receipted by that run, which receipts all 73 theorems of that head; same tree as the head). That label belongs to the theorem, not to the declaration here, which stays `specified`, and not to any claim row; a kernel-checked theorem is evidence about its Prop only, and alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. ### `Specifications/Field.lean` (12 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | | `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | @@ -73,7 +73,7 @@ ladder; it is `specified`. ### `Specifications/Lifetime.lean` (24 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | | `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | @@ -102,32 +102,32 @@ ladder; it is `specified`. ### `Specifications/Side24.lean` (28 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---| -| `ThetaSummable` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.thetaSummable†` | +| `ThetaSummable` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.thetaSummable` | | `IsUnitTuple` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `DerivativePairingBound` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.derivativePairingBound†` | -| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.derivativeBoundAtLeastOne†` | -| `DerivativeBoundAtZero` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.derivativeBoundAtZero†` | -| `LatticeImageSumBound` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.latticeImageSumBound†` | +| `DerivativePairingBound` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.derivativePairingBound` | +| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.derivativeBoundAtLeastOne` | +| `DerivativeBoundAtZero` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.derivativeBoundAtZero` | +| `LatticeImageSumBound` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.latticeImageSumBound` | | `SuccessiveTermRatio` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.successiveTermRatio` | | `ExponentialBounds` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.exponentialBounds` | | `ImageBound` | def_prop | `side24-proof` | `side24-coefficient` | — | | `CovarianceOrdering` | def_prop | `side24-proof` | `side24-coefficient` | — | | `EntrywiseCovarianceBound` | def_prop | `side24-proof` | `side24-coefficient` | — | | `ReferenceCovarianceLowerBound` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.referenceOddAndPinBlocks†` | -| `DensityComparison` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.densityComparison†` | -| `IntegrandRatioBounds` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.integrandRatioBounds†` | -| `ReferencePinDensityProduct` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.referencePinDensityProduct†` | +| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.referenceOddAndPinBlocks` | +| `DensityComparison` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.densityComparison` | +| `IntegrandRatioBounds` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.integrandRatioBounds` | +| `ReferencePinDensityProduct` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.referencePinDensityProduct` | | `ReferenceCoefficientIdentity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.referenceCoefficientIdentity` | | `ConeMomentInterface` | structure | `side24-proof` | `side24-coefficient` | — | | `NegDefCone` | def_prop | `side24-proof` | `side24-coefficient` | — | -| `ConeMomentD1Identity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.coneMomentD1Identity†` | +| `ConeMomentD1Identity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.coneMomentD1Identity` | | `ConeMomentD2Identity` | def_prop | `side24-proof` | `side24-coefficient` | — | | `ConeMomentD2Algebra` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.coneMomentD2Algebra` | -| `TraceMoments` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.traceMoments†` | -| `RayleighSquareLaw` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.rayleighSquareLaw†` | +| `TraceMoments` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.traceMoments` | +| `RayleighSquareLaw` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.rayleighSquareLaw` | | `ElementaryConeIntegral` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Elementary.elementaryConeIntegral` | | `PeriodicCoefficientInterface` | structure | `side24-proof` | `side24-coefficient`, `math.side24-coefficient` | — | | `CoefficientEnclosure` | def_prop | `side24-proof` | `side24-coefficient`, `math.side24-coefficient` | — | @@ -135,7 +135,7 @@ ladder; it is `specified`. ### `Specifications/RN.lean` (28 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---| | `IsCriticalPoint` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | — | | `HessianHasInertia` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | — | @@ -162,13 +162,13 @@ ladder; it is `specified`. | `FourierCutoffPlanarSecondFactorialF4` | def_prop | `c6-fourier-cutoff` | `OA-C6-FOURIER-20260929-v1` | — | | `PalmRouteFactorialMomentsQ` | def_prop | `c6-palm-route` | `CL-C6-PALM-20260929-v1` | — | | `PalmRouteSecondFactorialTheta` | def_prop | `c6-palm-route` | `CL-C6-PALM-20260929-v1` | — | -| `RareClusterPoissonObstructionO` | def_prop | `c6-rare-cluster` | `OA-C6-RARE-CLUSTER-20260929-v1` | `RNCluster.rareClusterPoissonObstructionO†` | +| `RareClusterPoissonObstructionO` | def_prop | `c6-rare-cluster` | `OA-C6-RARE-CLUSTER-20260929-v1` | `RNCluster.rareClusterPoissonObstructionO` | | `RemoteCollisionSecondFactorialD` | def_prop | `remote-collision` | `math.rn-region.witness-collision`, `CL-D5-REMOTE-COLLISION-20260928-v1` | — | | `WitnessCollisionNearPinPairsTarget` | def_prop | `d5-reconciliation` | `math.rn-region.witness-collision` | — | ### `Specifications/P15.lean` (58 registered declarations) -| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | registered under claim rows | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---| | `RealizedData` | structure | `p15-realized-covers` | `p15-realized-covers` | — | | `IsRealizedFamily` | def_prop | `p15-realized-covers` | `p15-realized-covers` | — | @@ -231,11 +231,11 @@ ladder; it is `specified`. ## Claims -171 claim rows: 10 `landing_claim`, 33 `graph_node`, 2 `open_obligation`, 95 `proof_index_result`, 7 `existing_lean_package`, 12 `main_experiment_proof`, 12 `exact_certificate`. Claim-level formal progress: 33 `specified` (a `def … : Prop` stating the claim itself is registered), 138 `none`; no research claim is `proved` or `kernel-checked`. 197 statement entries point at the 150 declarations of this package (all `specified`). 155 statement entries carry the label `proved` and are theorems of other packages, listed as the arithmetic skeleton or the Prop-level proof of the claim they sit under and never as the claim: 31 are the 31 Side24 arithmetic targets of main's `formal/` at origin/main `f6deeba7`; 41 are the 28 P15 arithmetic targets of main's `formal/` at `91cffbb5` (main PR #358, not yet on main; kernel evidence the hosted workspace-landing run 38096827443), each listed on every P15 row whose layer 0 source it anchors to; 83 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (73 theorems, 72 distinct Props, each listed on every row that registers its Prop; the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 as kernel-checked and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, which the receipt records as checked_commit: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; same trees as the heads; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems marked † (`Side24Analysis`, `Side24Interfaces` and `RNCluster`) were added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), and have no hosted receipt yet: the first hosted `lean-specification-proofs.yml` run on the round-4 branch is their first receipt). Exactly one Prop registered as stating a claim itself has such a theorem: `DemandOneRefutation` on rows `p15-price-boundary` and `math.p15-price-boundary`; their claim-level label stays `specified` (the register admits only none/specified at claim level, alignment of the Prop with the source paragraph is PENDING_INDEPENDENT_REVIEW, and a receipt is about the Prop only), as the rows' `not established` cells record. `disposition` is the verbatim LANDING_CLAIMS value or `—` (null). Nothing in these tables is a scientific status, a review verdict or an acceptance; alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. +171 claim rows: 10 `landing_claim`, 33 `graph_node`, 2 `open_obligation`, 95 `proof_index_result`, 7 `existing_lean_package`, 12 `main_experiment_proof`, 12 `exact_certificate`. Claim-level formal progress: 33 `specified` (a `def … : Prop` stating the claim itself is registered), 138 `none`; no research claim is `proved` or `kernel-checked`. 197 statement entries point at the 150 declarations of this package (all `specified`). 155 statement entries carry the label `proved` and are theorems of other packages, listed as the arithmetic skeleton or the Prop-level proof of the claim they sit under and never as the claim: 31 are the 31 Side24 arithmetic targets of main's `formal/` at origin/main `f6deeba7`; 41 are the 28 P15 arithmetic targets of main's `formal/` at `91cffbb5` (main PR #358, not yet on main; kernel evidence the hosted workspace-landing run 38096827443), each listed on every P15 row whose layer 0 source it anchors to; 83 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (73 theorems, 72 distinct Props, each listed on every row that registers its Prop; the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 as kernel-checked and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, which the receipt records as checked_commit: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; same trees as the heads; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems of `Side24Analysis`, `Side24Interfaces` and `RNCluster`, added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), are kernel-checked by the hosted run 38180705952 on PR head 48bfcbae (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, the head of PR #423; receipt theorem_count 73, leanchecker PASS; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd) and were first receipted by that run, which receipts all 73 theorems of that head; same tree as the head). Exactly one Prop registered as stating a claim itself has such a theorem: `DemandOneRefutation` on rows `p15-price-boundary` and `math.p15-price-boundary`; their claim-level label stays `specified` (the register admits only none/specified at claim level, alignment of the Prop with the source paragraph is PENDING_INDEPENDENT_REVIEW, and a receipt is about the Prop only), as the rows' `not established` cells record. `disposition` is the verbatim LANDING_CLAIMS value or `—` (null). Nothing in these tables is a scientific status, a review verdict or an acceptance; alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. | claim_id | kind | disposition (LANDING_CLAIMS, verbatim) | formal_progress | layer 0 source | statements | |---|---|---|---|---|---| -| `side24-coefficient` | landing_claim | REVIEWED_SCOPED | `specified` | `side24-proof` | 28 specified: `ThetaSummable`, `IsUnitTuple`, `DerivativePairingBound`, `DerivativeBoundAtLeastOne`, `DerivativeBoundAtZero`, `LatticeImageSumBound`, `SuccessiveTermRatio`, `ExponentialBounds`, `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound`, `ReferenceOddAndPinBlocks`, `DensityComparison`, `IntegrandRatioBounds`, `ReferencePinDensityProduct`, `ReferenceCoefficientIdentity`, `ConeMomentInterface`, `NegDefCone`, `ConeMomentD1Identity`, `ConeMomentD2Identity`, `ConeMomentD2Algebra`, `TraceMoments`, `RayleighSquareLaw`, `ElementaryConeIntegral`, `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison`; 27 proved (main, origin/main f6deeba7): `pairing_sum_six`, `pairing_sum_le_six`, `pairing_terms_six`, `sixth_moment_double_factorial`, `lattice_shell_constant`, `geometric_ratio_constants`, `period_exponent`, `image_constant_value`, `exp_taylor_partial_sum_gt_ten`, `covariance_relative_bound`, `odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`, `exponent_bounds`, `exponent_values`, `ratio_constant_ordering`, `density_comparison_below_reported`, `conditional_third_derivative_variance`, `transverse_covariance_entries_thirds`, `cone_moment_m1_thirds`, `trace_and_traceless_variances`, `fourth_moment_of_trace`, `exponential_moment_of_trace`, `cone_moment_m2_algebra`, `cube_root_simplification`, `pin_determinant_and_joint_dimension`, `hessian_block_eigenvalues`; 17 proved (proof packet): `Side24Elementary.exponentialBounds`, `Side24Elementary.successiveTermRatio`, `Side24Elementary.coneMomentD2Algebra`, `Side24Elementary.referenceCoefficientIdentity`, `Side24Elementary.elementaryConeIntegral`, `Side24Analysis.thetaSummable†`, `Side24Analysis.derivativePairingBound†`, `Side24Analysis.derivativeBoundAtLeastOne†`, `Side24Analysis.derivativeBoundAtZero†`, `Side24Analysis.referenceOddAndPinBlocks†`, `Side24Analysis.referencePinDensityProduct†`, `Side24Analysis.integrandRatioBounds†`, `Side24Analysis.latticeImageSumBound†`, `Side24Analysis.densityComparison†`, `Side24Interfaces.coneMomentD1Identity†`, `Side24Interfaces.traceMoments†`, `Side24Interfaces.rayleighSquareLaw†` | +| `side24-coefficient` | landing_claim | REVIEWED_SCOPED | `specified` | `side24-proof` | 28 specified: `ThetaSummable`, `IsUnitTuple`, `DerivativePairingBound`, `DerivativeBoundAtLeastOne`, `DerivativeBoundAtZero`, `LatticeImageSumBound`, `SuccessiveTermRatio`, `ExponentialBounds`, `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound`, `ReferenceOddAndPinBlocks`, `DensityComparison`, `IntegrandRatioBounds`, `ReferencePinDensityProduct`, `ReferenceCoefficientIdentity`, `ConeMomentInterface`, `NegDefCone`, `ConeMomentD1Identity`, `ConeMomentD2Identity`, `ConeMomentD2Algebra`, `TraceMoments`, `RayleighSquareLaw`, `ElementaryConeIntegral`, `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison`; 27 proved (main, origin/main f6deeba7): `pairing_sum_six`, `pairing_sum_le_six`, `pairing_terms_six`, `sixth_moment_double_factorial`, `lattice_shell_constant`, `geometric_ratio_constants`, `period_exponent`, `image_constant_value`, `exp_taylor_partial_sum_gt_ten`, `covariance_relative_bound`, `odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`, `exponent_bounds`, `exponent_values`, `ratio_constant_ordering`, `density_comparison_below_reported`, `conditional_third_derivative_variance`, `transverse_covariance_entries_thirds`, `cone_moment_m1_thirds`, `trace_and_traceless_variances`, `fourth_moment_of_trace`, `exponential_moment_of_trace`, `cone_moment_m2_algebra`, `cube_root_simplification`, `pin_determinant_and_joint_dimension`, `hessian_block_eigenvalues`; 17 proved (proof packet): `Side24Elementary.exponentialBounds`, `Side24Elementary.successiveTermRatio`, `Side24Elementary.coneMomentD2Algebra`, `Side24Elementary.referenceCoefficientIdentity`, `Side24Elementary.elementaryConeIntegral`, `Side24Analysis.thetaSummable`, `Side24Analysis.derivativePairingBound`, `Side24Analysis.derivativeBoundAtLeastOne`, `Side24Analysis.derivativeBoundAtZero`, `Side24Analysis.referenceOddAndPinBlocks`, `Side24Analysis.referencePinDensityProduct`, `Side24Analysis.integrandRatioBounds`, `Side24Analysis.latticeImageSumBound`, `Side24Analysis.densityComparison`, `Side24Interfaces.coneMomentD1Identity`, `Side24Interfaces.traceMoments`, `Side24Interfaces.rayleighSquareLaw` | | `lifetime-remainder` | landing_claim | REVIEWED_SCOPED | `specified` | `lifetime-remainder` | 4 specified: `RemainderBoundedR1`, `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `IsLifetimeDensityVersion` | | `rn-count-interface` | landing_claim | HOLD_WITH_DOMAIN | `none` | `rn-count-interface` | 5 specified: `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5`; 5 proved (proof packet): `RNProbability.conditionalMeanIdentityN3`, `RNProbability.holderCountBoundN1`, `RNProbability.holderConsequenceN2`, `RNProbability.uniformCounterexampleN4`, `RNProbability.exponentialTailLayerCakeN5` | | `rn-fixed-remote-window` | landing_claim | HOLD_WITH_DOMAIN | `specified` | `rn-fixed-remote-window` | 5 specified: `PinnedLawInterface`, `IsCriticalPoint`, `HessianHasInertia`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | @@ -289,7 +289,7 @@ ladder; it is `specified`. | `CL-D5-DIMENSION-LIFT-20260929-v1` | proof_index_result | — | `specified` | `d5-dimension-lift` | 4 specified: `DimensionLiftPuncturedPinBallP`, `DimensionLiftCompactCollarC`, `DimensionLiftIntermediateShellsI`, `DimensionLiftGlobalWindowG` | | `OA-C6-FOURIER-20260929-v1` | proof_index_result | — | `none` | `c6-fourier-cutoff` | 1 specified: `FourierCutoffPlanarSecondFactorialF4` | | `CL-C6-PALM-20260929-v1` | proof_index_result | — | `specified` | `c6-palm-route` | 2 specified: `PalmRouteFactorialMomentsQ`, `PalmRouteSecondFactorialTheta` | -| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | `c6-rare-cluster` | 1 specified: `RareClusterPoissonObstructionO`; 1 proved (proof packet): `RNCluster.rareClusterPoissonObstructionO†` | +| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | `c6-rare-cluster` | 1 specified: `RareClusterPoissonObstructionO`; 1 proved (proof packet): `RNCluster.rareClusterPoissonObstructionO` | | `CL-D5-REMOTE-COLLISION-20260928-v1` | proof_index_result | — | `none` | `remote-collision` | 1 specified: `RemoteCollisionSecondFactorialD` | | `cumulative-transfer-correction` | proof_index_result | — | `none` | `cumulative-transfer-correction` | — | | `hist.D0-RN-UNIF-historical-predicates` | open_obligation | — | `none` | `downstream-graph` | — | @@ -364,7 +364,8 @@ Packets listed by PROOF_INDEX by id only (82 rows, kind `proof_index_result`, fo PR head 00d2ac6e, actual tested merge c4c66bb8…; both with base b2d70d77… and each with the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194…), are receipted by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96…, the head of PR #422, and first by run 38149524759: PR head 88c21058, actual tested merge 3306233b…, same base; the 13 theorems - added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…, receipt theorem_count 60) have no hosted receipt yet, marked †). + of `Side24Analysis`, `Side24Interfaces` and `RNCluster`, added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…, receipt theorem_count 60), are receipted by run 38180705952: PR head 48bfcbae, + actual tested merge 9833d6d9… with base 12e072d6…, the head of PR #423, receipt theorem_count 73, and were first receipted by that run). The 150 declarations here remain `specified`; a kernel-checked theorem is evidence about its Prop only, never about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; no claim-level label is `proved`, including the one row (`p15-price-boundary`) whose diff --git a/specifications/lean_statements_20261010/REGISTER.json b/specifications/lean_statements_20261010/REGISTER.json index 6e3f670b5..c8bb36ca1 100644 --- a/specifications/lean_statements_20261010/REGISTER.json +++ b/specifications/lean_statements_20261010/REGISTER.json @@ -885,7 +885,7 @@ "kind": "theorem", "formal_progress": "proved", "anchor": "The variable R^2 has density e^(-z/2)/2.", - "does_not_claim": "the informal Section 1 display \"The variable R^2 has density e^(-z/2)/2\" of the source as the source's object, nor its review and acceptance (the alignment of the interface fields `x`, `y`, `P`, `sxy_law` and of `gaussianReal` with the source's `x, y ~ N(0, 1)` is PENDING_INDEPENDENT_REVIEW); from the Prop's own `Does not claim:` line, \"the law (statement only)\": the law is proved here only for the formal objects of the Prop, as the distribution function `P(R^2 ≤ a)` for `a ≥ 0`, not as a density and not for the source's variable; and \"measurability of the event (if the event is not measurable, `P` is its outer measure)\": measurability of the formal event `{x^2 + y^2 ≤ a}` is proved here (`Aux.rl_measurableSet_disk`, `Measure.map_apply_of_aemeasurable`), so the outer-measure reading does not arise for this event, but nothing is established about any other event; that any inhabitant of `ConeMomentInterface` exists, or that its laws are the conditional law `A | V = 0` of the reference field (the laws are hypothesis fields of the given `I`, not derived here); `ConeMomentD2Identity` or any cone moment; any enclosure of the coefficient; the statement module's own text (the statement packet is not edited here); the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." + "does_not_claim": "the informal Section 1 display \"The variable R^2 has density e^(-z/2)/2\" of the source as the source's object, nor its review and acceptance (the alignment of the interface fields `x`, `y`, `P`, `sxy_law` and of `gaussianReal` with the source's `x, y ~ N(0, 1)` is PENDING_INDEPENDENT_REVIEW); from the Prop's own `Does not claim:` line, \"the law (statement only)\": the law is proved here only for the formal objects of the Prop, as the distribution function `P(R^2 ≤ a)` for `a ≥ 0`, not as a density and not for the source's variable; and \"measurability of the event (if the event is not measurable, `P` is its outer measure)\": the formal event `{x^2 + y^2 ≤ a}` is the preimage of the closed disk, a measurable set (`Aux.rl_measurableSet_disk`), under the a.e.