From c6ce025652f76be6e580fa24947380253450cd3b Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 12 Oct 2026 02:19:19 +0000 Subject: [PATCH 1/2] Proof packet round 5: theorems of ReferenceCovarianceLowerBound and ConeMomentD2Identity (no hosted receipt yet) One forward commit on the round-4 head be014794 (draft PR #434, readback PASS 5485768387). It adds two proof modules whose inventory theorems have exactly the types of two registered Props of specifications/lean_statements_20261010/Specifications/Side24.lean. These are the last two Props that the round-4 triage rated PROVABLE_WITH_WORK ("weeks"). It moves no claim-level label. SpecificationProofs/Side24Covariance.lean (282 lines, sha256 08317c62...): referenceCovarianceLowerBound : UniversalLaw.Spec.Side24.ReferenceCovarianceLowerBound, for d = 2 or 3 as the Prop states it. Odd-order contractions of Kinf vanish at 0 because Kinf is even (ContinuousLinearMap.iteratedFDeriv_comp_right with -id). The even orders reuse the round-4 helpers rb_order_two, rb_order_four and rb_order_six. Every block of Cref is then explicit, and Cref - I/3 is diagonal(0, 2/3...; 7/6; 5/3...) + (2/3) v v^T + w w^T, a sum of positive semidefinite matrices. Helpers cv_* in Aux. SpecificationProofs/Side24ConeMoment.lean (485 lines, sha256 08d53d0a...): coneMomentD2Identity (I) : UniversalLaw.Spec.Side24.ConeMomentD2Identity I, for every inhabitant of ConeMomentInterface. The route: a.e.-measurability of (s, x, y) and integral_map along sxy_law, the cone identity NegDefCone (s,x,y) <-> s < 0 and x^2+y^2 < s^2 (determinant and a Lagrange identity), Fubini with the bound s^4, and the inner planar Gaussian integral by polar coordinates and z = r^2, reduced to the packet theorem Side24Elementary.elementaryConeIntegral. Evenness halves the outer integral, and the moments of N(0, 5/3) close it. Helpers cm_* in Aux. Each module was written and adversarially verified in its own lane on 2026-10-12: a kernel verifier compiled it at the pin with -DwarningAsError=true and checked #print axioms within {propext, Classical.choice, Quot.sound} and that the header type is the registered constant; a style verifier found docstring inaccuracies, fixed in one round and delta-verified. Same account; organizational-independence credit 0. SpecificationProofs.lean imports the two modules. The proof packet's source check now counts 75 theorems over 74 registered Props in 13 modules (TriangleNeedsThree is proved twice). REGISTER.json (blob fa96260c, from c8bb36ca): two new `proved` statement entries on side24-coefficient. Each carries its Prop's registered anchor and its docstring's "Does not establish" field as does_not_claim. The side24-coefficient kernel_evidence, not_established and next_step are regenerated with a pending tier: the two theorems were added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964c... with base 12e072d6..., theorem_count 73). The round-4 triage sentence records that both are proved in round 5 (verdict and effort kept verbatim). math.side24-coefficient's landing listing is updated. The inventory parenthetical of the other nine proof-packet kernel_evidence notes moves 73/11/72 -> 75/13/74. In total 28 string leaves change. Every formal_progress is unchanged (138 none, 33 specified), and every other field is byte-identical. The statement packet README/SCOPE (74 of 141 Props with a theorem, dagger for "no hosted receipt yet") and the proof packet README/SCOPE are re-rendered. Checks (local, unverified worktree, records of computations): both packets' check.py --allow-dirty pass; test_check.py passes in normal and -O modes (97 and 70 tests). Not established: anything about an informal claim. No hosted receipt for the two theorems yet. The coefficient c_{d,24} is not enclosed. CovarianceOrdering, EntrywiseCovarianceBound and every BLOCKED row keep their verdicts. No inhabitant of ConeMomentInterface is constructed, and D_2 is not identified with the parent's D_u. Alignment PENDING_INDEPENDENT_REVIEW; scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01Cz7WZybv8znP64SpPj6sWY --- .../lean_statement_proofs_20261011/README.md | 31 +- .../lean_statement_proofs_20261011/SCOPE.md | 2 +- .../SpecificationProofs.lean | 2 + .../SpecificationProofs/Side24ConeMoment.lean | 485 ++++++++++++++++++ .../SpecificationProofs/Side24Covariance.lean | 282 ++++++++++ .../lean_statements_20261010/README.md | 30 +- .../lean_statements_20261010/REGISTER.json | 46 +- .../lean_statements_20261010/SCOPE.md | 30 +- 8 files changed, 855 insertions(+), 53 deletions(-) create mode 100644 specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24ConeMoment.lean create mode 100644 specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Covariance.lean diff --git a/specifications/lean_statement_proofs_20261011/README.md b/specifications/lean_statement_proofs_20261011/README.md index 4c8163d92..63765554c 100644 --- a/specifications/lean_statement_proofs_20261011/README.md +++ b/specifications/lean_statement_proofs_20261011/README.md @@ -43,8 +43,8 @@ register gives it. ## Registered Props without a theorem here, and why -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 +Inventory at this commit: 75 theorems, 74 of the 141 registered `def … : Prop` declarations (`TriangleNeedsThree` +is proved twice). The other 67 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. 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; @@ -60,7 +60,14 @@ head of PR #423, with the same tree as the head; job proof-evidence: fresh build 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. +run, is evidence about no commit; later commits need their own run. The 2 theorems of the two round-5 modules +(`Side24Covariance`, `Side24ConeMoment`), marked † below, have **no hosted receipt yet**: they were added after PR head +`be014794`, the last head with a hosted receipt (run 38187538744, event pull_request on PR #434; the actually tested +checkout is the synthetic merge `79d3964c…` of that head into base `12e072d6…`; receipt theorem_count 73, `leanchecker` +PASS; artifact `lean-specification-proofs-38187538744-1`, sha256 `60c9be0b…` as GitHub reports it), which does not name +them. Locally, `lake env lean -DwarningAsError=true` exited 0 on each module at the pin and `#print axioms` stayed +within `propext`, `Classical.choice`, `Quot.sound`, on an unverified worktree, which is evidence about no commit; the +next hosted `lean-specification-proofs.yml` run is their first receipt. | group | Props (names without `UniversalLaw.Spec..`) | why no theorem here | |---|---|---| @@ -70,11 +77,13 @@ run, is evidence about no commit; later commits need their own run. | 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 | `NegDefCone` (predicate), `CovarianceOrdering`, `EntrywiseCovarianceBound` | `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). | | 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)`; 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. | +| Side24 covariance and density, formerly listed here (round 5) | — | `ReferenceCovarianceLowerBound` has a theorem in `Side24Covariance` since round 5 (`Side24Covariance.referenceCovarianceLowerBound†`; † = no hosted receipt yet, see above). The PROVABLE_WITH_WORK (weeks) reason this table gave for it is superseded by the theorem, and the module records that two of its details were not load-bearing: no order-five general-direction level of the peeling is needed, and no quadratic-form expansion through `Matrix.posSemidef_iff_dotProduct_mulVec`. Route (the module docstring): the odd-order contractions of the even kernel `Kinf d` vanish at `0` in arbitrary directions (`ContinuousLinearMap.iteratedFDeriv_comp_right`, `ContinuousMultilinearMap.map_smul_univ`), the orders two, four and six are the round-4 helpers `Side24Analysis.Aux.rb_order_two`, `rb_order_four`, `rb_order_six` (imported, not re-proved), and in an orthonormal frame with `u = R 0` the matrix `Cref - I/3` is entrywise `diagonal δ + (2/3) v vᵀ + w wᵀ` (`v = e_{G_0} - (9/2) e_t`, `w` the indicator of the diagonal Hessian coordinates), each summand positive semidefinite; the helper `Aux.cv_psd` (not an inventory theorem) holds for every `d > 0`, the inventory theorem has exactly the Prop's type, for `d = 2` or `3`, and its proof does not use that hypothesis; the `IsHermitian` half follows from the decomposition, not from any symmetry of `iteratedFDeriv`. Not established: that `Cref` is the covariance of any random vector; symmetry of `Cref d u R` for a general `u` or frame; main's `UniversalLaw.Side24.odd_block_eigenvalues`, `odd_block_shifted_minors` and `smaller_eigenvalue_exceeds_third` are neither imported, cited nor relied on, and no minor, no eigenvalue `8 ± sqrt 58` and no comparison of an eigenvalue with `1/3` is computed here. | +| Side24 interface statements | `CoefficientEnclosure`, `ReferenceComparison` (over `PeriodicCoefficientInterface`) | `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)`; 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` (by these three; it has its own theorem since round 5, next row); any cone moment of the `K24` field. `ConeMomentD2Algebra`, `ElementaryConeIntegral` and `ReferenceCoefficientIdentity` (proved earlier) need no interface. | +| Side24 interface statements, formerly listed here (round 5) | — | `ConeMomentD2Identity` has a theorem in `Side24ConeMoment` since round 5 (`Side24ConeMoment.coneMomentD2Identity†`; † = no hosted receipt yet, see above), for a GIVEN `I : ConeMomentInterface`. The PROVABLE_WITH_WORK (weeks) reason this table gave for it is superseded by the theorem. Route (the module docstring): transport along the law field `sxy_law` (`integral_map` after `Side24Interfaces.Aux.tm_aemeasurable_triple`), Fubini with the bound `\|coneIntegrand q\| ≤ q.1^4`, the cone identification `NegDefCone (s, x, y) ↔ s < 0 ∧ x² + y² < s²`, the inner planar Gaussian integral `s⁴ - 4s² + 8 - 8 exp(-s²/2)` for `s < 0` by polar coordinates and the substitution `z = r²` onto `elementaryConeIntegral`, the halving of the even bracket under `gaussianReal 0 (5/3)`, and the moments `25/3`, `5/3`, `√(3/8)` from `Side24Interfaces`; of the law fields only `sxy_law` is used (with the instances `mΩ`, `isProb`); `A` and `A_law` are not used, and `RayleighSquareLaw` is not invoked. Not established: that `ConeMomentInterface` has an inhabitant, or that its laws are the conditional law `A \| V = 0` of the reference field; that `D_2` is the parent's cone moment `D_u` of (15.1) for the reference law; the source's Rayleigh integration route as the source's statements (for the formal objects the proof follows the shape of the source's Section 1 computation, so it is not a cross-check of it by a different route); any enclosure of the coefficient. | +| 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 67 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`) and the 8 P15 predicates it did not re-examine; its 2 PROVABLE_WITH_WORK Props (`ReferenceCovarianceLowerBound`, `ConeMomentD2Identity`) have theorems since round 5 (marked †, no hosted receipt yet). | | 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 @@ -88,9 +97,11 @@ 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; 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) + Side24Interfaces, Side24Covariance, Side24ConeMoment; RN into + RNProbability, RNCluster (each list in import order; the root + imports the eight modules of rounds 1-3, then Side24Analysis, + Side24Interfaces, RNCluster (round 4) and Side24Covariance, + Side24ConeMoment (round 5)) ``` 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 fbe2f3c17..f18a5cac3 100644 --- a/specifications/lean_statement_proofs_20261011/SCOPE.md +++ b/specifications/lean_statement_proofs_20261011/SCOPE.md @@ -31,7 +31,7 @@ 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`, 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) | +| Side24 (`Specifications.Side24`, modules Side24Elementary, Side24Analysis, Side24Interfaces, Side24Covariance, Side24ConeMoment) | 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; the reference covariance lower bound: `Cref d (R 0) R - (1/3) • 1` positive semidefinite, its `IsHermitian` half included, for every orthonormal frame `R` and `d = 2` or `3`, as the Prop states it (the proof does not use that hypothesis; only the helper `Aux.cv_psd`, not an inventory theorem, is stated for every `0 < d`)†; 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, and the `m = 2` cone moment `∫ coneIntegrand (s, x, y) ∂I.P = (1/2)(25/3 - 20/3 + 8 - 8 √(3/8))`† from the law field `sxy_law` alone (not `A_law`); all as the Props state them, for the formal objects (`E d`, `Kinf`, `latticeVec`, `derivContraction`, `gaussDensity`, `jointCov`, `Cref`, `coneIntegrand`, the interface structures); † = no hosted receipt yet (added after PR head `be014794`, the last head with a hosted receipt: run 38187538744, actual tested merge `79d3964c…` with base `12e072d6…`) | 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`); 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 or `Cref` is the covariance of a random object; symmetry of `Cref d u R` for a general `u` or frame; the odd-block minors and eigenvalues `8 ± sqrt 58` (main's theorems, neither imported nor relied on here); that `D_2` is the parent's cone moment `D_u` of (15.1) for the reference law, and the source's Rayleigh integration route as the source's statements; 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 | diff --git a/specifications/lean_statement_proofs_20261011/SpecificationProofs.lean b/specifications/lean_statement_proofs_20261011/SpecificationProofs.lean index 32f0e0c1f..c4b2cb100 100644 --- a/specifications/lean_statement_proofs_20261011/SpecificationProofs.lean +++ b/specifications/lean_statement_proofs_20261011/SpecificationProofs.lean @@ -9,3 +9,5 @@ import SpecificationProofs.P15Interpolation import SpecificationProofs.Side24Analysis import SpecificationProofs.Side24Interfaces import SpecificationProofs.RNCluster +import SpecificationProofs.Side24Covariance +import SpecificationProofs.Side24ConeMoment diff --git a/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24ConeMoment.lean b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24ConeMoment.lean new file mode 100644 index 000000000..ec1a96b2d --- /dev/null +++ b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24ConeMoment.lean @@ -0,0 +1,485 @@ +import Specifications.Side24 +import SpecificationProofs.Side24Interfaces +import SpecificationProofs.Side24Elementary + +/-! +# SIDE24 cone moment: the `m = 2` identity `ConeMomentD2Identity` over `ConeMomentInterface` +(statement module `Specifications.Side24`) + +Scientific effect: NONE. Organizational-independence credit: 0 (Anthropic / Claude). + +The one theorem outside the `Aux` namespace has as its type exactly the registered `Prop` +`UniversalLaw.Spec.Side24.ConeMomentD2Identity` 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 that `Prop` is evidence about that +`Prop` only, for the formal objects it names (`ConeMomentInterface`, `transverseMatrix`, +`NegDefCone`, `coneIntegrand`, Mathlib's `gaussianReal` and Bochner integral): it is not evidence +about the informal Layer 0 note (SIDE24-COEFFICIENT-D23-20260924-v1, register row +`side24-coefficient`), which it does not make 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; the theorem holds for every inhabitant and uses +only the law field `sxy_law` and the instances `mΩ`, `isProb` (the latter through +`tm_aemeasurable_triple`); the fields `A` and `A_law` are not used. + +Provenance: round-5 lane for `ConeMomentD2Identity`, selected by the round-4 triage. Helper lemmas +live in `UniversalLaw.Proofs.Side24ConeMoment.Aux` with the tag `cm_`; the gate inventories only +the theorem outside that namespace. The module reuses, without restating them, the public helpers +of the round-4 module `SpecificationProofs.Side24Interfaces` (`tm_aemeasurable_triple`, +`tm_fourth_moment`, `tm_exp_moment`, `c1_second_moment_gaussianReal`, +`c1_integrable_sq_gaussianReal`, `rl_pdf_mul`, `rl_measurableSet_radial`, `rl_Ioi_inter_eq`, +`rl_angular_integral`) and the inventory theorem `elementaryConeIntegral` of +`SpecificationProofs.Side24Elementary`. The mathematics: + +* Cone identification (`cm_cone_iff`): `NegDefCone (s, x, y)` holds iff `s < 0` and + `x^2 + y^2 < s^2`. Forward: `det (-A) = det A = s^2 - x^2 - y^2 > 0` (`Matrix.PosDef.det_pos`, + `cm_det_transverse`) and the diagonal entries `-s - x`, `-s + x` are positive + (`Matrix.PosDef.diag_pos`). Converse: `-A` is symmetric and its quadratic form at `v = (a, b)` + is `-s (a^2 + b^2) - x (a^2 - b^2) - 2 y a b`, which is positive for `v ≠ 0` by the Lagrange + identity `(x (a^2 - b^2) + 2 y a b)^2 + (2 x a b - y (a^2 - b^2))^2 = (x^2 + y^2) (a^2 + b^2)^2` + (`Matrix.PosDef.of_dotProduct_mulVec_pos`, `cm_quad_pos`). Hence `coneIntegrand` is the + indicator of `{s < 0 ∧ x^2 + y^2 < s^2}` times `(s^2 - x^2 - y^2)^2` + (`cm_coneIntegrand_eq`), a measurable function bounded by `s^4`. +* Transport and Fubini: the triple `(s, x, y)` is a.e.