diff --git a/.github/workflows/lean-specifications.yml b/.github/workflows/lean-specifications.yml new file mode 100644 index 000000000..9e20e91aa --- /dev/null +++ b/.github/workflows/lean-specifications.yml @@ -0,0 +1,60 @@ +name: Lean statement specifications (statements only, no proof, no scientific promotion) + +'on': + pull_request: + paths: + - 'specifications/lean_statements_20261010/**' + - '.github/workflows/lean-specifications.yml' + - 'formal/lake-manifest.json' + workflow_dispatch: + +permissions: + contents: read + +jobs: + statement-evidence: + runs-on: ubuntu-latest + timeout-minutes: 45 + env: + LEAN_NUM_THREADS: 2 + PYTHONDONTWRITEBYTECODE: '1' + steps: + - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 + with: + persist-credentials: false + fetch-depth: 0 + - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 + with: + python-version: '3.11.16' + - name: Unit tests in both Python modes and the source check (no Lean) + run: | + set -euo pipefail + P=specifications/lean_statements_20261010 + python -B -S "$P/test_check.py" + python -B -O -S "$P/test_check.py" + python -B -S "$P/check.py" + - name: Install pinned Lean and retrieve the Mathlib cache + uses: leanprover/lean-action@f061402b660e0c34644504b324e830f2991d4865 + with: + lake-package-directory: specifications/lean_statements_20261010 + auto-config: 'false' + build: 'false' + test: 'false' + lint: 'false' + use-mathlib-cache: 'true' + use-github-cache: 'false' + - name: Lean-driving unit tests on dependency-free fixture packages (environment audit and its controls, autoImplicit off) + run: python -B -S specifications/lean_statements_20261010/test_check.py LeanHelpers + - name: "Fresh build, leanchecker, #check and #print axioms of every registered statement, environment audit, negative controls, receipt" + run: python -B -S specifications/lean_statements_20261010/check.py --execute + - name: Tracked tree unchanged + run: git diff --exit-code + - name: Preserve logs and receipt + if: always() + uses: actions/upload-artifact@043fb46d1a93c77aae656e7c1c64a875d1fc6a0a + with: + name: lean-specifications-${{ github.run_id }}-${{ github.run_attempt }} + path: specifications/lean_statements_20261010/.lake/evidence/ + include-hidden-files: true + if-no-files-found: error + retention-days: 14 diff --git a/specifications/lean_statements_20261010/.gitignore b/specifications/lean_statements_20261010/.gitignore new file mode 100644 index 000000000..ae7367994 --- /dev/null +++ b/specifications/lean_statements_20261010/.gitignore @@ -0,0 +1,2 @@ +.lake/ +__pycache__/ diff --git a/specifications/lean_statements_20261010/README.md b/specifications/lean_statements_20261010/README.md new file mode 100644 index 000000000..7d4732f4b --- /dev/null +++ b/specifications/lean_statements_20261010/README.md @@ -0,0 +1,383 @@ +# Lean statement specifications (2026-10-10) + +Scientific effect: **NONE**. Organizational-independence credit of this packet: 0 (Anthropic / +Claude, delegated AI work). Alignment status of every statement: `PENDING_INDEPENDENT_REVIEW`. +Formal progress of every declaration here: `specified`. No claim, premise, obligation or +disposition changes because this packet exists, builds, or is green in CI. + +## What this packet is + +A **statement-only** Lean 4 + Mathlib package. Each module declares, as `def … : Prop` and +interface `structure`s, the mathematical objects and results that the byte-pinned Layer 0 texts +under `sources/` describe. Two independent stages hold the modules to "statements only". The +source scan (`check.py`, no Lean) refuses `theorem`, `lemma`, `example`, `instance`, `axiom`, +`sorry` and the rest of its lexical list as whole tokens of comment-stripped code, refuses string +literals, guillemet names, `#` commands and attributes outright, requires every command keyword +at column 0 (so `open … in` and indented commands are refused), refuses `section` and `variable` +(no module needs them; a `variable (h : False)` would add a hypothesis the scan cannot see), requires an +explicit result type on every definition, refuses proof-shaped result types, `Sort` and `_root_`. +It is a fail-closed pre-Lean stage, not a Lean lexer. The authority is the environment audit of +`check.py --execute`: Lean's own `ConstantInfo` of every constant the five modules add refuses +any theorem, axiom, opaque constant, instance or definition whose type is a proposition beyond +what `structure` and `def` elaboration generate, refuses any transitive axiom outside `propext`, +`Classical.choice`, `Quot.sound`, and requires Lean's statement definitions (type syntactically +`∀ …, Prop`, read without reduction, so a `Set`-valued definition is auxiliary exactly as the +scan's `: Prop` rule has it), structures and auxiliary definitions to equal the register and the +scan exactly. A `def … : Prop` that +elaborates is a precise statement of what would have to be proved. It is not L5 of the #95 +ladder; it is `specified`. + +| file | role | +|---|---| +| `Specifications.lean` | root; imports exactly the five modules below, in this order | +| `Specifications/Field.lean` | shared objects of the periodized Gaussian lifetime track (namespace `UniversalLaw.Spec.Field`) | +| `Specifications/Lifetime.lean` | lifetime parent Theorems A/B/C, remainder, planar laws (`UniversalLaw.Spec.Lifetime`) | +| `Specifications/Side24.lean` | SIDE24 coefficient objects and enclosure statements (`UniversalLaw.Spec.Side24`) | +| `Specifications/RN.lean` | RN count interface, fixed-window and collision statements (`UniversalLaw.Spec.RN`) | +| `Specifications/P15.lean` | P15 realized covers, price boundary, price budget, full price (`UniversalLaw.Spec.P15`) | +| `REGISTER.json` | the statement register: claim rows, their Lean statements, formal-progress labels, what is not established, next step | +| `sources/` | byte-pinned Layer 0 copies (`sources/SOURCES.json`: repository, commit, path, blob, bytes, sha256) | +| `check.py`, `test_check.py`, `replay.sh` | the checker, its unit tests and negative controls, the local replay | +| `lakefile.toml`, `lean-toolchain`, `lake-manifest.json` | Lean `leanprover/lean4:v4.34.1`, Mathlib `d13f23b723b8a846827a245b89c10fc7d3f11612`, lock identical to `formal/lake-manifest.json`; `autoImplicit = false` so an unbound name in a header is an error, never a silent implicit parameter | +| `.github/workflows/lean-specifications.yml` | CI: tests, source check, `lean-action` with the Mathlib cache, `check.py --execute`, artifact of `.lake/evidence/` | + +## Declarations + +150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified`; 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. + +### `Specifications/Field.lean` (12 registered declarations) + +| declaration | kind | source | registered under claim rows | +|---|---|---|---| +| `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `GaussianFieldInterface` | structure | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsCriticalPoint` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `HessianHasInertia` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsNondegMax` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsIndexSaddle` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsCandidatePair` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsMorseDistinct` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `SuperlevelBarInterface` | structure | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsFiniteBar` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsLifetimeDensityVersion` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | + +### `Specifications/Lifetime.lean` (24 registered declarations) + +| declaration | kind | source | registered under claim rows | +|---|---|---|---| +| `IsLPeriodic` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsSmooth` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `GaussianFieldInterface` | structure | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsCriticalPoint` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `HessianHasInertia` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsNondegMax` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsIndexSaddle` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsCandidatePair` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsMorseDistinct` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `SuperlevelBarInterface` | structure | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsFiniteBar` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime` | +| `IsLifetimeDensityVersion` | def_prop | `lifetime-parent` | `lifetime-remainder`, `math.uniform-matrix-cap-lifetime`, `D1-THM-B` | +| `IsWindowPair` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime`, `D1-THM-B` | +| `IsWindowElderPair` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime`, `D1-THM-B` | +| `IsUnrestrictedPair` | def_prop | `lifetime-parent` | `lifetime-remainder`, `math.uniform-matrix-cap-lifetime`, `D1-THM-C` | +| `PairLawInterface` | structure | `lifetime-parent` | `math.uniform-matrix-cap-lifetime`, `D1-THM-A` | +| `ParentUniformSelectionA` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime`, `D1-THM-A` | +| `ParentCompactWindowDensityB` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime`, `D1-THM-B` | +| `IsUnrestrictedLeadingCoefficient` | def_prop | `lifetime-parent` | `lifetime-remainder`, `math.uniform-matrix-cap-lifetime`, `D1-THM-C` | +| `ParentUnrestrictedLeadingDensityC` | def_prop | `lifetime-parent` | `math.uniform-matrix-cap-lifetime`, `D1-THM-C` | +| `RemainderBoundedR1` | def_prop | `lifetime-remainder` | `lifetime-remainder`, `math.lifetime-remainder` | +| `PlanarElderLowerE2` | def_prop | `elder-lower-density-gap` | `OA-ELDER-LOWER-DENSITY-GAP-20260928-v1` | +| `PlanarTwoSidedRateE4` | def_prop | `elder-lower-density-gap` | `OA-ELDER-LOWER-DENSITY-GAP-20260928-v1` | +| `PlanarDensityGapE11` | def_prop | `elder-lower-density-gap` | `OA-ELDER-LOWER-DENSITY-GAP-20260928-v1` | + +### `Specifications/Side24.lean` (28 registered declarations) + +| declaration | kind | source | registered under claim rows | +|---|---|---|---| +| `ThetaSummable` | def_prop | `side24-proof` | `side24-coefficient` | +| `IsUnitTuple` | def_prop | `side24-proof` | `side24-coefficient` | +| `DerivativePairingBound` | def_prop | `side24-proof` | `side24-coefficient` | +| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | `side24-coefficient` | +| `DerivativeBoundAtZero` | def_prop | `side24-proof` | `side24-coefficient` | +| `LatticeImageSumBound` | def_prop | `side24-proof` | `side24-coefficient` | +| `SuccessiveTermRatio` | def_prop | `side24-proof` | `side24-coefficient` | +| `ExponentialBounds` | def_prop | `side24-proof` | `side24-coefficient` | +| `ImageBound` | def_prop | `side24-proof` | `side24-coefficient` | +| `CovarianceOrdering` | def_prop | `side24-proof` | `side24-coefficient` | +| `EntrywiseCovarianceBound` | def_prop | `side24-proof` | `side24-coefficient` | +| `ReferenceCovarianceLowerBound` | def_prop | `side24-proof` | `side24-coefficient` | +| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | `side24-coefficient` | +| `DensityComparison` | def_prop | `side24-proof` | `side24-coefficient` | +| `IntegrandRatioBounds` | def_prop | `side24-proof` | `side24-coefficient` | +| `ReferencePinDensityProduct` | def_prop | `side24-proof` | `side24-coefficient` | +| `ReferenceCoefficientIdentity` | def_prop | `side24-proof` | `side24-coefficient` | +| `ConeMomentInterface` | structure | `side24-proof` | `side24-coefficient` | +| `NegDefCone` | def_prop | `side24-proof` | `side24-coefficient` | +| `ConeMomentD1Identity` | def_prop | `side24-proof` | `side24-coefficient` | +| `ConeMomentD2Identity` | def_prop | `side24-proof` | `side24-coefficient` | +| `ConeMomentD2Algebra` | def_prop | `side24-proof` | `side24-coefficient` | +| `TraceMoments` | def_prop | `side24-proof` | `side24-coefficient` | +| `RayleighSquareLaw` | def_prop | `side24-proof` | `side24-coefficient` | +| `ElementaryConeIntegral` | def_prop | `side24-proof` | `side24-coefficient` | +| `PeriodicCoefficientInterface` | structure | `side24-proof` | `side24-coefficient`, `math.side24-coefficient` | +| `CoefficientEnclosure` | def_prop | `side24-proof` | `side24-coefficient`, `math.side24-coefficient` | +| `ReferenceComparison` | def_prop | `side24-proof` | `side24-coefficient`, `math.side24-coefficient` | + +### `Specifications/RN.lean` (28 registered declarations) + +| declaration | kind | source | registered under claim rows | +|---|---|---|---| +| `IsCriticalPoint` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | +| `HessianHasInertia` | def_prop | `lifetime-parent` | `rn-fixed-remote-window` | +| `HolderCountBoundN1` | def_prop | `rn-count-interface` | `rn-count-interface`, `math.rn-count-interface` | +| `HolderConsequenceN2` | def_prop | `rn-count-interface` | `rn-count-interface`, `math.rn-count-interface` | +| `ConditionalMeanIdentityN3` | def_prop | `rn-count-interface` | `rn-count-interface`, `math.rn-count-interface` | +| `UniformCounterexampleN4` | def_prop | `rn-count-interface` | `rn-count-interface`, `math.rn-count-interface` | +| `ExponentialTailLayerCakeN5` | def_prop | `rn-count-interface` | `rn-count-interface`, `math.rn-count-interface` | +| `PinnedLawInterface` | structure | `rn-fixed-remote-window` | `rn-fixed-remote-window`, `math.rn-fixed-remote-window` | +| `FixedRemoteMeanMeasureA2` | def_prop | `rn-fixed-remote-window` | `rn-fixed-remote-window`, `math.rn-fixed-remote-window`, `math.rn-region.fixed-remote`, `math.d5-component.remote-window-proof` | +| `FixedRemoteCountBoundA3` | def_prop | `rn-fixed-remote-window` | `rn-fixed-remote-window`, `math.rn-fixed-remote-window`, `math.rn-region.fixed-remote`, `math.d5-component.remote-window-proof` | +| `FixedAnnulusWindowCandidateA2` | def_prop | `rn-fixed-annulus-window` | `rn-fixed-annulus-window`, `math.rn-fixed-annulus-window`, `math.rn-region.fixed-annulus-window` | +| `AnnulusBridgeAllHeight2` | def_prop | `rn-annulus-bridge` | `math.rn-region.mesoscopic-scaled-annulus`, `math.d5-component.annulus-proof` | +| `D5aPuncturedPinDisks` | def_prop | `d5-reconciliation` | `math.rn-region.pin-collision`, `math.d5-pin-neighborhood-first-moment`, `math.d5-component.punctured-pin-proof` | +| `D5bMicrodisk` | def_prop | `d5-reconciliation` | `math.rn-region.pin-collision`, `math.d5-pin-neighborhood-first-moment`, `math.d5-component.punctured-pin-proof` | +| `D5cCollarC1` | def_prop | `d5-reconciliation` | `math.d5-pin-neighborhood-first-moment`, `math.d5-component.collar-proof` | +| `D5dSummedPinNeighbourhoodC2` | def_prop | `d5-reconciliation` | `math.rn-region.mesoscopic-scaled-annulus`, `math.rn-region.pin-collision`, `math.d5-pin-neighborhood-first-moment` | +| `D5eIntermediateShellsI3I4` | def_prop | `d5-reconciliation` | `math.rn-region.intermediate-r-to-rho`, `math.d5-pin-neighborhood-first-moment`, `math.d5-component.intermediate-window-proof` | +| `D5fGlobalWindowFirstMomentI5` | def_prop | `d5-reconciliation` | `math.d5-pin-neighborhood-first-moment`, `math.d5-component.intermediate-window-proof` | +| `DimensionLiftPuncturedPinBallP` | def_prop | `d5-dimension-lift` | `CL-D5-DIMENSION-LIFT-20260929-v1` | +| `DimensionLiftCompactCollarC` | def_prop | `d5-dimension-lift` | `CL-D5-DIMENSION-LIFT-20260929-v1` | +| `DimensionLiftIntermediateShellsI` | def_prop | `d5-dimension-lift` | `CL-D5-DIMENSION-LIFT-20260929-v1` | +| `DimensionLiftGlobalWindowG` | def_prop | `d5-dimension-lift` | `CL-D5-DIMENSION-LIFT-20260929-v1` | +| `FourierCutoffPlanarSecondFactorialF4` | def_prop | `c6-fourier-cutoff` | `OA-C6-FOURIER-20260929-v1` | +| `PalmRouteFactorialMomentsQ` | def_prop | `c6-palm-route` | `CL-C6-PALM-20260929-v1` | +| `PalmRouteSecondFactorialTheta` | def_prop | `c6-palm-route` | `CL-C6-PALM-20260929-v1` | +| `RareClusterPoissonObstructionO` | def_prop | `c6-rare-cluster` | `OA-C6-RARE-CLUSTER-20260929-v1` | +| `RemoteCollisionSecondFactorialD` | def_prop | `remote-collision` | `math.rn-region.witness-collision`, `CL-D5-REMOTE-COLLISION-20260928-v1` | +| `WitnessCollisionNearPinPairsTarget` | def_prop | `d5-reconciliation` | `math.rn-region.witness-collision` | + +### `Specifications/P15.lean` (58 registered declarations) + +| declaration | kind | source | registered under claim rows | +|---|---|---|---| +| `RealizedData` | structure | `p15-realized-covers` | `p15-realized-covers` | +| `IsRealizedFamily` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `IsGood` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `IsKDecomposable` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `IsGeneratorCover` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `IsPaletteFeasible` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `PaletteNumberAttained` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `IsMinimalForbidden` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `IsTransversal` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `MinimalForbiddenCharacterisation` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `LocalRestriction` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `LocalDecomposability` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `LocalObstructionIsFullBlock` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `LocalPriceChain` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `IndependentBlockHazard` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `FullBlockCoverIff` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `CoverCostHazardBound` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `WholeGroundChromaticNumber` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `ElementaryPaletteFormula` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `TriangleNeedsThree` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `Benchmark816IsRealized` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `Benchmark816Palette` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `Benchmark816Cover` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `Benchmark816WholeGround` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `Benchmark816MinimalForbiddenCounts` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `Benchmark816Price` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `Benchmark816UnionBound` | def_prop | `p15-realized-covers` | `p15-realized-covers` | +| `TwoCoordGeneratorPrices` | def_prop | `p15-price-boundary` | `p15-price-boundary`, `math.p15-price-boundary` | +| `DemandOneRefutation` | def_prop | `p15-price-boundary` | `p15-price-boundary`, `math.p15-price-boundary` | +| `LogOnePlusBelowIdentity` | def_prop | `p15-price-boundary` | `p15-price-boundary` | +| `LogRatioBound` | def_prop | `p15-price-budget` | `p15-price-budget-restricted` | +| `RatioExponentBound` | def_prop | `p15-price-budget` | `p15-price-budget-restricted` | +| `RestrictedLocalBudget` | def_prop | `p15-price-budget` | `p15-price-budget-restricted` | +| `RestrictedTransformedPriceBudget` | def_prop | `p15-price-budget` | `p15-price-budget-restricted` | +| `RhoTestSufficient` | def_prop | `p15-price-budget` | `p15-price-budget-restricted` | +| `CapTestAtTwoThirds` | def_prop | `p15-price-budget` | `p15-price-budget-restricted` | +| `Benchmark816TransformedRange` | def_prop | `p15-price-budget` | `p15-price-budget-restricted` | +| `Benchmark816RationalPrice` | def_prop | `p15-price-budget` | `p15-price-budget-restricted` | +| `FullTransformedPriceBudget` | def_prop | `p15-full-price` | `p15-full-price`, `math.p15-full-price` | +| `RhoStarEnclosure` | def_prop | `p15-full-price` | `p15-full-price`, `math.p15-full-price` | +| `RhoStarBelowSixSevenths` | def_prop | `p15-full-price` | `p15-full-price` | +| `IsProperDownset` | def_prop | `p15-full-price` | `p15-full-price` | +| `HazardStarPositive` | def_prop | `p15-full-price` | `p15-full-price` | +| `CoordinatewiseConcavity` | def_prop | `p15-full-price` | `p15-full-price` | +| `HazardProductBound` | def_prop | `p15-full-price` | `p15-full-price` | +| `HazardPhiBound` | def_prop | `p15-full-price` | `p15-full-price` | +| `FullGroundPriceBound` | def_prop | `p15-full-price` | `p15-full-price` | +| `PhiAtPStar` | def_prop | `p15-full-price` | `p15-full-price` | +| `FullGroundBudgetIff` | def_prop | `p15-full-price` | `p15-full-price` | +| `CapacityHazardIsBinomial` | def_prop | `p15-full-price` | `p15-full-price` | +| `OddMajorityIdentity` | def_prop | `p15-full-price` | `p15-full-price` | +| `CapacityWorstCase` | def_prop | `p15-full-price` | `p15-full-price` | +| `HStarExceedsOne` | def_prop | `p15-full-price` | `p15-full-price` | +| `LocalPriceBoundFull` | def_prop | `p15-full-price` | `p15-full-price` | +| `GlobalAssemblyHazard` | def_prop | `p15-full-price` | `p15-full-price` | +| `SharpnessOfRhoStar` | def_prop | `p15-full-price` | `p15-full-price`, `math.p15-full-price` | +| `DemandOneBoundary` | def_prop | `p15-full-price` | `p15-full-price` | +| `SixSeventhsCertificate` | def_prop | `p15-full-price` | `p15-full-price` | + +## 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`); 31 statement entries are kernel-proved arithmetic theorems of main's `formal/` package (label `proved`: the 31 Side24 targets at origin/main `f6deeba7`; main's 28 P15 arithmetic targets are NOT listed because they exist only in this programme's uncommitted main worktree, on no commit), listed as the arithmetic skeleton of the claim they sit under and never as the claim. `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): `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` | +| `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` | +| `rn-fixed-remote-window` | landing_claim | HOLD_WITH_DOMAIN | `specified` | `rn-fixed-remote-window` | 5 specified: `PinnedLawInterface`, `IsCriticalPoint`, `HessianHasInertia`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | +| `rn-fixed-annulus-window` | landing_claim | REVIEWED_SCOPED | `specified` | `rn-fixed-annulus-window` | 1 specified: `FixedAnnulusWindowCandidateA2` | +| `p15-realized-covers` | landing_claim | HOLD_WITH_DOMAIN | `specified` | `p15-realized-covers` | 27 specified: `RealizedData`, `IsRealizedFamily`, `IsGood`, `IsKDecomposable`, `IsGeneratorCover`, `IsPaletteFeasible`, `PaletteNumberAttained`, `IsMinimalForbidden`, `IsTransversal`, `MinimalForbiddenCharacterisation`, `LocalRestriction`, `LocalDecomposability`, `LocalObstructionIsFullBlock`, `LocalPriceChain`, `IndependentBlockHazard`, `FullBlockCoverIff`, `CoverCostHazardBound`, `WholeGroundChromaticNumber`, `ElementaryPaletteFormula`, `TriangleNeedsThree`, `Benchmark816IsRealized`, `Benchmark816Palette`, `Benchmark816Cover`, `Benchmark816WholeGround`, `Benchmark816MinimalForbiddenCounts`, `Benchmark816Price`, `Benchmark816UnionBound` | +| `p15-price-boundary` | landing_claim | EXACT_COUNTEREXAMPLE | `specified` | `p15-price-boundary` | 3 specified: `TwoCoordGeneratorPrices`, `DemandOneRefutation`, `LogOnePlusBelowIdentity` | +| `p15-price-budget-restricted` | landing_claim | HOLD_WITH_DOMAIN | `specified` | `p15-price-budget` | 8 specified: `LogRatioBound`, `RatioExponentBound`, `RestrictedLocalBudget`, `RestrictedTransformedPriceBudget`, `RhoTestSufficient`, `CapTestAtTwoThirds`, `Benchmark816TransformedRange`, `Benchmark816RationalPrice` | +| `p15-full-price` | landing_claim | REVIEWED_SCOPED | `specified` | `p15-full-price` | 20 specified: `FullTransformedPriceBudget`, `RhoStarEnclosure`, `RhoStarBelowSixSevenths`, `IsProperDownset`, `HazardStarPositive`, `CoordinatewiseConcavity`, `HazardProductBound`, `HazardPhiBound`, `FullGroundPriceBound`, `PhiAtPStar`, `FullGroundBudgetIff`, `CapacityHazardIsBinomial`, `OddMajorityIdentity`, `CapacityWorstCase`, `HStarExceedsOne`, `LocalPriceBoundFull`, `GlobalAssemblyHazard`, `SharpnessOfRhoStar`, `DemandOneBoundary`, `SixSeventhsCertificate` | +| `downstream-hard-gate` | landing_claim | ENGINEERING_HOLD | `none` | `downstream-graph` | — | +| `math.uniform-matrix-cap-lifetime` | graph_node | — | `specified` | `lifetime-parent` | 32 specified: `IsLPeriodic`, `IsSmooth`, `GaussianFieldInterface`, `IsCriticalPoint`, `HessianHasInertia`, `IsNondegMax`, `IsIndexSaddle`, `IsCandidatePair`, `IsMorseDistinct`, `SuperlevelBarInterface`, `IsFiniteBar`, `IsLifetimeDensityVersion`, `IsLPeriodic`, `IsSmooth`, `GaussianFieldInterface`, `IsCriticalPoint`, `HessianHasInertia`, `IsNondegMax`, `IsIndexSaddle`, `IsCandidatePair`, `IsMorseDistinct`, `SuperlevelBarInterface`, `IsFiniteBar`, `IsLifetimeDensityVersion`, `IsWindowPair`, `IsWindowElderPair`, `IsUnrestrictedPair`, `PairLawInterface`, `ParentUniformSelectionA`, `ParentCompactWindowDensityB`, `IsUnrestrictedLeadingCoefficient`, `ParentUnrestrictedLeadingDensityC` | +| `math.d1-component.congruence-erratum` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d1-component.section9-replacement-v1_1` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d1-component.marked-cylinder-cap` | graph_node | — | `none` | `marked-cylinder-cap` | — | +| `math.d1-component.reconciliation-record` | graph_node | — | `none` | `downstream-graph` | — | +| `math.lifetime-remainder` | graph_node | — | `specified` | `lifetime-remainder` | 1 specified: `RemainderBoundedR1` | +| `math.side24-coefficient` | graph_node | — | `specified` | `side24-proof` | 3 specified: `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison` | +| `math.rn-count-interface` | graph_node | — | `none` | `rn-count-interface` | 5 specified: `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5` | +| `math.rn-fixed-remote-window` | graph_node | — | `specified` | `rn-fixed-remote-window` | 3 specified: `PinnedLawInterface`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | +| `math.rn-region.fixed-remote` | graph_node | — | `specified` | `downstream-graph` | 2 specified: `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | +| `math.rn-region.mesoscopic-scaled-annulus` | graph_node | — | `specified` | `d5-reconciliation` | 2 specified: `D5dSummedPinNeighbourhoodC2`, `AnnulusBridgeAllHeight2` | +| `math.rn-region.pin-collision` | graph_node | — | `specified` | `d5-reconciliation` | 3 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk`, `D5dSummedPinNeighbourhoodC2` | +| `math.rn-region.intermediate-r-to-rho` | graph_node | — | `specified` | `d5-reconciliation` | 1 specified: `D5eIntermediateShellsI3I4` | +| `math.rn-region.witness-collision` | open_obligation | — | `specified` | `d5-reconciliation` | 2 specified: `WitnessCollisionNearPinPairsTarget`, `RemoteCollisionSecondFactorialD` | +| `math.rn-mesoscopic-reduction` | graph_node | — | `none` | `downstream-graph` | — | +| `math.p15-full-price` | graph_node | — | `specified` | `p15-full-price` | 3 specified: `FullTransformedPriceBudget`, `RhoStarEnclosure`, `SharpnessOfRhoStar` | +| `math.p15-price-boundary` | graph_node | — | `specified` | `p15-price-boundary` | 2 specified: `TwoCoordGeneratorPrices`, `DemandOneRefutation` | +| `math.rn-selector-region-crosswalk` | graph_node | — | `none` | `downstream-graph` | — | +| `math.rn-fixed-annulus-window` | graph_node | — | `specified` | `rn-fixed-annulus-window` | 1 specified: `FixedAnnulusWindowCandidateA2` | +| `math.rn-region.fixed-annulus-window` | graph_node | — | `specified` | `downstream-graph` | 1 specified: `FixedAnnulusWindowCandidateA2` | +| `math.d5-pin-neighborhood-first-moment` | graph_node | — | `specified` | `d5-reconciliation` | 6 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk`, `D5cCollarC1`, `D5dSummedPinNeighbourhoodC2`, `D5eIntermediateShellsI3I4`, `D5fGlobalWindowFirstMomentI5` | +| `math.d5-component.punctured-pin-proof` | graph_node | — | `specified` | `d5-reconciliation` | 2 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk` | +| `math.d5-component.punctured-pin-review-111` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d5-component.punctured-pin-continuum-review` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d5-component.punctured-pin-continuum-check` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d5-component.collar-review` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d5-component.annulus-review` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d5-component.i5-review-xai` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d5-component.i5-review-claude` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d5-component.offpin-second-moment-review` | graph_node | — | `none` | `downstream-graph` | — | +| `math.d5-component.collar-proof` | graph_node | — | `specified` | `d5-reconciliation` | 1 specified: `D5cCollarC1` | +| `math.d5-component.annulus-proof` | graph_node | — | `specified` | `rn-annulus-bridge` | 1 specified: `AnnulusBridgeAllHeight2` | +| `math.d5-component.intermediate-window-proof` | graph_node | — | `specified` | `d5-reconciliation` | 2 specified: `D5eIntermediateShellsI3I4`, `D5fGlobalWindowFirstMomentI5` | +| `math.d5-component.remote-window-proof` | graph_node | — | `specified` | `rn-fixed-remote-window` | 2 specified: `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | +| `D1-THM-A` | proof_index_result | — | `specified` | `lifetime-parent` | 2 specified: `PairLawInterface`, `ParentUniformSelectionA` | +| `D1-THM-B` | proof_index_result | — | `specified` | `lifetime-parent` | 4 specified: `IsWindowPair`, `IsWindowElderPair`, `IsLifetimeDensityVersion`, `ParentCompactWindowDensityB` | +| `D1-THM-C` | proof_index_result | — | `specified` | `lifetime-parent` | 3 specified: `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `ParentUnrestrictedLeadingDensityC` | +| `OA-ELDER-LOWER-DENSITY-GAP-20260928-v1` | proof_index_result | — | `specified` | `elder-lower-density-gap` | 3 specified: `PlanarElderLowerE2`, `PlanarTwoSidedRateE4`, `PlanarDensityGapE11` | +| `two-scale-s6-s21` | proof_index_result | — | `none` | `—` | — | +| `axial-density` | proof_index_result | — | `none` | `—` | — | +| `fixed-transverse-chart` | proof_index_result | — | `none` | `—` | — | +| `CL-D5-DIMENSION-LIFT-20260929-v1` | proof_index_result | — | `specified` | `d5-dimension-lift` | 4 specified: `DimensionLiftPuncturedPinBallP`, `DimensionLiftCompactCollarC`, `DimensionLiftIntermediateShellsI`, `DimensionLiftGlobalWindowG` | +| `OA-C6-FOURIER-20260929-v1` | proof_index_result | — | `none` | `c6-fourier-cutoff` | 1 specified: `FourierCutoffPlanarSecondFactorialF4` | +| `CL-C6-PALM-20260929-v1` | proof_index_result | — | `specified` | `c6-palm-route` | 2 specified: `PalmRouteFactorialMomentsQ`, `PalmRouteSecondFactorialTheta` | +| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | `c6-rare-cluster` | 1 specified: `RareClusterPoissonObstructionO` | +| `CL-D5-REMOTE-COLLISION-20260928-v1` | proof_index_result | — | `none` | `remote-collision` | 1 specified: `RemoteCollisionSecondFactorialD` | +| `cumulative-transfer-correction` | proof_index_result | — | `none` | `cumulative-transfer-correction` | — | +| `hist.D0-RN-UNIF-historical-predicates` | open_obligation | — | `none` | `downstream-graph` | — | +| `lean:main-formal-UniversalLaw` | existing_lean_package | — | `none` | `side24-enclosure` | 