-measurable coordinate map `(x, y)` (`Aux.rl_aemeasurable_xy`); `Measure.map_apply_of_aemeasurable` transports the measure along that map, so `P` of the event equals the disk measure under the law. The preimage is thereby null-measurable for `I.P` (`AEMeasurable.nullMeasurableSet_preimage`); `MeasurableSet` of the sample-space preimage is not asserted (`ConeMomentInterface` assumes neither that `I.P` is complete nor that the coordinate fields are measurable), and nothing is established about any other event; that any inhabitant of `ConeMomentInterface` exists, or that its laws are the conditional law `A | V = 0` of the reference field (the laws are hypothesis fields of the given `I`, not derived here); `ConeMomentD2Identity` or any cone moment; any enclosure of the coefficient; the statement module's own text (the statement packet is not edited here); the disposition of `side24-coefficient` (REVIEWED_SCOPED), which is transcribed in the register, not derived here; nonauthor review or acceptance of SIDE24-COEFFICIENT-D23-20260924-v1." } ], "kernel_evidence": [ @@ -896,12 +896,12 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); theorems added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60) have no hosted receipt yet: the next hosted lean-specification-proofs.yml run is their first receipt", - "note": "17 theorems of this row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral, Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). 12 (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them. `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); the theorems added after PR head 12e072d6 (modules Side24Analysis, Side24Interfaces, RNCluster) are receipted by run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 73, worktree CLEAN) and were first receipted by that run; base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434; the merge has the same tree as its head; all 73 theorems of PR head 48bfcbae receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd)", + "note": "17 theorems of this row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral, Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). 12 (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 and kernel-checked by the hosted run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 73, worktree CLEAN; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd); they were first receipted by that run (PR #434, event pull_request); base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434, and the merge has the same tree as its head. All 73 theorems of PR head 48bfcbae, including every theorem of the tiers above, are receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], - "not_established": "Nothing here encloses the coefficient. CoefficientEnclosure and ReferenceComparison are statements for any object satisfying PeriodicCoefficientInterface; the interface takes the cone moment D_u of (15.1) as a GIVEN field, so c_{d,24} is not uniquely determined in Lean and no inhabitant is constructed. The 27 main theorems listed as `proved` are the integer/rational arithmetic skeleton of PROOF.md sections 1-4 (coefficient sums, shell constants, Taylor partial sum, block minors, cone-moment algebra), not the Gaussian conditioning, the Rayleigh cone integration, the Stirling log-gamma step (section 5, not specified), the interval arithmetic of coefficient.py, or the enclosure itself. Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, LatticeImageSumBound, SuccessiveTermRatio, ExponentialBounds, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (17 entries below, label `proved`; 5 of them (Side24Elementary.successiveTermRatio, Side24Elementary.exponentialBounds, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.coneMomentD2Algebra, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.latticeImageSumBound, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.densityComparison, Side24Analysis.integrandRatioBounds, Side24Analysis.referencePinDensityProduct, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them). What those theorems prove, for the formal objects of the statement module only: elementary real inequalities (ExponentialBounds), the term ratio (SuccessiveTermRatio), the D2 cone-moment algebra (ConeMomentD2Algebra), the reference-coefficient identity as stated (ReferenceCoefficientIdentity; its left side is not identified with the parent formula (15.2) at the reference law: rotational invariance of K_infty is not specified) and the elementary cone integral (ElementaryConeIntegral); summability of the formal theta family n |-> exp(-|z + 24 n|^2/2) over Fin d -> Z for every d (ThetaSummable: the statement entry's does_not_claim \"that the family is summable (true, but not shown here)\" describes the statement packet, which is not edited; the summability is now shown in the proof packet, for the formal family, by comparison with a product of one-dimensional Gaussian sums); the Section 2 derivative bounds for q <= 6 as the Props state them (DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero: a Hermite-type recursion on the operator norms of iteratedFDeriv, no closed form of the Gaussian derivatives, the contraction at a unit tuple bounded by the operator norm; main's pairing_sum_six, pairing_sum_le_six and sixth_moment_double_factorial are neither cited nor relied on (the 1 + 15 + 45 + 15 = 76 arithmetic is recomputed inline by norm_num); nothing for q > 6); the lattice image sum (LatticeImageSumBound: summability and the bound 1458 e^(-288), via 432 e^(-288); the source's shell count 729 sum_j j^9 e^(-288 j^2) is not reproduced and not specified); the five reference block identities and the pin density product for the formal matrices (ReferenceOddAndPinBlocks, ReferencePinDensityProduct: derivative contractions of Kinf at 0 in an orthonormal frame; none of these matrices is shown to be the covariance of any random vector or field); pure real arithmetic in epsilon and the exponents a, b for d = 2, 3 (IntegrandRatioBounds; that the source's integrand ratio equals these power quotients is not specified); Loewner determinant and inverse monotonicity for positive-definite real matrices (DensityComparison; the cone-moment consequence (1 +/- epsilon)^m D0 and the scaling a^m are not specified); and, over ConeMomentInterface, the m = 1 cone moment 4/3, the trace moments 5/3, 25/3, sqrt(3/8) and the distribution function P(R^2 <= a) = integral_0^a e^(-z/2)/2 dz for a >= 0 (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw: proved for every inhabitant from the law fields A_law, sxy_law and isProb alone; no inhabitant is constructed, the laws are not derived as the conditional law A | V = 0 of the reference field, and the Rayleigh law is proved as a distribution function, not as a density). None of them is the Gaussian conditioning, ConeMomentD2Identity, ImageBound, the covariance ordering or any enclosure of the coefficient, and the claim-level label stays `specified` (claim-level vocabulary none/specified; a theorem of a Prop is evidence about that Prop, for the formal objects, not acceptance of the informal theorem; alignment PENDING_INDEPENDENT_REVIEW). Theorem-shaped Props of this row still `specified` with no theorem: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison; the parametrised predicates IsUnitTuple, NegDefCone are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited.", - "next_step": "Math- Mathlib lane: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); the 12 theorems of this row added after PR head 12e072d6 (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) await the next hosted lean-specification-proofs.yml run, their first receipt; the other theorems of this row have a hosted receipt: the 5 of PR head 00d2ac6e (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); each of those receipted theorems corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, ReferenceOddAndPinBlocks, IntegrandRatioBounds, ReferencePinDensityProduct, ConeMomentD1Identity, TraceMoments PROVABLE_NOW and LatticeImageSumBound, DensityComparison, RayleighSquareLaw PROVABLE_WITH_WORK: proved in round 4 (see entries); ReferenceCovarianceLowerBound PROVABLE_WITH_WORK (weeks); ConeMomentD2Identity PROVABLE_WITH_WORK (weeks); ImageBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum; the global version is inapplicable to lattice translates; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), iteratedFDeriv_tsum_apply (SmoothSeries.lean)); CovarianceOrdering BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: local term-wise iterated differentiation of the lattice sum, the d-dimensional Gaussian tail bound and higher-order symmetry of iteratedFDeriv; display (3) needs only the crude norm_iteratedFDeriv_comp_le, with a factor about 8 to spare; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), hasFDerivAt_tsum_of_isPreconnected (SmoothSeries.lean)); EntrywiseCovarianceBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum for the term-wise iterated differentiation of the lattice sum (the same blocker as CovarianceOrdering; the finding records no further shape observation); present but insufficient: norm_iteratedFDeriv_comp_le (Bounds.lean), iteratedFDeriv_tsum (SmoothSeries.lean)); CoefficientEnclosure BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); ReferenceComparison BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); IsUnitTuple, NegDefCone PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. The enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md) and an interface that determines c_{d,24} (the cone moment is a given field; ConeMomentD2Identity for the reference law under ConeMomentInterface is the open Rayleigh-integration step). Proof packet README: for the Props proved in round 4 (ConeMomentD1Identity, DensityComparison, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, DerivativePairingBound, IntegrandRatioBounds, LatticeImageSumBound, RayleighSquareLaw, ReferenceOddAndPinBlocks, ReferencePinDensityProduct, ThetaSummable, TraceMoments) the informal reasons the README gave were found not load-bearing by the triage and are superseded by their theorems; the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure and ReferenceComparison stand (for the last two the triage confirms the README's reason as the precise missing ingredient) (the proof-packet maintainers' regeneration, not this register's). Independent reviewer: alignment verdicts on the 28 Side24 statements, in particular of the 17 Props of this row that have a theorem of exactly their type, with their anchors (ThetaSummable with the source's R^d, 24 Z^d and K_24; the derivative Props with Section 2; the interface Props with the source's A ~ N(0, 8/3), s ~ N(0, 5/3), x, y ~ N(0, 1))." + "not_established": "Nothing here encloses the coefficient. CoefficientEnclosure and ReferenceComparison are statements for any object satisfying PeriodicCoefficientInterface; the interface takes the cone moment D_u of (15.1) as a GIVEN field, so c_{d,24} is not uniquely determined in Lean and no inhabitant is constructed. The 27 main theorems listed as `proved` are the integer/rational arithmetic skeleton of PROOF.md sections 1-4 (coefficient sums, shell constants, Taylor partial sum, block minors, cone-moment algebra), not the Gaussian conditioning, the Rayleigh cone integration, the Stirling log-gamma step (section 5, not specified), the interval arithmetic of coefficient.py, or the enclosure itself. Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, LatticeImageSumBound, SuccessiveTermRatio, ExponentialBounds, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (17 entries below, label `proved`; 5 of them (Side24Elementary.successiveTermRatio, Side24Elementary.exponentialBounds, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.coneMomentD2Algebra, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.latticeImageSumBound, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.densityComparison, Side24Analysis.integrandRatioBounds, Side24Analysis.referencePinDensityProduct, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 and kernel-checked by the hosted run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 73, worktree CLEAN; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd); they were first receipted by that run (PR #434, event pull_request); base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434, and the merge has the same tree as its head). What those theorems prove, for the formal objects of the statement module only: elementary real inequalities (ExponentialBounds), the term ratio (SuccessiveTermRatio), the D2 cone-moment algebra (ConeMomentD2Algebra), the reference-coefficient identity as stated (ReferenceCoefficientIdentity; its left side is not identified with the parent formula (15.2) at the reference law: rotational invariance of K_infty is not specified) and the elementary cone integral (ElementaryConeIntegral); summability of the formal theta family n |-> exp(-|z + 24 n|^2/2) over Fin d -> Z for every d (ThetaSummable: the statement entry's does_not_claim \"that the family is summable (true, but not shown here)\" describes the statement packet, which is not edited; the summability is now shown in the proof packet, for the formal family, by comparison with a product of one-dimensional Gaussian sums); the Section 2 derivative bounds for q <= 6 as the Props state them (DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero: a Hermite-type recursion on the operator norms of iteratedFDeriv, no closed form of the Gaussian derivatives, the contraction at a unit tuple bounded by the operator norm; main's pairing_sum_six, pairing_sum_le_six and sixth_moment_double_factorial are neither cited nor relied on (the 1 + 15 + 45 + 15 = 76 arithmetic is recomputed inline by norm_num); nothing for q > 6); the lattice image sum (LatticeImageSumBound: summability and the bound 1458 e^(-288), via 432 e^(-288); the source's shell count 729 sum_j j^9 e^(-288 j^2) is not reproduced and not specified); the five reference block identities and the pin density product for the formal matrices (ReferenceOddAndPinBlocks, ReferencePinDensityProduct: derivative contractions of Kinf at 0 in an orthonormal frame; none of these matrices is shown to be the covariance of any random vector or field); pure real arithmetic in epsilon and the exponents a, b for d = 2, 3 (IntegrandRatioBounds; that the source's integrand ratio equals these power quotients is not specified); Loewner determinant and inverse monotonicity for positive-definite real matrices (DensityComparison; the cone-moment consequence (1 +/- epsilon)^m D0 and the scaling a^m are not specified); and, over ConeMomentInterface, the m = 1 cone moment 4/3, the trace moments 5/3, 25/3, sqrt(3/8) and the distribution function P(R^2 <= a) = integral_0^a e^(-z/2)/2 dz for a >= 0 (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw: proved for every inhabitant from the law fields A_law, sxy_law and isProb alone; no inhabitant is constructed, the laws are not derived as the conditional law A | V = 0 of the reference field, and the Rayleigh law is proved as a distribution function, not as a density). None of them is the Gaussian conditioning, ConeMomentD2Identity, ImageBound, the covariance ordering or any enclosure of the coefficient, and the claim-level label stays `specified` (claim-level vocabulary none/specified; a theorem of a Prop is evidence about that Prop, for the formal objects, not acceptance of the informal theorem; alignment PENDING_INDEPENDENT_REVIEW). Theorem-shaped Props of this row still `specified` with no theorem: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison; the parametrised predicates IsUnitTuple, NegDefCone are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited.", + "next_step": "Math- Mathlib lane: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); every theorem of this row has a hosted receipt: the 12 added after PR head 12e072d6 (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) first by run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (receipt theorem_count 73); the 5 of PR head 00d2ac6e (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); each of those 5 theorems corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60); every theorem of this row receipted on PR head 48bfcbae by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, ReferenceOddAndPinBlocks, IntegrandRatioBounds, ReferencePinDensityProduct, ConeMomentD1Identity, TraceMoments PROVABLE_NOW and LatticeImageSumBound, DensityComparison, RayleighSquareLaw PROVABLE_WITH_WORK: proved in round 4 (see entries); ReferenceCovarianceLowerBound PROVABLE_WITH_WORK (weeks); ConeMomentD2Identity PROVABLE_WITH_WORK (weeks); ImageBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum; the global version is inapplicable to lattice translates; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), iteratedFDeriv_tsum_apply (SmoothSeries.lean)); CovarianceOrdering BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: local term-wise iterated differentiation of the lattice sum and higher-order symmetry of iteratedFDeriv (the 2026-10-11 triage table, kept as the historical record on the Math- reading branch claude/round4-triage-table-20261011, commit 43a975ec, file reviews/round4_triage_20261011/triage_table.json, also listed the d-dimensional Gaussian tail bound, since discharged by latticeImageSumBound, and the derivative constants, since supplied by the derivative theorems); display (3) needs only the crude norm_iteratedFDeriv_comp_le, with a factor about 8 to spare; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), hasFDerivAt_tsum_of_isPreconnected (SmoothSeries.lean)); EntrywiseCovarianceBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum for the term-wise iterated differentiation of the lattice sum (the same blocker as CovarianceOrdering; the finding records no further shape observation); present but insufficient: norm_iteratedFDeriv_comp_le (Bounds.lean), iteratedFDeriv_tsum (SmoothSeries.lean)); CoefficientEnclosure BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); ReferenceComparison BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); IsUnitTuple, NegDefCone PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. The enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md) and an interface that determines c_{d,24} (the cone moment is a given field; ConeMomentD2Identity for the reference law under ConeMomentInterface is the open Rayleigh-integration step). Proof packet README: for the Props proved in round 4 (ConeMomentD1Identity, DensityComparison, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, DerivativePairingBound, IntegrandRatioBounds, LatticeImageSumBound, RayleighSquareLaw, ReferenceOddAndPinBlocks, ReferencePinDensityProduct, ThetaSummable, TraceMoments) the informal reasons the README gave were found not load-bearing by the triage and are superseded by their theorems; the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure and ReferenceComparison stand (for the last two the triage confirms the README's reason as the precise missing ingredient) (the proof-packet maintainers' regeneration, not this register's). Independent reviewer: alignment verdicts on the 28 Side24 statements, in particular of the 17 Props of this row that have a theorem of exactly their type, with their anchors (ThetaSummable with the source's R^d, 24 Z^d and K_24; the derivative Props with Section 2; the interface Props with the source's A ~ N(0, 8/3), s ~ N(0, 5/3), x, y ~ N(0, 1))." }, { "claim_id": "lifetime-remainder", @@ -1060,8 +1060,8 @@ "kernel_evidence": [ { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", - "note": "5 theorems of this row (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); all 73 theorems of PR head 48bfcbae receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd)", + "note": "5 theorems of this row (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). All 73 theorems of PR head 48bfcbae, including every theorem of the tiers above, are receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "No Prop states Theorem N as a whole, so the claim-level label stays none: (N1)-(N5) (Hoelder bound, its power-law consequence, the conditional-mean identity, the uniform counterexample, the exponential-tail layer cake) are stated as abstract probability-space Props and each has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (5 entries below, label `proved`; the hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked them (receipt formalization_status kernel-checked, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound})); (N6) (the full-pin weighted Kac-Rice expression for E_Q^W N_B) and (N7) (the enclosure-to-rate implication under Z_r >= z_* r^2) are NOT stated, and they are the Gaussian numerator interface that the disposition reason calls \"a target rather than a discharged global RN theorem\". The five theorems prove the Props for the abstract probability spaces and counts they quantify over: nothing about the Gaussian model is stated or proved; no RN rate, no 24-jet enclosure, no ENV-RESCOV discharge; D3-LEMMA-RN-UNIF stays not closed. Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -1704,8 +1704,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", - "note": "20 theorems of this row (P15Elementary.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Price, P15Structure.elementaryPaletteFormula, P15Structure.triangleNeedsThree, P15Structure.benchmark816Palette, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.benchmark816WholeGround, P15Structure.minimalForbiddenCharacterisation, P15Structure.benchmark816MinimalForbiddenCounts, P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Covers.benchmark816Cover, P15Covers.localRestriction, P15Covers.paletteNumberAttained, P15Covers.benchmark816UnionBound, P15Budget.coverCostHazardBound) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 13 of them (P15Elementary.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Price, P15Structure.elementaryPaletteFormula, P15Structure.triangleNeedsThree, P15Structure.benchmark816Palette, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.benchmark816WholeGround, P15Structure.minimalForbiddenCharacterisation, P15Structure.benchmark816MinimalForbiddenCounts) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 7 (P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Covers.benchmark816Cover, P15Covers.localRestriction, P15Covers.paletteNumberAttained, P15Covers.benchmark816UnionBound, P15Budget.coverCostHazardBound) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); all 73 theorems of PR head 48bfcbae receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd)", + "note": "20 theorems of this row (P15Elementary.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Price, P15Structure.elementaryPaletteFormula, P15Structure.triangleNeedsThree, P15Structure.benchmark816Palette, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.benchmark816WholeGround, P15Structure.minimalForbiddenCharacterisation, P15Structure.benchmark816MinimalForbiddenCounts, P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Covers.benchmark816Cover, P15Covers.localRestriction, P15Covers.paletteNumberAttained, P15Covers.benchmark816UnionBound, P15Budget.coverCostHazardBound) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 13 of them (P15Elementary.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Price, P15Structure.elementaryPaletteFormula, P15Structure.triangleNeedsThree, P15Structure.benchmark816Palette, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.benchmark816WholeGround, P15Structure.minimalForbiddenCharacterisation, P15Structure.benchmark816MinimalForbiddenCounts) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 7 (P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Covers.benchmark816Cover, P15Covers.localRestriction, P15Covers.paletteNumberAttained, P15Covers.benchmark816UnionBound, P15Budget.coverCostHazardBound) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). All 73 theorems of PR head 48bfcbae, including every theorem of the tiers above, are receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: PaletteNumberAttained, MinimalForbiddenCharacterisation, LocalRestriction, LocalDecomposability, LocalObstructionIsFullBlock, LocalPriceChain, IndependentBlockHazard, FullBlockCoverIff, CoverCostHazardBound, WholeGroundChromaticNumber, ElementaryPaletteFormula, TriangleNeedsThree, Benchmark816IsRealized, Benchmark816Palette, Benchmark816Cover, Benchmark816WholeGround, Benchmark816MinimalForbiddenCounts, Benchmark816Price, Benchmark816UnionBound (20 entries below, label `proved`; 13 of them (P15Structure.minimalForbiddenCharacterisation, P15Structure.localDecomposability, P15Structure.localObstructionIsFullBlock, P15Structure.fullBlockCoverIff, P15Structure.wholeGroundChromaticNumber, P15Structure.elementaryPaletteFormula, P15Elementary.triangleNeedsThree, P15Structure.triangleNeedsThree, P15Structure.benchmark816IsRealized, P15Structure.benchmark816Palette, P15Structure.benchmark816WholeGround, P15Structure.benchmark816MinimalForbiddenCounts, P15Structure.benchmark816Price) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 7 of them (P15Covers.paletteNumberAttained, P15Covers.localRestriction, P15Hazard.localPriceChain, P15Hazard.independentBlockHazard, P15Budget.coverCostHazardBound, P15Covers.benchmark816Cover, P15Covers.benchmark816UnionBound) added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN)); among them Proposition P1 (MinimalForbiddenCharacterisation), both halves of Theorem P2 (FullBlockCoverIff, display (P7), and CoverCostHazardBound, display (P8)), the local statements (P2)-(P5) and the benchmark Props including the union bound (P11). Theorem-shaped Props of this row still `specified` with no theorem: none; the parametrised predicates IsRealizedFamily, IsGood, IsKDecomposable, IsGeneratorCover, IsPaletteFeasible, IsMinimalForbidden, IsTransversal are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. Theorem P2 is proved as the two Props state it (for the formal RealizedData record and the formal downset, prices 0 <= c_v <= p_v), which is a statement about those Props, not acceptance of the informal theorem: the claim-level label stays `specified` (this register admits only none/specified at claim level), alignment of RealizedData / IsRealizedFamily / downset / coverCost / mu with the informal objects is PENDING_INDEPENDENT_REVIEW, and the price range p_v < c_v <= phi(p_v) is a different claim row. The 12 main P15 arithmetic theorems listed as `proved` (finite arithmetic of Section 6: 409, 2454, 15, 500616, 419743994415, 816, 815*3 < 2448, 818, 409p = 1/100, the (P11) rational bound, positivity of 6/40900^409, the elementary instances of (P9)) are at commit 91cffbb5 of main PR #358, not on main; they do not count the minimal forbidders, show the palette is 816 or evaluate 6/40900^409, and the proof-packet theorems that do (Benchmark816MinimalForbiddenCounts, Benchmark816Palette, Benchmark816Price) do so for the formal RealizedData record, whose identification with the informal six-block construction is an alignment question. Nothing is asserted for prices p_v < c_v <= phi(p_v) (row p15-price-budget-restricted), for arbitrary P15 inputs or for the prize (original_prize_closed stays false). Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -1839,8 +1839,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", - "note": "3 theorems of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.logOnePlusBelowIdentity, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 3 of them (P15Elementary.twoCoordGeneratorPrices, P15Elementary.logOnePlusBelowIdentity, P15Elementary.demandOneRefutation) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); all 73 theorems of PR head 48bfcbae receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd)", + "note": "3 theorems of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.logOnePlusBelowIdentity, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 3 of them (P15Elementary.twoCoordGeneratorPrices, P15Elementary.logOnePlusBelowIdentity, P15Elementary.demandOneRefutation) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). All 73 theorems of PR head 48bfcbae, including every theorem of the tiers above, are receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "DemandOneRefutation states the refutation (the two-coordinate data form a realized family with K_H(d) = 1, obstruction {X}, admissible prices 2/3, mu_p(D) = 3/4, cover cost 4/9 > 1/3 > log(4/3)). DemandOneRefutation, TwoCoordGeneratorPrices and LogOnePlusBelowIdentity each have a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011 (listed below as `proved`, the label of a committed proof package; the hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked them (receipt formalization_status kernel-checked, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound})); the two logarithm inequalities phi(1/2) = log 2 > 2/3 and log(4/3) < 1/3 are conjuncts of DemandOneRefutation and are proved inside its theorem from Mathlib's Real.log bounds. The claim-level label stays `specified`, not `proved`: this register admits only none/specified at claim level (no research claim of this programme is proved), whether DemandOneRefutation says what the source paragraph says is the independent reviewer's alignment question (PENDING_INDEPENDENT_REVIEW), the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked the Prop and nothing more, and a kernel-checked Prop is evidence about that Prop only. This is the one row of the register where a theorem proves the Prop registered as stating the claim itself; it is recorded here for the operator and not acted on. The 6 main P15 arithmetic theorems listed as `proved` (good sets, obstruction, 3/4 over denominator 4, prices 9,6,6,4 over 9, 3*4 > 9) are at commit 91cffbb5 of main PR #358, not on main; they are finite arithmetic, not the refutation. The counterexample does not refute d_i >= 2 successors or original P15-B (the disposition reason; the theorem's docstring says the same). Disposition EXACT_COUNTEREXAMPLE is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -2037,8 +2037,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", - "note": "8 theorems of this row (P15Elementary.logRatioBound, P15Elementary.capTestAtTwoThirds, P15Elementary.ratioExponentBound, P15Structure.benchmark816RationalPrice, P15Budget.restrictedLocalBudget, P15Budget.rhoTestSufficient, P15Budget.restrictedTransformedPriceBudget, P15Budget.benchmark816TransformedRange) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 4 of them (P15Elementary.logRatioBound, P15Elementary.capTestAtTwoThirds, P15Elementary.ratioExponentBound, P15Structure.benchmark816RationalPrice) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 4 (P15Budget.restrictedLocalBudget, P15Budget.rhoTestSufficient, P15Budget.restrictedTransformedPriceBudget, P15Budget.benchmark816TransformedRange) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); all 73 theorems of PR head 48bfcbae receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd)", + "note": "8 theorems of this row (P15Elementary.logRatioBound, P15Elementary.capTestAtTwoThirds, P15Elementary.ratioExponentBound, P15Structure.benchmark816RationalPrice, P15Budget.restrictedLocalBudget, P15Budget.rhoTestSufficient, P15Budget.restrictedTransformedPriceBudget, P15Budget.benchmark816TransformedRange) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 4 of them (P15Elementary.logRatioBound, P15Elementary.capTestAtTwoThirds, P15Elementary.ratioExponentBound, P15Structure.benchmark816RationalPrice) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 4 (P15Budget.restrictedLocalBudget, P15Budget.rhoTestSufficient, P15Budget.restrictedTransformedPriceBudget, P15Budget.benchmark816TransformedRange) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). All 73 theorems of PR head 48bfcbae, including every theorem of the tiers above, are receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "RestrictedTransformedPriceBudget states (T1) for every realized family with a_i >= 1, d_i >= 2, p_v <= 1/4, c_v <= phi(p_v) and is proved as stated in the proof packet; that is evidence about the Prop (for the formal RealizedData record, downset, coverCost and mu), not acceptance of the informal theorem: the claim-level label stays `specified` (claim-level vocabulary none/specified; alignment PENDING_INDEPENDENT_REVIEW). Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: LogRatioBound, RatioExponentBound, RestrictedLocalBudget, RestrictedTransformedPriceBudget, RhoTestSufficient, CapTestAtTwoThirds, Benchmark816TransformedRange, Benchmark816RationalPrice (8 entries below, label `proved`; 4 of them (P15Elementary.logRatioBound, P15Elementary.ratioExponentBound, P15Elementary.capTestAtTwoThirds, P15Structure.benchmark816RationalPrice) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 4 of them (P15Budget.restrictedLocalBudget, P15Budget.restrictedTransformedPriceBudget, P15Budget.rhoTestSufficient, P15Budget.benchmark816TransformedRange) added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN)). Theorem-shaped Props of this row still `specified` with no theorem: none. The intermediate ratio steps of (T3) are proved inside the theorems that use them; the integral representation behind (T2), the admissibility integral and the Section 5 checker program are not specified. The 3 main P15 arithmetic theorems listed as `proved` (the a = 1, n = 3 instance 16/27, finite instances of the (T3) bound and the rational identity 81801/3345620000) are at commit 91cffbb5 of main PR #358, not on main; they are not (T1). The dependency on claim p15-realized-covers is discharged only to the extent that row records. Disposition HOLD_WITH_DOMAIN (\"independent theorem review not yet source-bound in manifest\") is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -2478,8 +2478,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); the theorems added after PR head 79731b96 (module P15Interpolation) are receipted by run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 60, worktree CLEAN) and were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422; same trees as the heads; corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", - "note": "19 theorems of this row (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.sixSeventhsCertificate, P15Elementary.hStarExceedsOne, P15Elementary.phiAtPStar, P15Elementary.hazardStarPositive, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.demandOneBoundary, P15Elementary.capacityHazardIsBinomial, P15Elementary.sharpnessOfRhoStar, P15Hazard.globalAssemblyHazard, P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull, P15Interpolation.fullTransformedPriceBudget) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 8 of them (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.sixSeventhsCertificate, P15Elementary.hStarExceedsOne, P15Elementary.phiAtPStar, P15Elementary.hazardStarPositive, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.demandOneBoundary) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 3 (P15Elementary.capacityHazardIsBinomial, P15Elementary.sharpnessOfRhoStar, P15Hazard.globalAssemblyHazard) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). 8 (P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull, P15Interpolation.fullTransformedPriceBudget) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head. The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); the theorems added after PR head 79731b96 (module P15Interpolation) are receipted by run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 60, worktree CLEAN) and were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422; same trees as the heads; corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); all 73 theorems of PR head 48bfcbae receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd)", + "note": "19 theorems of this row (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.sixSeventhsCertificate, P15Elementary.hStarExceedsOne, P15Elementary.phiAtPStar, P15Elementary.hazardStarPositive, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.demandOneBoundary, P15Elementary.capacityHazardIsBinomial, P15Elementary.sharpnessOfRhoStar, P15Hazard.globalAssemblyHazard, P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull, P15Interpolation.fullTransformedPriceBudget) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 8 of them (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.sixSeventhsCertificate, P15Elementary.hStarExceedsOne, P15Elementary.phiAtPStar, P15Elementary.hazardStarPositive, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.demandOneBoundary) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). 3 (P15Elementary.capacityHazardIsBinomial, P15Elementary.sharpnessOfRhoStar, P15Hazard.globalAssemblyHazard) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). 8 (P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull, P15Interpolation.fullTransformedPriceBudget) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head. The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). All 73 theorems of PR head 48bfcbae, including every theorem of the tiers above, are receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "FullTransformedPriceBudget states (F2) and is proved as stated; RhoStarEnclosure states (F3) and is proved as stated; SharpnessOfRhoStar states the attainment clause and is proved as stated. The claim-level label stays `specified` (claim-level vocabulary none/specified; a theorem of a Prop is evidence about that Prop, for the formal objects, not acceptance of Theorem F; alignment PENDING_INDEPENDENT_REVIEW). Theorem F is proved as FullTransformedPriceBudget states it: for the formal RealizedData record with a_i >= 1, d_i >= 2, every p in [0,1]^X, every 0 <= c_v <= phi(p_v) and the same K >= K_H(d), coverCost <= cappedHazard rho_star mu, through the coordinatewise concavity (CoordinatewiseConcavity, a theorem of exactly that Prop), (F6)-(F8), (F12) and (F13); that is evidence about those Props, not acceptance of the informal theorem. RhoStarEnclosure is proved by a rational certificate (Real.exp_bound series enclosures for e and for exp with explicit remainders, compared by norm_num), not by the source's full_price.py Fraction procedure, which remains unspecified. Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: FullTransformedPriceBudget, RhoStarEnclosure, RhoStarBelowSixSevenths, HazardStarPositive, CoordinatewiseConcavity, HazardProductBound, HazardPhiBound, FullGroundPriceBound, PhiAtPStar, FullGroundBudgetIff, CapacityHazardIsBinomial, OddMajorityIdentity, CapacityWorstCase, HStarExceedsOne, LocalPriceBoundFull, GlobalAssemblyHazard, SharpnessOfRhoStar, DemandOneBoundary, SixSeventhsCertificate (19 entries below, label `proved`; 8 of them (P15Elementary.rhoStarBelowSixSevenths, P15Elementary.hazardStarPositive, P15Elementary.phiAtPStar, P15Elementary.oddMajorityIdentity, P15Elementary.capacityWorstCase, P15Elementary.hStarExceedsOne, P15Elementary.demandOneBoundary, P15Elementary.sixSeventhsCertificate) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1); 3 of them (P15Elementary.capacityHazardIsBinomial, P15Hazard.globalAssemblyHazard, P15Elementary.sharpnessOfRhoStar) were added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN); 8 of them (P15Interpolation.fullTransformedPriceBudget, P15Interpolation.rhoStarEnclosure, P15Interpolation.coordinatewiseConcavity, P15Interpolation.hazardProductBound, P15Interpolation.hazardPhiBound, P15Interpolation.fullGroundPriceBound, P15Interpolation.fullGroundBudgetIff, P15Interpolation.localPriceBoundFull) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head). Theorem-shaped Props of this row still `specified` with no theorem: none; the parametrised predicates IsProperDownset are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. The derivative formulas (F4)-(F5) are not specified; CoordinatewiseConcavity, their consequence, is proved from the one-coordinate split of mu_p(A) (display (F4): affine in exp(-t_i) with coefficients 0 <= S_in <= S_out for a downset) and the second-derivative criterion for s |-> -log(a + b exp(-s)). SixSeventhsCertificate uses Mathlib's decimal bound on e in place of the source's series argument. The 7 main P15 arithmetic theorems listed as `proved` (finite arithmetic of the Section 6 certificate chain: adjacent 20-digit endpoints as Nat, 16/27 < 6/7, n >= 2a+1 instances, the rational orderings and the degree-5 Taylor partial sum) are at commit 91cffbb5 of main PR #358, not on main; they do not place rho_star = 1/(3 - log(3e - 2)) between the endpoints, do not prove e < 87/32 or 197/32 < exp(11/6) as real-number facts (the SixSeventhsCertificate theorem does, as that Prop states them), and are not Theorem F. The full_price.py Fraction procedure is not a kernel proof and is not specified. No arbitrary-downset or unrestricted P15 prize statement (original_prize_closed stays false). Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -2941,7 +2941,7 @@ "note": "The 31 Side24 targets at origin/main f6deeba7 are decide-proved integer or cross-multiplied rational facts (manifest label `proved`; `kernel-checked` only in a run receipt). Arithmetic skeleton only: formal/SCOPE.md says none of them is the coefficient enclosure." } ], - "not_established": "See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the \"evaluates_eq_15_2\" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. Of the three statements of this node, none has a theorem; stated, not proved: CoefficientEnclosure, ReferenceComparison. The 27 arithmetic theorems of main and the 17 proof-packet theorems of the landing row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral, Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw; none of them the enclosure; 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them) are listed on the landing row, never here and never as the node; the interface theorems among them (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw) hold for every inhabitant of ConeMomentInterface from its law fields alone and construct nothing. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited.", + "not_established": "See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the \"evaluates_eq_15_2\" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. Of the three statements of this node, none has a theorem; stated, not proved: CoefficientEnclosure, ReferenceComparison. The 27 arithmetic theorems of main and the 17 proof-packet theorems of the landing row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral, Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw; none of them the enclosure; 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 and kernel-checked by the hosted run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 73, worktree CLEAN; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd); they were first receipted by that run (PR #434, event pull_request); base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434, and the merge has the same tree as its head) are listed on the landing row, never here and never as the node; the interface theorems among them (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw) hold for every inhabitant of ConeMomentInterface from its law fields alone and construct nothing. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited.", "next_step": "As for row side24-coefficient. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): CoefficientEnclosure BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface elaborates, by doubling coneMoment and c24, so no theorem of its type can exist unless the interface is empty); ReferenceComparison BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, ReferenceComparison I) -> IsEmpty PeriodicCoefficientInterface elaborates likewise)." }, { @@ -3049,8 +3049,8 @@ "kernel_evidence": [ { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", - "note": "5 theorems of this row (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); all 73 theorems of PR head 48bfcbae receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd)", + "note": "5 theorems of this row (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 5 of them (RNProbability.conditionalMeanIdentityN3, RNProbability.holderCountBoundN1, RNProbability.holderConsequenceN2, RNProbability.uniformCounterexampleN4, RNProbability.exponentialTailLayerCakeN5) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). All 73 theorems of PR head 48bfcbae, including every theorem of the tiers above, are receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "See row rn-count-interface: (N1)-(N5) stated and each proved as a theorem of the proof packet (5 entries below, `proved`; kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)), (N6)-(N7) not stated, so no Prop states the interface as a whole and the node stays none; nothing about the Gaussian model is proved. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -3421,8 +3421,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); the theorems added after PR head 79731b96 (module P15Interpolation) are receipted by run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 60, worktree CLEAN) and were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422; same trees as the heads; corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", - "note": "3 theorems of this row (P15Elementary.sharpnessOfRhoStar, P15Interpolation.rhoStarEnclosure, P15Interpolation.fullTransformedPriceBudget) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 1 (P15Elementary.sharpnessOfRhoStar) was added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). 2 (P15Interpolation.rhoStarEnclosure, P15Interpolation.fullTransformedPriceBudget) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head. The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); the theorems added after PR head 79731b96 (module P15Interpolation) are receipted by run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 60, worktree CLEAN) and were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422; same trees as the heads; corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); all 73 theorems of PR head 48bfcbae receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd)", + "note": "3 theorems of this row (P15Elementary.sharpnessOfRhoStar, P15Interpolation.rhoStarEnclosure, P15Interpolation.fullTransformedPriceBudget) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 1 (P15Elementary.sharpnessOfRhoStar) was added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN). 2 (P15Interpolation.rhoStarEnclosure, P15Interpolation.fullTransformedPriceBudget) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head. The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). All 73 theorems of PR head 48bfcbae, including every theorem of the tiers above, are receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "See row p15-full-price. Of the three Props of this node, with a theorem of exactly their type in the proof packet: FullTransformedPriceBudget, RhoStarEnclosure, SharpnessOfRhoStar (3 entries below, label `proved`; 1 of them (P15Elementary.sharpnessOfRhoStar) was added after commit 00d2ac6e and kernel-checked by the hosted run 38121713840 (PR head 910809d6; actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, worktree CLEAN); 2 of them (P15Interpolation.fullTransformedPriceBudget, P15Interpolation.rhoStarEnclosure) were added after PR head 79731b96 and kernel-checked by the hosted run 38149888699 (PR head 0dcb0016; actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 60, worktree CLEAN; artifact lean-specification-proofs-38149888699-1, sha256 cdeb5c73aa03124ed0db16fa14d25fb03a8fa842457fce1e0c6565dbc5ee7ca8); they were first receipted by run 38149524759 (PR head 88c21058; actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with base 79731b961fac555f33be3cd90e379b246a5a80e3) (same Lean bytes; artifact lean-specification-proofs-38149524759-1, sha256 ddefbdcc63223d05fd8ed34addbeebc199e271525d87dead01109cddc529e836); base 79731b961fac555f33be3cd90e379b246a5a80e3 is the head of PR #422, and each merge has the same tree as its head); none of the three is left unproved. A theorem of SharpnessOfRhoStar is about the formal triple block and the formal cover cost; a theorem of FullTransformedPriceBudget is about the formal RealizedData record, downset, coverCost and mu; none of this is Theorem F, and the node-level label stays `specified`. The 7 main P15 arithmetic theorems of source p15-full-price (at commit 91cffbb5 of main PR #358, not on main) are listed here as the arithmetic skeleton, never as the node. The price-boundary refutation is a scope boundary (GRAPH edge boundary_counterexample_not_required_premise), not a hypothesis. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -3538,8 +3538,8 @@ }, { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690)", - "note": "2 theorems of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 2 of them (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38121713840 (PR head 910809d6; actual tested checkout the merge commit 5b198699308e0585904c21d12bf540cf2475d2f0 of that head into base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38121713840-1, sha256 640682d631d052f4903d66f506740ae8fa91d19185001e70a1502f404eb44175): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 52, worktree CLEAN; the 38 theorems of PR head 00d2ac6e were first receipted by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; artifact lean-specification-proofs-38116277597-1); corroborated on PR head d9b8dd33 by run 38122386459 (actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, same tree as the head; receipt formalization_status kernel-checked, theorem_count 52, worktree CLEAN; artifact lean-specification-proofs-38122386459-1, sha256 edd52190d6af5828e8229c161c3b77e87a540caf440070d66b1b7f415008ebc3); corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690); all 73 theorems of PR head 48bfcbae receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd)", + "note": "2 theorems of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 2 of them (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1). The 60 theorems of PR head 12e072d6 (every theorem of the tiers above) are corroborated by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690). All 73 theorems of PR head 48bfcbae, including every theorem of the tiers above, are receipted on that head by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd). `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], "not_established": "See row p15-price-boundary: TwoCoordGeneratorPrices and DemandOneRefutation (the two Props of this node) have theorems of exactly their type in the proof packet (listed as `proved`; kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)), the two logarithm inequalities included; the claim-level label stays `specified` for the reasons given there (claim-level vocabulary none/specified, alignment PENDING_INDEPENDENT_REVIEW, a Prop-level proof is evidence about the Prop only, whatever the hosted run 38116277597 receipted). The 6 main P15 arithmetic theorems of source p15-price-boundary (at commit 91cffbb5 of main PR #358, not on main) are listed as the arithmetic skeleton, never as the node. REFUTED is the graph classification of the extension, transcribed as source identity. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", @@ -4313,12 +4313,12 @@ "kernel_evidence": [ { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "receipt_or_workflow": "Hosted receipt: none yet. The proof-packet theorem of this row (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60, which does not name it); the next hosted lean-specification-proofs.yml run is the first receipt.", - "note": "1 theorem of this row (RNCluster.rareClusterPoissonObstructionO) is inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 1 (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them. `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." + "receipt_or_workflow": ".github/workflows/lean-specification-proofs.yml; hosted run 38180705952 (PR head 48bfcbae; actual tested checkout the merge commit 9833d6d9f9facc335e1a56e243c3144adefe6225 of that head into base 12e072d614dbe6e91b66d22a2ab55cf87860d693, the synthetic merge GitHub builds for a pull_request run, whose tree equals the head's and which the receipt records as checked_commit) (job proof-evidence, artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd): receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 73, worktree CLEAN; the proof-packet theorem of this row (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60, which does not name it) and was first receipted by run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225); base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434", + "note": "1 theorem of this row (RNCluster.rareClusterPoissonObstructionO) is inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 73 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster}.lean, 72 distinct registered Props, each theorem typed exactly as its registered Prop, applied to its own parameters). 1 (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 and kernel-checked by the hosted run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 73, worktree CLEAN; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd); it was first receipted by that run (PR #434, event pull_request); base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434, and the merge has the same tree as its head. `proved` is the label of a committed proof package; `kernel-checked` appears only in a run receipt. Evidence about the registered Props only: not about the informal claim, its alignment (PENDING_INDEPENDENT_REVIEW), review or acceptance; the claim-level label is unchanged; organizational-independence credit 0 (Anthropic / Claude)." } ], - "not_established": "Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified and the claim-level label stays `none` (the register admits only none/specified at claim level, and this row's claim is the packet, not Theorem O alone). RareClusterPoissonObstructionO has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (1 entry below, label `proved`; 1 of them (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them): it proves Theorem O's inequality as the Prop states it, for every family mu and constants a, A satisfying the Prop's own antecedents (0 < a, nonnegativity, summability and total mass one of each mu r, summable mean, (H1), (H2) on (0, r0]) and no others: the upper half with the witness lam = 0 (the Prop's existential form, not the source's infimum verbatim), the lower half through tail events {n >= 2} and {n >= 1}; 2a <= A is derived internally from the antecedents at r0, as in the source, and is neither a hypothesis nor a conclusion of the Prop. That theorem is evidence about the Prop, for the formal objects pmfTV, poissonPMF and the family mu : R -> N -> R, not acceptance of Theorem O; alignment of those objects with the source's dTV, Pois(lambda) and Law N_r is PENDING_INDEPENDENT_REVIEW. H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count (the Prop does not tie mu to any field; the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement lambda = mu, Theorems C and S, the factorial-moment matching, the independent-replica limit (7.1) and the conditional-law liminf remark are neither stated nor proved. For A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics; the hypotheses imply 2a <= A, so A > 0 is derivable and the Prop is not vacuous in general (the triage's example mu r = (1 - r^3) delta_0 + r^3 delta_2 with a = 1, A = 2 satisfies the antecedents for r <= 1). Disposition: none transcribed (disposition_transcribed null; the title's \"PROOF_INDEX reviewed scoped result\" is the source's wording, not a status decided here). Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", - "next_step": "State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); the 1 theorem of this row added after PR head 12e072d6 (RNCluster.rareClusterPoissonObstructionO) await the next hosted lean-specification-proofs.yml run, their first receipt. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RareClusterPoissonObstructionO PROVABLE_NOW: proved in round 4 (see entries). Independent reviewer: alignment of RareClusterPoissonObstructionO with Theorem O (3.1), in particular the existential form of the upper half and the (H1), (H2) antecedents on (0, r0]." + "not_established": "Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified and the claim-level label stays `none` (the register admits only none/specified at claim level, and this row's claim is the packet, not Theorem O alone). RareClusterPoissonObstructionO has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (1 entry below, label `proved`; 1 of them (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 and kernel-checked by the hosted run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 73, worktree CLEAN; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd); it was first receipted by that run (PR #434, event pull_request); base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434, and the merge has the same tree as its head): it proves Theorem O's inequality as the Prop states it, for every family mu and constants a, A satisfying the Prop's own antecedents (0 < a, nonnegativity, summability and total mass one of each mu r, summable mean, (H1), (H2) on (0, r0]) and no others: the upper half with the witness lam = 0 (the Prop's existential form, not the source's infimum verbatim), the lower half through tail events {n >= 2} and {n >= 1}; 2a <= A is derived internally from the antecedents at r0, as in the source, and is neither a hypothesis nor a conclusion of the Prop. That theorem is evidence about the Prop, for the formal objects pmfTV, poissonPMF and the family mu : R -> N -> R, not acceptance of Theorem O; alignment of those objects with the source's dTV, Pois(lambda) and Law N_r is PENDING_INDEPENDENT_REVIEW. H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count (the Prop does not tie mu to any field; the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement lambda = mu, Theorems C and S, the factorial-moment matching, the independent-replica limit (7.1) and the conditional-law liminf remark are neither stated nor proved. For A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics; the hypotheses imply 2a <= A, so A > 0 is derivable and the Prop is not vacuous in general (the triage's example mu r = (1 - r^3) delta_0 + r^3 delta_2 with a = 1, A = 2 satisfies the antecedents for r <= 1). Disposition: none transcribed (disposition_transcribed null; the title's \"PROOF_INDEX reviewed scoped result\" is the source's wording, not a status decided here). Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0.", + "next_step": "State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); the theorem of this row (RNCluster.rareClusterPoissonObstructionO, added after PR head 12e072d6) has a hosted receipt: first receipted by run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (receipt theorem_count 73). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RareClusterPoissonObstructionO PROVABLE_NOW: proved in round 4 (see entries). Independent reviewer: alignment of RareClusterPoissonObstructionO with Theorem O (3.1), in particular the existential form of the upper half and the (H1), (H2) antecedents on (0, r0]." }, { "claim_id": "CL-D5-REMOTE-COLLISION-20260928-v1", diff --git a/specifications/lean_statements_20261010/SCOPE.md b/specifications/lean_statements_20261010/SCOPE.md index 828e88406..4e8ab518a 100644 --- a/specifications/lean_statements_20261010/SCOPE.md +++ b/specifications/lean_statements_20261010/SCOPE.md @@ -7,8 +7,7 @@ Scientific effect: **NONE**. Formal progress: `specified` for every declaration. This package states; it does not prove. Proofs of some of its Props live in the sibling proof packet `specifications/lean_statement_proofs_20261011` (73 theorems typed exactly as 72 of the 141 Props; -the last column of the declaration tables and the `proved (proof packet …)` entries of the claims table, where † marks -a theorem added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…) with no hosted receipt yet); +the last column of the declaration tables and the `proved (proof packet …)` entries of the claims table); that packet depends on this one by path, never edits it, and its theorems change no label here and no claim-level label. The five modules under `Specifications/` declare, as `def … : Prop` and interface `structure`s in the namespaces `UniversalLaw.Spec.`, the @@ -60,11 +59,11 @@ a theorem between string literals, an indented Prop definition, a proof written ## Declarations -150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified` in this package; the 123 auxiliary definitions and abbreviations of the modules are inventory only and are not register statements. Names are given without the module namespace `UniversalLaw.Spec..`; the source column is the pinned source named by the docstring's `Source:` line. Anchors and "Not established" cells are copied verbatim from the Lean docstrings (`Anchor:` and `Does not claim:` lines) through REGISTER.json. The last column names the theorems of the sibling proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly the declaration (namespace `UniversalLaw.Proofs..`): 72 of the 141 Props have one (73 theorems; `TriangleNeedsThree` is proved twice), each listed in the register under the claim rows as a statement entry of kind `theorem` with label `proved` (the label of a committed proof package; `kernel-checked` appears only in a run receipt: the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, so each receipt records as checked_commit that merge and not the head: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; each merge has the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems marked † (`Side24Analysis`, `Side24Interfaces` and `RNCluster`) were added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), and have no hosted receipt yet: the first hosted `lean-specification-proofs.yml` run on the round-4 branch is their first receipt). That label belongs to the theorem, not to the declaration here, which stays `specified`, and not to any claim row; a kernel-checked theorem is evidence about its Prop only, and alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. +150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified` in this package; the 123 auxiliary definitions and abbreviations of the modules are inventory only and are not register statements. Names are given without the module namespace `UniversalLaw.Spec..`; the source column is the pinned source named by the docstring's `Source:` line. Anchors and "Not established" cells are copied verbatim from the Lean docstrings (`Anchor:` and `Does not claim:` lines) through REGISTER.json. The last column names the theorems of the sibling proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly the declaration (namespace `UniversalLaw.Proofs..`): 72 of the 141 Props have one (73 theorems; `TriangleNeedsThree` is proved twice), each listed in the register under the claim rows as a statement entry of kind `theorem` with label `proved` (the label of a committed proof package; `kernel-checked` appears only in a run receipt: the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, so each receipt records as checked_commit that merge and not the head: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; each merge has the same tree as its head; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems of `Side24Analysis`, `Side24Interfaces` and `RNCluster`, added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), are kernel-checked by the hosted run 38180705952 on PR head 48bfcbae (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, the head of PR #423; receipt theorem_count 73, leanchecker PASS; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd) and were first receipted by that run, which receipts all 73 theorems of that head; same tree as the head). That label belongs to the theorem, not to the declaration here, which stays `specified`, and not to any claim row; a kernel-checked theorem is evidence about its Prop only, and alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. ### `Specifications/Field.lean` (12 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance | anything about the field. | — | | `IsSmooth` | def_prop | `lifetime-parent` | In particular the field has a smooth version and all finite moments of its global derivative suprema exist. | existence of a smooth version, or moment bounds on derivative suprema. | — | @@ -81,7 +80,7 @@ a theorem between string literals, an indented Prop definition, a proof written ### `Specifications/Lifetime.lean` (24 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance | anything about the field. | — | | `IsSmooth` | def_prop | `lifetime-parent` | In particular the field has a smooth version and all finite moments of its global derivative suprema exist. | existence of a smooth version, or moment bounds on derivative suprema. | — | @@ -110,32 +109,32 @@ a theorem between string literals, an indented Prop definition, a proof written ### `Specifications/Side24.lean` (28 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---|---| -| `ThetaSummable` | def_prop | `side24-proof` | K_24(z) = sum_{n in Z^d} exp(-\|z+24n\|^2/2) | that the family is summable (true, but not shown here). | `Side24Analysis.thetaSummable†` | +| `ThetaSummable` | def_prop | `side24-proof` | K_24(z) = sum_{n in Z^d} exp(-\|z+24n\|^2/2) | that the family is summable (true, but not shown here). | `Side24Analysis.thetaSummable` | | `IsUnitTuple` | def_prop | `side24-proof` | For q<=6 unit directional differentiations of phi(x)=exp(-\|x\|^2/2), product-rule | anything. | — | -| `DerivativePairingBound` | def_prop | `side24-proof` | pairings bound the absolute derivative by | the bound (statement only); validity for `q > 6` (the source restricts to `q ≤ 6`); nothing here encloses the coefficient. | `Side24Analysis.derivativePairingBound†` | -| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | At \|x\|>=1 this is at most76 \|x\|^6 e^(-\|x\|^2/2). | the bound (statement only); the coefficient arithmetic `1+15+45+15=76` and `76` bounding the smaller orders are kernel-checked on main (`UniversalLaw.Side24.pairing_sum_six`, `pairing_sum_le_six`), not here; nothing here encloses the coefficient. | `Side24Analysis.derivativeBoundAtLeastOne†` | -| `DerivativeBoundAtZero` | def_prop | `side24-proof` | contraction has absolute value at most15. | the bound (statement only); `5!! = 15` is kernel-checked on main (`UniversalLaw.Side24.sixth_moment_double_factorial`), not here. | `Side24Analysis.derivativeBoundAtZero†` | -| `LatticeImageSumBound` | def_prop | `side24-proof` | <=1458 e^(-288). | the bound (statement only); the intermediate `729 sum_j j^9 e^(-288 j^2)` shell count is not specified; `27*27 = 729`, `729*2 = 1458` are kernel-checked on main (`UniversalLaw.Side24.lattice_shell_constant`), not here. | `Side24Analysis.latticeImageSumBound†` | +| `DerivativePairingBound` | def_prop | `side24-proof` | pairings bound the absolute derivative by | the bound (statement only); validity for `q > 6` (the source restricts to `q ≤ 6`); nothing here encloses the coefficient. | `Side24Analysis.derivativePairingBound` | +| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | At \|x\|>=1 this is at most76 \|x\|^6 e^(-\|x\|^2/2). | the bound (statement only); the coefficient arithmetic `1+15+45+15=76` and `76` bounding the smaller orders are kernel-checked on main (`UniversalLaw.Side24.pairing_sum_six`, `pairing_sum_le_six`), not here; nothing here encloses the coefficient. | `Side24Analysis.derivativeBoundAtLeastOne` | +| `DerivativeBoundAtZero` | def_prop | `side24-proof` | contraction has absolute value at most15. | the bound (statement only); `5!! = 15` is kernel-checked on main (`UniversalLaw.Side24.sixth_moment_double_factorial`), not here. | `Side24Analysis.derivativeBoundAtZero` | +| `LatticeImageSumBound` | def_prop | `side24-proof` | <=1458 e^(-288). | the bound (statement only); the intermediate `729 sum_j j^9 e^(-288 j^2)` shell count is not specified; `27*27 = 729`, `729*2 = 1458` are kernel-checked on main (`UniversalLaw.Side24.lattice_shell_constant`), not here. | `Side24Analysis.latticeImageSumBound` | | `SuccessiveTermRatio` | def_prop | `side24-proof` | successive terms have ratio at most512e^(-864)<1/2. | the bounds (statement only); `2^9 = 512`, `288*3 = 864` are kernel-checked on main (`UniversalLaw.Side24.geometric_ratio_constants`), not here. | `Side24Elementary.successiveTermRatio` | | `ExponentialBounds` | def_prop | `side24-proof` | hence e^(-288)<10^(-125) and the geometric-ratio inequality follows too. | the inequalities (statement only); the rational Taylor sum through degree 20 exceeding `10` is kernel-checked on main (the `UniversalLaw.Side24` Taylor-sum target in `Ledger.lean`), not here. | `Side24Elementary.exponentialBounds` | | `ImageBound` | def_prop | `side24-proof` | \|D^q K_24(0)-D^q K_infty(0)\| | the bound (statement only); differentiability of `K_24` (if `K_24` is not `C^q` the derivative is `0` in Lean); nothing here encloses the coefficient; nonauthor review or acceptance. | — | | `CovarianceOrdering` | def_prop | `side24-proof` | The inequalities are in the positive-semidefinite ordering and hold in every | the ordering (statement only); the passage to marginal and conditional covariances (not specified); `60E < epsilon` is kernel-checked on main (`UniversalLaw.Side24.covariance_relative_bound`), not here; nothing here encloses the coefficient; nonauthor review or acceptance. | — | | `EntrywiseCovarianceBound` | def_prop | `side24-proof` | Each covariance entry differs from the reference by at most2E, by (2). | the bound (statement only); the spectral-norm bound `20E` (not specified); nonauthor review or acceptance. | — | | `ReferenceCovarianceLowerBound` | def_prop | `side24-proof` | Thus C_ref >= I/3 and | the bound (statement only); the odd-block eigenvalues `8 ± sqrt 58` and the minors `2/3`, `7/9` are kernel-checked on main (`UniversalLaw.Side24.odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`), not here. | — | -| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | Var(t)=15, Cov(t,G)=(-3,0,...,0), | the identities (statement only); `15 - 9 = 6` is kernel-checked on main (`UniversalLaw.Side24.conditional_third_derivative_variance`), not here. | `Side24Analysis.referenceOddAndPinBlocks†` | -| `DensityComparison` | def_prop | `side24-proof` | [(1-epsilon)/(1+epsilon)]^(n/2) phi_((1-epsilon)C0) | the comparison (statement only); the cone-moment consequence `(1 ± epsilon)^m D0` and the scaling `a^m` of `h(A) = det(A)^2 1{A<0}` (not specified); nothing here encloses the coefficient; nonauthor review or acceptance. | `Side24Analysis.densityComparison†` | -| `IntegrandRatioBounds` | def_prop | `side24-proof` | ratio between1-13epsilon and | the bounds (statement only); that the integrand ratio of the source equals these power quotients (not specified); `13 ≤ 32`, `28 ≤ 32` and `32*10^(-108) < 10^(-106)` are kernel-checked on main (`UniversalLaw.Side24.ratio_constant_ordering`, `density_comparison_below_reported`), not here. | `Side24Analysis.integrandRatioBounds†` | -| `ReferencePinDensityProduct` | def_prop | `side24-proof` | p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3). | the identity (statement only); `det diag(3,1,…,1) = 3` is kernel-checked on main (`UniversalLaw.Side24.pin_determinant_and_joint_dimension`), not here. | `Side24Analysis.referencePinDensityProduct†` | +| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | Var(t)=15, Cov(t,G)=(-3,0,...,0), | the identities (statement only); `15 - 9 = 6` is kernel-checked on main (`UniversalLaw.Side24.conditional_third_derivative_variance`), not here. | `Side24Analysis.referenceOddAndPinBlocks` | +| `DensityComparison` | def_prop | `side24-proof` | [(1-epsilon)/(1+epsilon)]^(n/2) phi_((1-epsilon)C0) | the comparison (statement only); the cone-moment consequence `(1 ± epsilon)^m D0` and the scaling `a^m` of `h(A) = det(A)^2 1{A<0}` (not specified); nothing here encloses the coefficient; nonauthor review or acceptance. | `Side24Analysis.densityComparison` | +| `IntegrandRatioBounds` | def_prop | `side24-proof` | ratio between1-13epsilon and | the bounds (statement only); that the integrand ratio of the source equals these power quotients (not specified); `13 ≤ 32`, `28 ≤ 32` and `32*10^(-108) < 10^(-106)` are kernel-checked on main (`UniversalLaw.Side24.ratio_constant_ordering`, `density_comparison_below_reported`), not here. | `Side24Analysis.integrandRatioBounds` | +| `ReferencePinDensityProduct` | def_prop | `side24-proof` | p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3). | the identity (statement only); `det diag(3,1,…,1) = 3` is kernel-checked on main (`UniversalLaw.Side24.pin_determinant_and_joint_dimension`), not here. | `Side24Analysis.referencePinDensityProduct` | | `ReferenceCoefficientIdentity` | def_prop | `side24-proof` | This eliminates BOTH the angular and cone integrals for the reference expression. | the identity (statement only); that the left side is the parent formula (15.2) evaluated at the reference law (the angular integrand is constant by rotational invariance of `K_infty`, a fact not specified here); `6^2*2 = 3*24` is kernel-checked on main (`UniversalLaw.Side24.cube_root_simplification`), not here; nonauthor review or acceptance. | `Side24Elementary.referenceCoefficientIdentity` | | `ConeMomentInterface` | structure | `side24-proof` | Var(s)=5/3 and Var(x)=Var(y)=1. The negative-definite cone is s<-R, | that these laws are the conditional law `A \| V = 0` of the reference field (the source derives them; here they are hypotheses); the derivation of `Var(s) = 5/3` from the `(2/3) delta delta + …` formula (kernel-checked arithmetic on main, `UniversalLaw.Side24.trace_and_traceless_variances`). | — | | `NegDefCone` | def_prop | `side24-proof` | R=sqrt(x^2+y^2), and det A=s^2-R^2. | the source's reduction of the cone to `s < -R` (not specified; the cone is taken by its definition). | — | -| `ConeMomentD1Identity` | def_prop | `side24-proof` | D_1=E[A^2 1{A<0}]=4/3. | the identity (statement only); that `D_1` is the parent's cone moment `D_u` of (15.1) for the reference law; nothing here encloses the coefficient. | `Side24Interfaces.coneMomentD1Identity†` | +| `ConeMomentD1Identity` | def_prop | `side24-proof` | D_1=E[A^2 1{A<0}]=4/3. | the identity (statement only); that `D_1` is the parent's cone moment `D_u` of (15.1) for the reference law; nothing here encloses the coefficient. | `Side24Interfaces.coneMomentD1Identity` | | `ConeMomentD2Identity` | def_prop | `side24-proof` | D_2 = (1/2)[25/3-20/3+8-8sqrt(3/8)] | the identity (statement only); the Rayleigh integration route; that `D_2` is the parent's `D_u` for the reference law; nothing here encloses the coefficient; nonauthor review or acceptance. | — | | `ConeMomentD2Algebra` | def_prop | `side24-proof` | = 29/6-sqrt(6). | the identity (statement only); its rational part and `4 sqrt(3/8) = sqrt 6` are kernel-checked on main (`UniversalLaw.Side24.cone_moment_m2_algebra`), not here. | `Side24Elementary.coneMomentD2Algebra` | -| `TraceMoments` | def_prop | `side24-proof` | E s^2=5/3, E s^4=25/3 and | the identities (statement only); `3 (5/3)^2 = 25/3` and `1 + 5/3 = 8/3` are kernel-checked on main (`UniversalLaw.Side24.fourth_moment_of_trace`, `exponential_moment_of_trace`), not here. | `Side24Interfaces.traceMoments†` | -| `RayleighSquareLaw` | def_prop | `side24-proof` | The variable R^2 has density e^(-z/2)/2. | the law (statement only); measurability of the event (if the event is not measurable, `P` is its outer measure). | `Side24Interfaces.rayleighSquareLaw†` | +| `TraceMoments` | def_prop | `side24-proof` | E s^2=5/3, E s^4=25/3 and | the identities (statement only); `3 (5/3)^2 = 25/3` and `1 + 5/3 = 8/3` are kernel-checked on main (`UniversalLaw.Side24.fourth_moment_of_trace`, `exponential_moment_of_trace`), not here. | `Side24Interfaces.traceMoments` | +| `RayleighSquareLaw` | def_prop | `side24-proof` | The variable R^2 has density e^(-z/2)/2. | the law (statement only); measurability of the event (if the event is not measurable, `P` is its outer measure). | `Side24Interfaces.rayleighSquareLaw` | | `ElementaryConeIntegral` | def_prop | `side24-proof` | integral_0^a (a-z)^2 e^(-z/2) dz/2 = a^2-4a+8-8e^(-a/2). | the identity (statement only). | `Side24Elementary.elementaryConeIntegral` | | `PeriodicCoefficientInterface` | structure | `side24-proof` | Let c_d,24 be precisely its equation (15.2), using ordinary sphere area, ordered | existence of an object satisfying this interface; that `c24` is the finite-bar asymptotic constant (that reading is conditional on the parent's results, which are not touched); that the parent's "ordered maximum/saddle roles and the full height/gradient pin Jacobian" are encoded beyond the prefactor of (15.2). | — | | `CoefficientEnclosure` | def_prop | `side24-proof` | 0.07340691930603427103 < c_2,24 < 0.07340691930603427104, | the enclosures (statement only: nothing here encloses the coefficient); that an object satisfying the interface exists or is the coefficient of the source; the parent's results or the persistence interpretation; any finite-radius error estimate, unrestricted remainder or RN/24-jet statement; the arithmetic skeleton (endpoint adjacency, `d3 < d2`, endpoints in `(0, 1/10)`) is kernel-checked on main (31 `UniversalLaw.Side24` targets at origin/main f6deeba7), not here; nonauthor review or acceptance. | — | @@ -143,7 +142,7 @@ a theorem between string literals, an indented Prop definition, a proof written ### `Specifications/RN.lean` (28 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---|---| | `IsCriticalPoint` | def_prop | `lifetime-parent` | grad f(M)=grad f(S)=0. | differentiability of `g` at `x` (if `g` is not differentiable, `fderiv` is `0`). | — | | `HessianHasInertia` | def_prop | `lifetime-parent` | The index counts negative eigenvalues. | equivalence with an eigenvalue count for a non-symmetric form. | — | @@ -170,13 +169,13 @@ a theorem between string literals, an indented Prop definition, a proof written | `FourierCutoffPlanarSecondFactorialF4` | def_prop | `c6-fourier-cutoff` | E_Qr^W N(N-1) <= C r^3 log(1/r), | Theorem F (the cap `Psi_F` and its tail (F1)-(F2)); Theorem W for general `d` and `p` (F3); that `G_2` holds (the source takes it from the reviewed planar I5 composition); Proposition B; that the logarithm is sharp or removable; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. | — | | `PalmRouteFactorialMomentsQ` | def_prop | `c6-palm-route` | E_(Q_r^W)[ (N)_q ] <= C_q r^3. | the result or its proof (the cap `Psi` in `d` dimensions, the parametric Rouché tail, the marked Kac-Rice formula with Hölder insertion, the regimes R1-R4); the dependence on [C6L] and [DL] at their bytes; a lower bound for `q ≥ 3`; Corollary P; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. | — | | `PalmRouteSecondFactorialTheta` | def_prop | `c6-palm-route` | c r^3 <= E_(Q_r^W)[ N(N-1) ] <= C r^3, | the lower bounds' source ([EDL] (A4)) or the upper bounds' sources (Theorem Q, [DL] Corollary M_d); that the logarithm of the Fourier cutoff is removed as a reviewed fact; any statement about the positions of the pairs; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. | — | -| `RareClusterPoissonObstructionO` | def_prop | `c6-rare-cluster` | (a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3. | the result (Theorem O) or its proof; that `2a ≤ A` (derived in the source, not a hypothesis here); that H1 or H2 hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment matching or the independent-replica limit; the conditional-law liminf remark; that the upper half is the source's infimum bound verbatim: `∃ lam ≥ 0, dTV(Law N_r, Pois(lam)) ≤ A r^3` implies `inf_{lam ≥ 0} dTV ≤ A r^3` and is stronger unless the infimum is attained, and the existential form is used because the source's proof exhibits the witness `lam = 0` (the source: The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p); nonauthor review or acceptance. | `RNCluster.rareClusterPoissonObstructionO†` | +| `RareClusterPoissonObstructionO` | def_prop | `c6-rare-cluster` | (a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3. | the result (Theorem O) or its proof; that `2a ≤ A` (derived in the source, not a hypothesis here); that H1 or H2 hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment matching or the independent-replica limit; the conditional-law liminf remark; that the upper half is the source's infimum bound verbatim: `∃ lam ≥ 0, dTV(Law N_r, Pois(lam)) ≤ A r^3` implies `inf_{lam ≥ 0} dTV ≤ A r^3` and is stronger unless the infimum is attained, and the existential form is used because the source's proof exhibits the witness `lam = 0` (the source: The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p); nonauthor review or acceptance. | `RNCluster.rareClusterPoissonObstructionO` | | `RemoteCollisionSecondFactorialD` | def_prop | `remote-collision` | E_(Q_r^W) N(E)(N(E)-1) <= C r^5 \|E\|, | Theorem C (the ordered near-pair count) or its proof (Lemmas 1-5); Corollaries E and F; a torus-wide second factorial moment or `math.rn-region.witness-collision` in its full sense; pairs with a witness near the pins; equality of `N(E)` with the all-critical-point count `windowCountAll` (they agree only off the degenerate locus); any numerical constant; nonauthor review or acceptance. | — | | `WitnessCollisionNearPinPairsTarget` | def_prop | `d5-reconciliation` | A torus-wide second factorial moment `E_{Q_r^W} N(N−1)` for the window count, with pairs where at least | that this target holds, is attainable, or is implied by any reviewed source (the Palm route's Theorem Q is an author-side candidate whose `q = 2` case would imply it; that candidate is stated separately as `PalmRouteFactorialMomentsQ` and is not consumed here); that the lower obstruction `E N(N−1) ≥ 2c r³` is stated here; the optimal order; any status transition of the node. | — | ### `Specifications/P15.lean` (58 registered declarations) -| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; † = added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8…), no hosted receipt yet) | +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | proved by (proof packet, `proved`; 52 kernel-checked by hosted run 38121713840: PR head 910809d6, actual tested merge 5b198699… with base b2d70d77…; the 8 of P15Interpolation by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base 79731b96… (the head of PR #422); same trees as the heads; the 13 of Side24Analysis, Side24Interfaces and RNCluster by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6… (the head of PR #423), the merge having the same tree as PR head 48bfcbae; receipt theorem_count 73, artifact sha256 87309efa…) | |---|---|---|---|---|---| | `RealizedData` | structure | `p15-realized-covers` | Choose positive integers a_i,d_i and disjoint original-coordinate blocks X_i with | that the data satisfy the sources' side conditions (that is `IsRealizedFamily`). | — | | `IsRealizedFamily` | def_prop | `p15-realized-covers` | n_i=\|X_i\|=a_i d_i+1. | that any particular data record satisfies it. | — | @@ -239,11 +238,11 @@ a theorem between string literals, an indented Prop definition, a proof written ## Claims and what is not established -171 claim rows: 10 `landing_claim`, 33 `graph_node`, 2 `open_obligation`, 95 `proof_index_result`, 7 `existing_lean_package`, 12 `main_experiment_proof`, 12 `exact_certificate`. Claim-level formal progress: 33 `specified` (a `def … : Prop` stating the claim itself is registered), 138 `none`; no research claim is `proved` or `kernel-checked`. 197 statement entries point at the 150 declarations of this package (all `specified`). 155 statement entries carry the label `proved` and are theorems of other packages, listed as the arithmetic skeleton or the Prop-level proof of the claim they sit under and never as the claim: 31 are the 31 Side24 arithmetic targets of main's `formal/` at origin/main `f6deeba7`; 41 are the 28 P15 arithmetic targets of main's `formal/` at `91cffbb5` (main PR #358, not yet on main; kernel evidence the hosted workspace-landing run 38096827443), each listed on every P15 row whose layer 0 source it anchors to; 83 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (73 theorems, 72 distinct Props, each listed on every row that registers its Prop; the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 as kernel-checked and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, which the receipt records as checked_commit: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; same trees as the heads; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems marked † (`Side24Analysis`, `Side24Interfaces` and `RNCluster`) were added after PR head 12e072d6, the last head with a hosted receipt (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), and have no hosted receipt yet: the first hosted `lean-specification-proofs.yml` run on the round-4 branch is their first receipt). Exactly one Prop registered as stating a claim itself has such a theorem: `DemandOneRefutation` on rows `p15-price-boundary` and `math.p15-price-boundary`; their claim-level label stays `specified` (the register admits only none/specified at claim level, alignment of the Prop with the source paragraph is PENDING_INDEPENDENT_REVIEW, and a receipt is about the Prop only), as the rows' `not established` cells record. `disposition` is the verbatim LANDING_CLAIMS value or `—` (null). Nothing in these tables is a scientific status, a review verdict or an acceptance; alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. +171 claim rows: 10 `landing_claim`, 33 `graph_node`, 2 `open_obligation`, 95 `proof_index_result`, 7 `existing_lean_package`, 12 `main_experiment_proof`, 12 `exact_certificate`. Claim-level formal progress: 33 `specified` (a `def … : Prop` stating the claim itself is registered), 138 `none`; no research claim is `proved` or `kernel-checked`. 197 statement entries point at the 150 declarations of this package (all `specified`). 155 statement entries carry the label `proved` and are theorems of other packages, listed as the arithmetic skeleton or the Prop-level proof of the claim they sit under and never as the claim: 31 are the 31 Side24 arithmetic targets of main's `formal/` at origin/main `f6deeba7`; 41 are the 28 P15 arithmetic targets of main's `formal/` at `91cffbb5` (main PR #358, not yet on main; kernel evidence the hosted workspace-landing run 38096827443), each listed on every P15 row whose layer 0 source it anchors to; 83 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (73 theorems, 72 distinct Props, each listed on every row that registers its Prop; the hosted `lean-specification-proofs.yml` run 38121713840 receipted the 52 theorems of PR head 910809d6 as kernel-checked and run 38116277597 the first 38; a pull_request run checks out the synthetic merge of the PR head into the base, which the receipt records as checked_commit: run 38121713840 PR head 910809d6, actual tested merge 5b198699308e0585904c21d12bf540cf2475d2f0 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; run 38116277597 PR head 00d2ac6e, actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with the same base; run 38122386459 corroborates on PR head d9b8dd33, actual tested merge 866136cca66b46e6dbd3186b47c9340f7e93b4f6; same trees as the heads; the 8 theorems of `P15Interpolation`, added after PR head 79731b96 (run 38124262242, actual tested merge 318ff194eecc42c39a6b27c1d7bf03f30d6a34d9 with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2, receipt theorem_count 52), are kernel-checked by the hosted run 38149888699 on PR head 0dcb0016 (actual tested merge 4363b6928c62da242952631c6b0357d096c3b252 with base 79731b961fac555f33be3cd90e379b246a5a80e3, the head of PR #422; receipt theorem_count 60, leanchecker PASS) and were first receipted by run 38149524759 on PR head 88c21058 (actual tested merge 3306233bbdeadb15d9b4b07b94cc52a80ed8caeb with the same base); same trees as the heads; the 13 theorems of `Side24Analysis`, `Side24Interfaces` and `RNCluster`, added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60, corroborating the 60 theorems above on that head; artifact lean-specification-proofs-38150363063-1, sha256 bf0e0515e2035103b07d894e2a73eeb5b542885f6da16b98c228877a5e0c1690), are kernel-checked by the hosted run 38180705952 on PR head 48bfcbae (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, the head of PR #423; receipt theorem_count 73, leanchecker PASS; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd) and were first receipted by that run, which receipts all 73 theorems of that head; same tree as the head). Exactly one Prop registered as stating a claim itself has such a theorem: `DemandOneRefutation` on rows `p15-price-boundary` and `math.p15-price-boundary`; their claim-level label stays `specified` (the register admits only none/specified at claim level, alignment of the Prop with the source paragraph is PENDING_INDEPENDENT_REVIEW, and a receipt is about the Prop only), as the rows' `not established` cells record. `disposition` is the verbatim LANDING_CLAIMS value or `—` (null). Nothing in these tables is a scientific status, a review verdict or an acceptance; alignment of every statement is `PENDING_INDEPENDENT_REVIEW`. | claim_id | kind | disposition | formal_progress | statements | not established | next step | |---|---|---|---|---|---|---| -| `side24-coefficient` | landing_claim | REVIEWED_SCOPED | `specified` | 28 specified: `ThetaSummable`, `IsUnitTuple`, `DerivativePairingBound`, `DerivativeBoundAtLeastOne`, `DerivativeBoundAtZero`, `LatticeImageSumBound`, `SuccessiveTermRatio`, `ExponentialBounds`, `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound`, `ReferenceOddAndPinBlocks`, `DensityComparison`, `IntegrandRatioBounds`, `ReferencePinDensityProduct`, `ReferenceCoefficientIdentity`, `ConeMomentInterface`, `NegDefCone`, `ConeMomentD1Identity`, `ConeMomentD2Identity`, `ConeMomentD2Algebra`, `TraceMoments`, `RayleighSquareLaw`, `ElementaryConeIntegral`, `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison`; 27 proved (main, origin/main f6deeba7): `pairing_sum_six`, `pairing_sum_le_six`, `pairing_terms_six`, `sixth_moment_double_factorial`, `lattice_shell_constant`, `geometric_ratio_constants`, `period_exponent`, `image_constant_value`, `exp_taylor_partial_sum_gt_ten`, `covariance_relative_bound`, `odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`, `exponent_bounds`, `exponent_values`, `ratio_constant_ordering`, `density_comparison_below_reported`, `conditional_third_derivative_variance`, `transverse_covariance_entries_thirds`, `cone_moment_m1_thirds`, `trace_and_traceless_variances`, `fourth_moment_of_trace`, `exponential_moment_of_trace`, `cone_moment_m2_algebra`, `cube_root_simplification`, `pin_determinant_and_joint_dimension`, `hessian_block_eigenvalues`; 17 proved (proof packet): `Side24Elementary.exponentialBounds`, `Side24Elementary.successiveTermRatio`, `Side24Elementary.coneMomentD2Algebra`, `Side24Elementary.referenceCoefficientIdentity`, `Side24Elementary.elementaryConeIntegral`, `Side24Analysis.thetaSummable†`, `Side24Analysis.derivativePairingBound†`, `Side24Analysis.derivativeBoundAtLeastOne†`, `Side24Analysis.derivativeBoundAtZero†`, `Side24Analysis.referenceOddAndPinBlocks†`, `Side24Analysis.referencePinDensityProduct†`, `Side24Analysis.integrandRatioBounds†`, `Side24Analysis.latticeImageSumBound†`, `Side24Analysis.densityComparison†`, `Side24Interfaces.coneMomentD1Identity†`, `Side24Interfaces.traceMoments†`, `Side24Interfaces.rayleighSquareLaw†` | Nothing here encloses the coefficient. CoefficientEnclosure and ReferenceComparison are statements for any object satisfying PeriodicCoefficientInterface; the interface takes the cone moment D_u of (15.1) as a GIVEN field, so c_{d,24} is not uniquely determined in Lean and no inhabitant is constructed. The 27 main theorems listed as `proved` are the integer/rational arithmetic skeleton of PROOF.md sections 1-4 (coefficient sums, shell constants, Taylor partial sum, block minors, cone-moment algebra), not the Gaussian conditioning, the Rayleigh cone integration, the Stirling log-gamma step (section 5, not specified), the interval arithmetic of coefficient.py, or the enclosure itself. Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, LatticeImageSumBound, SuccessiveTermRatio, ExponentialBounds, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (17 entries below, label `proved`; 5 of them (Side24Elementary.successiveTermRatio, Side24Elementary.exponentialBounds, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.coneMomentD2Algebra, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.latticeImageSumBound, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.densityComparison, Side24Analysis.integrandRatioBounds, Side24Analysis.referencePinDensityProduct, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them). What those theorems prove, for the formal objects of the statement module only: elementary real inequalities (ExponentialBounds), the term ratio (SuccessiveTermRatio), the D2 cone-moment algebra (ConeMomentD2Algebra), the reference-coefficient identity as stated (ReferenceCoefficientIdentity; its left side is not identified with the parent formula (15.2) at the reference law: rotational invariance of K_infty is not specified) and the elementary cone integral (ElementaryConeIntegral); summability of the formal theta family n \|-> exp(-\|z + 24 n\|^2/2) over Fin d -> Z for every d (ThetaSummable: the statement entry's does_not_claim "that the family is summable (true, but not shown here)" describes the statement packet, which is not edited; the summability is now shown in the proof packet, for the formal family, by comparison with a product of one-dimensional Gaussian sums); the Section 2 derivative bounds for q <= 6 as the Props state them (DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero: a Hermite-type recursion on the operator norms of iteratedFDeriv, no closed form of the Gaussian derivatives, the contraction at a unit tuple bounded by the operator norm; main's pairing_sum_six, pairing_sum_le_six and sixth_moment_double_factorial are neither cited nor relied on (the 1 + 15 + 45 + 15 = 76 arithmetic is recomputed inline by norm_num); nothing for q > 6); the lattice image sum (LatticeImageSumBound: summability and the bound 1458 e^(-288), via 432 e^(-288); the source's shell count 729 sum_j j^9 e^(-288 j^2) is not reproduced and not specified); the five reference block identities and the pin density product for the formal matrices (ReferenceOddAndPinBlocks, ReferencePinDensityProduct: derivative contractions of Kinf at 0 in an orthonormal frame; none of these matrices is shown to be the covariance of any random vector or field); pure real arithmetic in epsilon and the exponents a, b for d = 2, 3 (IntegrandRatioBounds; that the source's integrand ratio equals these power quotients is not specified); Loewner determinant and inverse monotonicity for positive-definite real matrices (DensityComparison; the cone-moment consequence (1 +/- epsilon)^m D0 and the scaling a^m are not specified); and, over ConeMomentInterface, the m = 1 cone moment 4/3, the trace moments 5/3, 25/3, sqrt(3/8) and the distribution function P(R^2 <= a) = integral_0^a e^(-z/2)/2 dz for a >= 0 (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw: proved for every inhabitant from the law fields A_law, sxy_law and isProb alone; no inhabitant is constructed, the laws are not derived as the conditional law A \| V = 0 of the reference field, and the Rayleigh law is proved as a distribution function, not as a density). None of them is the Gaussian conditioning, ConeMomentD2Identity, ImageBound, the covariance ordering or any enclosure of the coefficient, and the claim-level label stays `specified` (claim-level vocabulary none/specified; a theorem of a Prop is evidence about that Prop, for the formal objects, not acceptance of the informal theorem; alignment PENDING_INDEPENDENT_REVIEW). Theorem-shaped Props of this row still `specified` with no theorem: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison; the parametrised predicates IsUnitTuple, NegDefCone are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited. | Math- Mathlib lane: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); the 12 theorems of this row added after PR head 12e072d6 (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) await the next hosted lean-specification-proofs.yml run, their first receipt; the other theorems of this row have a hosted receipt: the 5 of PR head 00d2ac6e (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); each of those receipted theorems corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, ReferenceOddAndPinBlocks, IntegrandRatioBounds, ReferencePinDensityProduct, ConeMomentD1Identity, TraceMoments PROVABLE_NOW and LatticeImageSumBound, DensityComparison, RayleighSquareLaw PROVABLE_WITH_WORK: proved in round 4 (see entries); ReferenceCovarianceLowerBound PROVABLE_WITH_WORK (weeks); ConeMomentD2Identity PROVABLE_WITH_WORK (weeks); ImageBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum; the global version is inapplicable to lattice translates; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), iteratedFDeriv_tsum_apply (SmoothSeries.lean)); CovarianceOrdering BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: local term-wise iterated differentiation of the lattice sum, the d-dimensional Gaussian tail bound and higher-order symmetry of iteratedFDeriv; display (3) needs only the crude norm_iteratedFDeriv_comp_le, with a factor about 8 to spare; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), hasFDerivAt_tsum_of_isPreconnected (SmoothSeries.lean)); EntrywiseCovarianceBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum for the term-wise iterated differentiation of the lattice sum (the same blocker as CovarianceOrdering; the finding records no further shape observation); present but insufficient: norm_iteratedFDeriv_comp_le (Bounds.lean), iteratedFDeriv_tsum (SmoothSeries.lean)); CoefficientEnclosure BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); ReferenceComparison BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); IsUnitTuple, NegDefCone PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. The enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md) and an interface that determines c_{d,24} (the cone moment is a given field; ConeMomentD2Identity for the reference law under ConeMomentInterface is the open Rayleigh-integration step). Proof packet README: for the Props proved in round 4 (ConeMomentD1Identity, DensityComparison, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, DerivativePairingBound, IntegrandRatioBounds, LatticeImageSumBound, RayleighSquareLaw, ReferenceOddAndPinBlocks, ReferencePinDensityProduct, ThetaSummable, TraceMoments) the informal reasons the README gave were found not load-bearing by the triage and are superseded by their theorems; the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure and ReferenceComparison stand (for the last two the triage confirms the README's reason as the precise missing ingredient) (the proof-packet maintainers' regeneration, not this register's). Independent reviewer: alignment verdicts on the 28 Side24 statements, in particular of the 17 Props of this row that have a theorem of exactly their type, with their anchors (ThetaSummable with the source's R^d, 24 Z^d and K_24; the derivative Props with Section 2; the interface Props with the source's A ~ N(0, 8/3), s ~ N(0, 5/3), x, y ~ N(0, 1)). | +| `side24-coefficient` | landing_claim | REVIEWED_SCOPED | `specified` | 28 specified: `ThetaSummable`, `IsUnitTuple`, `DerivativePairingBound`, `DerivativeBoundAtLeastOne`, `DerivativeBoundAtZero`, `LatticeImageSumBound`, `SuccessiveTermRatio`, `ExponentialBounds`, `ImageBound`, `CovarianceOrdering`, `EntrywiseCovarianceBound`, `ReferenceCovarianceLowerBound`, `ReferenceOddAndPinBlocks`, `DensityComparison`, `IntegrandRatioBounds`, `ReferencePinDensityProduct`, `ReferenceCoefficientIdentity`, `ConeMomentInterface`, `NegDefCone`, `ConeMomentD1Identity`, `ConeMomentD2Identity`, `ConeMomentD2Algebra`, `TraceMoments`, `RayleighSquareLaw`, `ElementaryConeIntegral`, `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison`; 27 proved (main, origin/main f6deeba7): `pairing_sum_six`, `pairing_sum_le_six`, `pairing_terms_six`, `sixth_moment_double_factorial`, `lattice_shell_constant`, `geometric_ratio_constants`, `period_exponent`, `image_constant_value`, `exp_taylor_partial_sum_gt_ten`, `covariance_relative_bound`, `odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`, `exponent_bounds`, `exponent_values`, `ratio_constant_ordering`, `density_comparison_below_reported`, `conditional_third_derivative_variance`, `transverse_covariance_entries_thirds`, `cone_moment_m1_thirds`, `trace_and_traceless_variances`, `fourth_moment_of_trace`, `exponential_moment_of_trace`, `cone_moment_m2_algebra`, `cube_root_simplification`, `pin_determinant_and_joint_dimension`, `hessian_block_eigenvalues`; 17 proved (proof packet): `Side24Elementary.exponentialBounds`, `Side24Elementary.successiveTermRatio`, `Side24Elementary.coneMomentD2Algebra`, `Side24Elementary.referenceCoefficientIdentity`, `Side24Elementary.elementaryConeIntegral`, `Side24Analysis.thetaSummable`, `Side24Analysis.derivativePairingBound`, `Side24Analysis.derivativeBoundAtLeastOne`, `Side24Analysis.derivativeBoundAtZero`, `Side24Analysis.referenceOddAndPinBlocks`, `Side24Analysis.referencePinDensityProduct`, `Side24Analysis.integrandRatioBounds`, `Side24Analysis.latticeImageSumBound`, `Side24Analysis.densityComparison`, `Side24Interfaces.coneMomentD1Identity`, `Side24Interfaces.traceMoments`, `Side24Interfaces.rayleighSquareLaw` | Nothing here encloses the coefficient. CoefficientEnclosure and ReferenceComparison are statements for any object satisfying PeriodicCoefficientInterface; the interface takes the cone moment D_u of (15.1) as a GIVEN field, so c_{d,24} is not uniquely determined in Lean and no inhabitant is constructed. The 27 main theorems listed as `proved` are the integer/rational arithmetic skeleton of PROOF.md sections 1-4 (coefficient sums, shell constants, Taylor partial sum, block minors, cone-moment algebra), not the Gaussian conditioning, the Rayleigh cone integration, the Stirling log-gamma step (section 5, not specified), the interval arithmetic of coefficient.py, or the enclosure itself. Props of this row with a theorem of exactly their type in the proof packet specifications/lean_statement_proofs_20261011: ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, LatticeImageSumBound, SuccessiveTermRatio, ExponentialBounds, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (17 entries below, label `proved`; 5 of them (Side24Elementary.successiveTermRatio, Side24Elementary.exponentialBounds, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.coneMomentD2Algebra, Side24Elementary.elementaryConeIntegral) are among the 38 theorems of PR head 00d2ac6e kernel-checked by the hosted run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2; job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound} for each, receipt formalization_status kernel-checked, worktree CLEAN; artifact lean-specification-proofs-38116277597-1); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.latticeImageSumBound, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.densityComparison, Side24Analysis.integrandRatioBounds, Side24Analysis.referencePinDensityProduct, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 and kernel-checked by the hosted run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 73, worktree CLEAN; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd); they were first receipted by that run (PR #434, event pull_request); base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434, and the merge has the same tree as its head). What those theorems prove, for the formal objects of the statement module only: elementary real inequalities (ExponentialBounds), the term ratio (SuccessiveTermRatio), the D2 cone-moment algebra (ConeMomentD2Algebra), the reference-coefficient identity as stated (ReferenceCoefficientIdentity; its left side is not identified with the parent formula (15.2) at the reference law: rotational invariance of K_infty is not specified) and the elementary cone integral (ElementaryConeIntegral); summability of the formal theta family n \|-> exp(-\|z + 24 n\|^2/2) over Fin d -> Z for every d (ThetaSummable: the statement entry's does_not_claim "that the family is summable (true, but not shown here)" describes the statement packet, which is not edited; the summability is now shown in the proof packet, for the formal family, by comparison with a product of one-dimensional Gaussian sums); the Section 2 derivative bounds for q <= 6 as the Props state them (DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero: a Hermite-type recursion on the operator norms of iteratedFDeriv, no closed form of the Gaussian derivatives, the contraction at a unit tuple bounded by the operator norm; main's pairing_sum_six, pairing_sum_le_six and sixth_moment_double_factorial are neither cited nor relied on (the 1 + 15 + 45 + 15 = 76 arithmetic is recomputed inline by norm_num); nothing for q > 6); the lattice image sum (LatticeImageSumBound: summability and the bound 1458 e^(-288), via 432 e^(-288); the source's shell count 729 sum_j j^9 e^(-288 j^2) is not reproduced and not specified); the five reference block identities and the pin density product for the formal matrices (ReferenceOddAndPinBlocks, ReferencePinDensityProduct: derivative contractions of Kinf at 0 in an orthonormal frame; none of these matrices is shown to be the covariance of any random vector or field); pure real arithmetic in epsilon and the exponents a, b for d = 2, 3 (IntegrandRatioBounds; that the source's integrand ratio equals these power quotients is not specified); Loewner determinant and inverse monotonicity for positive-definite real matrices (DensityComparison; the cone-moment consequence (1 +/- epsilon)^m D0 and the scaling a^m are not specified); and, over ConeMomentInterface, the m = 1 cone moment 4/3, the trace moments 5/3, 25/3, sqrt(3/8) and the distribution function P(R^2 <= a) = integral_0^a e^(-z/2)/2 dz for a >= 0 (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw: proved for every inhabitant from the law fields A_law, sxy_law and isProb alone; no inhabitant is constructed, the laws are not derived as the conditional law A \| V = 0 of the reference field, and the Rayleigh law is proved as a distribution function, not as a density). None of them is the Gaussian conditioning, ConeMomentD2Identity, ImageBound, the covariance ordering or any enclosure of the coefficient, and the claim-level label stays `specified` (claim-level vocabulary none/specified; a theorem of a Prop is evidence about that Prop, for the formal objects, not acceptance of the informal theorem; alignment PENDING_INDEPENDENT_REVIEW). Theorem-shaped Props of this row still `specified` with no theorem: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison; the parametrised predicates IsUnitTuple, NegDefCone are the vocabulary of the other Props, not closed statements (a theorem of their type would assert them for every argument, which is false), and are not targets. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited. | Math- Mathlib lane: ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); every theorem of this row has a hosted receipt: the 12 added after PR head 12e072d6 (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) first by run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (receipt theorem_count 73); the 5 of PR head 00d2ac6e (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) by run 38116277597 (actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); each of those 5 theorems corroborated on PR head 12e072d6 by run 38150363063 (actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3; receipt theorem_count 60); every theorem of this row receipted on PR head 48bfcbae by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): ThetaSummable, DerivativePairingBound, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, ReferenceOddAndPinBlocks, IntegrandRatioBounds, ReferencePinDensityProduct, ConeMomentD1Identity, TraceMoments PROVABLE_NOW and LatticeImageSumBound, DensityComparison, RayleighSquareLaw PROVABLE_WITH_WORK: proved in round 4 (see entries); ReferenceCovarianceLowerBound PROVABLE_WITH_WORK (weeks); ConeMomentD2Identity PROVABLE_WITH_WORK (weeks); ImageBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum; the global version is inapplicable to lattice translates; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), iteratedFDeriv_tsum_apply (SmoothSeries.lean)); CovarianceOrdering BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: local term-wise iterated differentiation of the lattice sum and higher-order symmetry of iteratedFDeriv (the 2026-10-11 triage table, kept as the historical record on the Math- reading branch claude/round4-triage-table-20261011, commit 43a975ec, file reviews/round4_triage_20261011/triage_table.json, also listed the d-dimensional Gaussian tail bound, since discharged by latticeImageSumBound, and the derivative constants, since supplied by the derivative theorems); display (3) needs only the crude norm_iteratedFDeriv_comp_le, with a factor about 8 to spare; present but insufficient: iteratedFDeriv_tsum (SmoothSeries.lean), hasFDerivAt_tsum_of_isPreconnected (SmoothSeries.lean)); EntrywiseCovarianceBound BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: no local (open-set) iteratedFDeriv_tsum for the term-wise iterated differentiation of the lattice sum (the same blocker as CovarianceOrdering; the finding records no further shape observation); present but insufficient: norm_iteratedFDeriv_comp_le (Bounds.lean), iteratedFDeriv_tsum (SmoothSeries.lean)); CoefficientEnclosure BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); ReferenceComparison BLOCKED_INTERFACE (probe-level: universal closure implies IsEmpty PeriodicCoefficientInterface); IsUnitTuple, NegDefCone PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. The enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md) and an interface that determines c_{d,24} (the cone moment is a given field; ConeMomentD2Identity for the reference law under ConeMomentInterface is the open Rayleigh-integration step). Proof packet README: for the Props proved in round 4 (ConeMomentD1Identity, DensityComparison, DerivativeBoundAtLeastOne, DerivativeBoundAtZero, DerivativePairingBound, IntegrandRatioBounds, LatticeImageSumBound, RayleighSquareLaw, ReferenceOddAndPinBlocks, ReferencePinDensityProduct, ThetaSummable, TraceMoments) the informal reasons the README gave were found not load-bearing by the triage and are superseded by their theorems; the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, ReferenceCovarianceLowerBound, ConeMomentD2Identity, CoefficientEnclosure and ReferenceComparison stand (for the last two the triage confirms the README's reason as the precise missing ingredient) (the proof-packet maintainers' regeneration, not this register's). Independent reviewer: alignment verdicts on the 28 Side24 statements, in particular of the 17 Props of this row that have a theorem of exactly their type, with their anchors (ThetaSummable with the source's R^d, 24 Z^d and K_24; the derivative Props with Section 2; the interface Props with the source's A ~ N(0, 8/3), s ~ N(0, 5/3), x, y ~ N(0, 1)). | | `lifetime-remainder` | landing_claim | REVIEWED_SCOPED | `specified` | 4 specified: `RemainderBoundedR1`, `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `IsLifetimeDensityVersion` | RemainderBoundedR1 states the shape of (R1) for versions of the densities under the interface structures; the result is not proved. Not stated: (R19)-(R20), the observation transform and determinant lemmas R3.1/R3.2, the cap-loss bound, the parent interfaces D1-A-E it imports, the marked-cylinder-cap support. The coefficient c enters only through IsUnrestrictedLeadingCoefficient (c > 0 and the leading-term role), not through the representation (15.2). No numerical C or ell_*, no second coefficient, no convergence of the remainder, no compact-window O(ell^(2/3)) difference without cutoffs. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (R19) (cumulative form) as a further Prop; transcribe the parent interfaces consumed (section 8 comparison, section 9 as replaced by v1.1, section 10, (14.1), (15.2)) as explicit hypotheses; independent reviewer: alignment verdict on RemainderBoundedR1 against (R1), including the "versions" clause and the exclusion of the essential global maximum. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemainderBoundedR1 BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean), ncard_le_ncard (Card.lean)); IsLifetimeDensityVersion, IsUnrestrictedLeadingCoefficient, IsUnrestrictedPair PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. | | `rn-count-interface` | landing_claim | HOLD_WITH_DOMAIN | `none` | 5 specified: `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5`; 5 proved (proof packet): `RNProbability.conditionalMeanIdentityN3`, `RNProbability.holderCountBoundN1`, `RNProbability.holderConsequenceN2`, `RNProbability.uniformCounterexampleN4`, `RNProbability.exponentialTailLayerCakeN5` | No Prop states Theorem N as a whole, so the claim-level label stays none: (N1)-(N5) (Hoelder bound, its power-law consequence, the conditional-mean identity, the uniform counterexample, the exponential-tail layer cake) are stated as abstract probability-space Props and each has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (5 entries below, label `proved`; the hosted lean-specification-proofs.yml run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked them (receipt formalization_status kernel-checked, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound})); (N6) (the full-pin weighted Kac-Rice expression for E_Q^W N_B) and (N7) (the enclosure-to-rate implication under Z_r >= z_* r^2) are NOT stated, and they are the Gaussian numerator interface that the disposition reason calls "a target rather than a discharged global RN theorem". The five theorems prove the Props for the abstract probability spaces and counts they quantify over: nothing about the Gaussian model is stated or proved; no RN rate, no 24-jet enclosure, no ENV-RESCOV discharge; D3-LEMMA-RN-UNIF stays not closed. Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | State (N7) as an implication with the Kac-Rice numerator as an interface field (the integral of (N6) as a given quantity) and Z_r >= z_* r^2 as a hypothesis; then the claim would be `specified`. The hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2) kernel-checked the five proved entries. Independent reviewer: alignment of the five Props with (N1)-(N5). | | `rn-fixed-remote-window` | landing_claim | HOLD_WITH_DOMAIN | `specified` | 5 specified: `PinnedLawInterface`, `IsCriticalPoint`, `HessianHasInertia`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | FixedRemoteMeanMeasureA2 and FixedRemoteCountBoundA3 state the shapes of displays (2) and (3) for any object satisfying PinnedLawInterface (the regression law Q_r, the weight W_r, the normalizer and the contact kernel Lambda_j are interface fields, gated on 0 < r < L); no inhabitant of the interface is constructed and the kernel is not identified with formula (13). Not stated: the observation transform U_r and target v_r, the coupling (5), the weight expansion (10)-(11), the Kac-Rice formula (12), the kernel formula (13), the unnormalized numerator bound (14), and the multi-witness Section 6 displays (15)-(16). Nothing is proved; no numerical C or r_*; no shrinking exclusion rho -> 0, no collisions, no global RN complement (the disposition reason). Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Section 6 ((15)-(16), separated m-tuples) as Props; record an inhabitant construction of PinnedLawInterface as a proof-lane target (Dirac mass at one admissible smooth path); independent reviewer: alignment of A2/A3 with displays (2)-(3), in particular the uniformity in b, k, R, j, E and the fixed rho. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point); HessianHasInertia, IsCriticalPoint PREDICATE_VOCABULARY: vocabulary of the other Props, not a target. | @@ -259,7 +258,7 @@ a theorem between string literals, an indented Prop definition, a proof written | `math.d1-component.marked-cylinder-cap` | graph_node | — | `none` | — | No Prop of this package states the deterministic cap theorem (unique transverse ridge, closed cap through S separating M from every older point, elder death partner of M is S under conditions (1)). The pinned bytes exist (sources/marked-cylinder-cap). Kernel evidence exists outside this package: Math- frontiers/cap_first_exit_lean_20261002 proves the topological global step and the analytic chain L1-L13, L16-L20 (torus_capHyp_blocks, torus_elder_death_level_blocks), but its 152 declarations are not registered here with anchors, its alignment review is NOT_CLAIMED, and it does not formalize that the field on G_r satisfies SP section 2, L14/L15, L23 or the persistence-pairing identification. The probabilistic corollary (2)-(3) and sections 6-8 are not consumed by D1 and not stated. | Register the CapFirstExit declarations here with their ALIGNMENT.md anchors against the pinned CAP bytes (statements of kind theorem, label proved, kernel_evidence cap-first-exit-lean.yml), or state the deterministic theorem as a def : Prop in Lifetime.lean so that the Math- lane's delta review has a target; independent (non-Claude) reviewer for the v1.5-v2.6 delta. | | `math.d1-component.reconciliation-record` | graph_node | — | `none` | — | A review/reading-rule record, not a mathematical statement: nothing to state in Lean beyond what its components say. The embedding-radius condition r < L/(4 sqrt 2) and the W1 wording are not transcribed into any hypothesis of the parent Props. Its ACCEPT verdicts are not re-derived; the OpenAI author-provider check it records is non-discharge by its own terms. | Add the embedding radius as an explicit hypothesis to ParentUniformSelectionA / PairLawInterface once the reading rule is pinned; no proof target. | | `math.lifetime-remainder` | graph_node | — | `specified` | 1 specified: `RemainderBoundedR1` | See row lifetime-remainder: (R1) is stated, not proved; the imports of parent interfaces D1-A-E are not transcribed; Theorem A's cubic bound (1.1) is not consumed by R and is not a hypothesis here. The GRAPH classification and review disposition are source identity only. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row lifetime-remainder. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemainderBoundedR1 BLOCKED_MISSING_MATHLIB (weeks; missing at Mathlib d13f23b7: Kac-Rice / Rice formula, Bulinskaya lemma, critical-point counting measurability; present but insufficient: ae_eq_of_forall_setIntegral_eq_of_sigmaFinite (AEEqOfIntegral.lean), ncard_le_ncard (Card.lean)). | -| `math.side24-coefficient` | graph_node | — | `specified` | 3 specified: `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison` | See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the "evaluates_eq_15_2" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. Of the three statements of this node, none has a theorem; stated, not proved: CoefficientEnclosure, ReferenceComparison. The 27 arithmetic theorems of main and the 17 proof-packet theorems of the landing row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral, Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw; none of them the enclosure; 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them) are listed on the landing row, never here and never as the node; the interface theorems among them (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw) hold for every inhabitant of ConeMomentInterface from its law fields alone and construct nothing. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited. | As for row side24-coefficient. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): CoefficientEnclosure BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface elaborates, by doubling coneMoment and c24, so no theorem of its type can exist unless the interface is empty); ReferenceComparison BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, ReferenceComparison I) -> IsEmpty PeriodicCoefficientInterface elaborates likewise). | +| `math.side24-coefficient` | graph_node | — | `specified` | 3 specified: `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison` | See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the "evaluates_eq_15_2" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. Of the three statements of this node, none has a theorem; stated, not proved: CoefficientEnclosure, ReferenceComparison. The 27 arithmetic theorems of main and the 17 proof-packet theorems of the landing row (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral, Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw; none of them the enclosure; 5 of them (Side24Elementary.exponentialBounds, Side24Elementary.successiveTermRatio, Side24Elementary.coneMomentD2Algebra, Side24Elementary.referenceCoefficientIdentity, Side24Elementary.elementaryConeIntegral) kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2); 12 of them (Side24Analysis.thetaSummable, Side24Analysis.derivativePairingBound, Side24Analysis.derivativeBoundAtLeastOne, Side24Analysis.derivativeBoundAtZero, Side24Analysis.referenceOddAndPinBlocks, Side24Analysis.referencePinDensityProduct, Side24Analysis.integrandRatioBounds, Side24Analysis.latticeImageSumBound, Side24Analysis.densityComparison, Side24Interfaces.coneMomentD1Identity, Side24Interfaces.traceMoments, Side24Interfaces.rayleighSquareLaw) were added after PR head 12e072d6 and kernel-checked by the hosted run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 73, worktree CLEAN; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd); they were first receipted by that run (PR #434, event pull_request); base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434, and the merge has the same tree as its head) are listed on the landing row, never here and never as the node; the interface theorems among them (ConeMomentD1Identity, TraceMoments, RayleighSquareLaw) hold for every inhabitant of ConeMomentInterface from its law fields alone and construct nothing. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. The round-4 triage elaborated this at probe level (a scratch file, not a receipt and not in any packet): (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface, and likewise for ReferenceComparison, by doubling coneMoment and c24; the label stays `specified` and no statement is edited. | As for row side24-coefficient. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): CoefficientEnclosure BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, CoefficientEnclosure I) -> IsEmpty PeriodicCoefficientInterface elaborates, by doubling coneMoment and c24, so no theorem of its type can exist unless the interface is empty); ReferenceComparison BLOCKED_INTERFACE (probe-level, not a receipt: (forall I, ReferenceComparison I) -> IsEmpty PeriodicCoefficientInterface elaborates likewise). | | `math.rn-count-interface` | graph_node | — | `none` | 5 specified: `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5`; 5 proved (proof packet): `RNProbability.conditionalMeanIdentityN3`, `RNProbability.holderCountBoundN1`, `RNProbability.holderConsequenceN2`, `RNProbability.uniformCounterexampleN4`, `RNProbability.exponentialTailLayerCakeN5` | See row rn-count-interface: (N1)-(N5) stated and each proved as a theorem of the proof packet (5 entries below, `proved`; kernel-checked by the hosted run 38116277597 (PR head 00d2ac6e; actual tested merge c4c66bb8cb26f0cd9bc03d52bb25f1b3925b272c with base b2d70d77876a10c365f5db5d3b6cafcf180af4e2)), (N6)-(N7) not stated, so no Prop states the interface as a whole and the node stays none; nothing about the Gaussian model is proved. Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | As for row rn-count-interface. | | `math.rn-fixed-remote-window` | graph_node | — | `specified` | 3 specified: `PinnedLawInterface`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | See row rn-fixed-remote-window: displays (2)-(3) stated over PinnedLawInterface, not proved; Section 6 not stated; no inhabitant of the interface. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point). | | `math.rn-region.fixed-remote` | graph_node | — | `specified` | 2 specified: `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | The region's mathematical content is the fixed-remote Theorem A at exactly this scope, stated by A2/A3 and not proved. The coverage classification itself (COVERED_BY_CANDIDATE, coverage_source) is graph bookkeeping and is not a Lean statement. No all-height bound on the remote region, no rho -> 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window; nothing region-specific to add. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FixedRemoteCountBoundA3 BLOCKED_INTERFACE (a Dirac inhabitant with one extra critical point on a null remote set fails both displays; the interface transcribes neither the Kac-Rice representation (12) nor any regularity of Q_r^W); FixedRemoteMeanMeasureA2 BLOCKED_INTERFACE (the kernel clause is a conclusion about the interface field I.Lambda, which no field constrains; with Lambda := 0 the clause 0 < cLambda <= I.Lambda fails at any remote point). | @@ -297,7 +296,7 @@ a theorem between string literals, an indented Prop definition, a proof written | `CL-D5-DIMENSION-LIFT-20260929-v1` | proof_index_result | — | `specified` | 4 specified: `DimensionLiftPuncturedPinBallP`, `DimensionLiftCompactCollarC`, `DimensionLiftIntermediateShellsI`, `DimensionLiftGlobalWindowG` | The four theorem shapes (1.2), (1.3), (1.5)-(1.6), (1.7) are stated over PinnedLawInterface for fixed d >= 2 and not proved. Not stated: display (1.4), Corollaries F_d (1.8) and M_d (1.9), the five devices D1-D5, the consumed sources ([LP] sections 2-5, (5.5), the planar templates, [RM], [RC], [EDL], [LOCAL]). No all-height intermediate or global O(r^3) count; no torus-wide second factorial moment; no elder selection; no uniformity in k, L, d; no numerical C, c, r_*, s_0; no recovery of historical RN/JETMOD/24-jet carriers. The reviews (Math-#141 OpenAI, source-exposed) are source identity; the 2026-10-05 erratum may amend these statements and was not read. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Read reviews/d5_dimension_lift_erratum_20261005 and amend the four Props if the displays changed; state Corollaries F_d and M_d; independent reviewer: alignment of the four Props with (1.2)-(1.7). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): BLOCKED_INTERFACE (interface under-determination, not statement defects; each refuted by a concrete Dirac-law inhabitant): DimensionLiftCompactCollarC (extra critical point at the midpoint, E = {0}, every d >= 2), DimensionLiftGlobalWindowG (extra window-height critical point at the midpoint), DimensionLiftIntermediateShellsI (extra window-height critical point at distance 8r from the midpoint), DimensionLiftPuncturedPinBallP (extra critical point in the punctured pin ball, null E). | | `OA-C6-FOURIER-20260929-v1` | proof_index_result | — | `none` | 1 specified: `FourierCutoffPlanarSecondFactorialF4` | No Prop states Theorem F (the cap functional Psi_F and its stretched-exponential tail (F1)-(F2)), Theorem W in general d and p (F3), or Proposition B; only the planar consequence (F4) is stated, GIVEN the hypothesis G_2 (E N <= C r^3), so the claim as a whole is not specified. Nothing is proved; the logarithm is neither shown sharp nor removable here; no numerical cutoff or parameter-uniform constants; the abstract barrier is not a Gaussian counterexample. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (F1)-(F2) with Psi_F as an interface field (a measurable functional dominating the total critical count) and (F3) for general d, p; then the claim would be `specified`. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): FourierCutoffPlanarSecondFactorialF4 BLOCKED_INTERFACE (the antecedent G_2 plus the interface fields admit a rare large cluster (two-point law, cluster size >= 1/r with probability r^3/n_r) with E N(N-1) of order r^2 while E N = r^3; Theorem W's Gaussian input is not transcribed). | | `CL-C6-PALM-20260929-v1` | proof_index_result | — | `specified` | 2 specified: `PalmRouteFactorialMomentsQ`, `PalmRouteSecondFactorialTheta` | PalmRouteFactorialMomentsQ states (1.1) (with r_* allowed to depend on q) and PalmRouteSecondFactorialTheta states (1.2) over PinnedLawInterface; not proved. Corollary P (size-biased law) and Sections 3-7 (the cap Psi, the parametric Rouche tail, the marked Kac-Rice formula with Hoelder insertion, regimes R1-R4) are not stated; the dependence on [C6L] and [DL] at their bytes is not transcribed; the lower bounds' source [EDL] (A4) is not stated. No lower bound for q >= 3; no distribution of N beyond factorial moments; no elder selection; no numerical constants; for d >= 3 conditional on [DL]. The GRAPH records that this global result is not a proof of the regional witness-collision node. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Corollary P; independent reviewer: alignment of the two Props with (1.1)-(1.2), including whether r_* may depend on q. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): PalmRouteFactorialMomentsQ BLOCKED_INTERFACE (a Dirac inhabitant whose single path has two window critical points off the pins gives E (N)_2 = 2 for all small r); PalmRouteSecondFactorialTheta BLOCKED_INTERFACE (the lower bounds c r^3 fail for an inhabitant with no window critical points off the pins and for one with infinitely many (windowCountAll uses Set.ncard, 0 on infinite sets)). | -| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | 1 specified: `RareClusterPoissonObstructionO`; 1 proved (proof packet): `RNCluster.rareClusterPoissonObstructionO†` | Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified and the claim-level label stays `none` (the register admits only none/specified at claim level, and this row's claim is the packet, not Theorem O alone). RareClusterPoissonObstructionO has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (1 entry below, label `proved`; 1 of them (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 (the last head with a hosted receipt: run 38150363063, actual tested merge 86a6e8c876f7904cadda2d006a56f8792dc5c1af with base 79731b961fac555f33be3cd90e379b246a5a80e3, receipt theorem_count 60): locally `lake env lean -DwarningAsError=true` exit 0 and `#print axioms` within {propext, Classical.choice, Quot.sound} on an unverified worktree, not evidence about any commit; the next hosted lean-specification-proofs.yml run is the first receipt for them): it proves Theorem O's inequality as the Prop states it, for every family mu and constants a, A satisfying the Prop's own antecedents (0 < a, nonnegativity, summability and total mass one of each mu r, summable mean, (H1), (H2) on (0, r0]) and no others: the upper half with the witness lam = 0 (the Prop's existential form, not the source's infimum verbatim), the lower half through tail events {n >= 2} and {n >= 1}; 2a <= A is derived internally from the antecedents at r0, as in the source, and is neither a hypothesis nor a conclusion of the Prop. That theorem is evidence about the Prop, for the formal objects pmfTV, poissonPMF and the family mu : R -> N -> R, not acceptance of Theorem O; alignment of those objects with the source's dTV, Pois(lambda) and Law N_r is PENDING_INDEPENDENT_REVIEW. H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count (the Prop does not tie mu to any field; the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement lambda = mu, Theorems C and S, the factorial-moment matching, the independent-replica limit (7.1) and the conditional-law liminf remark are neither stated nor proved. For A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics; the hypotheses imply 2a <= A, so A > 0 is derivable and the Prop is not vacuous in general (the triage's example mu r = (1 - r^3) delta_0 + r^3 delta_2 with a = 1, A = 2 satisfies the antecedents for r <= 1). Disposition: none transcribed (disposition_transcribed null; the title's "PROOF_INDEX reviewed scoped result" is the source's wording, not a status decided here). Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); the 1 theorem of this row added after PR head 12e072d6 (RNCluster.rareClusterPoissonObstructionO) await the next hosted lean-specification-proofs.yml run, their first receipt. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RareClusterPoissonObstructionO PROVABLE_NOW: proved in round 4 (see entries). Independent reviewer: alignment of RareClusterPoissonObstructionO with Theorem O (3.1), in particular the existential form of the upper half and the (H1), (H2) antecedents on (0, r0]. | +| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | 1 specified: `RareClusterPoissonObstructionO`; 1 proved (proof packet): `RNCluster.rareClusterPoissonObstructionO` | Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified and the claim-level label stays `none` (the register admits only none/specified at claim level, and this row's claim is the packet, not Theorem O alone). RareClusterPoissonObstructionO has a theorem of exactly its type in the proof packet specifications/lean_statement_proofs_20261011 (1 entry below, label `proved`; 1 of them (RNCluster.rareClusterPoissonObstructionO) was added after PR head 12e072d6 and kernel-checked by the hosted run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 73, worktree CLEAN; artifact lean-specification-proofs-38180705952-1, sha256 87309efad9b804c53c543b60fdb466de1f797971417363cc2e8fa1e10eac26bd); it was first receipted by that run (PR #434, event pull_request); base 12e072d614dbe6e91b66d22a2ab55cf87860d693 is the head of PR #423, the base of PR #434, and the merge has the same tree as its head): it proves Theorem O's inequality as the Prop states it, for every family mu and constants a, A satisfying the Prop's own antecedents (0 < a, nonnegativity, summability and total mass one of each mu r, summable mean, (H1), (H2) on (0, r0]) and no others: the upper half with the witness lam = 0 (the Prop's existential form, not the source's infimum verbatim), the lower half through tail events {n >= 2} and {n >= 1}; 2a <= A is derived internally from the antecedents at r0, as in the source, and is neither a hypothesis nor a conclusion of the Prop. That theorem is evidence about the Prop, for the formal objects pmfTV, poissonPMF and the family mu : R -> N -> R, not acceptance of Theorem O; alignment of those objects with the source's dTV, Pois(lambda) and Law N_r is PENDING_INDEPENDENT_REVIEW. H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count (the Prop does not tie mu to any field; the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement lambda = mu, Theorems C and S, the factorial-moment matching, the independent-replica limit (7.1) and the conditional-law liminf remark are neither stated nor proved. For A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics; the hypotheses imply 2a <= A, so A > 0 is derivable and the Prop is not vacuous in general (the triage's example mu r = (1 - r^3) delta_0 + r^3 delta_2 with a = 1, A = 2 satisfies the antecedents for r <= 1). Disposition: none transcribed (disposition_transcribed null; the title's "PROOF_INDEX reviewed scoped result" is the source's wording, not a status decided here). Nonauthor review and acceptance are not claimed by any statement or theorem; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet and of the proof packet is 0. | State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); the theorem of this row (RNCluster.rareClusterPoissonObstructionO, added after PR head 12e072d6) has a hosted receipt: first receipted by run 38180705952 (PR head 48bfcbae; actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693) (receipt theorem_count 73). Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RareClusterPoissonObstructionO PROVABLE_NOW: proved in round 4 (see entries). Independent reviewer: alignment of RareClusterPoissonObstructionO with Theorem O (3.1), in particular the existential form of the upper half and the (H1), (H2) antecedents on (0, r0]. | | `CL-D5-REMOTE-COLLISION-20260928-v1` | proof_index_result | — | `none` | 1 specified: `RemoteCollisionSecondFactorialD` | Only Corollary D (second factorial moment <= C r^5 \|E\| in the fixed remote region) is stated; Theorem C (ordered near-pair count), Corollaries E and F (event asymptotics, torus-wide lower bound) are not, so the packet is not specified as a whole. Nothing proved; pairs with a witness near the pins are excluded; no torus-wide second factorial moment; this is explicitly NOT the open witness-collision mechanism. Author Anthropic/Claude: same provider as this packet, independence credit 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Theorem C and Corollaries E, F; independent (non-Anthropic) reviewer for both the source and the Props. Round-4 triage (2026-10-11, same-account, credit 0, no label moves): RemoteCollisionSecondFactorialD BLOCKED_INTERFACE (a Dirac inhabitant with two prescribed remote critical points falsifies both displays; no field gives a.s. absence of index-j window pairs on null remote sets). | | `cumulative-transfer-correction` | proof_index_result | — | `none` | — | No Prop states the corrected cumulative theorem (sigma-finite (Z, lambda), alpha > -1, m > 0, ratio hypothesis h/(kappa r^m) -> 1, limit N(ell)/ell^beta -> (1/(alpha+1)) int B0 kappa^(-beta)), although its bytes are pinned. Nothing is proved; the correction does not license inferring a density asymptotic from a cumulative one; Section A of NOTE.md is outside both acceptances. | State the corrected theorem as a def : Prop over MeasureTheory.Measure Z (an abstract measure-space statement with no Gaussian input; a natural Mathlib-lane proof target afterwards). | | `hist.D0-RN-UNIF-historical-predicates` | open_obligation | — | `none` | — | No Lean statement: the historical numerical RN all-cell / 24-jet certification predicates have no in-tree source text to anchor (their carriers are ABSENT by fingerprint) and no Prop states any of them. The GRAPH records no complete proof where not explicitly superseded, and hist.lemma_closed is FALSE; this register changes none of that. The regional bypass (regional.fixed-annulus.high-jet-route, SUPERSEDED_NONBLOCKING) does not reopen or close these predicates in their original replay scopes; no numerical certificate exists in tree. | None for Lean until a source text exists; recovery of the absent carriers is a custody question (reviews/d0_custody_audit_20260926), not a formalization one. | @@ -442,5 +441,5 @@ about the registered Prop it names (and, for main's `formal/` targets, about a f identity), never about the claim row it sits under; `kernel-checked` is a run-receipt word (the proof packet's hosted run 38121713840 receipted the 52 theorems of PR head 910809d6: actual tested merge 5b198699… with base b2d70d77…, the synthetic merge a pull_request run checks out, same tree as the head; the 8 of `P15Interpolation` by run 38149888699: PR head 0dcb0016, actual tested merge 4363b692… with base -79731b96…, the head of PR #422; theorems added after PR head 12e072d6 (run 38150363063, actual tested merge 86a6e8c8… with base 79731b96…) await the first hosted -`lean-specification-proofs.yml` run on the round-4 branch, their first receipt). +79731b96…, the head of PR #422; the 13 of `Side24Analysis`, `Side24Interfaces` and `RNCluster` by run 38180705952: PR head 48bfcbae, actual tested merge 9833d6d9… with base +12e072d6…, the head of PR #423; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efa…).