-measurable (`tm_aemeasurable_triple`), so + `integral_map` and the law field `sxy_law` move the integral to + `N(0, 5/3) ⊗ (N(0, 1) ⊗ N(0, 1))`; the integrand is integrable there (bound `s^4`, + `memLp_id_gaussianReal'`, `Integrable.comp_fst`), so `integral_prod` gives an iterated integral. +* Inner integral (`cm_inner`): for `s ≥ 0` the slice is `0`; for `s < 0` the slice equals the + closed-disk integrand `1{x^2 + y^2 ≤ s^2} (s^2 - x^2 - y^2)^2` (equal on the circle, where the + square vanishes), whose planar Gaussian integral is `a^2 - 4a + 8 - 8 exp (-a/2)` at `a = s^2` + (`cm_gauss2_integral`): the product of the two `withDensity` laws is `withDensity` of the + product density (`prod_withDensity`, `integral_withDensity_eq_integral_toReal_smul`), polar + coordinates (`integral_comp_polarCoord_symm`, `setIntegral_prod_mul`), the angular integral + `2π`, and the radial integral by the substitution `z = r^2` + (`intervalIntegral.integral_comp_mul_deriv`) onto `elementaryConeIntegral`. +* Outer integral: the bracket `t^4 - 4t^2 + 8 - 8 exp (-t^2/2)` is even and integrable under + `N(0, 5/3)`, so its integral over `t < 0` is half its full integral (`gaussianReal_map_neg`, + `integral_add_compl`, `integral_Ici_eq_integral_Ioi`; `cm_setIntegral_Iio_even`), and the full + integral is `25/3 - 4 (5/3) + 8 - 8 sqrt (3/8)` from the moments `∫ t^4 = 25/3`, + `∫ t^2 = 5/3`, `∫ exp (-t^2/2) = sqrt (3/8)` (`cm_integral_bracket`). + +Remark recorded by the lane, for the maintainers: for the formal objects, the proof follows the +shape of the source's Section 1 computation for `m = 2`: the cone as `s < -R`, in the form +`s < 0 ∧ x^2 + y^2 < s^2` (`cm_cone_iff`); the truncated integral of `(a - z)^2` against the +density `e^(-z/2)/2` (`cm_radial_integral`, evaluated by `elementaryConeIntegral`, whose +registered anchor is the source's display of that integral); symmetry in `s` +(`cm_setIntegral_Iio_even`); and the three moments of `s` (`cm_integral_bracket`). It does not +invoke the registered `RayleighSquareLaw` or any statement of the law of `R^2`: the +`R^2`-density integral is obtained inline, by polar coordinates and the substitution `z = r^2` +(`cm_disk_integral`, `cm_radial_integral`), and is then evaluated by `elementaryConeIntegral`. +It also writes out steps the source does not write out (the transport along `sxy_law`, Fubini with +its integrability, and the closed disk in place of the open one, which changes nothing because +the integrand vanishes on the circle `x^2 + y^2 = s^2`). It is therefore not a cross-check of the +source's Rayleigh cone integration by a different route. Being authored by Anthropic / Claude at +organizational-independence credit 0, it is also not the nonauthor review that the source asks +for. That these formal steps are the source's steps is not established (alignment +PENDING_INDEPENDENT_REVIEW). This is a remark about a route, not about any claim; no register +label changes. +-/ + +namespace UniversalLaw.Proofs.Side24ConeMoment + +open MeasureTheory ProbabilityTheory UniversalLaw.Spec.Side24 + +namespace Aux + +open UniversalLaw.Proofs.Side24Interfaces.Aux + +/-! ## `cm_` (a): the negative-definite cone of the `m = 2` transverse matrix -/ + +/-- `det [[s + x, y], [y, s - x]] = s^2 - x^2 - y^2` (`Matrix.det_fin_two_of`). -/ +theorem cm_det_transverse (s x y : ℝ) : + (transverseMatrix s x y).det = s ^ 2 - x ^ 2 - y ^ 2 := by + unfold transverseMatrix + rw [Matrix.det_fin_two_of] + ring + +/-- Forward direction of the cone characterisation: positive definiteness of `-A` gives `s < 0` +(the two diagonal entries `-s - x`, `-s + x` are positive, `Matrix.PosDef.diag_pos`) and +`x^2 + y^2 < s^2` (`det (-A) = det A > 0`, `Matrix.PosDef.det_pos`, `Matrix.det_neg`). -/ +theorem cm_cone_forward (s x y : ℝ) (h : NegDefCone (s, x, y)) : + s < 0 ∧ x ^ 2 + y ^ 2 < s ^ 2 := by + unfold NegDefCone at h + have hdet := h.det_pos + rw [Matrix.det_neg, cm_det_transverse] at hdet + have hdiag := h.diag_pos (i := 0) + have hdiag' := h.diag_pos (i := 1) + simp [transverseMatrix] at hdiag hdiag' + simp only [Fintype.card_fin, even_two, Even.neg_pow, one_pow, one_mul] at hdet + constructor + · linarith + · linarith + +/-- A nonzero vector of `Fin 2 → ℝ` has positive squared norm `v 0 ^ 2 + v 1 ^ 2`. -/ +theorem cm_sq_add_sq_pos (v : Fin 2 → ℝ) (hv : v ≠ 0) : 0 < v 0 ^ 2 + v 1 ^ 2 := by + by_contra hle + rw [not_lt] at hle + have h0 : v 0 = 0 := by nlinarith [sq_nonneg (v 0), sq_nonneg (v 1)] + have h1 : v 1 = 0 := by nlinarith [sq_nonneg (v 0), sq_nonneg (v 1)] + apply hv + funext i + fin_cases i + · exact h0 + · exact h1 + +/-- The quadratic form of `-A` at `(a, b) ≠ 0` is positive inside the cone: by the Lagrange +identity `(x (a^2 - b^2) + 2 y a b)^2 + (2 x a b - y (a^2 - b^2))^2 = (x^2 + y^2) (a^2 + b^2)^2`, +`|x (a^2 - b^2) + 2 y a b| ≤ sqrt (x^2 + y^2) (a^2 + b^2) < -s (a^2 + b^2)` (`nlinarith`). -/ +theorem cm_quad_pos (s x y a b : ℝ) (hs : s < 0) (hr : x ^ 2 + y ^ 2 < s ^ 2) + (hab : 0 < a ^ 2 + b ^ 2) : + 0 < -s * (a ^ 2 + b ^ 2) - x * (a ^ 2 - b ^ 2) - 2 * y * a * b := by + have key : (x * (a ^ 2 - b ^ 2) + 2 * y * a * b) ^ 2 + (2 * x * a * b - y * (a ^ 2 - b ^ 2)) ^ 2 = + (x ^ 2 + y ^ 2) * (a ^ 2 + b ^ 2) ^ 2 := by ring + by_contra hle + rw [not_lt] at hle + have hL : -s * (a ^ 2 + b ^ 2) ≤ x * (a ^ 2 - b ^ 2) + 2 * y * a * b := by linarith + have hc : 0 < -s * (a ^ 2 + b ^ 2) := by nlinarith + have h1 : (-s * (a ^ 2 + b ^ 2)) ^ 2 ≤ (x * (a ^ 2 - b ^ 2) + 2 * y * a * b) ^ 2 := by + nlinarith + have h2 : (x ^ 2 + y ^ 2) * (a ^ 2 + b ^ 2) ^ 2 < s ^ 2 * (a ^ 2 + b ^ 2) ^ 2 := by + have : 0 < (a ^ 2 + b ^ 2) ^ 2 := by positivity + nlinarith + nlinarith [sq_nonneg (2 * x * a * b - y * (a ^ 2 - b ^ 2))] + +/-- Converse direction of the cone characterisation: for `s < 0` and `x^2 + y^2 < s^2`, `-A` is +symmetric and its quadratic form is positive on nonzero vectors +(`Matrix.PosDef.of_dotProduct_mulVec_pos`, `cm_quad_pos`). -/ +theorem cm_cone_backward (s x y : ℝ) (hs : s < 0) (hr : x ^ 2 + y ^ 2 < s ^ 2) : + NegDefCone (s, x, y) := by + unfold NegDefCone + refine Matrix.PosDef.of_dotProduct_mulVec_pos ?_ ?_ + · ext i j + fin_cases i <;> fin_cases j <;> simp [transverseMatrix] + · intro v hv + have hab := cm_sq_add_sq_pos v hv + have := cm_quad_pos s x y (v 0) (v 1) hs hr hab + simp [transverseMatrix, dotProduct, Matrix.mulVec, Fin.sum_univ_two] + nlinarith + +/-- The formal cone `NegDefCone q` is `q.1 < 0 ∧ q.2.1^2 + q.2.2^2 < q.1^2` (both directions +above). A statement about the formal definition only. -/ +theorem cm_cone_iff (q : ℝ × ℝ × ℝ) : + NegDefCone q ↔ q.1 < 0 ∧ q.2.1 ^ 2 + q.2.2 ^ 2 < q.1 ^ 2 := + ⟨fun h => cm_cone_forward q.1 q.2.1 q.2.2 h, + fun h => cm_cone_backward q.1 q.2.1 q.2.2 h.1 h.2⟩ + +/-! ## `cm_` (b): the planar Gaussian integral of the disk integrand -/ + +/-- The radial integral `∫_0^{sqrt a} r (a - r^2)^2 (2π)⁻¹ exp (-r^2/2) dr = +(2π)⁻¹ (a^2 - 4a + 8 - 8 exp (-a/2))`: the substitution `z = r^2` +(`intervalIntegral.integral_comp_mul_deriv`) turns it into `(2π)⁻¹` times the interval integral +of `ElementaryConeIntegral`, which `Side24Elementary.elementaryConeIntegral` evaluates. -/ +theorem cm_radial_integral (a : ℝ) (ha : 0 ≤ a) : + ∫ r in (0 : ℝ)..Real.sqrt a, + r * ((a - r ^ 2) ^ 2 * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2))) = + (2 * Real.pi)⁻¹ * (a ^ 2 - 4 * a + 8 - 8 * Real.exp (-a / 2)) := by + have hderiv : ∀ r ∈ Set.uIcc (0 : ℝ) (Real.sqrt a), + HasDerivAt (fun r : ℝ => r ^ 2) (2 * r) r := by + intro r _ + simpa using hasDerivAt_pow 2 r + have hcv := intervalIntegral.integral_comp_mul_deriv (a := 0) (b := Real.sqrt a) + (f := fun r : ℝ => r ^ 2) (f' := fun r => 2 * r) + (g := fun z : ℝ => (2 * Real.pi)⁻¹ * ((a - z) ^ 2 * Real.exp (-z / 2) / 2)) hderiv + (by fun_prop) (by fun_prop) + beta_reduce at hcv + have hpt : (fun r : ℝ => r * ((a - r ^ 2) ^ 2 * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2)))) = + fun r => ((fun z : ℝ => (2 * Real.pi)⁻¹ * ((a - z) ^ 2 * Real.exp (-z / 2) / 2)) ∘ + (fun r : ℝ => r ^ 2)) r * (2 * r) := by + funext r + simp only [Function.comp] + ring + rw [hpt, hcv, Real.sq_sqrt ha, show ((0 : ℝ)) ^ 2 = 0 by norm_num, + intervalIntegral.integral_const_mul, + UniversalLaw.Proofs.Side24Elementary.elementaryConeIntegral a ha] + +/-- In polar coordinates the disk integrand `p.1 • 1_D (a - |q|^2)^2 φ(q.1) φ(q.2)` at +`q = polarCoord.symm p` is the radial function `1_{r^2 ≤ a} r (a - r^2)^2 (2π)⁻¹ exp (-r^2/2)` of +`p.1` times the constant `1` of `p.2` (`rl_pdf_mul`, `cos^2 + sin^2 = 1`). -/ +theorem cm_polar_integrand (a : ℝ) (p : ℝ × ℝ) : + p.1 • Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ a} + (fun q => (a - q.1 ^ 2 - q.2 ^ 2) ^ 2 * + (gaussianPDFReal 0 1 q.1 * gaussianPDFReal 0 1 q.2)) (polarCoord.symm p) = + Set.indicator {r : ℝ | r ^ 2 ≤ a} + (fun r => r * ((a - r ^ 2) ^ 2 * ((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] + have hsq' : a - (p.1 * Real.cos p.2) ^ 2 - (p.1 * Real.sin p.2) ^ 2 = a - p.1 ^ 2 := by + rw [sub_sub, hsq] + simp only [polarCoord_symm_apply, Set.indicator, Set.mem_ofPred_eq, rl_pdf_mul, hsq, hsq', + smul_eq_mul, mul_one] + split_ifs <;> simp + +/-- The Lebesgue integral over the plane of `1{|q|^2 ≤ a} (a - |q|^2)^2 φ(q.1) φ(q.2)` (`φ` the +standard Gaussian density) is `a^2 - 4a + 8 - 8 exp (-a/2)`: polar coordinates +(`integral_comp_polarCoord_symm`), separation of variables (`setIntegral_prod_mul`), the angular +integral `2π` (`rl_angular_integral`) and the radial integral `cm_radial_integral`. -/ +theorem cm_disk_integral (a : ℝ) (ha : 0 ≤ a) : + ∫ p, Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ a} + (fun q => (a - q.1 ^ 2 - q.2 ^ 2) ^ 2 * + (gaussianPDFReal 0 1 q.1 * gaussianPDFReal 0 1 q.2)) p = + a ^ 2 - 4 * a + 8 - 8 * Real.exp (-a / 2) := by + have h2 : ∫ p, Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ a} + (fun q => (a - q.1 ^ 2 - q.2 ^ 2) ^ 2 * + (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 => (a - q.1 ^ 2 - q.2 ^ 2) ^ 2 * + (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 => (a - q.1 ^ 2 - q.2 ^ 2) ^ 2 * + (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 * ((a - r ^ 2) ^ 2 * ((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 => cm_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 * ((a - r ^ 2) ^ 2 * ((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 * ((a - r ^ 2) ^ 2 * ((2 * Real.pi)⁻¹ * Real.exp (-(r ^ 2) / 2)))) r = + (2 * Real.pi)⁻¹ * (a ^ 2 - 4 * a + 8 - 8 * 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), cm_radial_integral a ha] + rw [h2, h3] + refine h4.trans ?_ + rw [h5, rl_angular_integral] + field_simp + +/-- Under the standard planar Gaussian `gaussianReal 0 1 ⊗ gaussianReal 0 1`, the integral of +`1{|q|^2 ≤ a} (a - |q|^2)^2` is `a^2 - 4a + 8 - 8 exp (-a/2)`: the product of the two +`withDensity` laws is `withDensity` of the product density (`prod_withDensity`), the integral +against it is the Lebesgue integral of the density times the integrand +(`integral_withDensity_eq_integral_toReal_smul`, `toReal_gaussianPDF`), and `cm_disk_integral`. -/ +theorem cm_gauss2_integral (a : ℝ) (ha : 0 ≤ a) : + ∫ p, Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ a} (fun q => (a - q.1 ^ 2 - q.2 ^ 2) ^ 2) p + ∂((gaussianReal 0 1).prod (gaussianReal 0 1)) = + a ^ 2 - 4 * a + 8 - 8 * Real.exp (-a / 2) := by + have hmeas : Measurable (fun z : ℝ × ℝ => gaussianPDF 0 1 z.1 * gaussianPDF 0 1 z.2) := + ((measurable_gaussianPDF 0 1).comp measurable_fst).mul + ((measurable_gaussianPDF 0 1).comp measurable_snd) + 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, + integral_withDensity_eq_integral_toReal_smul hmeas + (Filter.Eventually.of_forall (fun _ => ENNReal.mul_lt_top gaussianPDF_lt_top gaussianPDF_lt_top)), + ← cm_disk_integral a ha] + congr 1 + funext p + simp only [ENNReal.toReal_mul, toReal_gaussianPDF, Set.indicator, Set.mem_ofPred_eq, smul_eq_mul] + split_ifs <;> ring + +/-! ## `cm_` (c): the bracket `t^4 - 4t^2 + 8 - 8 exp (-t^2/2)` under `N(0, 5/3)` -/ + +/-- `t ↦ t^4` is integrable against `gaussianReal 0 v` (`memLp_id_gaussianReal'` at `p = 4`, +`MemLp.integrable_norm_pow`, `|t|^4 = t^4`). -/ +theorem cm_integrable_pow_four (v : NNReal) : + Integrable (fun t : ℝ => t ^ 4) (gaussianReal 0 v) := by + have h := (memLp_id_gaussianReal' (μ := 0) (v := v) ((4 : ℕ) : ENNReal) (by simp)).integrable_norm_pow + (by norm_num) + refine h.congr (Filter.Eventually.of_forall fun x => ?_) + change ‖x‖ ^ 4 = x ^ 4 + rw [Real.norm_eq_abs, pow_abs, abs_of_nonneg (by positivity)] + +/-- `t ↦ exp (-t^2/2)` is integrable against `gaussianReal 0 v`: measurable and bounded by `1` +under a probability measure (`Integrable.mono'`). -/ +theorem cm_integrable_exp (v : NNReal) : + Integrable (fun t : ℝ => Real.exp (-(t ^ 2) / 2)) (gaussianReal 0 v) := by + refine (integrable_const (1 : ℝ)).mono' (by fun_prop : Measurable fun t : ℝ => + Real.exp (-(t ^ 2) / 2)).aestronglyMeasurable (Filter.Eventually.of_forall fun t => ?_) + rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _), Real.exp_le_one_iff] + have : 0 ≤ t ^ 2 := sq_nonneg t + linarith + +/-- The bracket `t^4 - 4t^2 + 8 - 8 exp (-t^2/2)` is integrable against `gaussianReal 0 v` +(sums and scalar multiples of integrable functions). -/ +theorem cm_integrable_bracket (v : NNReal) : + Integrable (fun t : ℝ => t ^ 4 - 4 * t ^ 2 + 8 - 8 * Real.exp (-(t ^ 2) / 2)) + (gaussianReal 0 v) := + ((((cm_integrable_pow_four v).sub ((c1_integrable_sq_gaussianReal v).const_mul 4)).add + (integrable_const 8)).sub ((cm_integrable_exp v).const_mul 8)) + +/-- The integral of the bracket against `gaussianReal 0 (5/3)` is +`25/3 - 20/3 + 8 - 8 sqrt (3/8)`: linearity, then the moments `∫ t^4 = 3 (5/3)^2` +(`tm_fourth_moment`), `∫ t^2 = 5/3` (`c1_second_moment_gaussianReal`) and +`∫ exp (-t^2/2) = sqrt (3/8)` (`tm_exp_moment`). -/ +theorem cm_integral_bracket : + ∫ t, (t ^ 4 - 4 * t ^ 2 + 8 - 8 * Real.exp (-(t ^ 2) / 2)) ∂(gaussianReal 0 (5 / 3 : NNReal)) = + 25 / 3 - 20 / 3 + 8 - 8 * Real.sqrt (3 / 8) := by + have h4 := cm_integrable_pow_four (5 / 3 : NNReal) + have h2 := c1_integrable_sq_gaussianReal (5 / 3 : NNReal) + have he := cm_integrable_exp (5 / 3 : NNReal) + have hA : Integrable (fun t : ℝ => t ^ 4 - 4 * t ^ 2) (gaussianReal 0 (5 / 3 : NNReal)) := + h4.sub (h2.const_mul 4) + have hB : Integrable (fun t : ℝ => t ^ 4 - 4 * t ^ 2 + 8) (gaussianReal 0 (5 / 3 : NNReal)) := + hA.add (integrable_const 8) + have hC : Integrable (fun t : ℝ => 8 * Real.exp (-(t ^ 2) / 2)) + (gaussianReal 0 (5 / 3 : NNReal)) := + he.const_mul 8 + have hD : Integrable (fun t : ℝ => 4 * t ^ 2) (gaussianReal 0 (5 / 3 : NNReal)) := + h2.const_mul 4 + rw [integral_sub hB hC, integral_add hA (integrable_const 8), integral_sub h4 hD, + integral_const_mul, integral_const_mul, integral_const, + tm_fourth_moment, c1_second_moment_gaussianReal, tm_exp_moment] + simp + norm_num + +/-- For an even, measurable, integrable `h` and `v ≠ 0`, the integral of `1_{Iio 0} h` against +`gaussianReal 0 v` is half the integral of `h`: the change of variables `t ↦ -t` +(`gaussianReal_map_neg`, `integral_map`) equates the integrals over `Iio 0` and `Ioi 0`, and +`integral_add_compl` with `integral_Ici_eq_integral_Ioi` (no atoms) adds them to the whole. -/ +theorem cm_setIntegral_Iio_even (v : NNReal) (hv : v ≠ 0) (h : ℝ → ℝ) (hm : Measurable h) + (heven : ∀ t, h (-t) = h t) (hint : Integrable h (gaussianReal 0 v)) : + ∫ t, Set.indicator (Set.Iio 0) h t ∂(gaussianReal 0 v) = + (1 / 2) * ∫ t, h t ∂(gaussianReal 0 v) := by + have : NullSingletonClass (gaussianReal 0 v) := nullSingletonClass_gaussianReal hv + set γ := gaussianReal 0 v with hγ + have hneg : γ.map (fun x => -x) = γ := by rw [hγ, gaussianReal_map_neg, neg_zero] + have hsym : ∫ t in Set.Iio (0 : ℝ), h t ∂γ = ∫ t in Set.Ioi (0 : ℝ), h t ∂γ := by + calc ∫ t in Set.Iio (0 : ℝ), h t ∂γ + = ∫ t, Set.indicator (Set.Iio (0 : ℝ)) h t ∂γ := + (integral_indicator measurableSet_Iio).symm + _ = ∫ t, Set.indicator (Set.Iio (0 : ℝ)) h t ∂(γ.map (fun x => -x)) := by rw [hneg] + _ = ∫ t, Set.indicator (Set.Iio (0 : ℝ)) h (-t) ∂γ := + integral_map measurable_neg.aemeasurable + (hm.indicator measurableSet_Iio).aestronglyMeasurable + _ = ∫ t, Set.indicator (Set.Ioi (0 : ℝ)) h t ∂γ := by + congr 1 + funext t + simp only [Set.indicator, Set.mem_Iio, Set.mem_Ioi, neg_lt_zero, heven] + _ = ∫ t in Set.Ioi (0 : ℝ), h t ∂γ := integral_indicator measurableSet_Ioi + have hsplit := integral_add_compl (μ := γ) (s := Set.Iio (0 : ℝ)) measurableSet_Iio hint + rw [Set.compl_Iio, integral_Ici_eq_integral_Ioi, ← hsym] at hsplit + rw [integral_indicator measurableSet_Iio] + linarith + +/-! ## `cm_` (d): the cone integrand on the product space -/ + +/-- `coneIntegrand q` is the indicator of `{q.1 < 0 ∧ q.2.1^2 + q.2.2^2 < q.1^2}` applied to +`(q.1^2 - q.2.1^2 - q.2.2^2)^2` (`cm_cone_iff`, `cm_det_transverse`). -/ +theorem cm_coneIntegrand_eq (q : ℝ × ℝ × ℝ) : + coneIntegrand q = Set.indicator {q : ℝ × ℝ × ℝ | q.1 < 0 ∧ q.2.1 ^ 2 + q.2.2 ^ 2 < q.1 ^ 2} + (fun q => (q.1 ^ 2 - q.2.1 ^ 2 - q.2.2 ^ 2) ^ 2) q := by + unfold coneIntegrand + simp only [Set.indicator, Set.mem_ofPred_eq, cm_cone_iff, cm_det_transverse] + +/-- `coneIntegrand` is measurable: by `cm_coneIntegrand_eq` it is a polynomial times the +indicator of a measurable set (`measurableSet_lt`, `Measurable.indicator`). -/ +theorem cm_measurable_coneIntegrand : Measurable coneIntegrand := by + have e : coneIntegrand = fun q => Set.indicator + {q : ℝ × ℝ × ℝ | q.1 < 0 ∧ q.2.1 ^ 2 + q.2.2 ^ 2 < q.1 ^ 2} + (fun q => (q.1 ^ 2 - q.2.1 ^ 2 - q.2.2 ^ 2) ^ 2) q := funext cm_coneIntegrand_eq + rw [e] + have h1 : MeasurableSet {q : ℝ × ℝ × ℝ | q.1 < 0} := + measurableSet_lt measurable_fst measurable_const + have h2 : MeasurableSet {q : ℝ × ℝ × ℝ | q.2.1 ^ 2 + q.2.2 ^ 2 < q.1 ^ 2} := + measurableSet_lt (by fun_prop) (by fun_prop) + exact Measurable.indicator (by fun_prop) (h1.inter h2) + +/-- `|coneIntegrand q| ≤ q.1^4`: inside the cone `0 ≤ q.1^2 - q.2.1^2 - q.2.2^2 ≤ q.1^2`, +outside it the integrand is `0`. -/ +theorem cm_coneIntegrand_bound (q : ℝ × ℝ × ℝ) : ‖coneIntegrand q‖ ≤ q.1 ^ 4 := by + rw [cm_coneIntegrand_eq] + by_cases h : q.1 < 0 ∧ q.2.1 ^ 2 + q.2.2 ^ 2 < q.1 ^ 2 + · rw [Set.indicator_of_mem + (show q ∈ {q : ℝ × ℝ × ℝ | q.1 < 0 ∧ q.2.1 ^ 2 + q.2.2 ^ 2 < q.1 ^ 2} from h), Real.norm_eq_abs, + abs_of_nonneg (sq_nonneg _)] + have h0 : 0 ≤ q.1 ^ 2 - q.2.1 ^ 2 - q.2.2 ^ 2 := by linarith [h.2] + have h1 : q.1 ^ 2 - q.2.1 ^ 2 - q.2.2 ^ 2 ≤ q.1 ^ 2 := by + nlinarith [sq_nonneg q.2.1, sq_nonneg q.2.2] + calc (q.1 ^ 2 - q.2.1 ^ 2 - q.2.2 ^ 2) ^ 2 ≤ (q.1 ^ 2) ^ 2 := pow_le_pow_left₀ h0 h1 2 + _ = q.1 ^ 4 := by ring + · rw [Set.indicator_of_notMem + (show q ∉ {q : ℝ × ℝ × ℝ | q.1 < 0 ∧ q.2.1 ^ 2 + q.2.2 ^ 2 < q.1 ^ 2} from h), norm_zero] + positivity + +/-- `coneIntegrand` is integrable against `N(0, 5/3) ⊗ (N(0, 1) ⊗ N(0, 1))`: it is dominated by +`q ↦ q.1^4`, integrable as a function of the first coordinate (`cm_integrable_pow_four`, +`Integrable.comp_fst`, `Integrable.mono'`). -/ +theorem cm_integrable_coneIntegrand : + Integrable coneIntegrand ((gaussianReal 0 (5 / 3 : NNReal)).prod + ((gaussianReal 0 1).prod (gaussianReal 0 1))) := + Integrable.mono' ((cm_integrable_pow_four (5 / 3 : NNReal)).comp_fst _) + cm_measurable_coneIntegrand.aestronglyMeasurable + (Filter.Eventually.of_forall cm_coneIntegrand_bound) + +/-- For `s < 0` the slice `p ↦ coneIntegrand (s, p)` is the closed-disk integrand +`1{|p|^2 ≤ s^2} (s^2 - |p|^2)^2` (on the circle `|p|^2 = s^2` both sides vanish). -/ +theorem cm_coneIntegrand_neg (s : ℝ) (p : ℝ × ℝ) (hs : s < 0) : + coneIntegrand (s, p) = Set.indicator {q : ℝ × ℝ | q.1 ^ 2 + q.2 ^ 2 ≤ s ^ 2} + (fun q => (s ^ 2 - q.1 ^ 2 - q.2 ^ 2) ^ 2) p := by + rw [cm_coneIntegrand_eq] + simp only [Set.indicator, Set.mem_ofPred_eq] + split_ifs with h1 h2 h2 + · rfl + · exact absurd h1.2.le h2 + · have he : p.1 ^ 2 + p.2 ^ 2 = s ^ 2 := le_antisymm h2 (not_lt.mp (fun h => h1 ⟨hs, h⟩)) + have he' : s ^ 2 - p.1 ^ 2 - p.2 ^ 2 = 0 := by linarith + rw [he'] + norm_num + · rfl + +/-- For `s ≥ 0` the slice `p ↦ coneIntegrand (s, p)` vanishes (the cone needs `s < 0`). -/ +theorem cm_coneIntegrand_nonneg (s : ℝ) (p : ℝ × ℝ) (hs : 0 ≤ s) : coneIntegrand (s, p) = 0 := by + rw [cm_coneIntegrand_eq, Set.indicator_of_notMem] + simp only [Set.mem_ofPred_eq, not_and] + intro h + linarith + +/-- The inner integral: `∫ coneIntegrand (s, ·) d(N(0, 1) ⊗ N(0, 1))` is +`1_{Iio 0}(s) (s^4 - 4s^2 + 8 - 8 exp (-s^2/2))` (`cm_coneIntegrand_neg` with +`cm_gauss2_integral` at `a = s^2`, and `cm_coneIntegrand_nonneg`). -/ +theorem cm_inner (s : ℝ) : + ∫ p, coneIntegrand (s, p) ∂((gaussianReal 0 1).prod (gaussianReal 0 1)) = + Set.indicator (Set.Iio 0) + (fun t : ℝ => t ^ 4 - 4 * t ^ 2 + 8 - 8 * Real.exp (-(t ^ 2) / 2)) s := by + rcases lt_or_ge s 0 with hs | hs + · rw [Set.indicator_of_mem (Set.mem_Iio.mpr hs)] + refine (integral_congr_ae (Filter.Eventually.of_forall + (fun p => cm_coneIntegrand_neg s p hs))).trans ?_ + rw [cm_gauss2_integral (s ^ 2) (sq_nonneg s)] + ring + · rw [Set.indicator_of_notMem (by simpa using hs)] + refine (integral_congr_ae (Filter.Eventually.of_forall + (fun p => cm_coneIntegrand_nonneg s p hs))).trans ?_ + simp + +end Aux + +/-! ## The `m = 2` cone moment identity for every inhabitant of the interface -/ + +/-- Proves: `UniversalLaw.Spec.Side24.ConeMomentD2Identity` (claim `side24-coefficient`), for every inhabitant `I : ConeMomentInterface`. +Method: transport along the law field `I.sxy_law` (`integral_map` after `Side24Interfaces.Aux.tm_aemeasurable_triple`), Fubini (`integral_prod`, with integrability from the bound `|coneIntegrand q| ≤ q.1^4`, `Aux.cm_integrable_coneIntegrand`), the cone identification `NegDefCone (s, x, y) ↔ s < 0 ∧ x^2 + y^2 < s^2` and `det A = s^2 - x^2 - y^2` (`Aux.cm_cone_iff`, `Aux.cm_det_transverse`), the inner planar Gaussian integral `s^4 - 4s^2 + 8 - 8 exp (-s^2/2)` for `s < 0` and `0` for `s ≥ 0` (`Aux.cm_inner`: `prod_withDensity`, polar coordinates `integral_comp_polarCoord_symm`, the substitution `z = r^2` onto `Side24Elementary.elementaryConeIntegral`), the symmetry halving of the even bracket under `gaussianReal 0 (5/3)` (`Aux.cm_setIntegral_Iio_even`), and the moments `25/3`, `5/3`, `sqrt (3/8)` (`Aux.cm_integral_bracket`, from `Side24Interfaces.Aux.tm_fourth_moment`, `c1_second_moment_gaussianReal`, `tm_exp_moment`). +Does not establish: the informal identity `D_2 = E[det(A)^2 1{A<0}] = (1/2)[25/3 - 20/3 + 8 - 8 sqrt(3/8)]` of the source as the source's object (the alignment of `ConeMomentInterface`, `transverseMatrix`, `NegDefCone`, `coneIntegrand` and `gaussianReal` with the source's objects is PENDING_INDEPENDENT_REVIEW); the source's Rayleigh integration route or its reduction of the cone to `s < -R` as the source's statements (the helper `Aux.cm_cone_iff` characterises the formal `NegDefCone` only, and `RayleighSquareLaw` is not used); integrability of the sample-space integrand under `I.P` as a separate statement (the integral is transported by `integral_map`, which needs only a.e.-measurability); that `D_2` 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; the value `29/6 - sqrt 6` (`ConeMomentD2Algebra`, not used here); 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 coneMomentD2Identity (I : ConeMomentInterface) : UniversalLaw.Spec.Side24.ConeMomentD2Identity I := by + unfold UniversalLaw.Spec.Side24.ConeMomentD2Identity + let := I.mΩ + have := I.isProb + calc ∫ ω, coneIntegrand (I.s ω, I.x ω, I.y ω) ∂I.P + = ∫ q, coneIntegrand q ∂(Measure.map (fun ω => (I.s ω, I.x ω, I.y ω)) I.P) := + (integral_map (UniversalLaw.Proofs.Side24Interfaces.Aux.tm_aemeasurable_triple I) + Aux.cm_measurable_coneIntegrand.aestronglyMeasurable).symm + _ = ∫ q, coneIntegrand q ∂((gaussianReal 0 (5 / 3 : NNReal)).prod + ((gaussianReal 0 1).prod (gaussianReal 0 1))) := by rw [I.sxy_law] + _ = ∫ s, ∫ p, coneIntegrand (s, p) ∂((gaussianReal 0 1).prod (gaussianReal 0 1)) + ∂(gaussianReal 0 (5 / 3 : NNReal)) := + integral_prod _ Aux.cm_integrable_coneIntegrand + _ = ∫ s, Set.indicator (Set.Iio 0) + (fun t : ℝ => t ^ 4 - 4 * t ^ 2 + 8 - 8 * Real.exp (-(t ^ 2) / 2)) s + ∂(gaussianReal 0 (5 / 3 : NNReal)) := by + congr 1 + funext s + exact Aux.cm_inner s + _ = (1 / 2) * ∫ t, (t ^ 4 - 4 * t ^ 2 + 8 - 8 * Real.exp (-(t ^ 2) / 2)) + ∂(gaussianReal 0 (5 / 3 : NNReal)) := + Aux.cm_setIntegral_Iio_even (5 / 3 : NNReal) (by norm_num) _ (by fun_prop) + (fun t => by simp only [neg_sq, show (-t) ^ 4 = t ^ 4 by ring]) + (Aux.cm_integrable_bracket _) + _ = (1 / 2) * (25 / 3 - 20 / 3 + 8 - 8 * Real.sqrt (3 / 8)) := by + rw [Aux.cm_integral_bracket] + +end UniversalLaw.Proofs.Side24ConeMoment diff --git a/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Covariance.lean b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Covariance.lean new file mode 100644 index 000000000..46303750e --- /dev/null +++ b/specifications/lean_statement_proofs_20261011/SpecificationProofs/Side24Covariance.lean @@ -0,0 +1,282 @@ +import Specifications.Side24 +import SpecificationProofs.Side24Analysis + +/-! +# SIDE24 reference covariance lower bound: `C_ref - I/3` is positive semidefinite +(statement module `Specifications.Side24`) + +Scientific effect: NONE. Organizational-independence credit: 0 (Anthropic / Claude). + +The one theorem outside the `Aux` namespace has as its type exactly the registered `Prop` +`UniversalLaw.Spec.Side24.ReferenceCovarianceLowerBound` of `Specifications/Side24.lean` +(statement packet `specifications/lean_statements_20261010`, which is not modified here). A +kernel-checked proof of that `Prop` is evidence about that `Prop` only, for the formal objects +(`E d`, `Kinf`, `derivContraction`, `jetDirs`, `svecScale`, `jointCov`, `Cref`, and Mathlib's +`Matrix.PosSemidef`, which includes `IsHermitian`): 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 show that `Cref` is the covariance of any +random vector (no field is constructed), it says nothing about `C_24`, `CovarianceOrdering` or +`EntrywiseCovarianceBound`, it does not enclose the coefficient `c_{d,24}`, and it promotes, +closes, discharges or reclassifies nothing. + +Provenance: the round-5 lane for `ReferenceCovarianceLowerBound`, the `Prop` that the round-4 +triage (workflow `wf_a3531fd8-891`) classified PROVABLE_WITH_WORK. The contraction values of +orders two, four and six are the round-4 helpers `Side24Analysis.Aux.rb_order_two`, +`rb_order_four`, `rb_order_six`, imported, not re-proved. Helper lemmas live in +`UniversalLaw.Proofs.Side24Covariance.Aux`, tagged `cv_`; the gate inventories only the theorem +outside that namespace. The mathematics: + +* Odd orders vanish at `0` (`cv_iteratedFDeriv_odd`, `cv_odd_zero`): `Kinf d` is even, i.e. + `Kinf d ∘ (-id) = Kinf d`, so `ContinuousLinearMap.iteratedFDeriv_comp_right` gives + `D^n Kinf 0 [m] = D^n Kinf 0 [-m_1, …, -m_n] = (-1)^n D^n Kinf 0 [m]` + (`ContinuousMultilinearMap.map_smul_univ`), hence `0` for odd `n`, in arbitrary directions. No + general-direction evaluation of order three or five is needed. +* The entries of `Cref d u R` (`cv_GG`, `cv_Gt`, `cv_tG`, `cv_tt`, `cv_GH`, `cv_HG`, `cv_tH`, + `cv_Ht`, `cv_HH`), read off `jointCov` with the sign `(-1)^{|b|}` and the `svec` scaling: gradient + block `⟪R i, R j⟫`; `Cov(G_i, t) = Cov(t, G_i) = -3 ⟪u, u⟫ ⟪R i, u⟫`; `Var(t) = 15 ⟪u, u⟫^3`; + every gradient/Hessian and `t`/Hessian entry is an odd-order contraction, hence `0`; the + Hessian block is `s_p s_q` times the three pairings of order four. +* In an orthonormal frame with `u = R 0` (`cv_decomp`), entrywise, + `Cref - (1/3) I = diagonal δ + (2/3) v vᵀ + w wᵀ` with `δ = 0` at `G_0`, `2/3` at `G_i` + (`i ≠ 0`), `7/6` at `t`, `5/3` at every `svec` coordinate; `v = e_{G_0} - (9/2) e_t`; `w` the + indicator of the diagonal Hessian coordinates. On the odd block this reads as the certificate + `(2/3) a^2 - 6 a b + (44/3) b^2 = (2/3) (a - (9/2) b)^2 + (7/6) b^2` (an informal reading of + the entries, not a separate lemma). The minors `2/3` and `7/9` named in the statement-packet + docstring are quoted for comparison only (`7/6 = (7/9)/(2/3)`): main's + `UniversalLaw.Side24.odd_block_eigenvalues`, `odd_block_shifted_minors` and + `smaller_eigenvalue_exceeds_third` are neither imported, cited nor relied on; the entry + arithmetic (`15 - 1/3 = 7/6 + (2/3) (9/2)^2`, `-3 = (2/3) (-9/2)`) is discharged inline by + `simp`/`norm_num` in `cv_decomp`, and no minor, no eigenvalue `8 ± sqrt 58` and no comparison + of an eigenvalue with `1/3` is computed or stated here. On the Hessian block the decomposition + is `J + 2 I - I/3` on the diagonal coordinates and `2 - 1/3` on the off-diagonal ones (the `√2` + scaling squares to `2`). +* Each summand is positive semidefinite (`Matrix.PosSemidef.diagonal`, + `Matrix.posSemidef_vecMulVec_self_star`, `Matrix.PosSemidef.smul`, `Matrix.PosSemidef.add`), so + `cv_psd` holds for every `d` with `0 < d`; the inventory theorem specialises it, and the + hypothesis `d = 2 ∨ d = 3` of the `Prop` is not used by the proof (a remark about the formal + statement, not a change of it). The symmetry the `Prop` requires (the `IsHermitian` half of + `PosSemidef`, of `Cref d (R ⟨0, hd⟩) R - (1/3) • 1` for orthonormal `R` and `0 < d`) follows + from the entrywise decomposition `cv_decomp` into symmetric summands, not from any symmetry of + `iteratedFDeriv`. Symmetry of `Cref d u R` for general `u` and `R` is not stated here. + +Remark recorded by the lane, for the maintainers (nothing below is edited here): the proof-packet +README row "Side24 covariance and density" says, of this `Prop`, that "order 5 needs one more +general-direction level of the peeling" and names `Matrix.posSemidef_iff_dotProduct_mulVec` as the +route; neither is load-bearing here (the evenness argument above replaces the order-five level, +and the decomposition into manifestly positive semidefinite summands replaces the quadratic-form +expansion over the index type). This is a remark about an informal reason, not about any claim; +no register label changes. +-/ + +namespace UniversalLaw.Proofs.Side24Covariance + +open UniversalLaw.Spec.Side24 +open scoped RealInnerProductSpace + +namespace Aux + +/-! ## `cv_`: odd orders of `Kinf d` vanish at the origin -/ + +/-- `Kinf d` is even: `Kinf d ∘ (-id) = Kinf d` (`norm_neg`). -/ +theorem cv_kinf_comp_neg {d : ℕ} : + (Kinf d ∘ (-ContinuousLinearMap.id ℝ (E d))) = Kinf d := by + funext x + simp [Kinf] + +/-- Every odd-order iterated derivative of `Kinf d` at `0` vanishes, in arbitrary directions: +`ContinuousLinearMap.iteratedFDeriv_comp_right` with the evenness `cv_kinf_comp_neg` gives +`D^n Kinf 0 [m] = (-1)^n D^n Kinf 0 [m]` (`ContinuousMultilinearMap.map_smul_univ`). -/ +theorem cv_iteratedFDeriv_odd {d : ℕ} (n : ℕ) (hn : Odd n) (m : Fin n → E d) : + iteratedFDeriv ℝ n (Kinf d) 0 m = 0 := by + have h := (-ContinuousLinearMap.id ℝ (E d)).iteratedFDeriv_comp_right + (Side24Analysis.Aux.rb_kinf_contDiff (d := d)) (0 : E d) (i := n) (mod_cast le_top) + rw [cv_kinf_comp_neg] at h + have h2 := congrArg (fun T => T m) h + simp only [ContinuousMultilinearMap.compContinuousLinearMap_apply, map_zero] at h2 + have h3 : (fun i => (-ContinuousLinearMap.id ℝ (E d)) (m i)) = fun i => (-1 : ℝ) • m i := by + funext i + simp + rw [h3, ContinuousMultilinearMap.map_smul_univ] at h2 + simp only [Finset.prod_const, Finset.card_univ, Fintype.card_fin, hn.neg_one_pow] at h2 + simp at h2 + linarith + +/-- A contraction of `Kinf d` at `0` against a list of odd length vanishes. -/ +theorem cv_odd_zero {d : ℕ} (l : List (E d)) (hl : Odd l.length) : + derivContraction d (Kinf d) 0 l = 0 := + cv_iteratedFDeriv_odd _ hl _ + +/-! ## `cv_`: the entries of `Cref d u R` in a general frame -/ + +/-- Gradient/gradient entry: `(-1)^1 · (-⟪R i, R j⟫) = ⟪R i, R j⟫`. -/ +theorem cv_GG {d : ℕ} (u : E d) (R : Fin d → E d) (i j : Fin d) : + Cref d u R (Sum.inl i) (Sum.inl j) = ⟪R i, R j⟫ := by + simp [Cref, jointCov, svecScale, jetDirs, Side24Analysis.Aux.rb_order_two] + +/-- Gradient/`t` entry: `(-1)^3 · D^4 Kinf 0 [R i, u, u, u] = -3 ⟪u, u⟫ ⟪R i, u⟫`. -/ +theorem cv_Gt {d : ℕ} (u : E d) (R : Fin d → E d) (i : Fin d) (t : Unit) : + Cref d u R (Sum.inl i) (Sum.inr (Sum.inl t)) = -(3 * (⟪u, u⟫ * ⟪R i, u⟫)) := by + simp [Cref, jointCov, svecScale, jetDirs, Side24Analysis.Aux.rb_order_four] + ring + +/-- `t`/gradient entry: `(-1)^1 · D^4 Kinf 0 [u, u, u, R j] = -3 ⟪u, u⟫ ⟪u, R j⟫`. -/ +theorem cv_tG {d : ℕ} (u : E d) (R : Fin d → E d) (j : Fin d) (t : Unit) : + Cref d u R (Sum.inr (Sum.inl t)) (Sum.inl j) = -(3 * (⟪u, u⟫ * ⟪u, R j⟫)) := by + simp [Cref, jointCov, svecScale, jetDirs, Side24Analysis.Aux.rb_order_four] + ring + +/-- `t`/`t` entry: `(-1)^3 · D^6 Kinf 0 [u, …, u] = 15 ⟪u, u⟫^3`. -/ +theorem cv_tt {d : ℕ} (u : E d) (R : Fin d → E d) (t t' : Unit) : + Cref d u R (Sum.inr (Sum.inl t)) (Sum.inr (Sum.inl t')) = + 15 * (⟪u, u⟫ * ⟪u, u⟫ * ⟪u, u⟫) := by + simp [Cref, jointCov, svecScale, jetDirs, Side24Analysis.Aux.rb_order_six] + ring + +/-- Gradient/Hessian entry: an order-three contraction, hence `0`. -/ +theorem cv_GH {d : ℕ} (u : E d) (R : Fin d → E d) (i : Fin d) (q : SymIndex d) : + Cref d u R (Sum.inl i) (Sum.inr (Sum.inr q)) = 0 := by + simp only [Cref, jointCov, Matrix.of_apply, jetDirs, List.cons_append, List.nil_append] + rw [cv_odd_zero _ ⟨1, rfl⟩, mul_zero] + +/-- Hessian/gradient entry: an order-three contraction, hence `0`. -/ +theorem cv_HG {d : ℕ} (u : E d) (R : Fin d → E d) (p : SymIndex d) (j : Fin d) : + Cref d u R (Sum.inr (Sum.inr p)) (Sum.inl j) = 0 := by + simp only [Cref, jointCov, Matrix.of_apply, jetDirs, List.cons_append, List.nil_append] + rw [cv_odd_zero _ ⟨1, rfl⟩, mul_zero] + +/-- `t`/Hessian entry: an order-five contraction, hence `0`. -/ +theorem cv_tH {d : ℕ} (u : E d) (R : Fin d → E d) (t : Unit) (q : SymIndex d) : + Cref d u R (Sum.inr (Sum.inl t)) (Sum.inr (Sum.inr q)) = 0 := by + simp only [Cref, jointCov, Matrix.of_apply, jetDirs, List.cons_append, List.nil_append] + rw [cv_odd_zero _ ⟨2, rfl⟩, mul_zero] + +/-- Hessian/`t` entry: an order-five contraction, hence `0`. -/ +theorem cv_Ht {d : ℕ} (u : E d) (R : Fin d → E d) (p : SymIndex d) (t : Unit) : + Cref d u R (Sum.inr (Sum.inr p)) (Sum.inr (Sum.inl t)) = 0 := by + simp only [Cref, jointCov, Matrix.of_apply, jetDirs, List.cons_append, List.nil_append] + rw [cv_odd_zero _ ⟨2, rfl⟩, mul_zero] + +/-- Hessian/Hessian entry: `s_p s_q (-1)^2` times the three pairings of +`D^4 Kinf 0 [R p₁, R p₂, R q₁, R q₂]`. -/ +theorem cv_HH {d : ℕ} (u : E d) (R : Fin d → E d) (p q : SymIndex d) : + Cref d u R (Sum.inr (Sum.inr p)) (Sum.inr (Sum.inr q)) = + svecScale d (Sum.inr (Sum.inr p)) * svecScale d (Sum.inr (Sum.inr q)) * + (⟪R q.1.1, R q.1.2⟫ * ⟪R p.1.1, R p.1.2⟫ + ⟪R p.1.2, R q.1.2⟫ * ⟪R p.1.1, R q.1.1⟫ + + ⟪R p.1.1, R q.1.2⟫ * ⟪R p.1.2, R q.1.1⟫) := by + simp only [Cref, jointCov, Matrix.of_apply, jetDirs, List.cons_append, List.nil_append, + Side24Analysis.Aux.rb_order_four] + simp + +/-! ## `cv_`: the decomposition into positive semidefinite summands -/ + +/-- In an orthonormal frame with `u = R 0`, `Cref - (1/3) I` is, entrywise, a nonnegative +diagonal matrix plus `(2/3) v vᵀ` with `v = e_{G_0} - (9/2) e_t` plus `w wᵀ` with `w` the +indicator of the diagonal Hessian coordinates. -/ +theorem cv_decomp {d : ℕ} (hd : 0 < d) (R : Fin d → E d) (hR : Orthonormal ℝ R) : + Cref d (R ⟨0, hd⟩) R - (1 / 3 : ℝ) • (1 : Matrix (JointIndex d) (JointIndex d) ℝ) = + Matrix.diagonal (Sum.elim (fun i : Fin d => if i = ⟨0, hd⟩ then (0 : ℝ) else 2 / 3) + (Sum.elim (fun _ : Unit => (7 / 6 : ℝ)) (fun _ : SymIndex d => (5 / 3 : ℝ)))) + + (2 / 3 : ℝ) • Matrix.vecMulVec + (Sum.elim (fun i : Fin d => if i = ⟨0, hd⟩ then (1 : ℝ) else 0) + (Sum.elim (fun _ : Unit => (-9 / 2 : ℝ)) (fun _ : SymIndex d => (0 : ℝ)))) + (Sum.elim (fun i : Fin d => if i = ⟨0, hd⟩ then (1 : ℝ) else 0) + (Sum.elim (fun _ : Unit => (-9 / 2 : ℝ)) (fun _ : SymIndex d => (0 : ℝ)))) + + Matrix.vecMulVec + (Sum.elim (fun _ : Fin d => (0 : ℝ)) + (Sum.elim (fun _ : Unit => (0 : ℝ)) + (fun p : SymIndex d => if p.1.1 = p.1.2 then (1 : ℝ) else 0))) + (Sum.elim (fun _ : Fin d => (0 : ℝ)) + (Sum.elim (fun _ : Unit => (0 : ℝ)) + (fun p : SymIndex d => if p.1.1 = p.1.2 then (1 : ℝ) else 0))) := by + have hite := orthonormal_iff_ite.mp hR + ext a b + rcases a with i | t | p <;> rcases b with j | t' | q + · simp only [Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply, Matrix.one_apply, + Matrix.diagonal_apply, Matrix.vecMulVec_apply, cv_GG, hite, Sum.inl.injEq, Sum.elim_inl, + smul_eq_mul] + split_ifs <;> simp_all <;> norm_num + · rw [Matrix.sub_apply, cv_Gt, hite, hite] + simp [Matrix.add_apply, Matrix.smul_apply, Matrix.vecMulVec_apply] + split_ifs <;> norm_num + · simp [Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply, Matrix.vecMulVec_apply, cv_GH] + · rw [Matrix.sub_apply, cv_tG, hite, hite] + simp [Matrix.add_apply, Matrix.smul_apply, Matrix.vecMulVec_apply] + by_cases hj : j = ⟨0, hd⟩ + · subst hj + norm_num + · simp [hj, Ne.symm hj] + · rw [Matrix.sub_apply, cv_tt, hite] + simp [Matrix.add_apply, Matrix.smul_apply, Matrix.vecMulVec_apply] + norm_num + · simp [Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply, Matrix.vecMulVec_apply, cv_tH] + · simp [Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply, Matrix.vecMulVec_apply, cv_HG] + · simp [Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply, Matrix.vecMulVec_apply, cv_Ht] + · obtain ⟨⟨i, j⟩, hij⟩ := p + obtain ⟨⟨k, l⟩, hkl⟩ := q + simp [Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply, Matrix.one_apply, + Matrix.diagonal_apply, Matrix.vecMulVec_apply, cv_HH, hite, svecScale] + rw [Fin.le_def] at hij hkl + split_ifs <;> simp_all [Fin.ext_iff] <;> first | omega | norm_num + +/-- `Cref - (1/3) I` is positive semidefinite in every orthonormal frame with `u = R 0`, for every +`d > 0`: the three summands of `cv_decomp` are positive semidefinite. -/ +theorem cv_psd {d : ℕ} (hd : 0 < d) (R : Fin d → E d) (hR : Orthonormal ℝ R) : + (Cref d (R ⟨0, hd⟩) R - + (1 / 3 : ℝ) • (1 : Matrix (JointIndex d) (JointIndex d) ℝ)).PosSemidef := by + rw [cv_decomp hd R hR] + refine Matrix.PosSemidef.add (Matrix.PosSemidef.add ?_ ?_) ?_ + · refine Matrix.PosSemidef.diagonal fun a => ?_ + rcases a with i | t | p + · simp only [Sum.elim_inl, Pi.zero_apply] + split_ifs <;> norm_num + · simp only [Sum.elim_inr, Sum.elim_inl, Pi.zero_apply] + norm_num + · simp only [Sum.elim_inr, Pi.zero_apply] + norm_num + · refine Matrix.PosSemidef.smul ?_ (by norm_num) + simpa using Matrix.posSemidef_vecMulVec_self_star + (Sum.elim (fun i : Fin d => if i = ⟨0, hd⟩ then (1 : ℝ) else 0) + (Sum.elim (fun _ : Unit => (-9 / 2 : ℝ)) (fun _ : SymIndex d => (0 : ℝ)))) + · simpa using Matrix.posSemidef_vecMulVec_self_star + (Sum.elim (fun _ : Fin d => (0 : ℝ)) + (Sum.elim (fun _ : Unit => (0 : ℝ)) + (fun p : SymIndex d => if p.1.1 = p.1.2 then (1 : ℝ) else 0))) + +end Aux + +/-- Proves: `UniversalLaw.Spec.Side24.ReferenceCovarianceLowerBound` (claim +`side24-coefficient`), for the formal `Cref` of `Specifications.Side24`. +Method: odd-order contractions of the even kernel `Kinf d` vanish at `0` (`Aux.cv_odd_zero`, +via `ContinuousLinearMap.iteratedFDeriv_comp_right` and `ContinuousMultilinearMap.map_smul_univ`); +the orders two, four and six are the round-4 helpers `Side24Analysis.Aux.rb_order_two`, +`rb_order_four`, `rb_order_six`; in an orthonormal frame with `u = R 0`, +`Cref - I/3 = diagonal (0, 2/3, …, 2/3; 7/6; 5/3, …, 5/3) + (2/3) v vᵀ + w wᵀ` entrywise +(`Aux.cv_decomp`, `v = e_{G_0} - (9/2) e_t`, `w` the indicator of the diagonal Hessian +coordinates), and each summand is positive semidefinite (`Aux.cv_psd`, which holds for every +`d > 0`; the hypothesis `d = 2 ∨ d = 3` of the `Prop` is not used by the proof). +Does not establish: the informal Section 3 display "C_ref >= I/3" of the source as the source's +object (the alignment of `E d`, `Kinf`, `derivContraction`, `jetDirs`, `svecScale`, `jointCov` and +`Cref` with the source's `R^d`, `K_infty`, jet and joint covariance, and of Mathlib's +`Matrix.PosSemidef` (which includes `IsHermitian`) with the source's `>=` order, is +PENDING_INDEPENDENT_REVIEW); that `Cref` is the covariance of any random vector (no field is +constructed); symmetry of `Cref d u R` for general `u` or a general frame (only the `IsHermitian` +half of the `Prop`'s `PosSemidef`, at `u = R 0` in an orthonormal frame with `0 < d`, is proved); +the statement module's own `Does not claim:` text ("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"), which describes the statement packet and is not +edited here; main's `UniversalLaw.Side24.odd_block_eigenvalues`, `odd_block_shifted_minors` and +`smaller_eigenvalue_exceeds_third` are neither imported, cited nor relied on (the certificate's +entry arithmetic is discharged inline by `simp`/`norm_num` in `Aux.cv_decomp`), and no minor +(in particular not `7/9`), no eigenvalue `8 ± sqrt 58` and no comparison of the smaller +eigenvalue with `1/3` is computed or stated here; anything about `C_24`, `CovarianceOrdering` or +`EntrywiseCovarianceBound`; any enclosure of the coefficient `c_{d,24}`; 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. It promotes, closes, +discharges or reclassifies nothing. -/ +theorem referenceCovarianceLowerBound : UniversalLaw.Spec.Side24.ReferenceCovarianceLowerBound := by + intro d _ hd R hR + exact Aux.cv_psd hd R hR + +end UniversalLaw.Proofs.Side24Covariance diff --git a/specifications/lean_statements_20261010/README.md b/specifications/lean_statements_20261010/README.md index e421fa7f4..ff0584ebe 100644 --- a/specifications/lean_statements_20261010/README.md +++ b/specifications/lean_statements_20261010/README.md @@ -4,11 +4,12 @@ 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 (72 of the 141 Props, 73 theorems); the register lists them under the claim rows as +Props (74 of the 141 Props, 75 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 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), +PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6…, the head of PR #423, same tree as the head; the 2 of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964c…), +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. @@ -52,11 +53,11 @@ 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 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`. +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..`): 74 of the 141 Props have one (75 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; the 2 theorems marked † (`Side24Covariance` and `Side24ConeMoment`) were added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), and have no hosted receipt yet: the next hosted `lean-specification-proofs.yml` run 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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), no hosted receipt yet) | |---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | | `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | @@ -73,7 +74,7 @@ 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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), no hosted receipt yet) | |---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | | `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | @@ -102,7 +103,7 @@ 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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), no hosted receipt yet) | |---|---|---|---|---| | `ThetaSummable` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.thetaSummable` | | `IsUnitTuple` | def_prop | `side24-proof` | `side24-coefficient` | — | @@ -115,7 +116,7 @@ ladder; it is `specified`. | `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` | — | +| `ReferenceCovarianceLowerBound` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Covariance.referenceCovarianceLowerBound†` | | `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` | @@ -124,7 +125,7 @@ ladder; it is `specified`. | `ConeMomentInterface` | structure | `side24-proof` | `side24-coefficient` | — | | `NegDefCone` | def_prop | `side24-proof` | `side24-coefficient` | — | | `ConeMomentD1Identity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.coneMomentD1Identity` | -| `ConeMomentD2Identity` | def_prop | `side24-proof` | `side24-coefficient` | — | +| `ConeMomentD2Identity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24ConeMoment.coneMomentD2Identity†` | | `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` | @@ -135,7 +136,7 @@ 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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), no hosted receipt yet) | |---|---|---|---|---| | `IsCriticalPoint` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | — | | `HessianHasInertia` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | — | @@ -168,7 +169,7 @@ ladder; it is `specified`. ### `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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), no hosted receipt yet) | |---|---|---|---|---| | `RealizedData` | structure | `p15-realized-covers` | `p15-realized-covers` | — | | `IsRealizedFamily` | def_prop | `p15-realized-covers` | `p15-realized-covers` | — | @@ -231,11 +232,11 @@ 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 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`. +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`). 157 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; 85 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (75 theorems, 74 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; the 2 theorems marked † (`Side24Covariance` and `Side24ConeMoment`) were added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), and have no hosted receipt yet: the next hosted `lean-specification-proofs.yml` run 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`; 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`; 19 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`, `Side24Covariance.referenceCovarianceLowerBound†`, `Side24ConeMoment.coneMomentD2Identity†` | | `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` | @@ -355,7 +356,7 @@ 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 73 theorems whose type is exactly one of 72 registered Props (P15, Side24 + packet by path and holds 75 theorems whose type is exactly one of 74 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 @@ -365,7 +366,8 @@ Packets listed by PROOF_INDEX by id only (82 rows, kind `proof_index_result`, fo 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 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). + actual tested merge 9833d6d9… with base 12e072d6…, the head of PR #423, receipt theorem_count 73, and were first receipted by that run; the 2 theorems + of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…, receipt theorem_count 73), 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 c8bb36ca1..adbd01250 100644 --- a/specifications/lean_statements_20261010/REGISTER.json +++ b/specifications/lean_statements_20261010/REGISTER.json @@ -886,6 +886,24 @@ "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)\": 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." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24Covariance.lean", + "declaration": "UniversalLaw.Proofs.Side24Covariance.referenceCovarianceLowerBound", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "Thus C_ref >= I/3 and", + "does_not_claim": "the informal Section 3 display \"C_ref >= I/3\" of the source as the source's object (the alignment of `E d`, `Kinf`, `derivContraction`, `jetDirs`, `svecScale`, `jointCov` and `Cref` with the source's `R^d`, `K_infty`, jet and joint covariance, and of Mathlib's `Matrix.PosSemidef` (which includes `IsHermitian`) with the source's `>=` order, is PENDING_INDEPENDENT_REVIEW); that `Cref` is the covariance of any random vector (no field is constructed); symmetry of `Cref d u R` for general `u` or a general frame (only the `IsHermitian` half of the `Prop`'s `PosSemidef`, at `u = R 0` in an orthonormal frame with `0 < d`, is proved); the statement module's own `Does not claim:` text (\"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\"), which describes the statement packet and is not edited here; main's `UniversalLaw.Side24.odd_block_eigenvalues`, `odd_block_shifted_minors` and `smaller_eigenvalue_exceeds_third` are neither imported, cited nor relied on (the certificate's entry arithmetic is discharged inline by `simp`/`norm_num` in `Aux.cv_decomp`), and no minor (in particular not `7/9`), no eigenvalue `8 ± sqrt 58` and no comparison of the smaller eigenvalue with `1/3` is computed or stated here; anything about `C_24`, `CovarianceOrdering` or `EntrywiseCovarianceBound`; any enclosure of the coefficient `c_{d,24}`; 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. It promotes, closes, discharges or reclassifies nothing." + }, + { + "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", + "module": "SpecificationProofs/Side24ConeMoment.lean", + "declaration": "UniversalLaw.Proofs.Side24ConeMoment.coneMomentD2Identity", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "D_2 = (1/2)[25/3-20/3+8-8sqrt(3/8)]", + "does_not_claim": "the informal identity `D_2 = E[det(A)^2 1{A<0}] = (1/2)[25/3 - 20/3 + 8 - 8 sqrt(3/8)]` of the source as the source's object (the alignment of `ConeMomentInterface`, `transverseMatrix`, `NegDefCone`, `coneIntegrand` and `gaussianReal` with the source's objects is PENDING_INDEPENDENT_REVIEW); the source's Rayleigh integration route or its reduction of the cone to `s < -R` as the source's statements (the helper `Aux.cm_cone_iff` characterises the formal `NegDefCone` only, and `RayleighSquareLaw` is not used); integrability of the sample-space integrand under `I.P` as a separate statement (the integral is transported by `integral_map`, which needs only a.e.-measurability); that `D_2` 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; the value `29/6 - sqrt 6` (`ConeMomentD2Algebra`, not used here); 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." } ], "kernel_evidence": [ @@ -896,12 +914,12 @@ }, { "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); 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)." + "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); theorems added after PR head be014794 (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73) have no hosted receipt yet: the next hosted lean-specification-proofs.yml run is their first receipt", + "note": "19 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, Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). 2 (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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. 