4 proved (main): `d2_endpoints_adjacent`, `d3_endpoints_adjacent`, `d3_below_d2`, `endpoints_in_unit_tenth` | +| `lean:math-formal-ResearchFormalCoreR1` | existing_lean_package | — | `none` | `—` | — | +| `lean:math-companion-d2_moment_bridge_20261006` | existing_lean_package | — | `none` | `—` | — | +| `lean:math-companion-d2_square_schur_20261006` | existing_lean_package | — | `none` | `—` | — | +| `lean:math-companion-d2_atom_integral_20261006` | existing_lean_package | — | `none` | `—` | — | +| `lean:math-frontier-cap_first_exit_lean_20261002` | existing_lean_package | — | `none` | `marked-cylinder-cap` | — | +| `lean:math-w3scratch-side24-proposal` | existing_lean_package | — | `none` | `—` | — | +| `main:experiments/universality/finite_gaussian_contact` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/universality/finite_candidate_lifetime` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/universality/finite_fold_chart` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/universality/finite_weighted_fold_window` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/universality/finite_contact_genericity` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/universality/finite_elder_transfer` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/universality/finite_h0_lifetime` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/universality/fold_structural_universality` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/universality/periodized_gaussian_h0` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/universality/iid_short_bar_process` | main_experiment_proof | — | `none` | `—` | — | +| `main:experiments/periodic_h0/spectral_shrinking_bins` | main_experiment_proof | — | `none` | `—` | — | +| `main:unmerged-codex-prs-330-350` | main_experiment_proof | — | `none` | `—` | — | +| `cert:main-periodic_h0-exact_h0_1` | exact_certificate | — | `none` | `—` | — | +| `cert:main-periodic_h0-certificate8` | exact_certificate | — | `none` | `—` | — | +| `cert:main-periodic_h0-nodal1` | exact_certificate | — | `none` | `—` | — | +| `cert:main-periodic_h0-hessian1` | exact_certificate | — | `none` | `—` | — | +| `cert:main-periodic_h0-gaussian_tail1` | exact_certificate | — | `none` | `—` | — | +| `cert:main-periodic_h0-gaussian_coupling1` | exact_certificate | — | `none` | `—` | — | +| `cert:main-periodic_h0-word_polynomial1` | exact_certificate | — | `none` | `—` | — | +| `cert:main-periodic_h0-finite_count_loss1` | exact_certificate | — | `none` | `—` | — | +| `cert:main-periodic_h0-count_confidence1` | exact_certificate | — | `none` | `—` | — | +| `cert:main-universality-fold_controls` | exact_certificate | — | `none` | `—` | — | +| `cert:math-side24_v1-coefficient.py` | exact_certificate | — | `none` | `side24-enclosure` | — | +| `cert:math-full_price-full_price.py` | exact_certificate | — | `none` | `p15-full-price` | — | + +Packets listed by PROOF_INDEX by id only (82 rows, kind `proof_index_result`, formal progress `none`, no statements, bytes not pinned here; each row's `not_established` says the statement was not extracted and no verdict is implied): + +`CL-SIDE24-PFV-20260929-v1`, `OA-SIDE24-REMAINDER-20260929-v1`, `CL-SIDE24-PERIODIZATION-REMAINDER-20260929-v1`, `OA-CONTACT-ANGULAR-TAIL-20260925-v1`, `OA-PIN-MICRO-COV-20260926-v1`, `OA-D5-MICRODISK-20260926-v1`, `D5-PIN-MICRODISK-20260927-v1`, `OA-BF-SIX-PIN-HESSIAN-20260926-v1`, `OA-CONTACT-KERNEL-SUBSTITUTE-20260926-v1`, `OA-D5-SHORT-EDGE-20260928-v1`, `CL-D5-PIN-DISK-20260928-v1`, `OA-D5-REMOTE-SINGLETON-LAW-20260928-v1`, `OA-REMOTE-HEIGHT-DECOUPLING-20260928-v1`, `OA-WINDOW-MULTIPLICITY-LOCAL`, `OA-WINDOW-MULTIPLICITY-REMOTE`, `OA-WINDOW-MULTIPLICITY-DISTANCE`, `OA-WINDOW-MULTIPLICITY-HEIGHT`, `OA-REMOTE-INVERSE-SEPARATION-20260928-v1`, `OA-ELDER-DIMENSION-LIFT-20260928-v1`, `CL-ELDER-LOWER-ALL-D-20260929-v1.1`, `CL-C6-FACTORIAL-20260929-v1`, `CL-C6-SHARPENED-20260929-v1`, `OA-C6-COUNT-CAP-BOUNDARY-20260929-v1`, `OA-LOCAL-CLUSTER-TIGHTNESS-20260929-v1`, `OA-REMOTE-MIXED-INVERSE-20260929-v1`, `CL-D1-UNRESTRICTED-DIFFERENCE-20260929-v3`, `CL-C7-TOTAL-BOUNDED-20260929-v1.2`, `OA-C7-ZERO-GAP-LIMIT-20260929-v1`, `OA-SARD-ROBUST-CHARTS-20260929-v1`, `OA-SARD-A2-FIXED-FRAME-20260929-v2`, `packet:sard_g_a2_regularity_20260929`, `packet:sard_g_pinned_transfer_20260929`, `OA-P15-BIPARTITE-OVERLAP-20260928-v1`, `OA-P15-TAIL-LOAD-20260928-v1`, `packet:planar_cubic_cluster_20260929`, `packet:c6_marked_fourier_20260929`, `packet:c6_cluster_law_20260929`, `packet:cluster_exponential_weight_20260930`, `packet:spectral_cluster_closure_20260929`, `packet:lifetime_moment_boundary_20260930`, `packet:radial_rate_20260930`, `packet:extreme_cluster_sampling_20260930`, `packet:fixed_r_inverse_lifetime_20260930`, `packet:local_elder_geometry_20260930`, `packet:concave_fibre_elder_20260930`, `packet:elder_partner_extremes_20260930`, `packet:remainder_vanishing_20260930`, `packet:elder_selected_extremes_20260930`, `packet:remainder_rate_20260930`, `packet:far_elder_rate_20260930`, `packet:short_bar_endpoint_law_20261001`, `packet:c2_finite_jet_transfer_20261001`, `packet:cusp_second_order_20261001`, `packet:candidate_third_order_20261001`, `packet:elder_third_order_20261001`, `packet:third_order_rate_20261001`, `packet:elder_cusp_parity_20261002`, `packet:separated_fold_multiplicity_20261002`, `packet:subpin_collision_20261002`, `packet:soft_rejected_pairs_20261002`, `packet:soft_fold_limit_20261002`, `packet:soft_closed_form_20261002`, `packet:cusp_torus_transfer_20261001`, `packet:moving_remote_collision_20261003`, `packet:ordinary_short_bar_occurrence_20261001`, `packet:replacement_bar_occurrence_20261001`, `packet:strict_unique_replacement_coefficient_20261003`, `packet:unique_replacement_bar_intensity_20261001`, `packet:planar_rejected_endpoint_margin_20261003/A4`, `packet:planar_soft_layer_chain_20261003/C91`, `packet:planar_soft_layer_chain_20261003/C92`, `packet:planar_soft_layer_chain_20261003/C93`, `packet:planar_soft_layer_chain_20261003/C94`, `packet:planar_soft_layer_chain_20261003/C95`, `packet:planar_soft_layer_chain_20261003/C96`, `packet:planar_soft_layer_chain_20261003/C97`, `packet:planar_soft_layer_chain_20261003/C98`, `packet:planar_soft_layer_chain_20261003/C99`, `packet:planar_soft_layer_chain_20261003/C101`, `packet:planar_soft_layer_chain_20261003/C102`, `packet:planar_soft_layer_chain_20261003/C103`, `packet:planar_soft_layer_chain_20261003/C124`. + +## What this packet is not + +* Not a proof of anything. Every registered declaration is a definition or a structure. The + source scan refuses `theorem`, `lemma`, `example`, `instance`, `axiom`, `sorry`, `native_decide`, + `unsafe`, `opaque`, `partial`, `macro`, `notation`, `set_option` and the rest of the lexical list + as whole tokens of comment-stripped code; the environment audit of `--execute` refuses, by + Lean's `ConstantInfo`, any theorem, axiom, opaque constant, instance or definition whose type is + a proposition that `structure`/`def` elaboration did not generate, and the build log is refused + on `declaration uses` (sorry) and on Lean's "is a proposition; use `theorem`" linter. +* Not a receipt from a local run. `--execute --allow-dirty` was run once locally (2026-10-10, worktree + UNVERIFIED: not evidence about any commit): `lake build` of the five modules passed against the + pinned Mathlib `d13f23b7`; `lake env leanchecker Specifications` was killed by the sandbox's memory + cgroup (exit 137 at 13.4 GB resident) before finishing, so `execute()` failed closed at that step + and wrote no receipt. The later stages were then run with the same functions over the fresh oleans: + `#check` of all 150 registered declarations, `#print axioms` (149 within `{propext, + Classical.choice, Quot.sound}`, 1 within `{propext, Quot.sound}`), the environment audit (558 + constants, 0 refused: 141 statements, 9 structures, 123 auxiliary, 285 generated, equal to the + register and the scan) and every negative control. The first receipt for the real packet is the + hosted `lean-specifications.yml` run; its `leanchecker` step over `import Mathlib` has the + `cap-first-exit-lean.yml` precedent (about 30 minutes on a 16 GB runner) within the 45-minute + budget, and that budget and the runner's memory are untimed here. +* Not an alignment review. Whether a Prop says what its anchor says is the independent reviewer's + question (DESIGN honesty checklist). The register only records `PENDING_INDEPENDENT_REVIEW`. +* 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 a kernel receipt for the research claims. `--execute` shows that the statements elaborate + and depend on no axiom beyond `propext`, `Classical.choice`, `Quot.sound`; the receipt says + `formalization_status: "specified"` and nothing stronger. +* Not evidence from a dirty worktree. A receipt produced with `--allow-dirty` records the + worktree as unverified and is not evidence about any commit. + +## How to run + +```bash +# source mode (the optional positional `source` names the same default mode): identity against +# HEAD, lexical refusals, root imports, declaration inventory vs REGISTER.json and back, +# vocabulary, verbatim anchors (at least 16 characters), pinned bytes, lock, toolchain +python3 -B -S specifications/lean_statements_20261010/check.py +python3 -B -S specifications/lean_statements_20261010/check.py source + +# unit tests and negative controls, both interpreter modes (no Lean needed) +python3 -B -S specifications/lean_statements_20261010/test_check.py +python3 -B -O -S specifications/lean_statements_20261010/test_check.py + +# execution: fresh lake build, leanchecker, #check and #print axioms of every registered +# declaration, the environment audit of every constant the modules add, the negative controls +# (source-side, Lean-side and of the audit itself), receipt under .lake/evidence/ (needs the +# pinned toolchain and the Mathlib cache or a source build of Mathlib d13f23b7) +export PATH="$HOME/.elan/bin:$PATH" +python3 -B -S specifications/lean_statements_20261010/check.py --execute + +# everything in order +bash specifications/lean_statements_20261010/replay.sh +``` + +Environment: Lean `leanprover/lean4:v4.34.1` (release commit `5045d0056413266e57c625dcd7c365b10e377c52`), +Mathlib `d13f23b723b8a846827a245b89c10fc7d3f11612`, full dependency lock in `lake-manifest.json` +(identical to `formal/lake-manifest.json` except the package name). CI uses +`leanprover/lean-action@f061402b660e0c34644504b324e830f2991d4865` with `use-mathlib-cache: 'true'`. +Without the Mathlib cache (hosts `cache.mathlib.org`, `release.lean-lang.org`, +`reservoir.lean-lang.org`), Mathlib must be built from source, which takes hours. diff --git a/specifications/lean_statements_20261010/REGISTER.json b/specifications/lean_statements_20261010/REGISTER.json new file mode 100644 index 000000000..4bcbd16e0 --- /dev/null +++ b/specifications/lean_statements_20261010/REGISTER.json @@ -0,0 +1,4953 @@ +{ + "schema_version": 1, + "object": "LEAN-STATEMENT-REGISTER-20261010-v1", + "scientific_effect": "NONE", + "scientific_status_authority": false, + "alignment_status": "PENDING_INDEPENDENT_REVIEW", + "meaning": "Inventory of formal statements per registered claim; formal-progress labels only. A row records where a Lean statement (def : Prop or interface structure) exists for a claim and what it does not establish. Statements in this package are at most `specified`; `proved` may appear only for theorems of other packages listed under statements, `kernel-checked` only in a run receipt. Dispositions are transcribed verbatim from LANDING_CLAIMS.json and never re-derived. Nothing here is a scientific status, a review verdict, an acceptance or a promotion; alignment of every statement with its source is PENDING_INDEPENDENT_REVIEW.", + "toolchain": { + "lean": "leanprover/lean4:v4.34.1", + "lean_commit": "5045d0056413266e57c625dcd7c365b10e377c52", + "mathlib": "d13f23b723b8a846827a245b89c10fc7d3f11612" + }, + "vocabulary": { + "formal_progress": [ + "none", + "specified", + "proved", + "kernel-checked" + ] + }, + "sources": [ + { + "id": "lifetime-parent", + "repository": "d6g8k5htny-coder/Math-", + "commit": "c39164254ad1529ed49543743598d5dcb24bab98", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "local_copy": "sources/lifetime-parent/UNIFORM_MATRIX_CAP_AND_LIFETIME.md" + }, + { + "id": "marked-cylinder-cap", + "repository": "d6g8k5htny-coder/Math-", + "commit": "c39164254ad1529ed49543743598d5dcb24bab98", + "path": 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c_{2,24}, c_{3,24} (SIDE24-COEFFICIENT-D23-20260924-v1), heading \"Theorem\" of coefficients/side24_v1/PROOF.md", + "layer0": { + "source_id": "side24-proof", + "heading": "Theorem" + }, + "disposition_transcribed": "REVIEWED_SCOPED", + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ThetaSummable", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "K_24(z) = sum_{n in Z^d} exp(-|z+24n|^2/2)", + "does_not_claim": "that the family is summable (true, but not shown here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.IsUnitTuple", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "For q<=6 unit directional differentiations of phi(x)=exp(-|x|^2/2), product-rule", + "does_not_claim": "anything." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.DerivativePairingBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "pairings bound the absolute derivative by", + "does_not_claim": "the bound (statement only); validity for `q > 6` (the source restricts to `q ≤ 6`); nothing here encloses the coefficient." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.DerivativeBoundAtLeastOne", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "At |x|>=1 this is at most76 |x|^6 e^(-|x|^2/2).", + "does_not_claim": "the bound (statement only); the coefficient arithmetic `1+15+45+15=76` and `76` bounding the smaller orders are kernel-checked on main (`UniversalLaw.Side24.pairing_sum_six`, `pairing_sum_le_six`), not here; nothing here encloses the coefficient." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.DerivativeBoundAtZero", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "contraction has absolute value at most15.", + "does_not_claim": "the bound (statement only); `5!! = 15` is kernel-checked on main (`UniversalLaw.Side24.sixth_moment_double_factorial`), not here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.LatticeImageSumBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "<=1458 e^(-288).", + "does_not_claim": "the bound (statement only); the intermediate `729 sum_j j^9 e^(-288 j^2)` shell count is not specified; `27*27 = 729`, `729*2 = 1458` are kernel-checked on main (`UniversalLaw.Side24.lattice_shell_constant`), not here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.SuccessiveTermRatio", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "successive terms have ratio at most512e^(-864)<1/2.", + "does_not_claim": "the bounds (statement only); `2^9 = 512`, `288*3 = 864` are kernel-checked on main (`UniversalLaw.Side24.geometric_ratio_constants`), not here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ExponentialBounds", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "hence e^(-288)<10^(-125) and the geometric-ratio inequality follows too.", + "does_not_claim": "the inequalities (statement only); the rational Taylor sum through degree 20 exceeding `10` is kernel-checked on main (the `UniversalLaw.Side24` Taylor-sum target in `Ledger.lean`), not here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ImageBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "|D^q K_24(0)-D^q K_infty(0)|", + "does_not_claim": "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." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.CovarianceOrdering", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The inequalities are in the positive-semidefinite ordering and hold in every", + "does_not_claim": "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." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.EntrywiseCovarianceBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Each covariance entry differs from the reference by at most2E, by (2).", + "does_not_claim": "the bound (statement only); the spectral-norm bound `20E` (not specified); nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ReferenceCovarianceLowerBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Thus C_ref >= I/3 and", + "does_not_claim": "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." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ReferenceOddAndPinBlocks", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Var(t)=15, Cov(t,G)=(-3,0,...,0),", + "does_not_claim": "the identities (statement only); `15 - 9 = 6` is kernel-checked on main (`UniversalLaw.Side24.conditional_third_derivative_variance`), not here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.DensityComparison", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "[(1-epsilon)/(1+epsilon)]^(n/2) phi_((1-epsilon)C0)", + "does_not_claim": "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." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.IntegrandRatioBounds", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "ratio between1-13epsilon and", + "does_not_claim": "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." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ReferencePinDensityProduct", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3).", + "does_not_claim": "the identity (statement only); `det diag(3,1,…,1) = 3` is kernel-checked on main (`UniversalLaw.Side24.pin_determinant_and_joint_dimension`), not here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ReferenceCoefficientIdentity", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "This eliminates BOTH the angular and cone integrals for the reference expression.", + "does_not_claim": "the identity (statement only); that the left side is the parent formula (15.2) evaluated at the reference law (the angular integrand is constant by rotational invariance of `K_infty`, a fact not specified here); `6^2*2 = 3*24` is kernel-checked on main (`UniversalLaw.Side24.cube_root_simplification`), not here; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ConeMomentInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "Var(s)=5/3 and Var(x)=Var(y)=1. The negative-definite cone is s<-R,", + "does_not_claim": "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`)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.NegDefCone", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "R=sqrt(x^2+y^2), and det A=s^2-R^2.", + "does_not_claim": "the source's reduction of the cone to `s < -R` (not specified; the cone is taken by its definition)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ConeMomentD1Identity", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "D_1=E[A^2 1{A<0}]=4/3.", + "does_not_claim": "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." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ConeMomentD2Identity", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "D_2 = (1/2)[25/3-20/3+8-8sqrt(3/8)]", + "does_not_claim": "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." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ConeMomentD2Algebra", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " = 29/6-sqrt(6).", + "does_not_claim": "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." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.TraceMoments", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E s^2=5/3, E s^4=25/3 and", + "does_not_claim": "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." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.RayleighSquareLaw", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The variable R^2 has density e^(-z/2)/2.", + "does_not_claim": "the law (statement only); measurability of the event (if the event is not measurable, `P` is its outer measure)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ElementaryConeIntegral", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "integral_0^a (a-z)^2 e^(-z/2) dz/2 = a^2-4a+8-8e^(-a/2).", + "does_not_claim": "the identity (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.PeriodicCoefficientInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "Let c_d,24 be precisely its equation (15.2), using ordinary sphere area, ordered", + "does_not_claim": "existence of an object satisfying this interface; that `c24` is the finite-bar asymptotic constant (that reading is conditional on the parent's results, which are not touched); that the parent's \"ordered maximum/saddle roles and the full height/gradient pin Jacobian\" are encoded beyond the prefactor of (15.2)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.CoefficientEnclosure", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "0.07340691930603427103 < c_2,24 < 0.07340691930603427104,", + "does_not_claim": "the enclosures (statement only: nothing here encloses the coefficient); that an object satisfying the interface exists or is the coefficient of the source; the parent's results or the persistence interpretation; any finite-radius error estimate, unrestricted remainder or RN/24-jet statement; the arithmetic skeleton (endpoint adjacency, `d3 < d2`, endpoints in `(0, 1/10)`) is kernel-checked on main (31 `UniversalLaw.Side24` targets at origin/main f6deeba7), not here; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ReferenceComparison", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "|c_d,24/c_d,ref -1| < 10^(-106), d=2,3.", + "does_not_claim": "the bound (statement only); equality of the periodic and reference constants (the source says it is NOT claimed); `32*10^(-108) < 10^(-106)` is kernel-checked on main (`UniversalLaw.Side24.density_comparison_below_reported`), not here; nothing here encloses the coefficient." + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.pairing_sum_six", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "The coefficient sum for q=6 is\n1+15+45+15=76", + "does_not_claim": "the derivative bound 76|x|^6 e^(-|x|^2/2) itself" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.pairing_sum_le_six", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "it bounds the smaller orders as well", + "does_not_claim": "the analytic statement the arithmetic is attached to; the coefficient bound; the parent theorem" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.pairing_terms_six", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "1+15+45+15=76 and it bounds the smaller orders", + "does_not_claim": "the analytic statement the arithmetic is attached to; the coefficient bound; the parent theorem" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.sixth_moment_double_factorial", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "At zero, every such\ncontraction has absolute value at most15.", + "does_not_claim": "that Gaussian derivative contractions at zero are bounded by 15" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.lattice_shell_constant", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "<=729 sum_{j>=1} j^9 e^(-288j^2)\n <=1458 e^(-288).", + "does_not_claim": "the lattice-point count or the geometric tail sum" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.geometric_ratio_constants", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "successive terms have ratio at most512e^(-864)<1/2", + "does_not_claim": "that the ratio is below 1/2" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.period_exponent", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "sum_{n!=0} |n|^6 e^(-288|n|^2)", + "does_not_claim": "the analytic statement the arithmetic is attached to; the coefficient bound; the parent theorem" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.image_constant_value", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "E := 1458*(76*24^6+15)*10^(-125)\n = 21175738586478 *10^(-125).", + "does_not_claim": "the inequality |D^q K_24(0)-D^q K_infty(0)| < E" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.exp_taylor_partial_sum_gt_ten", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "The code proves e^(288/125)>10 using the positive rational Taylor sum through20", + "does_not_claim": "that the partial sum is below exp(288/125) (positivity of the omitted terms is analysis)" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.covariance_relative_bound", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "epsilon=10^(-108), since60E= I/3" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.odd_block_eigenvalues", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "[[1,-3],[-3,15]], whose eigenvalues are8 +/- sqrt(58)", + "does_not_claim": "the analytic statement the arithmetic is attached to; the coefficient bound; the parent theorem" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.odd_block_shifted_minors", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "subtract I/3 and use first principal minor2/3 and determinant7/9", + "does_not_claim": "Sylvester's criterion" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.smaller_eigenvalue_exceeds_third", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "Its smaller eigenvalue exceeds\n1/3", + "does_not_claim": "monotonicity of the square root" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.exponent_bounds", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "For d=2,3, a+2b<14 and2a+b<13.", + "does_not_claim": "the log/exp estimates that follow" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.exponent_values", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "a=m+2/3+n/2, b=d+n/2.", + "does_not_claim": "the analytic statement the arithmetic is attached to; the coefficient bound; the parent theorem" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.ratio_constant_ordering", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "ratio between1-13epsilon and\n1+28epsilon, hence safely between1-32epsilon and1+32epsilon", + "does_not_claim": "the elementary log/exp bounds" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Ledger.lean", + "declaration": "UniversalLaw.Side24.density_comparison_below_reported", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "|c_d,24/c_d,ref -1| < 10^(-106), d=2,3.", + "does_not_claim": "the multiplicative density comparison" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.conditional_third_derivative_variance", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "Var(t)=15, Cov(t,G)=(-3,0,...,0)", + "does_not_claim": "the Gaussian conditioning formula or the covariance derivatives" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.transverse_covariance_entries_thirds", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "Cov(A_ij,A_kl | V=0)\n = (2/3)delta_ij delta_kl + delta_ik delta_jl + delta_il delta_jk.", + "does_not_claim": "the delta formula itself" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.cone_moment_m1_thirds", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "D_1=E[A^2 1{A<0}]=4/3.", + "does_not_claim": "centred Gaussian symmetry" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.trace_and_traceless_variances", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "Var(s)=5/3 and Var(x)=Var(y)=1.", + "does_not_claim": "independence beyond zero covariance, which is Gaussian" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.fourth_moment_of_trace", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "E s^2=5/3, E s^4=25/3", + "does_not_claim": "the Gaussian fourth-moment identity" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.exponential_moment_of_trace", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "E exp(-s^2/2)=sqrt(3/8)", + "does_not_claim": "the Gaussian exponential-moment identity" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.cone_moment_m2_algebra", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "D_2 = (1/2)[25/3-20/3+8-8sqrt(3/8)]\n = 29/6-sqrt(6).", + "does_not_claim": "the integral identity or the Rayleigh cone integration" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.cube_root_simplification", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "simplifying 6^(2/3)/24^(1/3)=(3/2)^(1/3)", + "does_not_claim": "the analytic statement the arithmetic is attached to; the coefficient bound; the parent theorem" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.pin_determinant_and_joint_dimension", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "Its dimension is at most10.", + "does_not_claim": "the analytic statement the arithmetic is attached to; the coefficient bound; the parent theorem" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/ConeMoments.lean", + "declaration": "UniversalLaw.Side24.hessian_block_eigenvalues", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "The reference Hessian block has eigenvalues d+2 on trace and2 on traceless matrices", + "does_not_claim": "the spectral decomposition" + } + ], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "receipt_or_workflow": ".github/workflows/formal-verification.yml (tools/formal_gate_check.py --run-lean inside the required aggregate `verify`); receipt formal/.lake/formal-evidence/receipt.json is a workflow artifact, none is committed in tree", + "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": "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. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed.", + "next_step": "Formal lane (Math- Mathlib package, by the lane's admission rules): construct the conditional law A_u | V_u = 0 and prove ConeMomentD1Identity / ConeMomentD2Identity for the reference law under ConeMomentInterface; then prove ReferenceCoefficientIdentity; the enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md). Independent reviewer: alignment verdicts on the 28 Side24 statements." + }, + { + "claim_id": "lifetime-remainder", + "kind": "landing_claim", + "title": "D2 Theorem R, bounded unrestricted short-lifetime remainder (LIFETIME-BOUNDED-REMAINDER-20260924-v1), heading \"Theorem R\"", + "layer0": { + "source_id": "lifetime-remainder", + "heading": "Theorem R" + }, + "disposition_transcribed": "REVIEWED_SCOPED", + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.RemainderBoundedR1", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "|nu_cand(ell)-c ell^(-1/3)| <= C,", + "does_not_claim": "the result or its proof (quadratic determinant cancellation (R7), coupling (R4), cap-loss bound (R12)); convergence of the remainder; a second coefficient; numerical `C` or `ell_*`; the compact-window `O(ell^(2/3))` difference without cutoffs; the parent interfaces it imports (D1-A-E); nonauthor review or acceptance. If no `c` satisfies the parent property the statement holds vacuously; that is a transcription of its dependency on (1.3), not a claim." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsUnrestrictedPair", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume.", + "does_not_claim": "anything about the off-diagonal (far) population separately (Section 14 of the source)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsUnrestrictedLeadingCoefficient", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "There is a positive finite constant c_{d,L} such that", + "does_not_claim": "the explicit Gaussian/angular representation (15.2) of `c_{d,L}`; uniqueness of `c` (which follows from the asymptotics but is not stated); existence of versions." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsLifetimeDensityVersion", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "These densities admit versions with", + "does_not_claim": "existence of a version; the weighted Kac-Rice identity; the radial ledger (10.2). Measurability of the pair count as a function of the sample is REQUIRED by the predicate (through `Integrable`, hence `AEStronglyMeasurable`), not established here." + } + ], + "kernel_evidence": [], + "not_established": "RemainderBoundedR1 states the shape of (R1) for versions of the densities under the interface structures; the result is not proved. Not stated: (R19)-(R20), the observation transform and determinant lemmas R3.1/R3.2, the cap-loss bound, the parent interfaces D1-A-E it imports, the marked-cylinder-cap support. The coefficient c enters only through IsUnrestrictedLeadingCoefficient (c > 0 and the leading-term role), not through the representation (15.2). No numerical C or ell_*, no second coefficient, no convergence of the remainder, no compact-window O(ell^(2/3)) difference without cutoffs. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State (R19) (cumulative form) as a further Prop; transcribe the parent interfaces consumed (section 8 comparison, section 9 as replaced by v1.1, section 10, (14.1), (15.2)) as explicit hypotheses; independent reviewer: alignment verdict on RemainderBoundedR1 against (R1), including the \"versions\" clause and the exclusion of the essential global maximum." + }, + { + "claim_id": "rn-count-interface", + "kind": "landing_claim", + "title": "D4 RN count interface (RN-COUNT-INTERFACE-20260924-v1; LANDING heading \"Theorem N\", the file carries displays (N1)-(N7))", + "layer0": { + "source_id": "rn-count-interface", + "heading": "Theorem N" + }, + "disposition_transcribed": "HOLD_WITH_DOMAIN", + "formal_progress": "none", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.HolderCountBoundN1", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E[N_r 1_Er] <= (E N_r^p)^(1/p) P(E_r)^(1-1/p).", + "does_not_claim": "that any count of the Gaussian model satisfies the moment hypothesis; the result or its proof; any rate in `r`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.HolderConsequenceN2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E[N_r 1_Er] <= A^(1/p) C^(1-1/p)", + "does_not_claim": "that the exponent can be rounded up to `3`; that any Gaussian count satisfies the hypotheses; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.ConditionalMeanIdentityN3", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E[N_r 1_Er]=q E[N_r | E_r].", + "does_not_claim": "the equivalence with a bounded conditional mean, the lower comparison `q ≥ c r^3`, or anything about the Gaussian model; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.UniformCounterexampleN4", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E N_r/r^3=ceil(log(1/r)) -> infinity.", + "does_not_claim": "the explicit bound `sup_(t>=1) e^(-3t)(t+1)^p` (only existence of a `p`-dependent bound); the one-`p` sharpness construction `N_r=ceil(r^(-3/p))1_Er`; the exponential-tail remark after (N5); that the construction says anything about the Gaussian field; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.ExponentialTailLayerCakeN5", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "= B q[1+log(A/q)].", + "does_not_claim": "the intermediate integral identity `integral_0^infinity min(q,Ae^(-t/B))dt`; that the logarithmic loss is attained (the source's remark about (N4)); anything about the Gaussian model; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + } + ], + "kernel_evidence": [], + "not_established": "No Prop states Theorem N as a whole: (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; (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\". Nothing about the Gaussian model is stated or proved; no RN rate, no 24-jet enclosure, no ENV-RESCOV discharge. Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State (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`. (N1)-(N3), (N5) are candidates for proofs in the Math- Mathlib lane (Hoelder, layer cake). Independent reviewer: alignment of the five Props with (N1)-(N5)." + }, + { + "claim_id": "rn-fixed-remote-window", + "kind": "landing_claim", + "title": "D4 Theorem A, fixed-remote mean measure (RN-FIXED-REMOTE-WINDOW-20260924-v1), heading \"Theorem A\"", + "layer0": { + "source_id": "rn-fixed-remote-window", + "heading": "Theorem A" + }, + "disposition_transcribed": "HOLD_WITH_DOMAIN", + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.PinnedLawInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "Q_r denotes continuous Gaussian regression on these 2(d+1) observations.", + "does_not_claim": "that `Q_r` is the Gaussian regression of any field; that `W_r` equals `F_d(H_M) F_(d-1)(H_S)`; the normalizer floor `Z_r >= z_* r^2`; that `Λ` is given by (13); existence or uniqueness of an object satisfying the interface (an inhabitant is not asserted; the consistency remark above is not stated); that `0 < r < L` is the sources' unspecified `small r` threshold." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.IsCriticalPoint", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "grad f(M)=grad f(S)=0.", + "does_not_claim": "differentiability of `g` at `x` (if `g` is not differentiable, `fderiv` is `0`)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.HessianHasInertia", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The index counts negative eigenvalues.", + "does_not_claim": "equivalence with an eigenvalue count for a non-symmetric form." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedRemoteMeanMeasureA2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "| E_(Q_r^W) N_(r,j)(E)/(k r^3)", + "does_not_claim": "the result or its proof (uniform remote nondegeneracy, coupling (5), weight expansion (10), normalizer (11)); identification of `I.Λ` with the kernel (13); the total-variation convergence remark; the multi-witness Section 6; any numerical `C` or `r_*`; uniformity as `ρ → 0`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedRemoteCountBoundA3", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_(r,j)(E) <= C k r^3 |E|,", + "does_not_claim": "a matching lower bound on the event probability (the source says this is NOT one); the unnormalized numerator bound (14); the result or its proof; any numerical constant; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "FixedRemoteMeanMeasureA2 and FixedRemoteCountBoundA3 state the shapes of displays (2) and (3) for any object satisfying PinnedLawInterface (the regression law Q_r, the weight W_r, the normalizer and the contact kernel Lambda_j are interface fields, gated on 0 < r < L); no inhabitant of the interface is constructed and the kernel is not identified with formula (13). Not stated: the observation transform U_r and target v_r, the coupling (5), the weight expansion (10)-(11), the Kac-Rice formula (12), the kernel formula (13), the unnormalized numerator bound (14), and the multi-witness Section 6 displays (15)-(16). Nothing is proved; no numerical C or r_*; no shrinking exclusion rho -> 0, no collisions, no global RN complement (the disposition reason). Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State Section 6 ((15)-(16), separated m-tuples) as Props; record an inhabitant construction of PinnedLawInterface as a proof-lane target (Dirac mass at one admissible smooth path); independent reviewer: alignment of A2/A3 with displays (2)-(3), in particular the uniformity in b, k, R, j, E and the fixed rho." + }, + { + "claim_id": "rn-fixed-annulus-window", + "kind": "landing_claim", + "title": "D5 fixed-annulus height-window candidate (D5-FIXED-ANNULUS-STITCH-20260925-v1, PR22), heading \"Candidate theorem\"", + "layer0": { + "source_id": "rn-fixed-annulus-window", + "heading": "Candidate theorem" + }, + "disposition_transcribed": "REVIEWED_SCOPED", + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedAnnulusWindowCandidateA2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_Qr^W N_j(r E0) <= C*k*r^3*area(E0),", + "does_not_claim": "the result or its proof (contact vector (A10), eigenfloor (A12), cutoff (A13), three-Hessian estimate (A18), inner-strip bound (A26)); the all-index sum or the Markov event bound; an all-height assertion; uniformity in `L`, as `k → 0`, as `B0 → ∞`, or for `d ≥ 3`; any numerical constant; nonauthor review or acceptance (the landing disposition REVIEWED_SCOPED is transcribed by the register, not here)." + } + ], + "kernel_evidence": [], + "not_established": "FixedAnnulusWindowCandidateA2 states the shape of (A2) (d = 2, fixed L, fixed scaled annulus, compact positive gaps, between-pin height window) over PinnedLawInterface; the result is not proved. Not stated: the six-pin regression (A3)-(A5), normalizer (A8), contact vector (A10)-(A12), cutoff (A13)-(A14), density (A17), Hessian bounds (A18)-(A20), intensities (A21)-(A26), the TWO_SCALE_ADDENDUM S6-S21 support (not pinned here), the eight reviewed interfaces as separate statements. No all-height assertion, no positive leading coefficient, no probability lower bound, no uniformity as k -> 0, B0 -> infinity, in L or for d >= 3; no global D5/RN/JETMOD transfer. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Pin TWO_SCALE_ADDENDUM.md and state (S3)-(S5) and (S20)-(S21); independent reviewer: alignment of FixedAnnulusWindowCandidateA2 with (A2), checking that the height window is strict and the constants depend only on L, A0, B0 and the mark compacts." + }, + { + "claim_id": "p15-realized-covers", + "kind": "landing_claim", + "title": "P15 realized covers (P15-REALIZED-COVERS-20260924-v1; LANDING heading \"Theorem P\", the file carries Proposition P1 and Theorem P2 with displays (P1)-(P11))", + "layer0": { + "source_id": "p15-realized-covers", + "heading": "Theorem P" + }, + "disposition_transcribed": "HOLD_WITH_DOMAIN", + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.RealizedData", + "kind": "structure", + "formal_progress": "specified", + "anchor": "Choose positive integers a_i,d_i and disjoint original-coordinate blocks X_i with", + "does_not_claim": "that the data satisfy the sources' side conditions (that is `IsRealizedFamily`)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.IsRealizedFamily", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " n_i=|X_i|=a_i d_i+1.", + "does_not_claim": "that any particular data record satisfies it." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.IsGood", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "D={U subset X: |U intersect X_i|<=a_i for every i,", + "does_not_claim": "that `D` is decreasing or contains the empty set (facts of the sources, not stated separately)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.IsKDecomposable", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "write I_k(D) for sets partitionable into at most k members of D", + "does_not_claim": "equivalence with coverings by at most `k` good sets (true for a decreasing family, not stated)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.IsGeneratorCover", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "A generator cover G of O_k(D) means every obstructed set contains some g in G.", + "does_not_claim": "anything." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.IsPaletteFeasible", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " intersection_(i in e) P_i = empty, for every e in H. (P6)", + "does_not_claim": "upward closure in `K` (true, not stated)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.PaletteNumberAttained", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "By discarding excess labels within blocks, one may take |P_i|=d_i.", + "does_not_claim": "the statement (not proved here; disjoint palettes of sizes `d_i` are a witness in the sources' argument)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.IsMinimalForbidden", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The inclusion-minimal forbidden ORIGINAL sets are exactly", + "does_not_claim": "anything." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.IsTransversal", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "(b) for each e in H, EVERY transversal {x_i:i in e}, with x_i in X_i.", + "does_not_claim": "anything." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.MinimalForbiddenCharacterisation", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "(a) every (a_i+1)-subset of a single X_i;", + "does_not_claim": "the characterisation (statement only); its completeness clause is what the source calls \"completeness, not just existence of some witnesses\"." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.LocalRestriction", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "D_i=D intersect 2^(X_i)={S subset X_i: |S|<=a_i}. (P2)", + "does_not_claim": "the identity (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.LocalDecomposability", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " I_k(D_i)={S:|S|<=a_i k}.", + "does_not_claim": "the identity (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.LocalObstructionIsFullBlock", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " G_i={X_i} (P3)", + "does_not_claim": "the statement (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.LocalPriceChain", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " =P(U_i=X_i)<=q_i<=phi(q_i), (P4)", + "does_not_claim": "the chain (statement only); the identity `prod p_v = P(U_i = X_i)` is the definition of the weight." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.IndependentBlockHazard", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " <=-log(product_i(1-q_i))<=-log mu_p(D). (P5)", + "does_not_claim": "the inequalities (statement only); independence of crossing events (\"Crossing events are not declared independent\")." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.FullBlockCoverIff", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " K>=K_H(d). (P7)", + "does_not_claim": "the criterion (statement only); optimality of any other cover (\"It does not prove that no different cover with the same budget works at fewer colors\"); unrestricted prize optimality; disposition transcribed from LANDING_CLAIMS for p15-realized-covers: HOLD_WITH_DOMAIN; nonauthor review and acceptance are not claimed here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.CoverCostHazardBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "covercost_c(O_K(D)) <= min(1,-log mu_p(D)). (P8)", + "does_not_claim": "the bound (statement only); anything for `c_v > p_v`; the unrestricted P15 prize; original P15-B; disposition transcribed from LANDING_CLAIMS for p15-realized-covers: HOLD_WITH_DOMAIN; nonauthor review and acceptance are not claimed here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.WholeGroundChromaticNumber", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "the chromatic number of the ENTIRE ground set X is K_H(d+1), because ceil(n_i/a_i)=d_i+1.", + "does_not_claim": "the statement (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.ElementaryPaletteFormula", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " K_H(d)=max(max_i d_i, ceil(sum_i d_i/s)). (P9)", + "does_not_claim": "the formula (statement only); the matroid palette formula of Section 5 (not specified)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.TriangleNeedsThree", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "at unit demands needs3 colors, although a formula considering only size-two edge loads would give2.", + "does_not_claim": "the value (statement only; finite and decidable in principle)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.Benchmark816IsRealized", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Every D-color class contains at most one vertex per block and occupies at most three blocks.", + "does_not_claim": "the statement (finite, decidable in principle; not proved here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.Benchmark816Palette", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " K_H(408,...,408)=816.", + "does_not_claim": "the value (statement only); the arithmetic `max(408, ceil(2448/3)) = 816` is a core-Lean target on main, not here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.Benchmark816Cover", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "At816 colors it covers the obstruction; at815 it does not, because omitting one vertex from each block leaves2448 vertices, each color holds at most3, and815*3<2448.", + "does_not_claim": "the statement (finite over `2^2454` subsets, not decidable in practice); `815*3 < 2448` is a core-Lean target on main." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.Benchmark816WholeGround", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The whole ground set, unlike the cover-avoiding sets, needs818 colors, since2454/3=818.", + "does_not_claim": "the statement (statement only); `2454/3 = 818` is a core-Lean target on main." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.Benchmark816MinimalForbiddenCounts", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " 15*409^4=419743994415 crossing transversal quadruples. (P10)", + "does_not_claim": "the counts (statement only); the evaluations `500616` and `419743994415` are core-Lean targets on main; the source stresses these are \"not a claim that the program enumerated more than four hundred billion sets\"." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.Benchmark816Price", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " 6/(40900^409)>0.", + "does_not_claim": "the identity (statement only; the six blocks are distinct members of the image)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.Benchmark816UnionBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " =2449227/8180000000 < 3/10000. (P11)", + "does_not_claim": "the bound (statement only; \"only a union bound, not an equality of the global bad probability\"); the two rational identities are core-Lean targets on main." + } + ], + "kernel_evidence": [], + "not_established": "MinimalForbiddenCharacterisation (Proposition P1), FullBlockCoverIff (Theorem P2, (P7)) and CoverCostHazardBound ((P8)) state the results for every realized family; none is proved here. No `proved` statement is listed: main's 12 P15 arithmetic theorems for this claim (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, and the elementary instances of (P9)) exist only in this programme's uncommitted main worktree, on no commit, and are not listed until a commit and a hosted formal-verification.yml run exist; even then they would not count the minimal forbidders, show the palette is 816 or evaluate 6/40900^409. The benchmark Props Benchmark816Cover/WholeGround/MinimalForbiddenCounts/Price/UnionBound quantify over 2^2454 subsets and are not decidable in practice; PaletteNumberAttained (nonemptiness of the feasible palette set) is stated, not proved. Nothing is asserted for prices p_v < c_v <= phi(p_v), 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; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Proof lane: TriangleNeedsThree, Benchmark816IsRealized, LocalRestriction and LocalDecomposability are finite or elementary and are the first kernel targets; PaletteNumberAttained via the disjoint-palette witness. Independent reviewer: alignment of MinimalForbiddenCharacterisation with Proposition P1 (completeness clause) and of FullBlockCoverIff with (P7). main: commit and push the P15 modules (hosted formal-verification.yml green), then list the P15 arithmetic theorems here as `proved` statements citing the 40-hex commit and the workflow run; until then no P15 theorem is listed." + }, + { + "claim_id": "p15-price-boundary", + "kind": "landing_claim", + "title": "P15 price boundary exact counterexample (P15-PRICE-BOUNDARY-20260924-v1), heading \"Counterexample\"", + "layer0": { + "source_id": "p15-price-boundary", + "heading": "Counterexample" + }, + "disposition_transcribed": "EXACT_COUNTEREXAMPLE", + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.TwoCoordGeneratorPrices", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The possible generator costs are1,2/3,2/3,4/9 for the empty", + "does_not_claim": "the values (statement only; as integers over `9` they are core-Lean targets on main)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.DemandOneRefutation", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " covercost_c(O_1(D))=4/9 > 1/3 > log(4/3).", + "does_not_claim": "the statement (statement only; the finite parts are decidable in principle, the two logarithm inequalities are analysis); that this contradicts original P15-B (\"It does not contradict the original P15-B amalgamation\"); anything about the `408`-scale or the unrestricted prize; the integral representation `log 2 = 2 integral_0^(1/3) 1/(1-t^2) dt` is not specified; disposition transcribed from LANDING_CLAIMS for p15-price-boundary: EXACT_COUNTEREXAMPLE; nonauthor review and acceptance are not claimed here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.LogOnePlusBelowIdentity", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "using log(1+x)0.", + "does_not_claim": "the inequality (statement only; the source proves it, this module does not)." + } + ], + "kernel_evidence": [], + "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)); it is not proved. No `proved` statement is listed: main's 6 P15 arithmetic theorems for this claim (the finite arithmetic: good sets, obstruction, 3/4 over denominator 4, prices 9,6,6,4 over 9, 3*4 > 9) exist only in this programme's uncommitted main worktree, on no commit, and are not listed until a commit and a hosted formal-verification.yml run exist; the two logarithm inequalities phi(1/2) = log 2 > 2/3 and log(4/3) < 1/3 are analysis and are not proved anywhere; log(1+x) < x (LogOnePlusBelowIdentity) is stated only. The counterexample does not refute d_i >= 2 successors or original P15-B (the disposition reason). Disposition EXACT_COUNTEREXAMPLE is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Math- Mathlib lane: prove the two logarithm inequalities (Real.log bounds) and the finite conjuncts of DemandOneRefutation; this is the smallest complete research statement in the packet that could move from `specified` to `proved`. Independent reviewer: alignment of DemandOneRefutation with the source paragraph. main: commit and push the P15 modules (hosted formal-verification.yml green), then list the P15 arithmetic theorems here as `proved` statements citing the 40-hex commit and the workflow run; until then no P15 theorem is listed." + }, + { + "claim_id": "p15-price-budget-restricted", + "kind": "landing_claim", + "title": "P15 restricted transformed-price extension, Theorem T1 (P15-TRANSFORMED-PRICE-BUDGET-20260924-v1), heading \"Theorem T1\"", + "layer0": { + "source_id": "p15-price-budget", + "heading": "Theorem T1" + }, + "disposition_transcribed": "HOLD_WITH_DOMAIN", + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.LogRatioBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " -log(1-p)=integral_0^p dt/(1-t) <= p/(1-p). (T2)", + "does_not_claim": "the inequality (statement only); the integral representation is not specified." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.RatioExponentBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " <= (4/3)*(4/9)^a <=16/27. (T3)", + "does_not_claim": "the inequalities (statement only); the preceding ratio steps of (T3) (price over `prod_{v in S} p_v`) are stated through their consequence `RestrictedLocalBudget`; the case `a = 1, n = 3` giving `16/27` is a core-Lean target on main." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.RestrictedLocalBudget", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " price({X}) <= (16/27)q <= phi(q). (T4)", + "does_not_claim": "the bound (statement only); the source notes \"The extra demand hypothesis d>=2 is load-bearing\" and that `p_v <= 1/4` \"is a sufficient range, not an asserted sharp threshold\"." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.RestrictedTransformedPriceBudget", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " covercost_c(O_K(D)) <= min(1,(16/27)*[-log mu_p(D)]). (T1)", + "does_not_claim": "the bound (statement only); any retraction of the demand-one refutation; the unrestricted prize; new classical coloring theory; disposition transcribed from LANDING_CLAIMS for p15-price-budget-restricted: HOLD_WITH_DOMAIN; nonauthor review (\"nonauthor review open\") and acceptance are not claimed here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.RhoTestSufficient", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "If rho<=1 the same argument proves price({X})<=q for that block.", + "does_not_claim": "the statement (statement only); anything when `rho > 1` (the source: a block with `rho > 1` \"must not be labeled infeasible\")." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.CapTestAtTwoThirds", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "or if q>=2/3 (then phi(q)=1, since log3>1)", + "does_not_claim": "the statement (statement only); the integral proof of `log 3 > 1` is not specified; the checker program of that section is not specified." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.Benchmark816TransformedRange", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " 0<=c_v<=log(40900/40899),", + "does_not_claim": "the statement (statement only); \"No large witness enumeration\" is involved, nor any claim about earlier P15 inputs." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.Benchmark816RationalPrice", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " 6*log(40900/40899)^409 <=6/(40899^409). (T7)", + "does_not_claim": "the statements (statement only); the rational identity `1/40900 + (1/40900)^2/2 = 81801/3345620000` is exact arithmetic not re-proved here; admissibility rests on `integral_0^p 1/(1-t) dt > integral_0^p (1+t) dt`, which is not specified." + } + ], + "kernel_evidence": [], + "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); not proved. (T2), the exponent arithmetic of (T3), (T4), the per-block sufficient test (T5), the cap test and the Section 4 benchmark statements are stated only; the intermediate ratio steps of (T3), the integral representation behind (T2) and the admissibility integral are not specified. No `proved` statement is listed: main's 3 P15 arithmetic theorems for this claim (the a = 1, n = 3 instance 16/27, finite instances of the (T3) bound and the rational identity 81801/3345620000) exist only in this programme's uncommitted main worktree, on no commit, and are not listed until a commit and a hosted formal-verification.yml run exist; they are not (T1) in any case. The dependency on claim p15-realized-covers is not discharged here. 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; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Math- Mathlib lane: LogRatioBound (-log(1-p) <= p/(1-p)) and RatioExponentBound are provable from Mathlib Real.log facts and are the first targets; independent reviewer: alignment of RestrictedTransformedPriceBudget with (T1) including the per-coordinate price hypothesis and the K >= K_H(d) palette. main: commit and push the P15 modules (hosted formal-verification.yml green), then list the P15 arithmetic theorems here as `proved` statements citing the 40-hex commit and the workflow run; until then no P15 theorem is listed." + }, + { + "claim_id": "p15-full-price", + "kind": "landing_claim", + "title": "D6 Theorem F, P15 full transformed-price budget with sharp factor rho_star (P15-FULL-TRANSFORMED-PRICE-20260924-v1), heading \"Theorem F\"", + "layer0": { + "source_id": "p15-full-price", + "heading": "Theorem F" + }, + "disposition_transcribed": "REVIEWED_SCOPED", + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.FullTransformedPriceBudget", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " covercost_c(O_K(D)) <= min(1,rho_star[-log mu_p(D)]). (F2)", + "does_not_claim": "the bound (statement only); optimality of `rho_star` (that is `SharpnessOfRhoStar`); any unrestricted downset or prize statement (the source: \"palette size remains K_H(d), which can depend on H and d\"); the nonauthor review recorded in LANDING_CLAIMS (REVIEWED_SCOPED) is not reproduced or extended here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.RhoStarEnclosure", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " 0.84547981724898672067 < rho_star", + "does_not_claim": "the enclosure (statement only; the source's `full_price.py` Fraction procedure is not a kernel proof and is not specified); this is the next full-result target named in main's `docs/FORMAL_VERIFICATION_ROLLOUT_20260927.md`, still at `specified`." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.RhoStarBelowSixSevenths", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Thus 6/7 is a simpler valid rational factor.", + "does_not_claim": "the inequality (statement only); `16/27 < 6/7` is a core-Lean target on main." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.IsProperDownset", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Let A be any proper decreasing family on n>=1 coordinates containing the empty set.", + "does_not_claim": "anything." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.HazardStarPositive", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The probability is positive on this compact cube because the empty set is admitted and every p_i(t)<1.", + "does_not_claim": "the statement (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.CoordinatewiseConcavity", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Only SEPARATE coordinatewise concavity is asserted; joint concavity is not needed or claimed.", + "does_not_claim": "the statement (statement only); joint concavity; the derivative formulas (F4)-(F5) themselves (not specified; concavity is their consequence)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.HazardProductBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " F_A(t) >= (product_i t_i) H_A. (F6)", + "does_not_claim": "the inequality (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.HazardPhiBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " -log mu_p(A) >= H_A product_i phi(p_i). (F7)", + "does_not_claim": "the inequality (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.FullGroundPriceBound", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " product_i c_i <= min(1, [-log mu_p(A)]/H_A). (F8)", + "does_not_claim": "the inequality (statement only); that the full-ground generator covers any obstruction (\"It does not assert that this generator alone covers an arbitrary obstruction\")." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.PhiAtPStar", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "at every p_i=p_star and every c_i=1, the two sides of (F7) are equal", + "does_not_claim": "the identity (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.FullGroundBudgetIff", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "a full-ground generator universally meets the capped hazard budget precisely when H_A>=1", + "does_not_claim": "the equivalence (statement only); coverage of arbitrary `A` (\"it does not infer coverage of arbitrary A\")." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.CapacityHazardIsBinomial", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " H_(n,a)=-log P(Bin(n,p_star)<=a). (F9)", + "does_not_claim": "the identity (statement only; a finite identity for each `n`, general in `n`)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.OddMajorityIdentity", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " =p(1-2p)P(S=a)<0. (F10)", + "does_not_claim": "the identities (statement only; polynomial identities in `p`, a future kernel target for each fixed `a` and by induction in general)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.CapacityWorstCase", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " =3-log(3e-2)=h_star>1. (F11)", + "does_not_claim": "the statements (statement only); the source stresses \"it is not a finite-grid extrapolation\"." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.HStarExceedsOne", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " e^2-(3e-2)=(e-1)(e-2)>0,", + "does_not_claim": "the statements (statement only)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.LocalPriceBoundFull", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " price({X_i}) <= rho_star [-log mu_p(D_i)], (F12)", + "does_not_claim": "the bound (statement only); `price <= q_i` (\"The inequality price<=q_i is NOT required and is in general stronger than what is proved here\")." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.GlobalAssemblyHazard", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " sum_i [-log mu_p(D_i)] <= -log mu_p(D). (F13)", + "does_not_claim": "the inequalities (statement only; \"without independence of crossing failures\")." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.SharpnessOfRhoStar", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Replacing rho_star by any smaller constant in (F2) would make its right side strictly less than one.", + "does_not_claim": "the statements (statement only; the finite parts are decidable in principle); a necessity claim for multi-block families with some demand one (the source: \"other constraints may change its budget\")." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.DemandOneBoundary", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " mu_p(D)=1-p_star^(a+1)>1-p_star=exp(-1),", + "does_not_claim": "the statements (statement only); any retraction of original P15-B (\"whose local-budget premise is missing in that case\")." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.SixSeventhsCertificate", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Therefore 3e-2<197/327/6 and rho_star<6/7.", + "does_not_claim": "the statements (statement only); the two exact rational identities are exact arithmetic not re-proved here; the series and geometric-tail bounds behind `e < 31967/11760` are not specified." + } + ], + "kernel_evidence": [], + "not_established": "FullTransformedPriceBudget states (F2), RhoStarEnclosure states (F3), SharpnessOfRhoStar states the attainment clause; none is proved. The hazard interpolation lemma is stated as separate coordinatewise concavity (CoordinatewiseConcavity) and its consequences (F6)-(F8); the derivative formulas (F4)-(F5) are not specified. No `proved` statement is listed: main's 7 P15 arithmetic theorems for this claim (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) exist only in this programme's uncommitted main worktree, on no commit, and are not listed until a commit and a hosted formal-verification.yml run exist; they would not place rho_star = 1/(3 - log(3e - 2)) between the endpoints, prove e < 87/32 or 197/32 < exp(11/6), or be 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; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Math- Mathlib lane: RhoStarBelowSixSevenths / SixSeventhsCertificate (Real.exp and Real.log bounds e < 31967/11760 < 87/32, 197/32 < exp(11/6)) are the next full-theorem target named in main docs/FORMAL_VERIFICATION_ROLLOUT_20260927.md; then HStarExceedsOne and PhiAtPStar. Independent reviewer: alignment of FullTransformedPriceBudget with (F2), in particular the quantification over ALL p in [0,1]^X and the SAME palette K >= K_H(d). main: commit and push the P15 modules (hosted formal-verification.yml green), then list the P15 arithmetic theorems here as `proved` statements citing the 40-hex commit and the workflow run; until then no P15 theorem is listed." + }, + { + "claim_id": "downstream-hard-gate", + "kind": "landing_claim", + "title": "Downstream hard gate, engineering integrity control (DOWNSTREAM-HARD-GATE-20260925-v1; LANDING statement_path frontiers/downstream_gate_20260925/README.md \"What it enforces\", not pinned here; GRAPH.json is pinned)", + "layer0": { + "source_id": "downstream-graph", + "heading": "DOWNSTREAM-HARD-GATE-20260925-v1 (GRAPH.json; the README statement_path is not among the pinned sources)" + }, + "disposition_transcribed": "ENGINEERING_HOLD", + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "An engineering control, not a mathematical claim: there is no theorem to state in Lean and none is stated. The gate's terminal classifications and non-discharge tokens are transcribed in the graph_node rows of this register only as source identity. Disposition ENGINEERING_HOLD (\"repository-wide external controlling-source binding remains unresolved\") is transcribed, not re-derived.", + "next_step": "No Lean target. The control's semantics live in Math- frontiers/downstream_gate_20260925 and its workflow; the register only records that every GRAPH math.* node has a row here." + }, + { + "claim_id": "math.uniform-matrix-cap-lifetime", + "kind": "graph_node", + "title": "GRAPH node math.uniform-matrix-cap-lifetime (D1 candidate, PROVED_REVIEWED, controlling false): parent UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1, Theorems A, B, C accepted at existential scope with the four reading-rule component nodes", + "layer0": { + "source_id": "lifetime-parent", + "heading": "Theorem A (uniform selection candidate); Theorem B (compact-mark density candidate); Theorem C (unrestricted finite-lifetime leading density candidate)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.IsLPeriodic", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance", + "does_not_claim": "anything about the field." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.IsSmooth", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "In particular the field has a smooth version and all finite moments of its global derivative suprema exist.", + "does_not_claim": "existence of a smooth version, or moment bounds on derivative suprema." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.GaussianFieldInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance", + "does_not_claim": "existence or uniqueness in law of the field; the Fourier representation and positive spectrum of Section 2; almost-sure Morse genericity of Section 8." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.IsCriticalPoint", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "grad f(M)=grad f(S)=0.", + "does_not_claim": "differentiability of `g` at `x` (if `g` is not differentiable, `fderiv` is `0`)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.HessianHasInertia", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The index counts negative eigenvalues.", + "does_not_claim": "equivalence with an eigenvalue count for a non-symmetric form." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.IsNondegMax", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue.", + "does_not_claim": "that every local maximum of a sample path is nondegenerate (Morse genericity is Section 8 of the source)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.IsIndexSaddle", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue.", + "does_not_claim": "that such a saddle is a merging saddle of any superlevel bar." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.IsCandidatePair", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume.", + "does_not_claim": "positivity of the height gap (added separately where the source requires it)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.IsMorseDistinct", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Countable exhaustion now proves that Q is almost surely Morse with distinct critical values.", + "does_not_claim": "that sample paths are almost surely Morse (Section 8 of the source)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.SuperlevelBarInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint.", + "does_not_claim": "Borel measurability of the elder event (Section 8 / the Section 9 replacement), identification with a persistence-module elder rule, or any behaviour off the Morse distinct-value locus." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.IsFiniteBar", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "each finite superlevel H0 bar has exactly one local-maximum birth and one index-(d-1) merging-saddle death.", + "does_not_claim": "the identification of these pairs with bars of a persistence module; that the count of such pairs equals the finite-bar count off the Morse locus." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Field.lean", + "declaration": "UniversalLaw.Spec.Field.IsLifetimeDensityVersion", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "These densities admit versions with", + "does_not_claim": "existence of a version; the weighted Kac-Rice identity; the radial ledger (10.2). Measurability of the pair count as a function of the sample is REQUIRED by the predicate (through `Integrable`, hence `AEStronglyMeasurable`), not established here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsLPeriodic", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance", + "does_not_claim": "anything about the field." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsSmooth", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "In particular the field has a smooth version and all finite moments of its global derivative suprema exist.", + "does_not_claim": "existence of a smooth version, or moment bounds on derivative suprema." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.GaussianFieldInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance", + "does_not_claim": "existence or uniqueness in law of the field; the Fourier representation and positive spectrum of Section 2; almost-sure Morse genericity of Section 8." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsCriticalPoint", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "grad f(M)=grad f(S)=0.", + "does_not_claim": "differentiability of `g` at `x` (if `g` is not differentiable, `fderiv` is `0`)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.HessianHasInertia", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The index counts negative eigenvalues.", + "does_not_claim": "equivalence with an eigenvalue count for a non-symmetric form." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsNondegMax", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue.", + "does_not_claim": "that every local maximum of a sample path is nondegenerate (Morse genericity is Section 8 of the source)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsIndexSaddle", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue.", + "does_not_claim": "that such a saddle is a merging saddle of any superlevel bar." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsCandidatePair", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume.", + "does_not_claim": "positivity of the height gap (added separately where the source requires it)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsMorseDistinct", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Countable exhaustion now proves that Q is almost surely Morse with distinct critical values.", + "does_not_claim": "that sample paths are almost surely Morse (Section 8 of the source)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.SuperlevelBarInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint.", + "does_not_claim": "Borel measurability of the elder event (Section 8 / the Section 9 replacement), identification with a persistence-module elder rule, or any behaviour off the Morse distinct-value locus." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsFiniteBar", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "each finite superlevel H0 bar has exactly one local-maximum birth and one index-(d-1) merging-saddle death.", + "does_not_claim": "the identification of these pairs with bars of a persistence module; that the count of such pairs equals the finite-bar count off the Morse locus." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsLifetimeDensityVersion", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "These densities admit versions with", + "does_not_claim": "existence of a version; the weighted Kac-Rice identity; the radial ledger (10.2). Measurability of the pair count as a function of the sample is REQUIRED by the predicate (through `Integrable`, hence `AEStronglyMeasurable`), not established here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsWindowPair", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Count ordered maximum/saddle pairs with 00, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume.", + "does_not_claim": "anything about the off-diagonal (far) population separately (Section 14 of the source)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.PairLawInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "Let p_r(b,k,R) be the Q^W probability that the **global ordinary superlevel elder death partner** of M is S.", + "does_not_claim": "existence of the regression law, the typed weight, the full normalizer `Z_r > 0`, or that `p` is the probability of any event under any measure declared here; that `p` depends on `G` or `I` in any way (no field mentions them; they only record which field and elder relation the law is meant to refer to)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.ParentUniformSelectionA", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "There exist r_*>0 and C= 3. The GRAPH classification PROVED_REVIEWED and the review disposition ACCEPT_AT_EXISTENTIAL_SCOPE are source identity, not re-derived; the five validity premises of the hardening-branch claim graph are not touched. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p or the field: the parent-shape Props are predicates on the interface objects, not closed statements; the universal closure of ParentUniformSelectionA over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed.", + "next_step": "Pin ERRATUM_CONGRUENCE.md, REPAIR.md (Section 9 v1.1) and the reconciliation record, and add the reading rule as explicit hypotheses or separate Props; state (15.2) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of the three parent Props against (1.1)-(1.3) with the honesty checklist (quantifier order, existential constants outside the universal quantifiers, \"versions\", compact marks, all frames)." + }, + { + "claim_id": "math.d1-component.congruence-erratum", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d1-component.congruence-erratum (amendment, PROVED_REVIEWED): D_r = diag(r^(-1/2), I) replaces the displayed congruence factor of parent section 5 (imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md, not pinned here)", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d1-component.congruence-erratum" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "No Lean statement: the erratum's source bytes are not pinned in this packet and no Prop states the corrected congruence D_r H_i D_r or det(D_r H_i D_r) = det(H_i)/r. The parent Props of this register do not transcribe section 5 at all, so the amendment has nothing to attach to yet.", + "next_step": "Pin ERRATUM_CONGRUENCE.md (blob 213594d6, 1782 B); state the corrected congruence as a matrix identity over Matrix (Fin d) (Fin d) R (a candidate kernel-proof target: block-matrix algebra)." + }, + { + "claim_id": "math.d1-component.section9-replacement-v1_1", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d1-component.section9-replacement-v1_1 (amendment, PROVED_REVIEWED): Borel elder mark in the weighted pair Kac-Rice identity (reviews/d1_section9_borel_repair_20260925/REPAIR.md, not pinned here)", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d1-component.section9-replacement-v1_1" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "No Lean statement: the repair's bytes are not pinned and no Prop states the weighted pair Kac-Rice identity (9.1) or its monotone-class extension to bounded Borel marks. In this register the marked Kac-Rice representation is deliberately NOT transcribed (densities are pinned only through IsLifetimeDensityVersion), so the Borel elder mark enters only as the ElderPartner field of SuperlevelBarInterface, whose measurability is disclaimed there.", + "next_step": "Pin REPAIR.md (blob fe9b9ce4, 9062 B); state (9.1)-(9.4) as Props over MeasureTheory.Measure on the pair domain with the Armentano-Azais-Leon identity as an interface hypothesis." + }, + { + "claim_id": "math.d1-component.marked-cylinder-cap", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d1-component.marked-cylinder-cap (deterministic_support, PROVED_REVIEWED): MARKED-CYLINDER-CAP-20260924-v1 deterministic separating-cap theorem, sections 1-5 consumed", + "layer0": { + "source_id": "marked-cylinder-cap", + "heading": "Deterministic theorem" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/Math- frontiers/cap_first_exit_lean_20261002 (CapFirstExit, 152 declarations, Lean v4.34.1 + Mathlib d13f23b7)", + "receipt_or_workflow": ".github/workflows/cap-first-exit-lean.yml (gate.py --execute: fresh lake build, leanchecker, #print axioms within propext/Classical.choice/Quot.sound)", + "note": "Kernel-checked topological and analytic steps of CAP sections 2-5 / SP L1-L22 (CapHyp, elder death level, ridge chain, torus transport). Not registered as statements here: its per-declaration anchors (ALIGNMENT.md) were not transcribed. MANIFEST: mathematical_acceptance false, alignment_review NOT_CLAIMED; v1.4 (53 decls) ALIGNED_SCOPED on PR #246, v1.5-v2.6 await delta review; author Anthropic/Claude, independence credit 0. Not formalized: SP section 2 setting as a property of the field, L14/L15, L23, P section 8, every probabilistic statement." + } + ], + "not_established": "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.", + "next_step": "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." + }, + { + "claim_id": "math.d1-component.reconciliation-record", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d1-component.reconciliation-record (reading_rule_record, PROVED_REVIEWED, author_provider Anthropic): reviews/d1_chain_reconciliation_20260928/ binding E1, E2, W1 and the embedding radius r < L/(4 sqrt 2) (not pinned here)", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d1-component.reconciliation-record" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "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.", + "next_step": "Add the embedding radius as an explicit hypothesis to ParentUniformSelectionA / PairLawInterface once the reading rule is pinned; no proof target." + }, + { + "claim_id": "math.lifetime-remainder", + "kind": "graph_node", + "title": "GRAPH node math.lifetime-remainder (D2 candidate, AUTHOR_SIDE_CANDIDATE, review_disposition ACCEPT by xAI/Grok, review_issue 67): Theorem R at its stated existential O(1)-remainder scope", + "layer0": { + "source_id": "lifetime-remainder", + "heading": "Theorem R" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.RemainderBoundedR1", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "|nu_cand(ell)-c ell^(-1/3)| <= C,", + "does_not_claim": "the result or its proof (quadratic determinant cancellation (R7), coupling (R4), cap-loss bound (R12)); convergence of the remainder; a second coefficient; numerical `C` or `ell_*`; the compact-window `O(ell^(2/3))` difference without cutoffs; the parent interfaces it imports (D1-A-E); nonauthor review or acceptance. If no `c` satisfies the parent property the statement holds vacuously; that is a transcription of its dependency on (1.3), not a claim." + } + ], + "kernel_evidence": [], + "not_established": "See row lifetime-remainder: (R1) is stated, not proved; the imports of parent interfaces D1-A-E are not transcribed; Theorem A's cubic bound (1.1) is not consumed by R and is not a hypothesis here. The GRAPH classification and review disposition are source identity only. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "As for row lifetime-remainder." + }, + { + "claim_id": "math.side24-coefficient", + "kind": "graph_node", + "title": "GRAPH node math.side24-coefficient (D3 candidate, AUTHOR_SIDE_CANDIDATE, review_disposition ACCEPT by xAI/Grok, review_issue 65): coefficient calculation for SIDE24 d=2,3; persistence reading via reconciled parent Theorem C", + "layer0": { + "source_id": "side24-proof", + "heading": "Theorem" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.PeriodicCoefficientInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "Let c_d,24 be precisely its equation (15.2), using ordinary sphere area, ordered", + "does_not_claim": "existence of an object satisfying this interface; that `c24` is the finite-bar asymptotic constant (that reading is conditional on the parent's results, which are not touched); that the parent's \"ordered maximum/saddle roles and the full height/gradient pin Jacobian\" are encoded beyond the prefactor of (15.2)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.CoefficientEnclosure", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "0.07340691930603427103 < c_2,24 < 0.07340691930603427104,", + "does_not_claim": "the enclosures (statement only: nothing here encloses the coefficient); that an object satisfying the interface exists or is the coefficient of the source; the parent's results or the persistence interpretation; any finite-radius error estimate, unrestricted remainder or RN/24-jet statement; the arithmetic skeleton (endpoint adjacency, `d3 < d2`, endpoints in `(0, 1/10)`) is kernel-checked on main (31 `UniversalLaw.Side24` targets at origin/main f6deeba7), not here; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Side24.lean", + "declaration": "UniversalLaw.Spec.Side24.ReferenceComparison", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "|c_d,24/c_d,ref -1| < 10^(-106), d=2,3.", + "does_not_claim": "the bound (statement only); equality of the periodic and reference constants (the source says it is NOT claimed); `32*10^(-108) < 10^(-106)` is kernel-checked on main (`UniversalLaw.Side24.density_comparison_below_reported`), not here; nothing here encloses the coefficient." + } + ], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "receipt_or_workflow": ".github/workflows/formal-verification.yml (tools/formal_gate_check.py --run-lean inside the required aggregate `verify`); receipt formal/.lake/formal-evidence/receipt.json is a workflow artifact, none is committed in tree", + "note": "The 31 Side24 targets at origin/main f6deeba7 are decide-proved integer or cross-multiplied rational facts (manifest label `proved`; `kernel-checked` only in a run receipt). Arithmetic skeleton only: formal/SCOPE.md says none of them is the coefficient enclosure." + } + ], + "not_established": "See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the \"evaluates_eq_15_2\" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. The 27 arithmetic theorems of main are listed on the landing row. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed.", + "next_step": "As for row side24-coefficient." + }, + { + "claim_id": "math.rn-count-interface", + "kind": "graph_node", + "title": "GRAPH node math.rn-count-interface (D4 candidate, AUTHOR_SIDE_CANDIDATE): probability-to-count interface; not an expected-count rate", + "layer0": { + "source_id": "rn-count-interface", + "heading": "Theorem N" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.HolderCountBoundN1", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E[N_r 1_Er] <= (E N_r^p)^(1/p) P(E_r)^(1-1/p).", + "does_not_claim": "that any count of the Gaussian model satisfies the moment hypothesis; the result or its proof; any rate in `r`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.HolderConsequenceN2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E[N_r 1_Er] <= A^(1/p) C^(1-1/p)", + "does_not_claim": "that the exponent can be rounded up to `3`; that any Gaussian count satisfies the hypotheses; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.ConditionalMeanIdentityN3", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E[N_r 1_Er]=q E[N_r | E_r].", + "does_not_claim": "the equivalence with a bounded conditional mean, the lower comparison `q ≥ c r^3`, or anything about the Gaussian model; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.UniformCounterexampleN4", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E N_r/r^3=ceil(log(1/r)) -> infinity.", + "does_not_claim": "the explicit bound `sup_(t>=1) e^(-3t)(t+1)^p` (only existence of a `p`-dependent bound); the one-`p` sharpness construction `N_r=ceil(r^(-3/p))1_Er`; the exponential-tail remark after (N5); that the construction says anything about the Gaussian field; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.ExponentialTailLayerCakeN5", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "= B q[1+log(A/q)].", + "does_not_claim": "the intermediate integral identity `integral_0^infinity min(q,Ae^(-t/B))dt`; that the logarithmic loss is attained (the source's remark about (N4)); anything about the Gaussian model; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here)." + } + ], + "kernel_evidence": [], + "not_established": "See row rn-count-interface: (N1)-(N5) stated, (N6)-(N7) not stated, so no Prop states the interface as a whole; nothing proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "As for row rn-count-interface." + }, + { + "claim_id": "math.rn-fixed-remote-window", + "kind": "graph_node", + "title": "GRAPH node math.rn-fixed-remote-window (D4 candidate, AUTHOR_SIDE_CANDIDATE, review_issue 76): cubic mean on fixed spatial exclusion and between-pin height window", + "layer0": { + "source_id": "rn-fixed-remote-window", + "heading": "Theorem A" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.PinnedLawInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "Q_r denotes continuous Gaussian regression on these 2(d+1) observations.", + "does_not_claim": "that `Q_r` is the Gaussian regression of any field; that `W_r` equals `F_d(H_M) F_(d-1)(H_S)`; the normalizer floor `Z_r >= z_* r^2`; that `Λ` is given by (13); existence or uniqueness of an object satisfying the interface (an inhabitant is not asserted; the consistency remark above is not stated); that `0 < r < L` is the sources' unspecified `small r` threshold." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedRemoteMeanMeasureA2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "| E_(Q_r^W) N_(r,j)(E)/(k r^3)", + "does_not_claim": "the result or its proof (uniform remote nondegeneracy, coupling (5), weight expansion (10), normalizer (11)); identification of `I.Λ` with the kernel (13); the total-variation convergence remark; the multi-witness Section 6; any numerical `C` or `r_*`; uniformity as `ρ → 0`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedRemoteCountBoundA3", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_(r,j)(E) <= C k r^3 |E|,", + "does_not_claim": "a matching lower bound on the event probability (the source says this is NOT one); the unnormalized numerator bound (14); the result or its proof; any numerical constant; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "See row rn-fixed-remote-window: displays (2)-(3) stated over PinnedLawInterface, not proved; Section 6 not stated; no inhabitant of the interface. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "As for row rn-fixed-remote-window." + }, + { + "claim_id": "math.rn-region.fixed-remote", + "kind": "graph_node", + "title": "GRAPH region math.rn-region.fixed-remote (D4, COVERED_BY_CANDIDATE, fingerprint \"dist>=rho and between-pin height\"): covered by rn-fixed-remote-window candidate only", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.rn-region.fixed-remote" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedRemoteMeanMeasureA2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "| E_(Q_r^W) N_(r,j)(E)/(k r^3)", + "does_not_claim": "the result or its proof (uniform remote nondegeneracy, coupling (5), weight expansion (10), normalizer (11)); identification of `I.Λ` with the kernel (13); the total-variation convergence remark; the multi-witness Section 6; any numerical `C` or `r_*`; uniformity as `ρ → 0`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedRemoteCountBoundA3", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_(r,j)(E) <= C k r^3 |E|,", + "does_not_claim": "a matching lower bound on the event probability (the source says this is NOT one); the unnormalized numerator bound (14); the result or its proof; any numerical constant; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "The region's mathematical content is the fixed-remote Theorem A at exactly this scope, stated by A2/A3 and not proved. The coverage classification itself (COVERED_BY_CANDIDATE, coverage_source) is graph bookkeeping and is not a Lean statement. No all-height bound on the remote region, no rho -> 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "As for row rn-fixed-remote-window; nothing region-specific to add." + }, + { + "claim_id": "math.rn-region.mesoscopic-scaled-annulus", + "kind": "graph_node", + "title": "GRAPH region math.rn-region.mesoscopic-scaled-annulus (D5, PROVED_REVIEWED at scope \"D5-d with R >= B (covers the reviewed fixed annulus)\", review_disposition ACCEPT_AT_STATED_SCOPE)", + "layer0": { + "source_id": "d5-reconciliation", + "heading": "section 2 table, row D5-d (C2)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5dSummedPinNeighbourhoodC2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_{Q_r^W} N_j(r B_R ∖ {M,S}) ≤ C_R r³", + "does_not_claim": "the composition `D5-a + D5-c at η = 1/4` as a proof; the \"collar to the reviewed annulus\" remark; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.AnnulusBridgeAllHeight2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_j(rE) <= C r^3 area(E), 0= B) is not a Lean statement. No d > 2, no numerical constants, no elder selection, no JETMOD or witness-collision closure. The eight reading-rule component nodes are source identity, not transcribed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Independent reviewer: alignment of D5dSummedPinNeighbourhoodC2 with row D5-d; proof lane: none yet (the composition D5-a + D5-c at eta = 1/4 is a proof, not a statement)." + }, + { + "claim_id": "math.rn-region.pin-collision", + "kind": "graph_node", + "title": "GRAPH region math.rn-region.pin-collision (D5, PROVED_REVIEWED at scope \"D5-a, D5-b, D5-d\", fingerprint \"near +-e1/2 scaled pin sites\")", + "layer0": { + "source_id": "d5-reconciliation", + "heading": "section 2 table, rows D5-a (P2), D5-b (microdisk), D5-d (C2)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5aPuncturedPinDisks", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_{Q_r^W} N_j(M + rE) ≤ C r³|E|", + "does_not_claim": "the P2 source (PUNCTURED_PIN_PROOF.md) or its proof; the P2 continuum review or the proposed register transitions of the reconciliation; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5bMicrodisk", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "the nested disks of radius `κr²` about each pin, for fixed `κ`: `O(r⁵)`", + "does_not_claim": "the P2 source or its nested-disk corollary; any numerical constant; uniformity in `κ`; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5dSummedPinNeighbourhoodC2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_{Q_r^W} N_j(r B_R ∖ {M,S}) ≤ C_R r³", + "does_not_claim": "the composition `D5-a + D5-c at η = 1/4` as a proof; the \"collar to the reviewed annulus\" remark; any numerical constant; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "The three row shapes are stated for any object satisfying PinnedLawInterface and are not proved; the P2 source (PUNCTURED_PIN_PROOF.md) is not pinned here, so the Props are anchored to the reconciliation's transcription of the rows, not to the proof's own bytes. Including the deterministic pin atoms would invalidate the vanishing bounds (the reconciliation's caveat); the Props puncture the disks accordingly. No d > 2, no numerical constants, no uniformity in kappa. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Pin PUNCTURED_PIN_PROOF.md and COLLAR_PROOF.md (reviews/d5_local_collar_20260928) and re-anchor D5a/D5b/D5c to their own displays; independent reviewer: alignment of the three Props with rows D5-a, D5-b, D5-d." + }, + { + "claim_id": "math.rn-region.intermediate-r-to-rho", + "kind": "graph_node", + "title": "GRAPH region math.rn-region.intermediate-r-to-rho (D5, PROVED_REVIEWED at scope \"D5-e (I3/I4, window I_r)\", fingerprint \"r<<|x|<0 mutual witness separation\"): the regional shrinking pin/witness-collision mechanism remains open; PROOF_INDEX open obligation", + "layer0": { + "source_id": "d5-reconciliation", + "heading": "section 5 (open obligation: torus-wide second factorial moment with near-pin witnesses)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.WitnessCollisionNearPinPairsTarget", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "A torus-wide second factorial moment `E_{Q_r^W} N(N−1)` for the window count, with pairs where at least", + "does_not_claim": "that this target holds, is attainable, or is implied by any reviewed source (the Palm route's Theorem Q is an author-side candidate whose `q = 2` case would imply it; that candidate is stated separately as `PalmRouteFactorialMomentsQ` and is not consumed here); that the lower obstruction `E N(N−1) ≥ 2c r³` is stated here; the optimal order; any status transition of the node." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.RemoteCollisionSecondFactorialD", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N(E)(N(E)-1) <= C r^5 |E|,", + "does_not_claim": "Theorem C (the ordered near-pair count) or its proof (Lemmas 1-5); Corollaries E and F; a torus-wide second factorial moment or `math.rn-region.witness-collision` in its full sense; pairs with a witness near the pins; equality of `N(E)` with the all-critical-point count `windowCountAll` (they agree only off the degenerate locus); any numerical constant; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "WitnessCollisionNearPinPairsTarget states the OPEN obligation as a target Prop, not as a result: nothing asserts that the target holds, is attainable, or follows from any reviewed source. RemoteCollisionSecondFactorialD states only the fixed-remote, fixed-eta part (Corollary D of remote_collision), which the GRAPH records as reviewed evidence and NOT as a proof of this node. The Palm route's Theorem Q (q = 2) would imply the target but is an author-side candidate stated separately (PalmRouteFactorialMomentsQ) and not consumed here; the c6_witness_collision reconciliation's proposed discharge is declarative with execution pending. The node stays OPEN_ACTIVE; this register cannot and does not change that. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "None for Lean beyond the target statement: the obligation needs a source-bound proof and nonauthor review in Math-; if Theorem Q is folded, re-anchor the target to the fold record." + }, + { + "claim_id": "math.rn-mesoscopic-reduction", + "kind": "graph_node", + "title": "GRAPH node math.rn-mesoscopic-reduction (D5 candidate, AUTHOR_SIDE_REDUCTION, source external:Math-PR7): legacy author-side PR7 route (RN-MESOSCOPIC-ANNULUS-REDUCTION-20260925-v1), regionally superseded/nonblocking for the reviewed fixed-annulus subregion, unresolved outside it", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.rn-mesoscopic-reduction" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "No Lean statement: the PR7 route's bytes (frontiers/rn_mesoscopic_20260925/PROOF.md) are not pinned here and its Lemma A (positive-definite covariance of the rows of (U_0, J_0(y)) on K_AB) is not stated. The supersession is a graph classification, not mathematics.", + "next_step": "Pin rn_mesoscopic_20260925/PROOF.md and state Lemma A (Matrix.PosDef lower bound on a compact annulus) if the route is ever revived; otherwise none." + }, + { + "claim_id": "math.p15-full-price", + "kind": "graph_node", + "title": "GRAPH node math.p15-full-price (D6 candidate, AUTHOR_SIDE_CANDIDATE, review_issue 74; notes \"review open\" are stale relative to LANDING_CLAIMS REVIEWED_SCOPED, a non-controlling discrepancy recorded by the claims reader)", + "layer0": { + "source_id": "p15-full-price", + "heading": "Theorem F" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.FullTransformedPriceBudget", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " covercost_c(O_K(D)) <= min(1,rho_star[-log mu_p(D)]). (F2)", + "does_not_claim": "the bound (statement only); optimality of `rho_star` (that is `SharpnessOfRhoStar`); any unrestricted downset or prize statement (the source: \"palette size remains K_H(d), which can depend on H and d\"); the nonauthor review recorded in LANDING_CLAIMS (REVIEWED_SCOPED) is not reproduced or extended here." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.RhoStarEnclosure", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " 0.84547981724898672067 < rho_star", + "does_not_claim": "the enclosure (statement only; the source's `full_price.py` Fraction procedure is not a kernel proof and is not specified); this is the next full-result target named in main's `docs/FORMAL_VERIFICATION_ROLLOUT_20260927.md`, still at `specified`." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.SharpnessOfRhoStar", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Replacing rho_star by any smaller constant in (F2) would make its right side strictly less than one.", + "does_not_claim": "the statements (statement only; the finite parts are decidable in principle); a necessity claim for multi-block families with some demand one (the source: \"other constraints may change its budget\")." + } + ], + "kernel_evidence": [], + "not_established": "See row p15-full-price: (F2), (F3) and sharpness stated, not proved; 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; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "As for row p15-full-price." + }, + { + "claim_id": "math.p15-price-boundary", + "kind": "graph_node", + "title": "GRAPH node math.p15-price-boundary (D6 candidate, REFUTED): unrestricted same-palette transformed-price extension is false", + "layer0": { + "source_id": "p15-price-boundary", + "heading": "Counterexample" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.TwoCoordGeneratorPrices", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "The possible generator costs are1,2/3,2/3,4/9 for the empty", + "does_not_claim": "the values (statement only; as integers over `9` they are core-Lean targets on main)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/P15.lean", + "declaration": "UniversalLaw.Spec.P15.DemandOneRefutation", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": " covercost_c(O_1(D))=4/9 > 1/3 > log(4/3).", + "does_not_claim": "the statement (statement only; the finite parts are decidable in principle, the two logarithm inequalities are analysis); that this contradicts original P15-B (\"It does not contradict the original P15-B amalgamation\"); anything about the `408`-scale or the unrestricted prize; the integral representation `log 2 = 2 integral_0^(1/3) 1/(1-t^2) dt` is not specified; disposition transcribed from LANDING_CLAIMS for p15-price-boundary: EXACT_COUNTEREXAMPLE; nonauthor review and acceptance are not claimed here." + } + ], + "kernel_evidence": [], + "not_established": "See row p15-price-boundary: the refutation is stated, its two logarithm inequalities are not proved anywhere; REFUTED is the graph classification of the extension, transcribed as source identity. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "As for row p15-price-boundary." + }, + { + "claim_id": "math.rn-selector-region-crosswalk", + "kind": "graph_node", + "title": "GRAPH inventory node math.rn-selector-region-crosswalk (D4, ENGINEERING_CONTROL): maps historical selectors onto fixed-remote vs open complements (frontiers/downstream_gate_20260925/SELECTOR_REGION.json, not pinned here)", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.rn-selector-region-crosswalk" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "An engineering inventory, not a mathematical claim; nothing to state in Lean and nothing stated.", + "next_step": "No Lean target." + }, + { + "claim_id": "math.rn-fixed-annulus-window", + "kind": "graph_node", + "title": "GRAPH node math.rn-fixed-annulus-window (D5 candidate, AUTHOR_SIDE_CANDIDATE, review_disposition ACCEPT by xAI/Grok, scientific_status_unchanged true): source-bound technical ACCEPT at the exact scoped theorem only", + "layer0": { + "source_id": "rn-fixed-annulus-window", + "heading": "Candidate theorem" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedAnnulusWindowCandidateA2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_Qr^W N_j(r E0) <= C*k*r^3*area(E0),", + "does_not_claim": "the result or its proof (contact vector (A10), eigenfloor (A12), cutoff (A13), three-Hessian estimate (A18), inner-strip bound (A26)); the all-index sum or the Markov event bound; an all-height assertion; uniformity in `L`, as `k → 0`, as `B0 → ∞`, or for `d ≥ 3`; any numerical constant; nonauthor review or acceptance (the landing disposition REVIEWED_SCOPED is transcribed by the register, not here)." + } + ], + "kernel_evidence": [], + "not_established": "See row rn-fixed-annulus-window: (A2) stated, not proved; no global D5/RN/JETMOD, k -> 0, d >= 3, pin-neighbourhood, intermediate-scale, collision or elder-selection closure. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "As for row rn-fixed-annulus-window." + }, + { + "claim_id": "math.rn-region.fixed-annulus-window", + "kind": "graph_node", + "title": "GRAPH region math.rn-region.fixed-annulus-window (D5, COVERED_BY_CANDIDATE, review_disposition ACCEPT_AT_SCOPED_SOURCE): covered by math.rn-fixed-annulus-window only at its exact declared scope", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.rn-region.fixed-annulus-window" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedAnnulusWindowCandidateA2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_Qr^W N_j(r E0) <= C*k*r^3*area(E0),", + "does_not_claim": "the result or its proof (contact vector (A10), eigenfloor (A12), cutoff (A13), three-Hessian estimate (A18), inner-strip bound (A26)); the all-index sum or the Markov event bound; an all-height assertion; uniformity in `L`, as `k → 0`, as `B0 → ∞`, or for `d ≥ 3`; any numerical constant; nonauthor review or acceptance (the landing disposition REVIEWED_SCOPED is transcribed by the register, not here)." + } + ], + "kernel_evidence": [], + "not_established": "The region's content is (A2) at its exact scope, stated and not proved; the coverage classification is graph bookkeeping, not a Lean statement; the superseded high-jet route (regional.fixed-annulus.high-jet-route, SUPERSEDED_NONBLOCKING) is not represented. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "As for row rn-fixed-annulus-window." + }, + { + "claim_id": "math.d5-pin-neighborhood-first-moment", + "kind": "graph_node", + "title": "GRAPH node math.d5-pin-neighborhood-first-moment (D5 candidate, PROVED_REVIEWED): D5-a..D5-f of RECONCILIATION.md section 2; existential constants; fixed T; compact marks with k >= k_- > 0; d = 2 (D5-RECONCILIATION-20260929-v2.1)", + "layer0": { + "source_id": "d5-reconciliation", + "heading": "section 2 table D5-a to D5-f" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5aPuncturedPinDisks", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_{Q_r^W} N_j(M + rE) ≤ C r³|E|", + "does_not_claim": "the P2 source (PUNCTURED_PIN_PROOF.md) or its proof; the P2 continuum review or the proposed register transitions of the reconciliation; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5bMicrodisk", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "the nested disks of radius `κr²` about each pin, for fixed `κ`: `O(r⁵)`", + "does_not_claim": "the P2 source or its nested-disk corollary; any numerical constant; uniformity in `κ`; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5cCollarC1", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "for Borel `E ⊂ C_{η,R}`, `η ≤ 1/4`, fixed `R ≥ 1`", + "does_not_claim": "the C1 source (COLLAR_PROOF.md) or its proof; the normalizer (P14) it imports; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5dSummedPinNeighbourhoodC2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_{Q_r^W} N_j(r B_R ∖ {M,S}) ≤ C_R r³", + "does_not_claim": "the composition `D5-a + D5-c at η = 1/4` as a proof; the \"collar to the reviewed annulus\" remark; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5eIntermediateShellsI3I4", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E N ≤ C r³(A_0^{−2} + ρ²)", + "does_not_claim": "the single-shell display (1.5) (stated in `DimensionLiftIntermediateShellsI`); the I3/I4 source (intermediate_window_20260928) or its proof; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5fGlobalWindowFirstMomentI5", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_{Q_r^W} N_{r,j}(T² ∖ {M,S}) ≤ C r³", + "does_not_claim": "the tiling composition `D5-d (R = 4) + I4 + remote Theorem A` as a proof; the lower bound's source (remote_collision Corollary F-) or the I5 source; any numerical constant; nonauthor review or acceptance; any register transition proposed by the reconciliation; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus." + } + ], + "kernel_evidence": [], + "not_established": "The six row shapes are stated for any object satisfying PinnedLawInterface (sample space Omega, gated pin regime 0 < r < L) and are not proved. The consumption graph (D5-d from D5-a + D5-c, D5-f from D5-d + I4 + remote Theorem A) is a proof structure and is not stated; the component sources PUNCTURED_PIN_PROOF.md, COLLAR_PROOF.md and intermediate_window PROOF.md are not pinned here, so D5a/b/c/e/f are anchored to the reconciliation's row text. The lower bound c r^3 <= P(A_r) of D5-f has its source in remote_collision Corollary F-, not pinned. No d > 2, no numerical constants, no uniformity in T, R, eta, k -> 0; no elder pairing; no RN/24-jet/JETMOD; the witness-collision region stays OPEN_ACTIVE. The STATUS.md note that D5 GRAPH transitions were not executed by that summary is a recorded discrepancy, not resolved here. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Pin the three component proofs and re-anchor; independent reviewer: alignment of the six Props with the section 2 table (all-height versus window I_r per row, exclusion of M and S)." + }, + { + "claim_id": "math.d5-component.punctured-pin-proof", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.punctured-pin-proof (proof, PROVED_REVIEWED, author_provider OpenAI): reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md (P2), not pinned here; review basis #111 (algebra ACCEPT, continuum HOLD) and the Anthropic continuum review (ACCEPT existential)", + "layer0": { + "source_id": "d5-reconciliation", + "heading": "section 2 table, rows D5-a (P2) and D5-b (microdisk)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5aPuncturedPinDisks", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_{Q_r^W} N_j(M + rE) ≤ C r³|E|", + "does_not_claim": "the P2 source (PUNCTURED_PIN_PROOF.md) or its proof; the P2 continuum review or the proposed register transitions of the reconciliation; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5bMicrodisk", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "the nested disks of radius `κr²` about each pin, for fixed `κ`: `O(r⁵)`", + "does_not_claim": "the P2 source or its nested-disk corollary; any numerical constant; uniformity in `κ`; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "The Props state rows D5-a and D5-b as the reconciliation transcribes P2, not P2 from its own bytes (not pinned); the slices P4, P9, P17, P20 and the continuum argument reviewed are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Pin PUNCTURED_PIN_PROOF.md (SHA-256 f972e50b… (GRAPH.json fingerprint)) and anchor D5a/D5b to its own displays." + }, + { + "claim_id": "math.d5-component.punctured-pin-review-111", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.punctured-pin-review-111 (review_record / review_checker, PROVED_REVIEWED): reviews/d5_punctured_pin_nonauthor_20260928/REVIEW.md (xAI), not pinned here", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d5-component.punctured-pin-review-111" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit.", + "next_step": "No Lean target. The proof it reviews is specified under its proof-component row." + }, + { + "claim_id": "math.d5-component.punctured-pin-continuum-review", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.punctured-pin-continuum-review (review_record / review_checker, PROVED_REVIEWED): reviews/d5_punctured_pin_continuum_claude_20260929/REVIEW.md (Anthropic), not pinned here", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d5-component.punctured-pin-continuum-review" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit.", + "next_step": "No Lean target. The proof it reviews is specified under its proof-component row." + }, + { + "claim_id": "math.d5-component.punctured-pin-continuum-check", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.punctured-pin-continuum-check (review_record / review_checker, PROVED_REVIEWED): reviews/d5_punctured_pin_continuum_claude_20260929/pp_continuum_check.py (Anthropic, review checker), not pinned here", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d5-component.punctured-pin-continuum-check" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit.", + "next_step": "No Lean target. The proof it reviews is specified under its proof-component row." + }, + { + "claim_id": "math.d5-component.collar-review", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.collar-review (review_record / review_checker, PROVED_REVIEWED): reviews/d5_collar_count_20260928/REVIEW.md (provider not named in record), not pinned here", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d5-component.collar-review" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit.", + "next_step": "No Lean target. The proof it reviews is specified under its proof-component row." + }, + { + "claim_id": "math.d5-component.annulus-review", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.annulus-review (review_record / review_checker, PROVED_REVIEWED): reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md (xAI), not pinned here", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d5-component.annulus-review" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit.", + "next_step": "No Lean target. The proof it reviews is specified under its proof-component row." + }, + { + "claim_id": "math.d5-component.i5-review-xai", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.i5-review-xai (review_record / review_checker, PROVED_REVIEWED): reviews/d5_i5_planar_f_20260928/REVIEW.md (xAI), not pinned here", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d5-component.i5-review-xai" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit.", + "next_step": "No Lean target. The proof it reviews is specified under its proof-component row." + }, + { + "claim_id": "math.d5-component.i5-review-claude", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.i5-review-claude (review_record / review_checker, PROVED_REVIEWED): reviews/d5_intermediate_window_claude_20260928/REVIEW.md (Anthropic), not pinned here", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d5-component.i5-review-claude" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit.", + "next_step": "No Lean target. The proof it reviews is specified under its proof-component row." + }, + { + "claim_id": "math.d5-component.offpin-second-moment-review", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.offpin-second-moment-review (review_record / review_checker, PROVED_REVIEWED): reviews/d5_offpin_second_moment_20260928/REVIEW.md (nonauthor; P_eta ACCEPT; component of math.rn-region.witness-collision), not pinned here", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.math.d5-component.offpin-second-moment-review" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit.", + "next_step": "No Lean target. The proof it reviews is specified under its proof-component row." + }, + { + "claim_id": "math.d5-component.collar-proof", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.collar-proof (proof, PROVED_REVIEWED, author_provider OpenAI): reviews/d5_local_collar_20260928/COLLAR_PROOF.md (C1), not pinned here", + "layer0": { + "source_id": "d5-reconciliation", + "heading": "section 2 table, row D5-c (C1)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5cCollarC1", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "for Borel `E ⊂ C_{η,R}`, `η ≤ 1/4`, fixed `R ≥ 1`", + "does_not_claim": "the C1 source (COLLAR_PROOF.md) or its proof; the normalizer (P14) it imports; any numerical constant; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "D5cCollarC1 states row D5-c as the reconciliation transcribes C1, not C1 from its own bytes (not pinned); the normalizer (P14) it imports and the corollary C2 are not stated separately; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Pin COLLAR_PROOF.md (SHA-256 794babe0… (GRAPH.json fingerprint)) and anchor D5cCollarC1 to display (C1)." + }, + { + "claim_id": "math.d5-component.annulus-proof", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.annulus-proof (proof, PROVED_REVIEWED, author_provider OpenAI; review basis xAI ACCEPT R1-R7): D5 all-height fixed-annulus bridge (D5-ALL-HEIGHT-ANNULUS-BRIDGE-20260925-v1, PR28); PROOF_INDEX reviewed scoped result", + "layer0": { + "source_id": "rn-annulus-bridge", + "heading": "Candidate theorem" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.AnnulusBridgeAllHeight2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_j(rE) <= C r^3 area(E), 0 0, 1 < A < B, all heights) over PinnedLawInterface; not proved. Not stated: displays (4)-(20), the R1-R7 review interfaces as separate statements, the deterministic three-Hessian lemma, the crossover factor min(1,(r/|v|)^6). No matching lower bound, no numerical constant or radius, no pin neighbourhoods, no k -> 0 uniformity, no d >= 3, no fixed-remote all-height bound, no all-scales RN theorem. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State the R1-R7 interfaces as Props; independent reviewer: alignment of AnnulusBridgeAllHeight2 with display (2), checking \"all heights\" and the uniformity over E and j." + }, + { + "claim_id": "math.d5-component.intermediate-window-proof", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.intermediate-window-proof (proof, PROVED_REVIEWED, author_provider OpenAI): frontiers/intermediate_window_20260928/PROOF.md (OA-D5-INTERMEDIATE-WINDOW-20260928-v1, I3/I4/I5), not pinned here", + "layer0": { + "source_id": "d5-reconciliation", + "heading": "section 2 table, rows D5-e (I3/I4) and D5-f (I5)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5eIntermediateShellsI3I4", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E N ≤ C r³(A_0^{−2} + ρ²)", + "does_not_claim": "the single-shell display (1.5) (stated in `DimensionLiftIntermediateShellsI`); the I3/I4 source (intermediate_window_20260928) or its proof; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.D5fGlobalWindowFirstMomentI5", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_{Q_r^W} N_{r,j}(T² ∖ {M,S}) ≤ C r³", + "does_not_claim": "the tiling composition `D5-d (R = 4) + I4 + remote Theorem A` as a proof; the lower bound's source (remote_collision Corollary F-) or the I5 source; any numerical constant; nonauthor review or acceptance; any register transition proposed by the reconciliation; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus." + } + ], + "kernel_evidence": [], + "not_established": "The Props state rows D5-e and D5-f as the reconciliation transcribes I3/I4/I5, not from the proof's own bytes (not pinned); the tiling N1 of the Claude review and the planar F+ lower bound source are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Pin intermediate_window_20260928/PROOF.md (SHA-256 b3eb9456… (GRAPH.json fingerprint)) and anchor the shell displays there." + }, + { + "claim_id": "math.d5-component.remote-window-proof", + "kind": "graph_node", + "title": "GRAPH reading-rule component math.d5-component.remote-window-proof (proof, PROVED_REVIEWED, author_provider OpenAI; review basis STATUS D4 main#76 external ACCEPT): the fixed-remote Theorem A consumed by the D5 tiling at rho = s_0", + "layer0": { + "source_id": "rn-fixed-remote-window", + "heading": "Theorem A" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedRemoteMeanMeasureA2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "| E_(Q_r^W) N_(r,j)(E)/(k r^3)", + "does_not_claim": "the result or its proof (uniform remote nondegeneracy, coupling (5), weight expansion (10), normalizer (11)); identification of `I.Λ` with the kernel (13); the total-variation convergence remark; the multi-witness Section 6; any numerical `C` or `r_*`; uniformity as `ρ → 0`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN is transcribed by the register, not here)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FixedRemoteCountBoundA3", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_(r,j)(E) <= C k r^3 |E|,", + "does_not_claim": "a matching lower bound on the event probability (the source says this is NOT one); the unnormalized numerator bound (14); the result or its proof; any numerical constant; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "See rows rn-fixed-remote-window and math.rn-fixed-remote-window; the D5 consumption at rho = s_0 is a proof step, not stated. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "As for row rn-fixed-remote-window." + }, + { + "claim_id": "D1-THM-A", + "kind": "proof_index_result", + "title": "D1 Theorem A (uniform selection candidate), parent UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1 display (1.1); PROOF_INDEX reviewed scoped result (STATUS.md ACCEPT, scoped; GRAPH node math.uniform-matrix-cap-lifetime)", + "layer0": { + "source_id": "lifetime-parent", + "heading": "Theorem A (uniform selection candidate)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.PairLawInterface", + "kind": "structure", + "formal_progress": "specified", + "anchor": "Let p_r(b,k,R) be the Q^W probability that the **global ordinary superlevel elder death partner** of M is S.", + "does_not_claim": "existence of the regression law, the typed weight, the full normalizer `Z_r > 0`, or that `p` is the probability of any event under any measure declared here; that `p` depends on `G` or `I` in any way (no field mentions them; they only record which field and elder relation the law is meant to refer to)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.ParentUniformSelectionA", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "There exist r_*>0 and C 0, C < infinity outside the universal quantifiers over 0 < r <= r_*, b in [bLo,bHi], k in [kLo,kHi] and all frames, for the interface field p of PairLawInterface. Not stated or proved: the full-normalizer floor Z_r >= z_* r^2 (5.5), the transverse-matrix boundary layer, the regression law and typed weight, the reading-rule amendments (E1, Section 9 v1.1, W1, r < L/(4 sqrt 2)), any numerical r_*, C, z_*. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. No globally uniform constant over unbounded b, k; no lower bound. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: ParentUniformSelectionA is a predicate on the interface objects, not a closed statement; its universal closure over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed.", + "next_step": "Independent reviewer: alignment of ParentUniformSelectionA with (1.1) under the honesty checklist; formal lane: none until the regression law is constructed." + }, + { + "claim_id": "D1-THM-B", + "kind": "proof_index_result", + "title": "D1 Theorem B (compact-mark density candidate), display (1.2) and the Section 11 coefficient c_{B,K}; PROOF_INDEX reviewed scoped result at existential scope", + "layer0": { + "source_id": "lifetime-parent", + "heading": "Theorem B (compact-mark density candidate)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsWindowPair", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "Count ordered maximum/saddle pairs with 0 0, the O(ell^(2/3)) difference) for compact B, K of positive length; not proved. Not stated: the exact coefficient formula (11.3), the radial ledger (10.2), the pushforward r = (ell/k)^(1/3); the distance cutoff is quantified existentially and not identified with the r_* of (1.1). The difference is NOT asserted for unrestricted densities; no numerical c_{B,K}, C_{B,K}; the historical H3 finite-band floor is not re-certified. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State (11.3) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of ParentCompactWindowDensityB with (1.2), including the positive-length requirement." + }, + { + "claim_id": "D1-THM-C", + "kind": "proof_index_result", + "title": "D1 Theorem C (unrestricted finite-lifetime leading density candidate), display (1.3) and coefficient (15.2); PROOF_INDEX reviewed scoped result, leading term only", + "layer0": { + "source_id": "lifetime-parent", + "heading": "Theorem C (unrestricted finite-lifetime leading density candidate)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsUnrestrictedPair", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume.", + "does_not_claim": "anything about the off-diagonal (far) population separately (Section 14 of the source)." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.IsUnrestrictedLeadingCoefficient", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "There is a positive finite constant c_{d,L} such that", + "does_not_claim": "the explicit Gaussian/angular representation (15.2) of `c_{d,L}`; uniqueness of `c` (which follows from the asymptotics but is not stated); existence of versions." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.ParentUnrestrictedLeadingDensityC", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "nu_cand^all(ell) ~ c_{d,L} ell^(-1/3),", + "does_not_claim": "the parent result or its proof (dominated convergence of Section 13, off-diagonal bound (14.1)); the representation (15.2); the compact-window `O(ell^(2/3))` difference for the unrestricted densities; any numerical enclosure of `c_{d,L}` (the SIDE24 coefficient track is separate); nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "ParentUnrestrictedLeadingDensityC states the shape of (1.3) (leading term only) with the coefficient as an abstract positive constant c in IsUnrestrictedLeadingCoefficient; not proved. The Gaussian/angular representation (15.2), the target-growth majorant (13.3)-(13.6), the integrability (13.5) and the off-diagonal bound (14.1) are not stated; the SIDE24 enclosure of c_{d,24} is a separate row. No unrestricted remainder or difference rate (that is Theorem R), no elementary closed form. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State (15.2) and (14.1) as Props; independent reviewer: alignment of ParentUnrestrictedLeadingDensityC with (1.3), in particular the exclusion of the essential global-maximum class." + }, + { + "claim_id": "OA-ELDER-LOWER-DENSITY-GAP-20260928-v1", + "kind": "proof_index_result", + "title": "Planar direct elder-failure lower bound (E2), two-sided rate (E4) and compact-window density gap (E11) (frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md; PROOF_INDEX navigation refresh #116 C1; disposition line of the pinned bytes: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR ANALYTIC REVIEW REQUIRED)", + "layer0": { + "source_id": "elder-lower-density-gap", + "heading": "Theorem E1 (direct lower bound)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/Lifetime.lean", + "declaration": "UniversalLaw.Spec.Lifetime.PlanarElderLowerE2", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "1-p_r(b,k,R) >= c r^3, 0= 3 (separate packets CL-ELDER-LOWER-ALL-D / OA-ELDER-DIMENSION-LIFT, not extracted here); no numerical c or r_*. The review verdict of reviews/d1_elder_lower_claude_20260928 is not transcribed here; the pinned bytes carry their own disposition line. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: PlanarElderLowerE2, PlanarTwoSidedRateE4 and PlanarDensityGapE11 are predicates on the interface objects, not closed statements; the universal closure of E2 and E4 over all inhabitants is false whenever the interfaces are inhabited (take p identically 1), and `specified` refers to the statement shape at the source's object, which is not constructed.", + "next_step": "Independent reviewer: record whether (E11)'s lower half should be stated without the (1.1) hypothesis (a narrower-scope verdict is the reviewer's to write); transcribe the review verdict once pinned." + }, + { + "claim_id": "two-scale-s6-s21", + "kind": "proof_index_result", + "title": "D5 two-scale addendum S6-S21 (D5-TWO-SCALE-20260925-v1, PR22; support dependency of rn-fixed-annulus-window; frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md, not pinned here); PROOF_INDEX reviewed scoped result", + "layer0": { + "source_id": null, + "heading": "TWO_SCALE_ADDENDUM.md \"Candidate conclusions\" (S3)-(S5), (S20)-(S21) (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "No Lean statement: the addendum is not pinned here and none of (S3)-(S5), (S20)-(S21) is stated; it enters the register only as the named support of FixedAnnulusWindowCandidateA2. Does not establish uniform O(r^3) on a fixed scaled-width annulus, anything at pin sites, near u = 0, r << |x| << rho, witness collisions, d >= 3, k_- = 0, uniform L.", + "next_step": "Pin TWO_SCALE_ADDENDUM.md (blob 89cae3a9, 15902 B) and state (S3)-(S5) over PinnedLawInterface with the Stecconi weighted Kac-Rice input as an interface field." + }, + { + "claim_id": "axial-density", + "kind": "proof_index_result", + "title": "Inner axial belt exponential conditional-gradient density bound (RN-INNER-AXIAL-DENSITY-20260925-v1, PR21; frontiers/axial_density_20260925/PROOF.md, not pinned here); PROOF_INDEX reviewed scoped result (density bound only)", + "layer0": { + "source_id": null, + "heading": "Candidate theorem (1): p_{grad f(X) under Q_r}(0) <= C r^(-5) exp(-c k^2/r^2) (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "No Lean statement: not pinned, not stated. The bound contains no pin-density factor, spatial Jacobian or Palm weight; nothing about the height-conditioned density, Hessian moments, the endpoint-weighted count, the full normalizer, the remaining thin belt, collisions, d = 3 or numerical constants.", + "next_step": "Pin axial_density_20260925/PROOF.md (blob a72418d7) and state (1) as a density bound over PinnedLawInterface with the conditional gradient density as an interface field." + }, + { + "claim_id": "fixed-transverse-chart", + "kind": "proof_index_result", + "title": "Corrected fixed-transverse chart cubic conditional count bound (reviews/downstream_boundary_20260925/TRANSVERSE_BOUND_CANDIDATE.md, byte import of commit 32b80ee0; PR16; not pinned here); PROOF_INDEX reviewed scoped result", + "layer0": { + "source_id": null, + "heading": "Candidate theorem: E_(Qr^W) N_j(rE) <= C k r^3 area(E) on the chart |v| >= eta (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "No Lean statement: not pinned, not stated. Constants are qualitative and not uniform as eta -> 0, B -> infinity or the gap interval approaches zero; no pin collisions, no intermediate distances, no d = 3, no positive leading coefficient, no probability lower bound.", + "next_step": "Pin TRANSVERSE_BOUND_CANDIDATE.md (blob 024d9277) and state the chart bound as a Prop parallel to FixedAnnulusWindowCandidateA2 with the extra |v| >= eta restriction." + }, + { + "claim_id": "CL-D5-DIMENSION-LIFT-20260929-v1", + "kind": "proof_index_result", + "title": "D5 in every fixed dimension: Theorems P_d, C_d, I_d, G_d and Corollaries F_d, M_d (frontiers/d5_dimension_lift_20260929/PROOF.md, Math-#141); PROOF_INDEX reviewed scoped result (separate packet, not folded into the planar graph); erratum packet reviews/d5_dimension_lift_erratum_20261005 exists and was not read", + "layer0": { + "source_id": "d5-dimension-lift", + "heading": "Theorem P_d (all-height punctured pin ball); Theorem C_d (compact collar); Theorem I_d (intermediate height-window shells); Theorem G_d (global height-window first moment)" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.DimensionLiftPuncturedPinBallP", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_j(M + r E) <= C r^3 integral_E omega_d(p, q) dp dq <= C' r^3.", + "does_not_claim": "the result or its proof (devices D1-D5, stable gradient frame, density and conditional moments, weighted Kac-Rice (4.10), crossover); the nested-ball `O(r^5)` corollary (stated for `d = 2` as `D5bMicrodisk`); the planar P2 identification; any numerical constant; uniformity in `d`; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.DimensionLiftCompactCollarC", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_j(r E) <= C r^3 |E|.", + "does_not_claim": "the consequence (1.4) for the scaled ball with pins removed (stated for `d = 2` as `D5dSummedPinNeighbourhoodC2`); the result or its proof; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.DimensionLiftIntermediateShellsI", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_(r,j)({ s <= |X| <= 2 s }) <= C r^3 [ (r/s)^2 + s^2 ],", + "does_not_claim": "the result or its proof (polynomial row rank, Euler suppression, shell ledger, dyadic sum); the planar I3/I4 identification; any numerical constant; nonauthor review or acceptance." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.DimensionLiftGlobalWindowG", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N_(r,j)( X minus {M, S} ) <= C r^3.", + "does_not_claim": "the tiling proof (P_d, C_d, I_d and the fixed-remote Theorem A); Corollaries F_d and M_d; the planar I5 identification; the `d = 3` SIDE24 application; any regional supersession of historical carriers; any numerical constant; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "The four theorem shapes (1.2), (1.3), (1.5)-(1.6), (1.7) are stated over PinnedLawInterface for fixed d >= 2 and not proved. Not stated: display (1.4), Corollaries F_d (1.8) and M_d (1.9), the five devices D1-D5, the consumed sources ([LP] sections 2-5, (5.5), the planar templates, [RM], [RC], [EDL], [LOCAL]). No all-height intermediate or global O(r^3) count; no torus-wide second factorial moment; no elder selection; no uniformity in k, L, d; no numerical C, c, r_*, s_0; no recovery of historical RN/JETMOD/24-jet carriers. The reviews (Math-#141 OpenAI, source-exposed) are source identity; the 2026-10-05 erratum may amend these statements and was not read. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "Read reviews/d5_dimension_lift_erratum_20261005 and amend the four Props if the displays changed; state Corollaries F_d and M_d; independent reviewer: alignment of the four Props with (1.2)-(1.7)." + }, + { + "claim_id": "OA-C6-FOURIER-20260929-v1", + "kind": "proof_index_result", + "title": "C6 Fourier cutoff: Theorem F (count cap), Theorem W (window consequence), Proposition B (frontiers/c6_fourier_cutoff_20260929/PROOF.md, Math-#148); PROOF_INDEX reviewed scoped result; STATUS.md ACCEPT, scoped (C6 row)", + "layer0": { + "source_id": "c6-fourier-cutoff", + "heading": "Theorem F (count cap, no G_d input); Theorem W (window consequence); Proposition B" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.FourierCutoffPlanarSecondFactorialF4", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_Qr^W N(N-1) <= C r^3 log(1/r),", + "does_not_claim": "Theorem F (the cap `Psi_F` and its tail (F1)-(F2)); Theorem W for general `d` and `p` (F3); that `G_2` holds (the source takes it from the reviewed planar I5 composition); Proposition B; that the logarithm is sharp or removable; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus." + } + ], + "kernel_evidence": [], + "not_established": "No Prop states Theorem F (the cap functional Psi_F and its stretched-exponential tail (F1)-(F2)), Theorem W in general d and p (F3), or Proposition B; only the planar consequence (F4) is stated, GIVEN the hypothesis G_2 (E N <= C r^3), so the claim as a whole is not specified. Nothing is proved; the logarithm is neither shown sharp nor removable here; no numerical cutoff or parameter-uniform constants; the abstract barrier is not a Gaussian counterexample. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State (F1)-(F2) with Psi_F as an interface field (a measurable functional dominating the total critical count) and (F3) for general d, p; then the claim would be `specified`." + }, + { + "claim_id": "CL-C6-PALM-20260929-v1", + "kind": "proof_index_result", + "title": "C6 sharp Palm route: Theorem Q (factorial moments, every fixed dimension), Corollary Theta, Corollary P (frontiers/c6_palm_route_20260929/PROOF.md, Math-#145); PROOF_INDEX reviewed scoped result (separate packet, not folded; does not resolve math.rn-region.witness-collision)", + "layer0": { + "source_id": "c6-palm-route", + "heading": "Theorem Q (factorial moments, every fixed dimension); Corollary Theta; Corollary P" + }, + "disposition_transcribed": null, + "formal_progress": "specified", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.PalmRouteFactorialMomentsQ", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W)[ (N)_q ] <= C_q r^3.", + "does_not_claim": "the result or its proof (the cap `Psi` in `d` dimensions, the parametric Rouché tail, the marked Kac-Rice formula with Hölder insertion, the regimes R1-R4); the dependence on [C6L] and [DL] at their bytes; a lower bound for `q ≥ 3`; Corollary P; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus." + }, + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.PalmRouteSecondFactorialTheta", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "c r^3 <= E_(Q_r^W)[ N(N-1) ] <= C r^3,", + "does_not_claim": "the lower bounds' source ([EDL] (A4)) or the upper bounds' sources (Theorem Q, [DL] Corollary M_d); that the logarithm of the Fourier cutoff is removed as a reviewed fact; any statement about the positions of the pairs; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus." + } + ], + "kernel_evidence": [], + "not_established": "PalmRouteFactorialMomentsQ states (1.1) (with r_* allowed to depend on q) and PalmRouteSecondFactorialTheta states (1.2) over PinnedLawInterface; not proved. Corollary P (size-biased law) and Sections 3-7 (the cap Psi, the parametric Rouche tail, the marked Kac-Rice formula with Hoelder insertion, regimes R1-R4) are not stated; the dependence on [C6L] and [DL] at their bytes is not transcribed; the lower bounds' source [EDL] (A4) is not stated. No lower bound for q >= 3; no distribution of N beyond factorial moments; no elder selection; no numerical constants; for d >= 3 conditional on [DL]. The GRAPH records that this global result is not a proof of the regional witness-collision node. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State Corollary P; independent reviewer: alignment of the two Props with (1.1)-(1.2), including whether r_* may depend on q." + }, + { + "claim_id": "OA-C6-RARE-CLUSTER-20260929-v1", + "kind": "proof_index_result", + "title": "C6 rare-cluster laws: Theorems O, C, S and the independent-replica limit (7.1), addendum (A1)-(A2) (frontiers/c6_rare_cluster_laws_20260929/PROOF.md, Math-#153); PROOF_INDEX reviewed scoped result (conditional packet; limiting cluster law unidentified)", + "layer0": { + "source_id": "c6-rare-cluster", + "heading": "Theorem O" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.RareClusterPoissonObstructionO", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "(a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3.", + "does_not_claim": "the result (Theorem O) or its proof; that `2a ≤ A` (derived in the source, not a hypothesis here); that H1 or H2 hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment matching or the independent-replica limit; the conditional-law liminf remark; that the upper half is the source's infimum bound verbatim: `∃ lam ≥ 0, dTV(Law N_r, Pois(lam)) ≤ A r^3` implies `inf_{lam ≥ 0} dTV ≤ A r^3` and is stronger unless the infimum is attained, and the existential form is used because the source's proof exhibits the witness `lam = 0` (the source: The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p); nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified. Nothing is proved; H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count; for A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); Theorem O is a candidate kernel target (finite total-variation arithmetic)." + }, + { + "claim_id": "CL-D5-REMOTE-COLLISION-20260928-v1", + "kind": "proof_index_result", + "title": "Witness collisions in the fixed-remote height window: Theorems C-F and Corollary D (frontiers/remote_collision_20260928/PROOF.md, author Anthropic/Claude; P_eta reviewed ACCEPT in reviews/d5_offpin_second_moment_20260928, GRAPH component offpin-second-moment-review); consumed by the D5 reconciliation (D5-f lower bound via Corollary F-)", + "layer0": { + "source_id": "remote-collision", + "heading": "1. Results (Theorems C-F, Corollary D)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [ + { + "package": "specifications/lean_statements_20261010", + "module": "Specifications/RN.lean", + "declaration": "UniversalLaw.Spec.RN.RemoteCollisionSecondFactorialD", + "kind": "def_prop", + "formal_progress": "specified", + "anchor": "E_(Q_r^W) N(E)(N(E)-1) <= C r^5 |E|,", + "does_not_claim": "Theorem C (the ordered near-pair count) or its proof (Lemmas 1-5); Corollaries E and F; a torus-wide second factorial moment or `math.rn-region.witness-collision` in its full sense; pairs with a witness near the pins; equality of `N(E)` with the all-critical-point count `windowCountAll` (they agree only off the degenerate locus); any numerical constant; nonauthor review or acceptance." + } + ], + "kernel_evidence": [], + "not_established": "Only Corollary D (second factorial moment <= C r^5 |E| in the fixed remote region) is stated; Theorem C (ordered near-pair count), Corollaries E and F (event asymptotics, torus-wide lower bound) are not, so the packet is not specified as a whole. Nothing proved; pairs with a witness near the pins are excluded; no torus-wide second factorial moment; this is explicitly NOT the open witness-collision mechanism. Author Anthropic/Claude: same provider as this packet, independence credit 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0.", + "next_step": "State Theorem C and Corollaries E, F; independent (non-Anthropic) reviewer for both the source and the Props." + }, + { + "claim_id": "cumulative-transfer-correction", + "kind": "proof_index_result", + "title": "Correction to the cumulative transfer statement (D2), corrected cumulative theorem (reviews/collision_mechanism_20260925/CUMULATIVE_TRANSFER_CORRECTION.md); PROOF_INDEX reviewed scoped result (this correction only; xAI ACCEPT in reviews/d2_cumulative_correction_20260925)", + "layer0": { + "source_id": "cumulative-transfer-correction", + "heading": "Corrected cumulative theorem" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "No Prop states the corrected cumulative theorem (sigma-finite (Z, lambda), alpha > -1, m > 0, ratio hypothesis h/(kappa r^m) -> 1, limit N(ell)/ell^beta -> (1/(alpha+1)) int B0 kappa^(-beta)), although its bytes are pinned. Nothing is proved; the correction does not license inferring a density asymptotic from a cumulative one; Section A of NOTE.md is outside both acceptances.", + "next_step": "State the corrected theorem as a def : Prop over MeasureTheory.Measure Z (an abstract measure-space statement with no Gaussian input; a natural Mathlib-lane proof target afterwards)." + }, + { + "claim_id": "CL-SIDE24-PFV-20260929-v1", + "kind": "proof_index_result", + "title": "SIDE24 periodization first variation, P_3(L) = -L^2(10L^4-147L^2+315)/105, P_3(24) = -620813376/35 (#144): Math- frontiers/side24_periodization_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/side24_periodization_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/side24_periodization_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-SIDE24-REMAINDER-20260929-v1", + "kind": "proof_index_result", + "title": "SIDE24 remainder |c_{d,24}/c_{d,ref} - 1 - P_d(24) e^{-288}| < 10^{-216} (#154): Math- coefficients/side24_remainder_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "coefficients/side24_remainder_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:coefficients/side24_remainder_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "CL-SIDE24-PERIODIZATION-REMAINDER-20260929-v1", + "kind": "proof_index_result", + "title": "SIDE24 periodization remainder (#156): Math- frontiers/side24_periodization_remainder_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/side24_periodization_remainder_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/side24_periodization_remainder_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-CONTACT-ANGULAR-TAIL-20260925-v1", + "kind": "proof_index_result", + "title": "contact angular tail (#35): Math- reviews/contact_angular_tail_20260925 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "reviews/contact_angular_tail_20260925 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:reviews/contact_angular_tail_20260925 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-PIN-MICRO-COV-20260926-v1", + "kind": "proof_index_result", + "title": "pin micro covariance (#60): Math- reviews/pin_micro_covariance_20260926 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "reviews/pin_micro_covariance_20260926 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:reviews/pin_micro_covariance_20260926 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-D5-MICRODISK-20260926-v1", + "kind": "proof_index_result", + "title": "D5 microdisk (#82): Math- reviews/d5_microdisk_20260926 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "reviews/d5_microdisk_20260926 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:reviews/d5_microdisk_20260926 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "D5-PIN-MICRODISK-20260927-v1", + "kind": "proof_index_result", + "title": "D5 pin microdisk (#90): Math- reviews/d5_pin_microdisk_20260927 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "reviews/d5_pin_microdisk_20260927 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:reviews/d5_pin_microdisk_20260927 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-BF-SIX-PIN-HESSIAN-20260926-v1", + "kind": "proof_index_result", + "title": "Bargmann-Fock six-pin Hessian certificate (+SCALAR) (#89): Math- certificates/bf_six_pin_hessian_20260926 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "certificates/bf_six_pin_hessian_20260926 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:certificates/bf_six_pin_hessian_20260926 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-CONTACT-KERNEL-SUBSTITUTE-20260926-v1", + "kind": "proof_index_result", + "title": "contact kernel substitute (#80): Math- reviews/contact_kernel_substitute_20260926 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "reviews/contact_kernel_substitute_20260926 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:reviews/contact_kernel_substitute_20260926 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-D5-SHORT-EDGE-20260928-v1", + "kind": "proof_index_result", + "title": "D5 short edge (#103): Math- reviews/d5_short_edge_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "reviews/d5_short_edge_20260928 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:reviews/d5_short_edge_20260928 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "CL-D5-PIN-DISK-20260928-v1", + "kind": "proof_index_result", + "title": "D5 pin disk, all directions (#104): Math- reviews/d5_pin_disk_all_directions_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "reviews/d5_pin_disk_all_directions_20260928 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:reviews/d5_pin_disk_all_directions_20260928 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-D5-REMOTE-SINGLETON-LAW-20260928-v1", + "kind": "proof_index_result", + "title": "D5 remote singleton law (#115): Math- frontiers/remote_singleton_law_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/remote_singleton_law_20260928 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/remote_singleton_law_20260928 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-REMOTE-HEIGHT-DECOUPLING-20260928-v1", + "kind": "proof_index_result", + "title": "remote height decoupling (#120): Math- frontiers/remote_height_decoupling_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/remote_height_decoupling_20260928 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/remote_height_decoupling_20260928 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-WINDOW-MULTIPLICITY-LOCAL", + "kind": "proof_index_result", + "title": "window multiplicity law, local (#116 C2; id abbreviated as listed by the navigation refresh): Math- frontiers/window_multiplicity_laws_20260928 (C2) at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/window_multiplicity_laws_20260928 (C2) (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/window_multiplicity_laws_20260928 (C2) at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-WINDOW-MULTIPLICITY-REMOTE", + "kind": "proof_index_result", + "title": "window multiplicity law, remote (#116 C3; id abbreviated as listed): Math- frontiers/window_multiplicity_laws_20260928 (C3) at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/window_multiplicity_laws_20260928 (C3) (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/window_multiplicity_laws_20260928 (C3) at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-WINDOW-MULTIPLICITY-DISTANCE", + "kind": "proof_index_result", + "title": "window multiplicity law, distance (#116 C4; id abbreviated as listed): Math- frontiers/window_multiplicity_laws_20260928 (C4) at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/window_multiplicity_laws_20260928 (C4) (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/window_multiplicity_laws_20260928 (C4) at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-WINDOW-MULTIPLICITY-HEIGHT", + "kind": "proof_index_result", + "title": "window multiplicity law, height (#116 C5; id abbreviated as listed): Math- frontiers/window_multiplicity_laws_20260928 (C5) at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/window_multiplicity_laws_20260928 (C5) (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/window_multiplicity_laws_20260928 (C5) at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-REMOTE-INVERSE-SEPARATION-20260928-v1", + "kind": "proof_index_result", + "title": "remote inverse separation (#128): Math- frontiers/remote_inverse_separation_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/remote_inverse_separation_20260928 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/remote_inverse_separation_20260928 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-ELDER-DIMENSION-LIFT-20260928-v1", + "kind": "proof_index_result", + "title": "elder lower bound 1 - p_r >= c r^3 in every fixed d (#125): Math- frontiers/elder_dimension_lift_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/elder_dimension_lift_20260928 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/elder_dimension_lift_20260928 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "CL-ELDER-LOWER-ALL-D-20260929-v1.1", + "kind": "proof_index_result", + "title": "elder lower bound, all d (#149): Math- frontiers/elder_lower_all_d_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/elder_lower_all_d_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/elder_lower_all_d_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "CL-C6-FACTORIAL-20260929-v1", + "kind": "proof_index_result", + "title": "C6 factorial moment, Lemma M and cap Psi (#140, v1.3): Math- frontiers/c6_factorial_moment_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/c6_factorial_moment_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/c6_factorial_moment_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "CL-C6-SHARPENED-20260929-v1", + "kind": "proof_index_result", + "title": "C6 sharpened (#142, v1.2): Math- frontiers/c6_sharpened_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/c6_sharpened_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/c6_sharpened_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-C6-COUNT-CAP-BOUNDARY-20260929-v1", + "kind": "proof_index_result", + "title": "C6 count cap boundary (#146): Math- frontiers/c6_count_cap_boundary_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/c6_count_cap_boundary_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/c6_count_cap_boundary_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-LOCAL-CLUSTER-TIGHTNESS-20260929-v1", + "kind": "proof_index_result", + "title": "local cluster tightness (#161): Math- frontiers/local_cluster_tightness_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/local_cluster_tightness_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/local_cluster_tightness_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-REMOTE-MIXED-INVERSE-20260929-v1", + "kind": "proof_index_result", + "title": "remote mixed inverse (#132): Math- frontiers/remote_mixed_inverse_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/remote_mixed_inverse_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/remote_mixed_inverse_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "CL-D1-UNRESTRICTED-DIFFERENCE-20260929-v3", + "kind": "proof_index_result", + "title": "Theorem U: the C7 unrestricted O(ell^{2/3}) rate is false (#133): Math- frontiers/unrestricted_selection_difference_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/unrestricted_selection_difference_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/unrestricted_selection_difference_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "CL-C7-TOTAL-BOUNDED-20260929-v1.2", + "kind": "proof_index_result", + "title": "C7 total bounded, Theorem K; read with ERRATUM_NORMALIZED_FORM.md and RATE_READING.md per the 2026-10-07 guidance (#150): Math- frontiers/c7_total_bounded_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/c7_total_bounded_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/c7_total_bounded_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-C7-ZERO-GAP-LIMIT-20260929-v1", + "kind": "proof_index_result", + "title": "C7 zero-gap limit, Theorem Z (#155): Math- frontiers/c7_zero_gap_limit_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/c7_zero_gap_limit_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/c7_zero_gap_limit_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-SARD-ROBUST-CHARTS-20260929-v1", + "kind": "proof_index_result", + "title": "SARD-G robust charts (#135; STATUS.md ACCEPT, scoped, qualitative successor; A1/A6 original formulation remains AMEND): Math- frontiers/sard_g_robust_charts_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/sard_g_robust_charts_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/sard_g_robust_charts_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-SARD-A2-FIXED-FRAME-20260929-v2", + "kind": "proof_index_result", + "title": "SARD-G A2 fixed frame (#137): Math- frontiers/sard_g_a2_parameter_20260929/v2/A2_FIXED_FRAME.md at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/sard_g_a2_parameter_20260929/v2/A2_FIXED_FRAME.md (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/sard_g_a2_parameter_20260929/v2/A2_FIXED_FRAME.md at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:sard_g_a2_regularity_20260929", + "kind": "proof_index_result", + "title": "SARD-G A2 regularity (reading rule #139 e947ade5): Math- frontiers/sard_g_a2_regularity_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/sard_g_a2_regularity_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/sard_g_a2_regularity_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:sard_g_pinned_transfer_20260929", + "kind": "proof_index_result", + "title": "SARD-G