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))." + "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, ReferenceCovarianceLowerBound, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (19 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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); the lower bound C_ref - I/3 positive semidefinite for the formal matrix Cref at u = R 0 in an orthonormal frame, for d = 2 or 3 as the Prop states it, its IsHermitian half included (ReferenceCovarianceLowerBound: the odd-order contractions of the even kernel Kinf vanish at 0, and Cref - I/3 decomposes entrywise into a diagonal matrix and two rank-one squares; the proof does not use the hypothesis d = 2 or d = 3, and only the helper Aux.cv_psd, not an inventory theorem, is stated for every 0 < d; Cref is not shown to be the covariance of any random vector, symmetry of Cref d u R for a general u or frame is not stated, and main's odd_block_eigenvalues, odd_block_shifted_minors and smaller_eigenvalue_exceeds_third are neither imported, cited nor relied on: no minor, no eigenvalue 8 +/- sqrt 58 and no comparison of an eigenvalue with 1/3 is computed); 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); and, over the same interface, the m = 2 cone moment, the integral of coneIntegrand (s, x, y) under I.P, equal to (1/2)(25/3 - 20/3 + 8 - 8 sqrt(3/8)) (ConeMomentD2Identity: proved for every inhabitant using, of the law fields, only sxy_law, with the instances mΩ and isProb; the fields A and A_law are not used; by transport along sxy_law, Fubini, the cone identification s < 0 and x^2 + y^2 < s^2, polar coordinates onto ElementaryConeIntegral and the symmetry halving under N(0, 5/3); RayleighSquareLaw is not used, the source's Rayleigh integration route is not reproduced as the source's statements, D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law, and the value 29/6 - sqrt 6 (ConeMomentD2Algebra) is not used). None of them is the Gaussian conditioning, 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, 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, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); the 2 theorems of this row added after PR head be014794 (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) await the next hosted lean-specification-proofs.yml run, their first receipt; the other theorems of this row have 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); each of those 17 theorems 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, ConeMomentD2Identity PROVABLE_WITH_WORK (weeks): proved in round 5 (2026-10-12; see entries, no hosted receipt yet); 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 now has a theorem for every inhabitant of ConeMomentInterface, with no hosted receipt yet, but no inhabitant is constructed and its laws are not derived as the conditional law A | V = 0 of the reference field, so D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law). 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; for ReferenceCovarianceLowerBound and ConeMomentD2Identity, proved in round 5 (no hosted receipt yet), the README's PROVABLE_WITH_WORK (weeks) reasons are superseded by their theorems (the ReferenceCovarianceLowerBound lane records that the order-five peeling step and the Matrix.posSemidef_iff_dotProduct_mulVec route the README named are not load-bearing); the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, 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 19 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); ReferenceCovarianceLowerBound with the Section 3 display C_ref >= I/3, with Mathlib's Matrix.PosSemidef, IsHermitian included, read as the source's order)." }, { "claim_id": "lifetime-remainder", @@ -1061,7 +1079,7 @@ { "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); 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)." + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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.", @@ -1705,7 +1723,7 @@ { "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); 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)." + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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.", @@ -1840,7 +1858,7 @@ { "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); 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)." + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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.", @@ -2038,7 +2056,7 @@ { "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); 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)." + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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.", @@ -2479,7 +2497,7 @@ { "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); 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)." + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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 +2959,7 @@ "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 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.", + "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 19 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, Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity; 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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, ConeMomentD2Identity) 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)." }, { @@ -3050,7 +3068,7 @@ { "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); 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)." + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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.", @@ -3422,7 +3440,7 @@ { "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); 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)." + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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.", @@ -3539,7 +3557,7 @@ { "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); 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)." + "note": "2 theorems of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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.", @@ -4314,7 +4332,7 @@ { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", "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)." + "note": "1 theorem of this row (RNCluster.rareClusterPoissonObstructionO) is inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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 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.", diff --git a/specifications/lean_statements_20261010/SCOPE.md b/specifications/lean_statements_20261010/SCOPE.md index 4e8ab518a..55fca1d25 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` (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); +`specifications/lean_statement_proofs_20261011` (75 theorems typed exactly as 74 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 be014794 (run 38187538744, actual tested merge 79d3964c…) 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..`): 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`. +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..`): 74 of the 141 Props have one (75 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; the 2 theorems marked † (`Side24Covariance` and `Side24ConeMoment`) were added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), and have no hosted receipt yet: the next hosted `lean-specification-proofs.yml` run 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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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. | — | @@ -109,7 +110,7 @@ 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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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). | `Side24Analysis.thetaSummable` | | `IsUnitTuple` | def_prop | `side24-proof` | For q<=6 unit directional differentiations of phi(x)=exp(-\|x\|^2/2), product-rule | anything. | — | @@ -122,7 +123,7 @@ a theorem between string literals, an indented Prop definition, a proof written | `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. | — | +| `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. | `Side24Covariance.referenceCovarianceLowerBound†` | | `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` | @@ -131,7 +132,7 @@ a theorem between string literals, an indented Prop definition, a proof written | `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` | -| `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. | — | +| `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. | `Side24ConeMoment.coneMomentD2Identity†` | | `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` | @@ -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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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. | — | @@ -175,7 +176,7 @@ a theorem between string literals, an indented Prop definition, a proof written ### `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; 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…) | +| 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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. | — | @@ -238,11 +239,11 @@ 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 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`. +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`). 157 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; 85 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (75 theorems, 74 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; the 2 theorems marked † (`Side24Covariance` and `Side24ConeMoment`) were added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), and have no hosted receipt yet: the next hosted `lean-specification-proofs.yml` run 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`; 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)). | +| `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`; 19 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`, `Side24Covariance.referenceCovarianceLowerBound†`, `Side24ConeMoment.coneMomentD2Identity†` | 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, ReferenceCovarianceLowerBound, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (19 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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); the lower bound C_ref - I/3 positive semidefinite for the formal matrix Cref at u = R 0 in an orthonormal frame, for d = 2 or 3 as the Prop states it, its IsHermitian half included (ReferenceCovarianceLowerBound: the odd-order contractions of the even kernel Kinf vanish at 0, and Cref - I/3 decomposes entrywise into a diagonal matrix and two rank-one squares; the proof does not use the hypothesis d = 2 or d = 3, and only the helper Aux.cv_psd, not an inventory theorem, is stated for every 0 < d; Cref is not shown to be the covariance of any random vector, symmetry of Cref d u R for a general u or frame is not stated, and main's odd_block_eigenvalues, odd_block_shifted_minors and smaller_eigenvalue_exceeds_third are neither imported, cited nor relied on: no minor, no eigenvalue 8 +/- sqrt 58 and no comparison of an eigenvalue with 1/3 is computed); 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); and, over the same interface, the m = 2 cone moment, the integral of coneIntegrand (s, x, y) under I.P, equal to (1/2)(25/3 - 20/3 + 8 - 8 sqrt(3/8)) (ConeMomentD2Identity: proved for every inhabitant using, of the law fields, only sxy_law, with the instances mΩ and isProb; the fields A and A_law are not used; by transport along sxy_law, Fubini, the cone identification s < 0 and x^2 + y^2 < s^2, polar coordinates onto ElementaryConeIntegral and the symmetry halving under N(0, 5/3); RayleighSquareLaw is not used, the source's Rayleigh integration route is not reproduced as the source's statements, D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law, and the value 29/6 - sqrt 6 (ConeMomentD2Algebra) is not used). None of them is the Gaussian conditioning, 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, 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, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); the 2 theorems of this row added after PR head be014794 (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) await the next hosted lean-specification-proofs.yml run, their first receipt; the other theorems of this row have 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); each of those 17 theorems 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, ConeMomentD2Identity PROVABLE_WITH_WORK (weeks): proved in round 5 (2026-10-12; see entries, no hosted receipt yet); 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 now has a theorem for every inhabitant of ConeMomentInterface, with no hosted receipt yet, but no inhabitant is constructed and its laws are not derived as the conditional law A \| V = 0 of the reference field, so D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law). 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; for ReferenceCovarianceLowerBound and ConeMomentD2Identity, proved in round 5 (no hosted receipt yet), the README's PROVABLE_WITH_WORK (weeks) reasons are superseded by their theorems (the ReferenceCovarianceLowerBound lane records that the order-five peeling step and the Matrix.posSemidef_iff_dotProduct_mulVec route the README named are not load-bearing); the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, 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 19 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); ReferenceCovarianceLowerBound with the Section 3 display C_ref >= I/3, with Mathlib's Matrix.PosSemidef, IsHermitian included, read as the source's order). | | `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. | @@ -258,7 +259,7 @@ 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 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.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 19 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, Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity; 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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, ConeMomentD2Identity) 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). | @@ -442,4 +443,5 @@ identity), never about the claim row it sits under; `kernel-checked` is a run-re 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; 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…). +12e072d6…, the head of PR #423; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efa…; theorems added after PR head be014794 (run 38187538744, actual tested merge 79d3964c… with base 12e072d6…) await the next hosted +`lean-specification-proofs.yml` run, their first receipt). From 32fc98455128c9903741aece5f79744ffe928d31 Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 12 Oct 2026 03:10:08 +0000 Subject: [PATCH 2/2] Proof packet round 5: bind the two theorems to their first hosted receipt (run 38195810463) One forward commit on c6ce025 (draft PR #439, source review PASS 5486104267). It changes the register and the four packet documents only; no Lean byte changes. The first hosted receipt of the round-5 theorems is workflow lean-specification-proofs.yml run 38195810463, attempt 1, job 114638065311 (proof-evidence, success), event pull_request on PR #439 head c6ce025652f76be6e580fa24947380253450cd3b. Its checked_commit is the synthetic merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 of that head into base be01479435f98f34b802258ee9ec3c3578f37169 (the head of PR #434), with tree e723d6ab equal to the head's tree. The receipt says formalization_status kernel-checked, leanchecker PASS, theorem_count 75, covered_props 74, controls refused, worktree CLEAN. Artifact lean-specification-proofs-38195810463-1 (id 11723487528, 29 files, 45823 bytes), sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff. REGISTER.json changes 23 string leaves across 11 rows, in the shapes round 4 used at be014794 for run 38180705952: - In side24-coefficient, the pending tier for the two theorems becomes the receipted tier naming run 38195810463, PR head c6ce0256 and actual tested merge 1320c5b2... with base be014794.... - All ten proof-packet kernel_evidence rows gain one corroboration clause: all 75 theorems of PR head c6ce0256 are receipted on that head by that run. - The "await the next hosted run" and "no hosted receipt yet" clauses of side24-coefficient and math.side24-coefficient become the binding. Every statements array, every formal_progress (138 none, 33 specified) and every other field is byte-identical; no triage verdict or effort changes. The statement packet README/SCOPE are re-rendered with no dagger left, and the proof packet README/SCOPE bind the run. Correction to the c6ce025 commit message: it gives the REGISTER.json blob as fa96260c. That was the pre-fix-round output; the committed blob at c6ce025 is adbd0125. The files were correct; only that message line was stale. Checks (local, unverified worktree, records of computations): both packets' check.py --allow-dirty pass; test_check.py passes in normal and -O modes (97 and 70 tests). The binding was adversarially verified (register leaf diff, documents), same account, organizational-independence credit 0. Not established: anything about an informal claim. A receipt is a record of a computation about the registered Props for the formal objects only. Alignment PENDING_INDEPENDENT_REVIEW; scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01Cz7WZybv8znP64SpPj6sWY --- .../lean_statement_proofs_20261011/README.md | 23 ++++++---- .../lean_statement_proofs_20261011/SCOPE.md | 2 +- .../lean_statements_20261010/README.md | 27 +++++------ .../lean_statements_20261010/REGISTER.json | 46 +++++++++---------- .../lean_statements_20261010/SCOPE.md | 29 ++++++------ 5 files changed, 65 insertions(+), 62 deletions(-) diff --git a/specifications/lean_statement_proofs_20261011/README.md b/specifications/lean_statement_proofs_20261011/README.md index 63765554c..f4ee40262 100644 --- a/specifications/lean_statement_proofs_20261011/README.md +++ b/specifications/lean_statement_proofs_20261011/README.md @@ -61,13 +61,16 @@ intended (negative controls refused, positive controls accepted), worktree CLEAN 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. The 2 theorems of the two round-5 modules -(`Side24Covariance`, `Side24ConeMoment`), marked † below, have **no hosted receipt yet**: they were added after PR head -`be014794`, the last head with a hosted receipt (run 38187538744, event pull_request on PR #434; the actually tested -checkout is the synthetic merge `79d3964c…` of that head into base `12e072d6…`; receipt theorem_count 73, `leanchecker` -PASS; artifact `lean-specification-proofs-38187538744-1`, sha256 `60c9be0b…` as GitHub reports it), which does not name -them. Locally, `lake env lean -DwarningAsError=true` exited 0 on each module at the pin and `#print axioms` stayed -within `propext`, `Classical.choice`, `Quot.sound`, on an unverified worktree, which is evidence about no commit; the -next hosted `lean-specification-proofs.yml` run is their first receipt. +(`Side24Covariance`, `Side24ConeMoment`) are **receipted by the hosted `lean-specification-proofs.yml` run 38195810463** +(event pull_request on PR #439, head `c6ce0256`; the actually tested checkout, the receipt's `checked_commit`, is the +synthetic merge `1320c5b2…` of that head into base `be014794…`, the head of PR #434, with the same tree as the head; job +proof-evidence: fresh build, `leanchecker` PASS, receipt `formalization_status: "kernel-checked"`, theorem_count 75, +covered_props 74, every control classified as intended (negative controls refused, positive controls accepted), +worktree CLEAN; artifact `lean-specification-proofs-38195810463-1`, sha256 `d8abd42b…` as GitHub reports it), their +first receipt; the same run receipts all 75 theorems of that head. They were added after PR head `be014794`, +whose hosted run 38187538744 (actual tested merge `79d3964c…` with base `12e072d6…`; receipt theorem_count 73) +does not name them; their local `lake env lean -DwarningAsError=true` checks at the pin, with `#print axioms` +within `propext`, `Classical.choice`, `Quot.sound`, like any local run, are evidence about no commit. | group | Props (names without `UniversalLaw.Spec..