pinned transfer (reading rule #139 e947ade5): Math- frontiers/sard_g_pinned_transfer_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/sard_g_pinned_transfer_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/sard_g_pinned_transfer_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-P15-BIPARTITE-OVERLAP-20260928-v1", + "kind": "proof_index_result", + "title": "P15 bipartite overlap (#118; prize reconnaissance, HOLD): Math- frontiers/p15_bipartite_overlap_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/p15_bipartite_overlap_20260928 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/p15_bipartite_overlap_20260928 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "OA-P15-TAIL-LOAD-20260928-v1", + "kind": "proof_index_result", + "title": "P15 tail load (#122; prize reconnaissance, HOLD): Math- frontiers/p15_tail_load_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/p15_tail_load_20260928 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/p15_tail_load_20260928 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_cubic_cluster_20260929", + "kind": "proof_index_result", + "title": "planar cubic cluster (#158 aa288385): Math- frontiers/planar_cubic_cluster_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_cubic_cluster_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_cubic_cluster_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:c6_marked_fourier_20260929", + "kind": "proof_index_result", + "title": "C6 marked Fourier (#157 2479a7c2): Math- frontiers/c6_marked_fourier_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/c6_marked_fourier_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/c6_marked_fourier_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:c6_cluster_law_20260929", + "kind": "proof_index_result", + "title": "C6 cluster law (#159 98fdb549): Math- frontiers/c6_cluster_law_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/c6_cluster_law_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/c6_cluster_law_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:cluster_exponential_weight_20260930", + "kind": "proof_index_result", + "title": "cluster exponential weight (#164 7d0c89a5): Math- frontiers/cluster_exponential_weight_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/cluster_exponential_weight_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/cluster_exponential_weight_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:spectral_cluster_closure_20260929", + "kind": "proof_index_result", + "title": "spectral cluster closure (#162 358f2562): Math- frontiers/spectral_cluster_closure_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/spectral_cluster_closure_20260929 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/spectral_cluster_closure_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:lifetime_moment_boundary_20260930", + "kind": "proof_index_result", + "title": "lifetime moment boundary (#177 dda8991f): Math- frontiers/lifetime_moment_boundary_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/lifetime_moment_boundary_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/lifetime_moment_boundary_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:radial_rate_20260930", + "kind": "proof_index_result", + "title": "radial rate (#176 7ca3eb00): Math- frontiers/radial_rate_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/radial_rate_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/radial_rate_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:extreme_cluster_sampling_20260930", + "kind": "proof_index_result", + "title": "extreme cluster sampling (#169 f430fdec): Math- frontiers/extreme_cluster_sampling_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/extreme_cluster_sampling_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/extreme_cluster_sampling_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:fixed_r_inverse_lifetime_20260930", + "kind": "proof_index_result", + "title": "fixed-r inverse lifetime (#182 2a10ed34): Math- frontiers/fixed_r_inverse_lifetime_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/fixed_r_inverse_lifetime_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/fixed_r_inverse_lifetime_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:local_elder_geometry_20260930", + "kind": "proof_index_result", + "title": "local elder geometry (#170 fa2e9909): Math- frontiers/local_elder_geometry_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/local_elder_geometry_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/local_elder_geometry_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:concave_fibre_elder_20260930", + "kind": "proof_index_result", + "title": "concave fibre elder (#175 eb659bd3): Math- frontiers/concave_fibre_elder_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/concave_fibre_elder_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/concave_fibre_elder_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:elder_partner_extremes_20260930", + "kind": "proof_index_result", + "title": "elder partner extremes (#181 443212fb): Math- frontiers/elder_partner_extremes_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/elder_partner_extremes_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/elder_partner_extremes_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:remainder_vanishing_20260930", + "kind": "proof_index_result", + "title": "remainder vanishing (#191 95565474): Math- frontiers/remainder_vanishing_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/remainder_vanishing_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/remainder_vanishing_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:elder_selected_extremes_20260930", + "kind": "proof_index_result", + "title": "elder selected extremes (#179 6e5f36d7): Math- frontiers/elder_selected_extremes_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/elder_selected_extremes_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/elder_selected_extremes_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:remainder_rate_20260930", + "kind": "proof_index_result", + "title": "remainder rate (#198 124c37d9): Math- frontiers/remainder_rate_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/remainder_rate_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/remainder_rate_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:far_elder_rate_20260930", + "kind": "proof_index_result", + "title": "far elder rate (#187 c2f1270d): Math- frontiers/far_elder_rate_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/far_elder_rate_20260930 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/far_elder_rate_20260930 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:short_bar_endpoint_law_20261001", + "kind": "proof_index_result", + "title": "short bar endpoint law (#221 f166ef52): Math- frontiers/short_bar_endpoint_law_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/short_bar_endpoint_law_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/short_bar_endpoint_law_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:c2_finite_jet_transfer_20261001", + "kind": "proof_index_result", + "title": "C2 finite jet transfer (#232 7fe06b01): Math- frontiers/c2_finite_jet_transfer_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/c2_finite_jet_transfer_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/c2_finite_jet_transfer_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:cusp_second_order_20261001", + "kind": "proof_index_result", + "title": "cusp second order (#207 566b1a14): Math- frontiers/cusp_second_order_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/cusp_second_order_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/cusp_second_order_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:candidate_third_order_20261001", + "kind": "proof_index_result", + "title": "candidate third order (#218 fb6ee97b): Math- frontiers/candidate_third_order_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/candidate_third_order_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/candidate_third_order_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:elder_third_order_20261001", + "kind": "proof_index_result", + "title": "elder third order (#220 0d797780): Math- frontiers/elder_third_order_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/elder_third_order_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/elder_third_order_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:third_order_rate_20261001", + "kind": "proof_index_result", + "title": "third order rate (#229 6f74f7a7): Math- frontiers/third_order_rate_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/third_order_rate_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/third_order_rate_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:elder_cusp_parity_20261002", + "kind": "proof_index_result", + "title": "elder cusp parity (#240 7636cae5): Math- frontiers/elder_cusp_parity_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/elder_cusp_parity_20261002 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/elder_cusp_parity_20261002 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:separated_fold_multiplicity_20261002", + "kind": "proof_index_result", + "title": "separated fold multiplicity (#239 240a9bfa): Math- frontiers/separated_fold_multiplicity_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/separated_fold_multiplicity_20261002 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/separated_fold_multiplicity_20261002 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:subpin_collision_20261002", + "kind": "proof_index_result", + "title": "subpin collision (#245 2d8616ef): Math- frontiers/subpin_collision_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/subpin_collision_20261002 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/subpin_collision_20261002 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:soft_rejected_pairs_20261002", + "kind": "proof_index_result", + "title": "soft rejected pairs (#242 54c6759e): Math- frontiers/soft_rejected_pairs_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/soft_rejected_pairs_20261002 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/soft_rejected_pairs_20261002 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:soft_fold_limit_20261002", + "kind": "proof_index_result", + "title": "soft fold limit (#243 fa8b0d21): Math- frontiers/soft_fold_limit_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/soft_fold_limit_20261002 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/soft_fold_limit_20261002 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:soft_closed_form_20261002", + "kind": "proof_index_result", + "title": "soft closed form (#244 07320089): Math- frontiers/soft_closed_form_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/soft_closed_form_20261002 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/soft_closed_form_20261002 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:cusp_torus_transfer_20261001", + "kind": "proof_index_result", + "title": "cusp torus transfer (3 Oct supplement): Math- frontiers/cusp_torus_transfer_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/cusp_torus_transfer_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/cusp_torus_transfer_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:moving_remote_collision_20261003", + "kind": "proof_index_result", + "title": "moving remote collision (3 Oct supplement): Math- frontiers/moving_remote_collision_20261003 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/moving_remote_collision_20261003 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/moving_remote_collision_20261003 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:ordinary_short_bar_occurrence_20261001", + "kind": "proof_index_result", + "title": "ordinary short bar occurrence (3 Oct supplement): Math- frontiers/ordinary_short_bar_occurrence_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/ordinary_short_bar_occurrence_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/ordinary_short_bar_occurrence_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:replacement_bar_occurrence_20261001", + "kind": "proof_index_result", + "title": "replacement bar occurrence (3 Oct supplement): Math- frontiers/replacement_bar_occurrence_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/replacement_bar_occurrence_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/replacement_bar_occurrence_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:strict_unique_replacement_coefficient_20261003", + "kind": "proof_index_result", + "title": "strict unique replacement coefficient (3 Oct supplement): Math- frontiers/strict_unique_replacement_coefficient_20261003 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/strict_unique_replacement_coefficient_20261003 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/strict_unique_replacement_coefficient_20261003 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:unique_replacement_bar_intensity_20261001", + "kind": "proof_index_result", + "title": "unique replacement bar intensity (3 Oct supplement): Math- frontiers/unique_replacement_bar_intensity_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/unique_replacement_bar_intensity_20261001 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/unique_replacement_bar_intensity_20261001 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_rejected_endpoint_margin_20261003/A4", + "kind": "proof_index_result", + "title": "planar rejected endpoint margin A4 (3 Oct supplement, nested): Math- frontiers/planar_rejected_endpoint_margin_20261003/A4 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_rejected_endpoint_margin_20261003/A4 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_rejected_endpoint_margin_20261003/A4 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C91", + "kind": "proof_index_result", + "title": "planar soft layer chain C91 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C91 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C91 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C91 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C92", + "kind": "proof_index_result", + "title": "planar soft layer chain C92 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C92 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C92 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C92 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C93", + "kind": "proof_index_result", + "title": "planar soft layer chain C93 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C93 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C93 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C93 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C94", + "kind": "proof_index_result", + "title": "planar soft layer chain C94 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C94 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C94 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C94 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C95", + "kind": "proof_index_result", + "title": "planar soft layer chain C95 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C95 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C95 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C95 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C96", + "kind": "proof_index_result", + "title": "planar soft layer chain C96 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C96 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C96 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C96 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C97", + "kind": "proof_index_result", + "title": "planar soft layer chain C97 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C97 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C97 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C97 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C98", + "kind": "proof_index_result", + "title": "planar soft layer chain C98 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C98 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C98 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C98 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C99", + "kind": "proof_index_result", + "title": "planar soft layer chain C99 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C99 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C99 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C99 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C101", + "kind": "proof_index_result", + "title": "planar soft layer chain C101 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C101 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C101 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C101 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C102", + "kind": "proof_index_result", + "title": "planar soft layer chain C102 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C102 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C102 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C102 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C103", + "kind": "proof_index_result", + "title": "planar soft layer chain C103 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C103 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C103 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C103 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "packet:planar_soft_layer_chain_20261003/C124", + "kind": "proof_index_result", + "title": "planar soft layer chain C124 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C124 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted", + "layer0": { + "source_id": null, + "heading": "frontiers/planar_soft_layer_chain_20261003/C124 (bytes not pinned in this packet; headline statement not extracted)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved.", + "next_step": "Read Math-:frontiers/planar_soft_layer_chain_20261003/C124 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted." + }, + { + "claim_id": "hist.D0-RN-UNIF-historical-predicates", + "kind": "open_obligation", + "title": "D0 historical RN-UNIF obligations recorded in GRAPH.json: hist.ENV-RESCOV, hist.ALLCELL-FDZ-Q4, hist.LOGQ-TAIL, hist.SYM-Fw-jet, hist.OBL-H5-JETMOD (OPEN_HISTORICAL), hist.CH-LIFT, hist.Piece-2-annulus (OPEN_ACTIVE), carriers hist.rnu_env.py / hist.CL_ANTHROPIC_BUNDLE_2026-09-17_v5.zip / hist.allcell_fdz_enclosures.json (BLOCKED_ABSENT), hist.lemma_closed FALSE; PROOF_INDEX \"Open obligations without a complete proof yet\"", + "layer0": { + "source_id": "downstream-graph", + "heading": "nodes.hist.* (ENV-RESCOV, ALLCELL-FDZ-Q4, LOGQ-TAIL, SYM-Fw-jet, CH-LIFT, Piece-2-annulus, OBL-H5-JETMOD, carriers, lemma_closed)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "No Lean statement: the historical numerical RN all-cell / 24-jet certification predicates have no in-tree source text to anchor (their carriers are ABSENT by fingerprint) and no Prop states any of them. The GRAPH records no complete proof where not explicitly superseded, and hist.lemma_closed is FALSE; this register changes none of that. The regional bypass (regional.fixed-annulus.high-jet-route, SUPERSEDED_NONBLOCKING) does not reopen or close these predicates in their original replay scopes; no numerical certificate exists in tree.", + "next_step": "None for Lean until a source text exists; recovery of the absent carriers is a custody question (reviews/d0_custody_audit_20260926), not a formalization one." + }, + { + "claim_id": "lean:main-formal-UniversalLaw", + "kind": "existing_lean_package", + "title": "main formal/ UniversalLaw (package side24-arithmetic-skeleton, core Lean v4.34.1, no Mathlib): 31 Side24 integer targets at origin/main f6deeba7 (Ledger 17, ConeMoments 10, Endpoints 4; author Anthropic/Claude via Cursor, 2026-09-27) plus 28 P15 targets staged uncommitted in this programme's worktree (PriceBoundary 6, RealizedCovers 12, FullPrice 10; on no commit, so not listed as statements here); manifest formalization_status proved, alignment_status PENDING_INDEPENDENT_REVIEW", + "layer0": { + "source_id": "side24-enclosure", + "heading": "ENCLOSURE.json (the four Endpoints targets anchor here; the other 55 targets are listed on the claim rows side24-coefficient, p15-price-boundary, p15-realized-covers, p15-price-budget-restricted, p15-full-price)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [ + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Endpoints.lean", + "declaration": "UniversalLaw.Side24.d2_endpoints_adjacent", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "\"2\": {\n \"lower\": \"0.07340691930603427103\",\n \"upper\": \"0.07340691930603427104\"", + "does_not_claim": "that c_2,24 lies between them" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Endpoints.lean", + "declaration": "UniversalLaw.Side24.d3_endpoints_adjacent", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "\"3\": {\n \"lower\": \"0.04177593184059834334\",\n \"upper\": \"0.04177593184059834335\"", + "does_not_claim": "that c_3,24 lies between them" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Endpoints.lean", + "declaration": "UniversalLaw.Side24.d3_below_d2", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "\"scientific_acceptance\": false", + "does_not_claim": "anything about the true coefficients" + }, + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "module": "UniversalLaw/Side24/Endpoints.lean", + "declaration": "UniversalLaw.Side24.endpoints_in_unit_tenth", + "kind": "theorem", + "formal_progress": "proved", + "anchor": "\"relative_periodization_bound\": \"1e-106\"", + "does_not_claim": "anything about the true coefficients" + } + ], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "receipt_or_workflow": ".github/workflows/formal-verification.yml (tools/formal_gate_check.py --run-lean inside the required aggregate `verify`); receipt formal/.lake/formal-evidence/receipt.json is a workflow artifact, none is committed in tree", + "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": "The package states no research claim as a Prop, so the claim-level label is none; its 31 theorems at origin/main f6deeba7 are finite integer/cross-multiplied rational facts proved by decide (label `proved` in the manifest, `kernel-checked` only in a run receipt). The four Endpoints theorems listed here show the published 20-digit endpoints are adjacent, ordered and below 1/10 as Nat; they do not place c_{2,24} or c_{3,24} between them and say nothing about the true coefficients (the module's own note). The 28 P15 arithmetic targets of this programme's main worktree are on no commit, are not listed in this register, and have only a local untrusted --run-lean receipt. main has NO mechanism for `specified`: its gate refuses any manifest label but `proved`, so statement-only work cannot live in this package. Alignment review by a non-Anthropic lane is an open ask (formal/reviews holds only the template).", + "next_step": "Commit and push the P15 modules so the hosted formal-verification.yml run produces a trusted receipt, then list them here as `proved` statements citing the 40-hex commit and the run; obtain a non-Anthropic alignment review of the 31 committed targets (59 once the P15 modules are committed); keep real-number statements in the Math- Mathlib lane." + }, + { + "claim_id": "lean:math-formal-ResearchFormalCoreR1", + "kind": "existing_lean_package", + "title": "Math- formal/ ResearchFormalCoreR1 (universal_law_formal, Lean v4.34.1 + Mathlib d13f23b7): 63 theorem targets in 8 modules (AlgebraV2 EC-005/008/010/011/014; ProbabilityCompanionsV2 P02-LM-008/009; MeasureBridge; WeightedLaw; MomentTail; MomentGenerality; WeightPerturbation; D2Schur); manifest formalization_status proved, alignment_status PENDING_INDEPENDENT_REVIEW; authors OpenAI GPT-6 Astra Pro (bridge, law, tail, perturbation, Schur), recovered GP-FOR-192 originals, Grok Bot agents 6/7 proposals (MomentGenerality)", + "layer0": { + "source_id": null, + "heading": null + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/Math- formal/ (ResearchFormalCoreR1)", + "receipt_or_workflow": ".github/workflows/formal-lean.yml via downstream-gate.yml (formal/gate.py --execute: version preflight, fresh lake build, leanchecker, #print axioms within propext/Classical.choice/Quot.sound, 5 negative controls, receipt.json in .lake/formal-evidence/); no success receipt is committed, formal/evidence/ holds three FAILURE logs", + "note": "Kernel-checked scalar identities, the Bochner Cauchy-Schwarz transfer, the normalized weighted law (withDensity), Markov fortieth-moment tails, L1 weight perturbation, the D2 Schur real algebra. formal/SCOPE.md: none of these targets is the SIDE24 coefficient enclosure, the elder-rule selection theorem or the joint extremum/merging-saddle law; all remain conditional at displayed hypotheses." + }, + { + "package": "d6g8k5htny-coder/main reviews/lean_probability_alignment_20261005", + "receipt_or_workflow": "REVIEW_272_20.md + alignment_272_20.json (statement-alignment record, Lean not re-executed)", + "note": "main reviews/lean_probability_alignment_20261005 (REVIEW_272_20.md, alignment_272_20.json): xAI/Grok 4.7 via Cursor statement-alignment ACCEPTED for exactly the 20 Math-#272 targets at Math- 270167900b71539d0e2135131a3a96f1f04289f6, Lean not re-executed, organizational-independence credit 0, Dylan's reading PENDING; STALE for the current 63-target manifest (manifest digest changed). The manifest value stays PENDING_INDEPENDENT_REVIEW." + } + ], + "not_established": "The package states no research claim of this register as a Prop and none of its statements can be listed here: its bound sources (P02-LM-008 proof.md.export.txt at 22876eda, MOMENT_TAIL.md, WEIGHT_PERTURBATION.md, the D2 small-L NOTE.md) are not among this packet's pinned sources, so no verbatim anchor is available. Not established by the package (its own does_not_establish): any concrete Gaussian or typed Palm law, moment bounds M, cW, cZ, lower normalizers, event tails, constant uniformity, GT5, P0.2, the parent persistence theorem, identification of weightedLaw with the program's conditioned law, D2 Schur in d >= 3, the coefficient formula (15.2). Alignment: ACCEPTED for 20 targets at an older head only, STALE for the current 63, credit 0.", + "next_step": "Pin the Math- formal sources in a later packet revision and register the 63 targets as `proved` statements with anchors; obtain a fresh alignment record for the 63-target manifest from a lane differing in provider, family and agent." + }, + { + "claim_id": "lean:math-companion-d2_moment_bridge_20261006", + "kind": "existing_lean_package", + "title": "Math- companions/d2_moment_bridge_20261006 (Lean companion compiled against formal/ via `lake env lean --root`, same toolchain and Mathlib; NOT in formal/manifest.json): 2 definitions + 16 theorems (D2MomentBridge: Bochner moments, Delta = integral of the cubic residual squared, Delta = 0 iff a.e. X = 0 or X^2 = m4/m2, strictness from two atoms); author OpenAI GPT-6 Astra Pro, 2026-10-06", + "layer0": { + "source_id": null, + "heading": null + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/Math- companions/d2_moment_bridge_20261006", + "receipt_or_workflow": ".github/workflows/d2-moment-bridge.yml (check.py: source custody, emit, negative controls with exactly `unsolved goals ⊢ False`, finish requiring a fresh core receipt kernel-checked at the same commit); receipts are workflow artifacts (30-day retention), none committed", + "note": "Axiom audits within propext/Classical.choice/Quot.sound; the companion pins the consumed core blobs by hash. Not independent scientific acceptance." + } + ], + "not_established": "States no research claim of this register; its sources (Math-#302 NOTE.md section 4 / C7, #326, #328, #343 comments) are not pinned here, so no statement can be anchored and listed. Does not establish: no uniform spectral floor, no concrete periodic probability law, no moment summability for that law, no trigonometric q map, no Gaussian/Palm identification, no interval quadrature, no numerical coefficient or persistence/cap theorem; the finite countermodel law is tested in Python, not constructed in Lean. Same GitHub account as every lane: organizational-independence credit 0.", + "next_step": "Pin the companion sources and register its theorems as `proved` statements in a later packet revision; a reviewer differing in provider, family and agent for alignment." + }, + { + "claim_id": "lean:math-companion-d2_square_schur_20261006", + "kind": "existing_lean_package", + "title": "Math- companions/d2_square_schur_20261006 (Lean companion compiled against formal/ via `lake env lean --root`, same toolchain and Mathlib; NOT in formal/manifest.json): 4 theorems (D2SquareSchur: weighted square identity and lower bound, upper-moment-free Schur lower bounds min(m4 - m2^2, 2 m2^2) and min(g0, 2 s0^2)) plus Contract.lean exact-type examples; author OpenAI GPT-6 Astra Pro, 2026-10-06", + "layer0": { + "source_id": null, + "heading": null + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/Math- companions/d2_square_schur_20261006", + "receipt_or_workflow": ".github/workflows/d2-square-schur.yml (check.py: source custody, emit, negative controls with exactly `unsolved goals ⊢ False`, finish requiring a fresh core receipt kernel-checked at the same commit); receipts are workflow artifacts (30-day retention), none committed", + "note": "Axiom audits within propext/Classical.choice/Quot.sound; the companion pins the consumed core blobs by hash. Not independent scientific acceptance." + } + ], + "not_established": "States no research claim of this register; its sources (Math-#302 NOTE.md section 4 / C7, #326, #328, #343 comments) are not pinned here, so no statement can be anchored and listed. Does not establish: no atom-measure integral lower bound, moment realizability, periodic spectral law, trigonometric parameterization, Gaussian/Palm covariance identification, quadrature or persistence theorem; the #328 Part B stronger proposal is not incorporated. Same GitHub account as every lane: organizational-independence credit 0.", + "next_step": "Pin the companion sources and register its theorems as `proved` statements in a later packet revision; a reviewer differing in provider, family and agent for alignment." + }, + { + "claim_id": "lean:math-companion-d2_atom_integral_20261006", + "kind": "existing_lean_package", + "title": "Math- companions/d2_atom_integral_20261006 (Lean companion compiled against formal/ via `lake env lean --root`, same toolchain and Mathlib; NOT in formal/manifest.json): 6 theorems (D2AtomIntegral: two-event and two-radius integral lower bounds, second-moment, fourth-gap, residual-moment and Schur floors from radius masses) plus Contract.lean; author