`) | why no theorem here | |---|---|---| @@ -79,11 +82,11 @@ next hosted `lean-specification-proofs.yml` run is their first receipt. | 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` | `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). | | 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 covariance and density, formerly listed here (round 5) | — | `ReferenceCovarianceLowerBound` has a theorem in `Side24Covariance` since round 5 (`Side24Covariance.referenceCovarianceLowerBound†`; † = no hosted receipt yet, see above). The PROVABLE_WITH_WORK (weeks) reason this table gave for it is superseded by the theorem, and the module records that two of its details were not load-bearing: no order-five general-direction level of the peeling is needed, and no quadratic-form expansion through `Matrix.posSemidef_iff_dotProduct_mulVec`. Route (the module docstring): the odd-order contractions of the even kernel `Kinf d` vanish at `0` in arbitrary directions (`ContinuousLinearMap.iteratedFDeriv_comp_right`, `ContinuousMultilinearMap.map_smul_univ`), the orders two, four and six are the round-4 helpers `Side24Analysis.Aux.rb_order_two`, `rb_order_four`, `rb_order_six` (imported, not re-proved), and in an orthonormal frame with `u = R 0` the matrix `Cref - I/3` is entrywise `diagonal δ + (2/3) v vᵀ + w wᵀ` (`v = e_{G_0} - (9/2) e_t`, `w` the indicator of the diagonal Hessian coordinates), each summand positive semidefinite; the helper `Aux.cv_psd` (not an inventory theorem) holds for every `d > 0`, the inventory theorem has exactly the Prop's type, for `d = 2` or `3`, and its proof does not use that hypothesis; the `IsHermitian` half follows from the decomposition, not from any symmetry of `iteratedFDeriv`. Not established: that `Cref` is the covariance of any random vector; symmetry of `Cref d u R` for a general `u` or frame; main's `UniversalLaw.Side24.odd_block_eigenvalues`, `odd_block_shifted_minors` and `smaller_eigenvalue_exceeds_third` are neither imported, cited nor relied on, and no minor, no eigenvalue `8 ± sqrt 58` and no comparison of an eigenvalue with `1/3` is computed here. | +| Side24 covariance and density, formerly listed here (round 5) | — | `ReferenceCovarianceLowerBound` has a theorem in `Side24Covariance` since round 5 (`Side24Covariance.referenceCovarianceLowerBound`; first receipted by run 38195810463: PR head `c6ce0256`, actual tested merge `1320c5b2…` with base `be014794…`, see above). The PROVABLE_WITH_WORK (weeks) reason this table gave for it is superseded by the theorem, and the module records that two of its details were not load-bearing: no order-five general-direction level of the peeling is needed, and no quadratic-form expansion through `Matrix.posSemidef_iff_dotProduct_mulVec`. Route (the module docstring): the odd-order contractions of the even kernel `Kinf d` vanish at `0` in arbitrary directions (`ContinuousLinearMap.iteratedFDeriv_comp_right`, `ContinuousMultilinearMap.map_smul_univ`), the orders two, four and six are the round-4 helpers `Side24Analysis.Aux.rb_order_two`, `rb_order_four`, `rb_order_six` (imported, not re-proved), and in an orthonormal frame with `u = R 0` the matrix `Cref - I/3` is entrywise `diagonal δ + (2/3) v vᵀ + w wᵀ` (`v = e_{G_0} - (9/2) e_t`, `w` the indicator of the diagonal Hessian coordinates), each summand positive semidefinite; the helper `Aux.cv_psd` (not an inventory theorem) holds for every `d > 0`, the inventory theorem has exactly the Prop's type, for `d = 2` or `3`, and its proof does not use that hypothesis; the `IsHermitian` half follows from the decomposition, not from any symmetry of `iteratedFDeriv`. Not established: that `Cref` is the covariance of any random vector; symmetry of `Cref d u R` for a general `u` or frame; main's `UniversalLaw.Side24.odd_block_eigenvalues`, `odd_block_shifted_minors` and `smaller_eigenvalue_exceeds_third` are neither imported, cited nor relied on, and no minor, no eigenvalue `8 ± sqrt 58` and no comparison of an eigenvalue with `1/3` is computed here. | | Side24 interface statements | `CoefficientEnclosure`, `ReferenceComparison` (over `PeriodicCoefficientInterface`) | `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)`; 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` (by these three; it has its own theorem since round 5, next row); any cone moment of the `K24` field. `ConeMomentD2Algebra`, `ElementaryConeIntegral` and `ReferenceCoefficientIdentity` (proved earlier) need no interface. | -| Side24 interface statements, formerly listed here (round 5) | — | `ConeMomentD2Identity` has a theorem in `Side24ConeMoment` since round 5 (`Side24ConeMoment.coneMomentD2Identity†`; † = no hosted receipt yet, see above), for a GIVEN `I : ConeMomentInterface`. The PROVABLE_WITH_WORK (weeks) reason this table gave for it is superseded by the theorem. Route (the module docstring): transport along the law field `sxy_law` (`integral_map` after `Side24Interfaces.Aux.tm_aemeasurable_triple`), Fubini with the bound `\|coneIntegrand q\| ≤ q.1^4`, the cone identification `NegDefCone (s, x, y) ↔ s < 0 ∧ x² + y² < s²`, the inner planar Gaussian integral `s⁴ - 4s² + 8 - 8 exp(-s²/2)` for `s < 0` by polar coordinates and the substitution `z = r²` onto `elementaryConeIntegral`, the halving of the even bracket under `gaussianReal 0 (5/3)`, and the moments `25/3`, `5/3`, `√(3/8)` from `Side24Interfaces`; of the law fields only `sxy_law` is used (with the instances `mΩ`, `isProb`); `A` and `A_law` are not used, and `RayleighSquareLaw` is not invoked. Not established: that `ConeMomentInterface` has an inhabitant, or that its laws are the conditional law `A \| V = 0` of the reference field; that `D_2` is the parent's cone moment `D_u` of (15.1) for the reference law; the source's Rayleigh integration route as the source's statements (for the formal objects the proof follows the shape of the source's Section 1 computation, so it is not a cross-check of it by a different route); any enclosure of the coefficient. | -| 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 67 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`) and the 8 P15 predicates it did not re-examine; its 2 PROVABLE_WITH_WORK Props (`ReferenceCovarianceLowerBound`, `ConeMomentD2Identity`) have theorems since round 5 (marked †, no hosted receipt yet). | +| Side24 interface statements, formerly listed here (round 5) | — | `ConeMomentD2Identity` has a theorem in `Side24ConeMoment` since round 5 (`Side24ConeMoment.coneMomentD2Identity`; first receipted by run 38195810463: PR head `c6ce0256`, actual tested merge `1320c5b2…` with base `be014794…`, see above), for a GIVEN `I : ConeMomentInterface`. The PROVABLE_WITH_WORK (weeks) reason this table gave for it is superseded by the theorem. Route (the module docstring): transport along the law field `sxy_law` (`integral_map` after `Side24Interfaces.Aux.tm_aemeasurable_triple`), Fubini with the bound `\|coneIntegrand q\| ≤ q.1^4`, the cone identification `NegDefCone (s, x, y) ↔ s < 0 ∧ x² + y² < s²`, the inner planar Gaussian integral `s⁴ - 4s² + 8 - 8 exp(-s²/2)` for `s < 0` by polar coordinates and the substitution `z = r²` onto `elementaryConeIntegral`, the halving of the even bracket under `gaussianReal 0 (5/3)`, and the moments `25/3`, `5/3`, `√(3/8)` from `Side24Interfaces`; of the law fields only `sxy_law` is used (with the instances `mΩ`, `isProb`); `A` and `A_law` are not used, and `RayleighSquareLaw` is not invoked. Not established: that `ConeMomentInterface` has an inhabitant, or that its laws are the conditional law `A \| V = 0` of the reference field; that `D_2` is the parent's cone moment `D_u` of (15.1) for the reference law; the source's Rayleigh integration route as the source's statements (for the formal objects the proof follows the shape of the source's Section 1 computation, so it is not a cross-check of it by a different route); any enclosure of the coefficient. | +| 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 67 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`) and the 8 P15 predicates it did not re-examine; its 2 PROVABLE_WITH_WORK Props (`ReferenceCovarianceLowerBound`, `ConeMomentD2Identity`) have theorems since round 5 (first receipted by run 38195810463: PR head `c6ce0256`, actual tested merge `1320c5b2…` with base `be014794…`, see above). | | 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 diff --git a/specifications/lean_statement_proofs_20261011/SCOPE.md b/specifications/lean_statement_proofs_20261011/SCOPE.md index f18a5cac3..18d320f4f 100644 --- a/specifications/lean_statement_proofs_20261011/SCOPE.md +++ b/specifications/lean_statement_proofs_20261011/SCOPE.md @@ -31,7 +31,7 @@ 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`, modules Side24Elementary, Side24Analysis, Side24Interfaces, Side24Covariance, Side24ConeMoment) | 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; the reference covariance lower bound: `Cref d (R 0) R - (1/3) • 1` positive semidefinite, its `IsHermitian` half included, for every orthonormal frame `R` and `d = 2` or `3`, as the Prop states it (the proof does not use that hypothesis; only the helper `Aux.cv_psd`, not an inventory theorem, is stated for every `0 < d`)†; 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, and the `m = 2` cone moment `∫ coneIntegrand (s, x, y) ∂I.P = (1/2)(25/3 - 20/3 + 8 - 8 √(3/8))`† from the law field `sxy_law` alone (not `A_law`); all as the Props state them, for the formal objects (`E d`, `Kinf`, `latticeVec`, `derivContraction`, `gaussDensity`, `jointCov`, `Cref`, `coneIntegrand`, the interface structures); † = no hosted receipt yet (added after PR head `be014794`, the last head with a hosted receipt: run 38187538744, actual tested merge `79d3964c…` with base `12e072d6…`) | 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`); 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 or `Cref` is the covariance of a random object; symmetry of `Cref d u R` for a general `u` or frame; the odd-block minors and eigenvalues `8 ± sqrt 58` (main's theorems, neither imported nor relied on here); that `D_2` is the parent's cone moment `D_u` of (15.1) for the reference law, and the source's Rayleigh integration route as the source's statements; 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) | +| Side24 (`Specifications.Side24`, modules Side24Elementary, Side24Analysis, Side24Interfaces, Side24Covariance, Side24ConeMoment) | 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; the reference covariance lower bound: `Cref d (R 0) R - (1/3) • 1` positive semidefinite, its `IsHermitian` half included, for every orthonormal frame `R` and `d = 2` or `3`, as the Prop states it (the proof does not use that hypothesis; only the helper `Aux.cv_psd`, not an inventory theorem, is stated for every `0 < d`); 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, and the `m = 2` cone moment `∫ coneIntegrand (s, x, y) ∂I.P = (1/2)(25/3 - 20/3 + 8 - 8 √(3/8))` from the law field `sxy_law` alone (not `A_law`); all as the Props state them, for the formal objects (`E d`, `Kinf`, `latticeVec`, `derivContraction`, `gaussDensity`, `jointCov`, `Cref`, `coneIntegrand`, the interface structures); the reference covariance lower bound and the `m = 2` cone moment (round 5) are first receipted by run 38195810463 (PR head `c6ce0256`, actual tested merge `1320c5b2…` with base `be014794…`; receipt theorem_count 75) | 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`); 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 or `Cref` is the covariance of a random object; symmetry of `Cref d u R` for a general `u` or frame; the odd-block minors and eigenvalues `8 ± sqrt 58` (main's theorems, neither imported nor relied on here); that `D_2` is the parent's cone moment `D_u` of (15.1) for the reference law, and the source's Rayleigh integration route as the source's statements; 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 | diff --git a/specifications/lean_statements_20261010/README.md b/specifications/lean_statements_20261010/README.md index ff0584ebe..00cad229a 100644 --- a/specifications/lean_statements_20261010/README.md +++ b/specifications/lean_statements_20261010/README.md @@ -8,8 +8,8 @@ Props (74 of the 141 Props, 75 theorems); the register lists them under the clai 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 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; the 2 of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964c…), -have no hosted receipt yet and are marked † in the tables), +PR head 48bfcbae, actual tested merge 9833d6d9… with base 12e072d6…, the head of PR #423, same tree as the head; the 2 of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), by run 38195810463: +PR head c6ce0256, actual tested merge 1320c5b2… with base be014794…, the head of PR #434, 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. @@ -53,11 +53,11 @@ 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..`): 74 of the 141 Props have one (75 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; the 2 theorems marked † (`Side24Covariance` and `Side24ConeMoment`) were added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), and have no hosted receipt yet: the next hosted `lean-specification-proofs.yml` run 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..`): 74 of the 141 Props have one (75 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; the 2 theorems of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794 (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), are kernel-checked by the hosted run 38195810463 on PR head c6ce0256 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169, the head of PR #434; receipt theorem_count 75, leanchecker PASS; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff) and were first receipted by that run, which receipts all 75 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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | | `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | @@ -74,7 +74,7 @@ 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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---| | `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | | `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | — | @@ -103,7 +103,7 @@ 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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---| | `ThetaSummable` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Analysis.thetaSummable` | | `IsUnitTuple` | def_prop | `side24-proof` | `side24-coefficient` | — | @@ -116,7 +116,7 @@ ladder; it is `specified`. | `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` | `Side24Covariance.referenceCovarianceLowerBound†` | +| `ReferenceCovarianceLowerBound` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Covariance.referenceCovarianceLowerBound` | | `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` | @@ -125,7 +125,7 @@ ladder; it is `specified`. | `ConeMomentInterface` | structure | `side24-proof` | `side24-coefficient` | — | | `NegDefCone` | def_prop | `side24-proof` | `side24-coefficient` | — | | `ConeMomentD1Identity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24Interfaces.coneMomentD1Identity` | -| `ConeMomentD2Identity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24ConeMoment.coneMomentD2Identity†` | +| `ConeMomentD2Identity` | def_prop | `side24-proof` | `side24-coefficient` | `Side24ConeMoment.coneMomentD2Identity` | | `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` | @@ -136,7 +136,7 @@ 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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---| | `IsCriticalPoint` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | — | | `HessianHasInertia` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | — | @@ -169,7 +169,7 @@ ladder; it is `specified`. ### `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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---| | `RealizedData` | structure | `p15-realized-covers` | `p15-realized-covers` | — | | `IsRealizedFamily` | def_prop | `p15-realized-covers` | `p15-realized-covers` | — | @@ -232,11 +232,11 @@ 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`). 157 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; 85 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (75 theorems, 74 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; the 2 theorems marked † (`Side24Covariance` and `Side24ConeMoment`) were added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), and have no hosted receipt yet: the next hosted `lean-specification-proofs.yml` run 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`). 157 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; 85 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (75 theorems, 74 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; the 2 theorems of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794 (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), are kernel-checked by the hosted run 38195810463 on PR head c6ce0256 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169, the head of PR #434; receipt theorem_count 75, leanchecker PASS; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff) and were first receipted by that run, which receipts all 75 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`; 19 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`, `Side24Covariance.referenceCovarianceLowerBound†`, `Side24ConeMoment.coneMomentD2Identity†` | +| `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`; 19 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`, `Side24Covariance.referenceCovarianceLowerBound`, `Side24ConeMoment.coneMomentD2Identity` | | `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` | @@ -367,7 +367,8 @@ Packets listed by PROOF_INDEX by id only (82 rows, kind `proof_index_result`, fo 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 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 2 theorems - of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…, receipt theorem_count 73), have no hosted receipt yet, marked †). + of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…, receipt theorem_count 73), are receipted by run 38195810463: PR head c6ce0256, + actual tested merge 1320c5b2… with base be014794…, the head of PR #434, receipt theorem_count 75, 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 adbd01250..05cc80c1e 100644 --- a/specifications/lean_statements_20261010/REGISTER.json +++ b/specifications/lean_statements_20261010/REGISTER.json @@ -914,12 +914,12 @@ }, { "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); 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); theorems added after PR head be014794 (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73) have no hosted receipt yet: the next hosted lean-specification-proofs.yml run is their first receipt", - "note": "19 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, Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). 2 (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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); the theorems added after PR head be014794 (modules Side24Covariance, Side24ConeMoment) are receipted by run 38195810463 (PR head c6ce0256; actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169) (artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff; receipt formalization_status kernel-checked, leanchecker PASS, theorem_count 75, worktree CLEAN) and were first receipted by that run; base be01479435f98f34b802258ee9ec3c3578f37169 is the head of PR #434, the base of PR #439; the merge has the same tree as its head; all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "note": "19 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, Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). 2 (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 and kernel-checked by the hosted run 38195810463 (PR head c6ce0256; actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 75, worktree CLEAN; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff); they were first receipted by that run (PR #439, event pull_request); base be01479435f98f34b802258ee9ec3c3578f37169 is the head of PR #434, the base of PR #439, and the merge has the same tree as its head. All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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, ReferenceCovarianceLowerBound, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (19 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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); the lower bound C_ref - I/3 positive semidefinite for the formal matrix Cref at u = R 0 in an orthonormal frame, for d = 2 or 3 as the Prop states it, its IsHermitian half included (ReferenceCovarianceLowerBound: the odd-order contractions of the even kernel Kinf vanish at 0, and Cref - I/3 decomposes entrywise into a diagonal matrix and two rank-one squares; the proof does not use the hypothesis d = 2 or d = 3, and only the helper Aux.cv_psd, not an inventory theorem, is stated for every 0 < d; Cref is not shown to be the covariance of any random vector, symmetry of Cref d u R for a general u or frame is not stated, and main's odd_block_eigenvalues, odd_block_shifted_minors and smaller_eigenvalue_exceeds_third are neither imported, cited nor relied on: no minor, no eigenvalue 8 +/- sqrt 58 and no comparison of an eigenvalue with 1/3 is computed); 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); and, over the same interface, the m = 2 cone moment, the integral of coneIntegrand (s, x, y) under I.P, equal to (1/2)(25/3 - 20/3 + 8 - 8 sqrt(3/8)) (ConeMomentD2Identity: proved for every inhabitant using, of the law fields, only sxy_law, with the instances mΩ and isProb; the fields A and A_law are not used; by transport along sxy_law, Fubini, the cone identification s < 0 and x^2 + y^2 < s^2, polar coordinates onto ElementaryConeIntegral and the symmetry halving under N(0, 5/3); RayleighSquareLaw is not used, the source's Rayleigh integration route is not reproduced as the source's statements, D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law, and the value 29/6 - sqrt 6 (ConeMomentD2Algebra) is not used). None of them is the Gaussian conditioning, 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, 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, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); the 2 theorems of this row added after PR head be014794 (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) await the next hosted lean-specification-proofs.yml run, their first receipt; the other theorems of this row have 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); each of those 17 theorems 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, ConeMomentD2Identity PROVABLE_WITH_WORK (weeks): proved in round 5 (2026-10-12; see entries, no hosted receipt yet); 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 now has a theorem for every inhabitant of ConeMomentInterface, with no hosted receipt yet, but no inhabitant is constructed and its laws are not derived as the conditional law A | V = 0 of the reference field, so D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law). 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; for ReferenceCovarianceLowerBound and ConeMomentD2Identity, proved in round 5 (no hosted receipt yet), the README's PROVABLE_WITH_WORK (weeks) reasons are superseded by their theorems (the ReferenceCovarianceLowerBound lane records that the order-five peeling step and the Matrix.posSemidef_iff_dotProduct_mulVec route the README named are not load-bearing); the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, 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 19 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); ReferenceCovarianceLowerBound with the Section 3 display C_ref >= I/3, with Mathlib's Matrix.PosSemidef, IsHermitian included, read as the source's order)." + "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, ReferenceCovarianceLowerBound, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (19 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 and kernel-checked by the hosted run 38195810463 (PR head c6ce0256; actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 75, worktree CLEAN; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff); they were first receipted by that run (PR #439, event pull_request); base be01479435f98f34b802258ee9ec3c3578f37169 is the head of PR #434, the base of PR #439, 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); the lower bound C_ref - I/3 positive semidefinite for the formal matrix Cref at u = R 0 in an orthonormal frame, for d = 2 or 3 as the Prop states it, its IsHermitian half included (ReferenceCovarianceLowerBound: the odd-order contractions of the even kernel Kinf vanish at 0, and Cref - I/3 decomposes entrywise into a diagonal matrix and two rank-one squares; the proof does not use the hypothesis d = 2 or d = 3, and only the helper Aux.cv_psd, not an inventory theorem, is stated for every 0 < d; Cref is not shown to be the covariance of any random vector, symmetry of Cref d u R for a general u or frame is not stated, and main's odd_block_eigenvalues, odd_block_shifted_minors and smaller_eigenvalue_exceeds_third are neither imported, cited nor relied on: no minor, no eigenvalue 8 +/- sqrt 58 and no comparison of an eigenvalue with 1/3 is computed); 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); and, over the same interface, the m = 2 cone moment, the integral of coneIntegrand (s, x, y) under I.P, equal to (1/2)(25/3 - 20/3 + 8 - 8 sqrt(3/8)) (ConeMomentD2Identity: proved for every inhabitant using, of the law fields, only sxy_law, with the instances mΩ and isProb; the fields A and A_law are not used; by transport along sxy_law, Fubini, the cone identification s < 0 and x^2 + y^2 < s^2, polar coordinates onto ElementaryConeIntegral and the symmetry halving under N(0, 5/3); RayleighSquareLaw is not used, the source's Rayleigh integration route is not reproduced as the source's statements, D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law, and the value 29/6 - sqrt 6 (ConeMomentD2Algebra) is not used). None of them is the Gaussian conditioning, 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, 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, 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 2 added after PR head be014794 (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) first by run 38195810463 (PR head c6ce0256; actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169) (receipt theorem_count 75); 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); each of those 17 theorems receipted on PR head 48bfcbae by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73); every theorem of this row receipted on PR head c6ce0256 by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75). 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, ConeMomentD2Identity PROVABLE_WITH_WORK (weeks): proved in round 5 (2026-10-12; see entries); 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 now has a theorem for every inhabitant of ConeMomentInterface, but no inhabitant is constructed and its laws are not derived as the conditional law A | V = 0 of the reference field, so D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law). 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; for ReferenceCovarianceLowerBound and ConeMomentD2Identity, proved in round 5, the README's PROVABLE_WITH_WORK (weeks) reasons are superseded by their theorems (the ReferenceCovarianceLowerBound lane records that the order-five peeling step and the Matrix.posSemidef_iff_dotProduct_mulVec route the README named are not load-bearing); the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, 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 19 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); ReferenceCovarianceLowerBound with the Section 3 display C_ref >= I/3, with Mathlib's Matrix.PosSemidef, IsHermitian included, read as the source's order)." }, { "claim_id": "lifetime-remainder", @@ -1078,8 +1078,8 @@ "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); 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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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)." + "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); all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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.", @@ -1722,8 +1722,8 @@ }, { "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); 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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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)." + "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); all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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.", @@ -1857,8 +1857,8 @@ }, { "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); 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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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)." + "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); all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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.", @@ -2055,8 +2055,8 @@ }, { "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); 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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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)." + "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); all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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.", @@ -2496,8 +2496,8 @@ }, { "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); 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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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)." + "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); all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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.", @@ -2959,7 +2959,7 @@ "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 19 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, Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity; 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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, ConeMomentD2Identity) 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 19 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, Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity; 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 and kernel-checked by the hosted run 38195810463 (PR head c6ce0256; actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 75, worktree CLEAN; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff); they were first receipted by that run (PR #439, event pull_request); base be01479435f98f34b802258ee9ec3c3578f37169 is the head of PR #434, the base of PR #439, 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, ConeMomentD2Identity) 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)." }, { @@ -3067,8 +3067,8 @@ "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); 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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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)." + "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); all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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.", @@ -3439,8 +3439,8 @@ }, { "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); 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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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)." + "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); all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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.", @@ -3556,8 +3556,8 @@ }, { "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); 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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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)." + "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); all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "note": "2 theorems of this row (P15Elementary.twoCoordGeneratorPrices, P15Elementary.demandOneRefutation) are inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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). All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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.", @@ -4331,8 +4331,8 @@ "kernel_evidence": [ { "package": "d6g8k5htny-coder/Math- specifications/lean_statement_proofs_20261011", - "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: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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)." + "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; all 75 theorems of PR head c6ce0256 receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff)", + "note": "1 theorem of this row (RNCluster.rareClusterPoissonObstructionO) is inventoried by the proof packet's check.py (source mode SOURCE_CHECK_PASS: 75 inventory theorems over SpecificationProofs/{P15Elementary,P15Structure,P15Hazard,P15Covers,P15Budget,Side24Elementary,RNProbability,P15Interpolation,Side24Analysis,Side24Interfaces,RNCluster,Side24Covariance,Side24ConeMoment}.lean, 74 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. All 75 theorems of PR head c6ce0256, including every theorem of the tiers above, are receipted on that head by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff). `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 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.", diff --git a/specifications/lean_statements_20261010/SCOPE.md b/specifications/lean_statements_20261010/SCOPE.md index 55fca1d25..e0b7b981d 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` (75 theorems typed exactly as 74 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 be014794 (run 38187538744, actual tested merge 79d3964c…) 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..`): 74 of the 141 Props have one (75 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; the 2 theorems marked † (`Side24Covariance` and `Side24ConeMoment`) were added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), and have no hosted receipt yet: the next hosted `lean-specification-proofs.yml` run 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..`): 74 of the 141 Props have one (75 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; the 2 theorems of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794 (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), are kernel-checked by the hosted run 38195810463 on PR head c6ce0256 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169, the head of PR #434; receipt theorem_count 75, leanchecker PASS; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff) and were first receipted by that run, which receipts all 75 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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---|---| | `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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---|---| | `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. | — | @@ -110,7 +109,7 @@ 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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---|---| | `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. | — | @@ -123,7 +122,7 @@ a theorem between string literals, an indented Prop definition, a proof written | `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. | `Side24Covariance.referenceCovarianceLowerBound†` | +| `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. | `Side24Covariance.referenceCovarianceLowerBound` | | `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` | @@ -132,7 +131,7 @@ a theorem between string literals, an indented Prop definition, a proof written | `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` | -| `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. | `Side24ConeMoment.coneMomentD2Identity†` | +| `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. | `Side24ConeMoment.coneMomentD2Identity` | | `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` | @@ -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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---|---| | `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. | — | @@ -176,7 +175,7 @@ a theorem between string literals, an indented Prop definition, a proof written ### `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; 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…; † = added after PR head be014794 (run 38187538744, actual tested merge 79d3964c…), 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…; the 2 of Side24Covariance and Side24ConeMoment by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base be014794… (the head of PR #434), the merge having the same tree as PR head c6ce0256; receipt theorem_count 75, artifact sha256 d8abd42b…) | |---|---|---|---|---|---| | `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`). 157 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; 85 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (75 theorems, 74 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; the 2 theorems marked † (`Side24Covariance` and `Side24ConeMoment`) were added after PR head be014794, the last head with a hosted receipt (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), and have no hosted receipt yet: the next hosted `lean-specification-proofs.yml` run 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`). 157 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; 85 are theorems of the proof packet `specifications/lean_statement_proofs_20261011` whose type is exactly a registered Prop of this package (75 theorems, 74 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; the 2 theorems of `Side24Covariance` and `Side24ConeMoment`, added after PR head be014794 (run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73, corroborating the 73 theorems above on that head; artifact lean-specification-proofs-38187538744-1, sha256 60c9be0b49d45931d6c82f04d73fb7e6b6e97d539e7c10a376c3e99e302f2617), are kernel-checked by the hosted run 38195810463 on PR head c6ce0256 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169, the head of PR #434; receipt theorem_count 75, leanchecker PASS; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff) and were first receipted by that run, which receipts all 75 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`; 19 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`, `Side24Covariance.referenceCovarianceLowerBound†`, `Side24ConeMoment.coneMomentD2Identity†` | 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, ReferenceCovarianceLowerBound, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (19 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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); the lower bound C_ref - I/3 positive semidefinite for the formal matrix Cref at u = R 0 in an orthonormal frame, for d = 2 or 3 as the Prop states it, its IsHermitian half included (ReferenceCovarianceLowerBound: the odd-order contractions of the even kernel Kinf vanish at 0, and Cref - I/3 decomposes entrywise into a diagonal matrix and two rank-one squares; the proof does not use the hypothesis d = 2 or d = 3, and only the helper Aux.cv_psd, not an inventory theorem, is stated for every 0 < d; Cref is not shown to be the covariance of any random vector, symmetry of Cref d u R for a general u or frame is not stated, and main's odd_block_eigenvalues, odd_block_shifted_minors and smaller_eigenvalue_exceeds_third are neither imported, cited nor relied on: no minor, no eigenvalue 8 +/- sqrt 58 and no comparison of an eigenvalue with 1/3 is computed); 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); and, over the same interface, the m = 2 cone moment, the integral of coneIntegrand (s, x, y) under I.P, equal to (1/2)(25/3 - 20/3 + 8 - 8 sqrt(3/8)) (ConeMomentD2Identity: proved for every inhabitant using, of the law fields, only sxy_law, with the instances mΩ and isProb; the fields A and A_law are not used; by transport along sxy_law, Fubini, the cone identification s < 0 and x^2 + y^2 < s^2, polar coordinates onto ElementaryConeIntegral and the symmetry halving under N(0, 5/3); RayleighSquareLaw is not used, the source's Rayleigh integration route is not reproduced as the source's statements, D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law, and the value 29/6 - sqrt 6 (ConeMomentD2Algebra) is not used). None of them is the Gaussian conditioning, 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, 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, CoefficientEnclosure, ReferenceComparison remain the targets of this row (the parametrised predicates IsUnitTuple, NegDefCone are vocabulary, not targets); the 2 theorems of this row added after PR head be014794 (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) await the next hosted lean-specification-proofs.yml run, their first receipt; the other theorems of this row have 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); each of those 17 theorems 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, ConeMomentD2Identity PROVABLE_WITH_WORK (weeks): proved in round 5 (2026-10-12; see entries, no hosted receipt yet); 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 now has a theorem for every inhabitant of ConeMomentInterface, with no hosted receipt yet, but no inhabitant is constructed and its laws are not derived as the conditional law A \| V = 0 of the reference field, so D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law). 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; for ReferenceCovarianceLowerBound and ConeMomentD2Identity, proved in round 5 (no hosted receipt yet), the README's PROVABLE_WITH_WORK (weeks) reasons are superseded by their theorems (the ReferenceCovarianceLowerBound lane records that the order-five peeling step and the Matrix.posSemidef_iff_dotProduct_mulVec route the README named are not load-bearing); the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, 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 19 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); ReferenceCovarianceLowerBound with the Section 3 display C_ref >= I/3, with Mathlib's Matrix.PosSemidef, IsHermitian included, read as the source's order). | +| `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`; 19 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`, `Side24Covariance.referenceCovarianceLowerBound`, `Side24ConeMoment.coneMomentD2Identity` | 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, ReferenceCovarianceLowerBound, ReferenceOddAndPinBlocks, DensityComparison, IntegrandRatioBounds, ReferencePinDensityProduct, ReferenceCoefficientIdentity, ConeMomentD1Identity, ConeMomentD2Identity, ConeMomentD2Algebra, TraceMoments, RayleighSquareLaw, ElementaryConeIntegral (19 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 and kernel-checked by the hosted run 38195810463 (PR head c6ce0256; actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 75, worktree CLEAN; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff); they were first receipted by that run (PR #439, event pull_request); base be01479435f98f34b802258ee9ec3c3578f37169 is the head of PR #434, the base of PR #439, 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); the lower bound C_ref - I/3 positive semidefinite for the formal matrix Cref at u = R 0 in an orthonormal frame, for d = 2 or 3 as the Prop states it, its IsHermitian half included (ReferenceCovarianceLowerBound: the odd-order contractions of the even kernel Kinf vanish at 0, and Cref - I/3 decomposes entrywise into a diagonal matrix and two rank-one squares; the proof does not use the hypothesis d = 2 or d = 3, and only the helper Aux.cv_psd, not an inventory theorem, is stated for every 0 < d; Cref is not shown to be the covariance of any random vector, symmetry of Cref d u R for a general u or frame is not stated, and main's odd_block_eigenvalues, odd_block_shifted_minors and smaller_eigenvalue_exceeds_third are neither imported, cited nor relied on: no minor, no eigenvalue 8 +/- sqrt 58 and no comparison of an eigenvalue with 1/3 is computed); 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); and, over the same interface, the m = 2 cone moment, the integral of coneIntegrand (s, x, y) under I.P, equal to (1/2)(25/3 - 20/3 + 8 - 8 sqrt(3/8)) (ConeMomentD2Identity: proved for every inhabitant using, of the law fields, only sxy_law, with the instances mΩ and isProb; the fields A and A_law are not used; by transport along sxy_law, Fubini, the cone identification s < 0 and x^2 + y^2 < s^2, polar coordinates onto ElementaryConeIntegral and the symmetry halving under N(0, 5/3); RayleighSquareLaw is not used, the source's Rayleigh integration route is not reproduced as the source's statements, D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law, and the value 29/6 - sqrt 6 (ConeMomentD2Algebra) is not used). None of them is the Gaussian conditioning, 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, 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, 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 2 added after PR head be014794 (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) first by run 38195810463 (PR head c6ce0256; actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169) (receipt theorem_count 75); 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); each of those 17 theorems receipted on PR head 48bfcbae by run 38180705952 (actual tested merge 9833d6d9f9facc335e1a56e243c3144adefe6225 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693; receipt theorem_count 73); every theorem of this row receipted on PR head c6ce0256 by run 38195810463 (actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169; receipt theorem_count 75). 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, ConeMomentD2Identity PROVABLE_WITH_WORK (weeks): proved in round 5 (2026-10-12; see entries); 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 now has a theorem for every inhabitant of ConeMomentInterface, but no inhabitant is constructed and its laws are not derived as the conditional law A \| V = 0 of the reference field, so D_2 is not identified with the parent's cone moment D_u of (15.1) for the reference law). 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; for ReferenceCovarianceLowerBound and ConeMomentD2Identity, proved in round 5, the README's PROVABLE_WITH_WORK (weeks) reasons are superseded by their theorems (the ReferenceCovarianceLowerBound lane records that the order-five peeling step and the Matrix.posSemidef_iff_dotProduct_mulVec route the README named are not load-bearing); the reasons for ImageBound, CovarianceOrdering, EntrywiseCovarianceBound, 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 19 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); ReferenceCovarianceLowerBound with the Section 3 display C_ref >= I/3, with Mathlib's Matrix.PosSemidef, IsHermitian included, read as the source's order). | | `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 19 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, Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity; 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 (the last head with a hosted receipt: run 38187538744, actual tested merge 79d3964cf70e2eadad5c9629eef8f4b47f218dd0 with base 12e072d614dbe6e91b66d22a2ab55cf87860d693, receipt theorem_count 73): 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, ConeMomentD2Identity) 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 19 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, Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity; 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; 2 of them (Side24Covariance.referenceCovarianceLowerBound, Side24ConeMoment.coneMomentD2Identity) were added after PR head be014794 and kernel-checked by the hosted run 38195810463 (PR head c6ce0256; actual tested merge 1320c5b217e491d301ea140e5cc25b07ff5a21b7 with base be01479435f98f34b802258ee9ec3c3578f37169) (job proof-evidence: fresh build, leanchecker PASS, `#print axioms` within {propext, Classical.choice, Quot.sound}, receipt formalization_status kernel-checked, theorem_count 75, worktree CLEAN; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42be0facd4248647c3c7eb67a264bd19b9b47c6128733b5b5fc93b0a6ff); they were first receipted by that run (PR #439, event pull_request); base be01479435f98f34b802258ee9ec3c3578f37169 is the head of PR #434, the base of PR #439, 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, ConeMomentD2Identity) 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). | @@ -443,5 +442,5 @@ identity), never about the claim row it sits under; `kernel-checked` is a run-re 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; 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…; theorems added after PR head be014794 (run 38187538744, actual tested merge 79d3964c… with base 12e072d6…) await the next hosted -`lean-specification-proofs.yml` run, their first receipt). +12e072d6…, the head of PR #423; receipt theorem_count 73; artifact lean-specification-proofs-38180705952-1, sha256 87309efa…; the 2 of `Side24Covariance` and `Side24ConeMoment` by run 38195810463: PR head c6ce0256, actual tested merge 1320c5b2… with base +be014794…, the head of PR #434; receipt theorem_count 75; artifact lean-specification-proofs-38195810463-1, sha256 d8abd42b…).