OpenAI GPT-6 Astra Pro, 2026-10-06", + "layer0": { + "source_id": null, + "heading": null + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/Math- companions/d2_atom_integral_20261006", + "receipt_or_workflow": ".github/workflows/d2-atom-integral.yml (check.py: source custody, emit, negative controls with exactly `unsolved goals ⊢ False`, finish requiring a fresh core receipt kernel-checked at the same commit); receipts are workflow artifacts (30-day retention), none committed", + "note": "Axiom audits within propext/Classical.choice/Quot.sound; the companion pins the consumed core blobs by hash. Not independent scientific acceptance." + } + ], + "not_established": "States no research claim of this register; its sources (Math-#302 NOTE.md section 4 / C7, #326, #328, #343 comments) are not pinned here, so no statement can be anchored and listed. Does not establish: no formal sharpness, no actual periodic spectral measure or its summability, no numerical values or radii, no uniform-L or family floors, no Gaussian/Palm conditioning, no coefficient integral, no persistence matching. Same GitHub account as every lane: organizational-independence credit 0.", + "next_step": "Pin the companion sources and register its theorems as `proved` statements in a later packet revision; a reviewer differing in provider, family and agent for alignment." + }, + { + "claim_id": "lean:math-frontier-cap_first_exit_lean_20261002", + "kind": "existing_lean_package", + "title": "Math- frontiers/cap_first_exit_lean_20261002 (CAP-FIRST-EXIT-LEAN-20261002-v1, v2.6 at commit 4013764): own lake package cap_first_exit, Lean v4.34.1 + Mathlib d13f23b7, CapFirstExit.lean 326607 B, 152 declarations (138 theorems, 13 defs, 1 structure CapHyp), sorry count 0, allowed axioms propext/Classical.choice/Quot.sound; MANIFEST mathematical_acceptance false, alignment_review NOT_CLAIMED; author Anthropic/Claude; pins_on_main: SP review (a653dbc7) and MARKED_CYLINDER_CAP_PROOF.md (0633aca3)", + "layer0": { + "source_id": "marked-cylinder-cap", + "heading": "Deterministic theorem (CAP sections 1-5; one of the frontier's two pins_on_main)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/Math- frontiers/cap_first_exit_lean_20261002 (CapFirstExit, 152 declarations, Lean v4.34.1 + Mathlib d13f23b7)", + "receipt_or_workflow": ".github/workflows/cap-first-exit-lean.yml (gate.py --execute: fresh lake build, leanchecker, #print axioms within propext/Classical.choice/Quot.sound)", + "note": "Kernel-checked topological and analytic steps of CAP sections 2-5 / SP L1-L22 (CapHyp, elder death level, ridge chain, torus transport). Not registered as statements here: its per-declaration anchors (ALIGNMENT.md) were not transcribed. MANIFEST: mathematical_acceptance false, alignment_review NOT_CLAIMED; v1.4 (53 decls) ALIGNED_SCOPED on PR #246, v1.5-v2.6 await delta review; author Anthropic/Claude, independence credit 0. Not formalized: SP section 2 setting as a property of the field, L14/L15, L23, P section 8, every probabilistic statement." + } + ], + "not_established": "The frontier states and kernel-checks the topological global step of D1 component III (maximin level, elder merge, elder death level, saddle cut, older peak on compact locally connected spaces, torus transport) and the analytic chain L1-L13, L16-L20 from f in C^4 and CAP (1), ending in torus_capHyp_blocks and torus_elder_death_level_blocks. None of its 152 declarations is listed here as a statement because their per-declaration anchors (ALIGNMENT.md sections 1-3) were not transcribed in this programme. Not established (its own does_not_establish): that the local field on G_r satisfies SP section 2; the identification of SP's coordinates/contact frame and P's field with the Lean objects; L14 (critical-point count), L15 (Morse indices); the definition of the H0 superlevel persistence pairing (only the level s is proved); L23, P section 8, Theorem A, measurability and Q^W(G_r^c) <= C r^3 (every probabilistic statement). Math-#193 D1 kernel formalization remains partially unsupplied. Same provider as this packet: independence credit 0; the author will not merge.", + "next_step": "Transcribe ALIGNMENT.md row by row and register the 152 declarations as `proved` statements anchored in the pinned CAP bytes (and a pinned SP review); non-Claude delta review of v1.5-v2.6." + }, + { + "claim_id": "lean:math-w3scratch-side24-proposal", + "kind": "existing_lean_package", + "title": "Math- reviews/w3_side24_source_proposal_20261008/W3Scratch (D2SchurLinks.lean, Side24Law.lean, Side24Poisson.lean; about 101 declarations): unregistered, unreviewed, source-only proposal (option B of main#229 6050818585); NOT part of ResearchFormalCoreR1, NO lakefile, built by NO workflow; author Grok Bot agent 13, 2026-10-07; one Grok-Bot-on-Grok-Bot reader verdict (AMEND README/comments only), credit 0", + "layer0": { + "source_id": null, + "heading": null + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "No kernel evidence exists for this package in any workflow or committed receipt: builds and axiom reports are author-reported, box-local, in a scratch copy of formal/ with an appended lean_lib, and gate.py --execute was never satisfied (namespace mismatch; a 140-target option-A variant hit the 900 s build timeout). Nothing here is reviewed, aligned or a discharge; the identification of the Poisson-dual law with the NOTE's spectral law (its question (a)) is open; the all-L positivity statement is not the source's six-L interval certificate; no status field changes. Its declarations are not listed here (unpinned sources, no workflow).", + "next_step": "If the proposal is adopted, register it under a lakefile and workflow in Math- first; only then could its statements enter a register with anchors and `proved` labels." + }, + { + "claim_id": "main:experiments/universality/finite_gaussian_contact", + "kind": "main_experiment_proof", + "title": "main experiments/universality/finite_gaussian_contact/PROOF.md + REVIEW.md (origin/main f6deeba7): Positive finite Gaussian spectra: the contact and regression interfaces (PR #319, c74aa739; contact covariance, small-separation regression, compact-mark normalizer for ten ideal Gaussian profiles at cutoffs 2, 3); prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/finite_gaussian_contact/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/finite_gaussian_contact/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/universality/finite_candidate_lifetime", + "kind": "main_experiment_proof", + "title": "main experiments/universality/finite_candidate_lifetime/PROOF.md + REVIEW.md (origin/main f6deeba7): Finite Gaussian candidate-pair lifetime theorem (PR #321, 4b94a1b1): finite continuous density C_loc ell^(-1/3), cumulative (3/2) C_loc t^(2/3) for a fixed positive full-grid ideal Gaussian law at cutoff >= 2; REVIEW notes \"literal C210/Lean source alignment remains open\"; prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/finite_candidate_lifetime/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/finite_candidate_lifetime/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/universality/finite_fold_chart", + "kind": "main_experiment_proof", + "title": "main experiments/universality/finite_fold_chart/PROOF.md + REVIEW.md (origin/main f6deeba7): Uniform derivative-controlled fold charts (PR #321, 4b94a1b1); prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/finite_fold_chart/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/finite_fold_chart/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/universality/finite_weighted_fold_window", + "kind": "main_experiment_proof", + "title": "main experiments/universality/finite_weighted_fold_window/PROOF.md + REVIEW.md (origin/main f6deeba7): Weighted removal of deterministic fold-chart windows (PR #323, d4432d11): uniform marginal family bound with probability >= 1 - epsilon under determinant-weighted pin regression; prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/finite_weighted_fold_window/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/finite_weighted_fold_window/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/universality/finite_contact_genericity", + "kind": "main_experiment_proof", + "title": "main experiments/universality/finite_contact_genericity/PROOF.md + REVIEW.md (origin/main f6deeba7): Contact-conditioned global genericity for a finite Gaussian field (PR #323, d4432d11): forced contact isolated, other critical points Morse, distinct values; prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/finite_contact_genericity/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/finite_contact_genericity/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/universality/finite_elder_transfer", + "kind": "main_experiment_proof", + "title": "main experiments/universality/finite_elder_transfer/PROOF.md + REVIEW.md (origin/main f6deeba7): Isolated cubic contact to an actual finite H0 elder bar (PR #326, c69cc727); prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/finite_elder_transfer/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/finite_elder_transfer/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/universality/finite_h0_lifetime", + "kind": "main_experiment_proof", + "title": "main experiments/universality/finite_h0_lifetime/PROOF.md + REVIEW.md (origin/main f6deeba7): Finite Gaussian H0 elder-bar lifetime intensity (PR #326, c69cc727): Borel selector counting every finite superlevel H0 bar once, density C_loc ell^(-1/3) for a fixed positive full-square ideal Gaussian law, cutoff >= 2; named by docs/site/research.html (9 Oct reading cut); prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/finite_h0_lifetime/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/finite_h0_lifetime/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/universality/fold_structural_universality", + "kind": "main_experiment_proof", + "title": "main experiments/universality/fold_structural_universality/PROOF.md + REVIEW.md (origin/main f6deeba7): Regular cubic folds and the structural H0 lifetime exponent (PR #329, 4ee141e6): dimension-independent exponent -1/3 under five explicit field-admission interfaces; prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/fold_structural_universality/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/fold_structural_universality/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/universality/periodized_gaussian_h0", + "kind": "main_experiment_proof", + "title": "main experiments/universality/periodized_gaussian_h0/PROOF.md + REVIEW.md (origin/main f6deeba7): Actual H0 lifetime intensity for the exact periodized Gaussian law (PR #329, 4ee141e6): exact periodized Gaussian H0 adapter for d >= 2, L > 0 discharging the five interfaces for its actual law; prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/periodized_gaussian_h0/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/periodized_gaussian_h0/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/universality/iid_short_bar_process", + "kind": "main_experiment_proof", + "title": "main experiments/universality/iid_short_bar_process/PROOF.md + REVIEW.md (origin/main f6deeba7): Marked iid-copy Poisson limit for actual short H0 bars (PR #329, 4ee141e6): superposition of independent whole copies of the fixed torus law; no spatial expanding-domain law, regional witness closure or global second-factorial bound; prose source proposal by OpenAI/Codex agents, same-provider PASS review, organizational-independence credit 0, scientific promotion NONE", + "layer0": { + "source_id": null, + "heading": "experiments/universality/iid_short_bar_process/PROOF.md (bytes not pinned in this packet; headline statement not extracted beyond the header)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row.", + "next_step": "Pin experiments/universality/iid_short_bar_process/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review." + }, + { + "claim_id": "main:experiments/periodic_h0/spectral_shrinking_bins", + "kind": "main_experiment_proof", + "title": "main experiments/periodic_h0/spectral_shrinking_bins/PROOF.md (C132-SPECTRAL-SHRINKING-BINS-v1, PR #264, f3808295): shrinking lifetime bins for coupled spectral samples; \"Author-side conditional analytic theorem; source-bound nonauthor review pending\"; authors OpenAI/Codex; credit 0; no human review claimed", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/spectral_shrinking_bins/PROOF.md (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "A conditional prose theorem under explicit input interfaces (spectral error, spatial interpolation, exceptional expected counts); not pinned here, not stated in Lean, review pending by its own header. It does not prove its imported interfaces anew or change their status; the C131 exact-sample shrinking-bin reading record (EXACT_SAMPLE_SHRINKING_BIN.md) is a reading record, not a claim, and has no row.", + "next_step": "Pin the bytes, extract the conditional theorem with its three input interfaces as structure fields, state as def : Prop; nonauthor review first." + }, + { + "claim_id": "main:unmerged-codex-prs-330-350", + "kind": "main_experiment_proof", + "title": "Unmerged codex/* PRs on main listed by number only (not on origin/main; stacked on base 1966a74): #330 spectral_gaussian_admission, #331 higher_homology_fold, #334 spectral_amplitude_crossover, #336 quartic_cusp_field, #340 two_parameter_cusp_population, #343 (+#344) exact_population_certificate, #346 cusp_analytic_guards, #349 (+#350) global_cusp_h0_selector; also unmerged cursor/intermediate-scale-bridge-ab2f (notes/intermediate_scale_20260926)", + "layer0": { + "source_id": null, + "heading": null + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Nothing on origin/main: these are open pull requests whose PROOF.md, guard and certificate files are not merged, not pinned and not stated; their RESULTS.md scope statuses record lean_kernel NOT_RUN and scientific_promotion NOT_RUN. Listing them here is inventory only; no verdict, no landing and no statement is implied.", + "next_step": "Revisit after merge: pin, extract, state. The exact Fraction certificates of #343/#349 (which import periodic_h0 sqrt/exp/log bound helpers) would be exact_certificate rows then." + }, + { + "claim_id": "cert:main-periodic_h0-exact_h0_1", + "kind": "exact_certificate", + "title": "main experiments/periodic_h0/exact_h0.py / exact_barcode_certificate.py (EXACT_H0.md) -> results/exact_h0_1/CERTIFICATE.json (origin/main f6deeba7; run by .github/workflows/periodic-h0-pilot.yml with --verify): Exact ordinary H0 barcode of an integer-valued periodic vertex cubical filtration for the frozen 1024x1024 dyadic grid (elder rule, ties, independent connectivity check, bin transfer); barcode.zero_count 1048506, computed_barcode_error 0, independent_connectivity_check true", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/results/exact_h0_1/CERTIFICATE.json scope field (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet.", + "next_step": "Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses." + }, + { + "claim_id": "cert:main-periodic_h0-certificate8", + "kind": "exact_certificate", + "title": "main experiments/periodic_h0/finite_certificate.py (FINITE_CERTIFICATE.md) -> results/certificate8/CERTIFICATE.json (origin/main f6deeba7; run by .github/workflows/periodic-h0-pilot.yml with --verify): Derivative bounds (spatial budget B) for exactly defined rounded dyadic finite Fourier polynomials, BITS = 128, Machin pi and integer square roots; exports sqrt_bounds used by other certifiers", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/results/certificate8/CERTIFICATE.json scope field (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet.", + "next_step": "Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses." + }, + { + "claim_id": "cert:main-periodic_h0-nodal1", + "kind": "exact_certificate", + "title": "main experiments/periodic_h0/nodal_certificate.py + nodal_core.py (NODAL_CERTIFICATE.md) -> results/nodal1/CERTIFICATE.json (origin/main f6deeba7; run by .github/workflows/periodic-h0-pilot.yml with --verify): Finite rounded-polynomial nodal error and abstract filtration stability bound for a 128-grid sample", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/results/nodal1/CERTIFICATE.json scope field (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet.", + "next_step": "Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses." + }, + { + "claim_id": "cert:main-periodic_h0-hessian1", + "kind": "exact_certificate", + "title": "main experiments/periodic_h0/hessian_grid_certificate.py + hessian_grid_core.py (HESSIAN_GRID.md) -> results/hessian1/CERTIFICATE.json (origin/main f6deeba7; run by .github/workflows/periodic-h0-pilot.yml with --verify): Sharper finite-field spatial and nodal bound via an exact Hessian grid (derivative_grid 256, samples 1024^2)", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/results/hessian1/CERTIFICATE.json scope field (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet.", + "next_step": "Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses." + }, + { + "claim_id": "cert:main-periodic_h0-gaussian_tail1", + "kind": "exact_certificate", + "title": "main experiments/periodic_h0/gaussian_tail_certificate.py + gaussian_tail.py (GAUSSIAN_TAIL.md; exports exp_neg_bounds) -> results/gaussian_tail1/CERTIFICATE.json (origin/main f6deeba7; run by .github/workflows/periodic-h0-pilot.yml with --verify): Ideal Gaussian omitted-mode (spectral tail) probability bound for the normalized periodized Bargmann-Fock field on (R/24Z)^2 at cutoff 24, with a conditional coupling contract; coupling to the frozen sample remains open", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/results/gaussian_tail1/CERTIFICATE.json scope field (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet.", + "next_step": "Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses." + }, + { + "claim_id": "cert:main-periodic_h0-gaussian_coupling1", + "kind": "exact_certificate", + "title": "main experiments/periodic_h0/gaussian_coupling_certificate.py + gaussian_coupling.py (GAUSSIAN_COUPLING.md) -> results/gaussian_coupling1/CERTIFICATE.json (origin/main f6deeba7; run by .github/workflows/periodic-h0-pilot.yml with --verify): Prospective 128-bit inverse-normal coupling J_i = floor(2^128 U_i) under an explicit ideal IID uniform-word input hypothesis (deterministic fixture, cutoff 1)", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/results/gaussian_coupling1/CERTIFICATE.json scope field (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet.", + "next_step": "Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses." + }, + { + "claim_id": "cert:main-periodic_h0-word_polynomial1", + "kind": "exact_certificate", + "title": "main experiments/periodic_h0/word_polynomial_certificate.py + word_polynomial.py (WORD_POLYNOMIAL.md) -> results/word_polynomial1/CERTIFICATE.json (origin/main f6deeba7; run by .github/workflows/periodic-h0-pilot.yml with --verify): Explicit finite words -> smooth polynomial -> exact H0 bin counts (grid 128, Hessian grid 32, cutoff 1, side 24); deterministic, no RNG", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/results/word_polynomial1/CERTIFICATE.json scope field (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet.", + "next_step": "Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses." + }, + { + "claim_id": "cert:main-periodic_h0-finite_count_loss1", + "kind": "exact_certificate", + "title": "main experiments/periodic_h0/finite_count_loss_certificate.py + finite_count_loss.py (FINITE_COUNT_LOSS.md) -> results/finite_count_loss1/CERTIFICATE.json (origin/main f6deeba7; run by .github/workflows/periodic-h0-pilot.yml with --verify): Finite-cutoff count-loss budgets for F64,K on the side-24 torus under the declared word-input law", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/results/finite_count_loss1/CERTIFICATE.json scope field (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet.", + "next_step": "Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses." + }, + { + "claim_id": "cert:main-periodic_h0-count_confidence1", + "kind": "exact_certificate", + "title": "main experiments/periodic_h0/count_confidence_certificate.py + count_confidence.py + mean_adaptive_confidence.py (exports log_bounds) -> results/count_confidence1/CERTIFICATE.json (origin/main f6deeba7; run by .github/workflows/periodic-h0-pilot.yml with --verify): Exact planning / finite-law controls for conditional count confidence (Hoeffding-type family risk, toy enumerations), K in {1, 24, 32}", + "layer0": { + "source_id": null, + "heading": "experiments/periodic_h0/results/count_confidence1/CERTIFICATE.json scope field (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet.", + "next_step": "Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses." + }, + { + "claim_id": "cert:main-universality-fold_controls", + "kind": "exact_certificate", + "title": "main experiments/universality/fold_controls.py (+ test_fold_controls.py; universality-controls workflow): exact toy pushforward, density power (alpha+1)/p - 1 and cumulative exponent (alpha+1)/p for dI = C r^alpha dr, ell = k r^p; cubic gap alone gives powers -2/3, -1/3, 0, +1 under different radial laws", + "layer0": { + "source_id": null, + "heading": "experiments/universality/fold_controls.py (bytes not pinned in this packet)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "Exact Fraction toy control (L2 evidence), by its own words \"no random-field admissibility or Lean claim\": not a stationary random-field counterexample, not a formal kernel proof, not stated in Lean.", + "next_step": "The exact pushforward exponent identities are elementary and could be stated and proved over ℚ in a proof package; low priority." + }, + { + "claim_id": "cert:math-side24_v1-coefficient.py", + "kind": "exact_certificate", + "title": "Math- coefficients/side24_v1/coefficient.py (+ test_coefficient.py, ENCLOSURE.json): outward rational enclosure of the SIDE24 coefficient c_{d,24} of eq. (15.2), d = 2, 3, SCALE 10^80, EPS 10^-108, Stirling remainder (DLMF 5.11); object SIDE24-COEFFICIENT-D23-20260924-v1, scientific_acceptance false; byte-pinned into main formal/sources/side24_v1 at Math-@9d7b6802", + "layer0": { + "source_id": "side24-enclosure", + "heading": "ENCLOSURE.json (\"2\": lower/upper, \"3\": lower/upper, \"scientific_acceptance\": false, \"relative_periodization_bound\": \"1e-106\")" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [ + { + "package": "d6g8k5htny-coder/main formal/ (UniversalLaw, side24-arithmetic-skeleton)", + "receipt_or_workflow": ".github/workflows/formal-verification.yml (tools/formal_gate_check.py --run-lean inside the required aggregate `verify`); receipt formal/.lake/formal-evidence/receipt.json is a workflow artifact, none is committed in tree", + "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": "L2 exact-arithmetic evidence: a Python outward-rounding procedure is not a kernel proof, and main's 31 Side24 theorems formalize only its integer arithmetic skeleton (not the interval arithmetic, not the Gamma / pi / exp enclosures, not the theta sums). The enclosure statement itself is CoefficientEnclosure (specified, not proved) on row side24-coefficient; ENCLOSURE.json says scientific_acceptance false.", + "next_step": "A proof-producing enclosure pipeline (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md): proof-producing Gamma(7/6), pi, exp and theta-sum enclosures in Mathlib, then CoefficientEnclosure under ConeMomentInterface." + }, + { + "claim_id": "cert:math-full_price-full_price.py", + "kind": "exact_certificate", + "title": "Math- frontiers/full_price_20260924/full_price.py (+ ASCII_3E_MINUS_2.md): exact Fraction procedure behind the rho_star enclosure (F3) and the Section 6 rational certificate chain; also frontiers/price_budget_20260924/price_budget.py and three_fronts_20260924/frontier_math.py (finite controls; prize reconnaissance, HOLD)", + "layer0": { + "source_id": "p15-full-price", + "heading": "0.84547981724898672067 < rho_star < 0.84547981724898672068 < 6/7. (F3)" + }, + "disposition_transcribed": null, + "formal_progress": "none", + "statements": [], + "kernel_evidence": [], + "not_established": "L2 exact-arithmetic evidence (stdlib Fraction) for finite rational facts; not a kernel proof. The FullPrice.lean module staged in this programme's uncommitted main worktree (on no commit; not listed in this register) targets only the adjacency of the endpoints as Nat, 16/27 < 6/7 and the rational orderings / Taylor partial sum of Section 6, not that rho_star = 1/(3 - log(3e - 2)) lies between the endpoints (RhoStarEnclosure, specified only). The finite suite \"supports algebra only\" (the source's own words); no prize status (original_prize_closed stays false).", + "next_step": "Prove RhoStarBelowSixSevenths / SixSeventhsCertificate in the Math- Mathlib lane (Real.exp / Real.log bounds); then RhoStarEnclosure via proof-producing exp/log enclosures." + } + ] +} diff --git a/specifications/lean_statements_20261010/SCOPE.md b/specifications/lean_statements_20261010/SCOPE.md new file mode 100644 index 000000000..6edc61286 --- /dev/null +++ b/specifications/lean_statements_20261010/SCOPE.md @@ -0,0 +1,435 @@ +# Scope of the statement-only Lean package + +Scientific effect: **NONE**. Formal progress: `specified` for every declaration. Alignment: +`PENDING_INDEPENDENT_REVIEW`. Organizational-independence credit: 0 (Anthropic / Claude). + +## Boundary + +This package states; it does not prove. The five modules under `Specifications/` declare, as +`def … : Prop` and interface `structure`s in the namespaces `UniversalLaw.Spec.`, the +objects and results of the byte-pinned Layer 0 texts in `sources/`. Anything the informal text +leaves informal (a Kac–Rice representation, "versions of the densities", the elder pairing, the +exact weighted law, the field itself) is a *field* of an interface structure whose docstring names +the source clause, never an axiom, and the Prop is stated for every object satisfying the +interface. Every specification docstring carries `Source:` (repository, commit, path, sha256), +`Anchor:` (a verbatim quote of the pinned bytes), `Interface:` (which inputs are given objects +whose defining property is a hypothesis field) and `Does not claim:`. + +The source scan (`check.py`, no Lean) refuses, in every module, as whole tokens of +comment-stripped code: `theorem`, `lemma`, `example`, `instance`, `axiom`, `sorry`, `admit`, +`native_decide`, `unsafe`, `opaque`, `partial`, `macro`, `notation`, `syntax`, `elab`, +`set_option`, `run_cmd`, `implemented_by`, `extern`, `decreasing_by`, `deriving`, `attribute`, +`include`, `omit`, `mutual`, `inductive`, `class`, `local`, `scoped`, `export`, `alias`, +`irreducible_def`, `Sort`, `_root_` and the rest of its list; string literals, guillemet names, +`#` commands and attributes anywhere in code; any command keyword off column 0 (so `open … in`, +two commands on one line and indented commands); any column-0 command other than `import`, +`namespace`, `end`, `open`, `def`, `noncomputable def`, `abbrev`, `structure` (`section` and +`variable`, which DESIGN 3b lists, are refused until a module needs them: a `variable (h : False)` +mentioned in a body would add a vacuating hypothesis the scan cannot see); imports other than +`Mathlib`; a definition without an explicit result +type; a Prop-valued result type not written exactly as `: Prop`; a proof-shaped result type (a +depth-zero relation, `True` or `False`); a declaration name with a `_`-prefixed component; a +registered-kind declaration without the four labelled docstring lines; a docstring or register +anchor shorter than 16 characters (a one-character anchor is verbatim in any source); a root module importing +anything but the five registered modules in order; unfilled double-angle-bracket table placeholders in README.md or SCOPE.md. +The scan is a fail-closed pre-Lean stage and not a Lean lexer. + +The authority is the environment audit of `check.py --execute` (`.lake/evidence/Environment.lean`): +Lean's own `ConstantInfo` of every constant the five modules add is classified as `structure`, +`statement` (type syntactically `Prop` or `∀ …, Prop`, read through `forallTelescope` without +reduction: a `Set`-valued definition, whose type unfolds to `… → Prop`, is `auxiliary` exactly as +the scan's `: Prop` rule has it, while the reducing `isProp` still decides "proof"), `auxiliary` +(any other definition) or `generated` +(projections, recursors, `casesOn`, `noConfusion`, `injEq`/`inj`/`sizeOf_spec`, `_sizeOf_*`, +`match_*`, `_proof_*`: tolerated only when range-less, flagged by the environment or an internal +name under its structure, and still axiom-checked); every other theorem, axiom, opaque or +quotient constant, non-structure inductive, instance, or definition whose type is a proposition +is refused, as is any transitive axiom outside `propext`, `Classical.choice`, `Quot.sound` for +any constant. The audit's `statement`, `structure` and `auxiliary` sets must equal the register +and the scan exactly. The build log is refused on `declaration uses` and on Lean's "is a +proposition; use `theorem`" linter. The lakefile pins `autoImplicit = false`, so an unbound name +in a header is an error rather than a silent implicit parameter. The audit is itself controlled +in every `--execute` run: an honest control file must classify exactly, and a control file hiding +a theorem between string literals, an indented Prop definition, a proof written as `def`, a +`sorry`, an `axiom` and an `instance` must be refused for exactly those reasons. + +## Declarations + +150 registered declarations (141 `def … : Prop`, 9 interface `structure`), every one with formal progress `specified`; 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. + +### `Specifications/Field.lean` (12 registered declarations) + +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | +|---|---|---|---|---| +| `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. | +| `GaussianFieldInterface` | structure | `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 | existence or uniqueness in law of the field; the Fourier representation and positive spectrum of Section 2; almost-sure Morse genericity of Section 8. | +| `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. | +| `IsNondegMax` | def_prop | `lifetime-parent` | A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue. | that every local maximum of a sample path is nondegenerate (Morse genericity is Section 8 of the source). | +| `IsIndexSaddle` | def_prop | `lifetime-parent` | A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue. | that such a saddle is a merging saddle of any superlevel bar. | +| `IsCandidatePair` | def_prop | `lifetime-parent` | let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume. | positivity of the height gap (added separately where the source requires it). | +| `IsMorseDistinct` | def_prop | `lifetime-parent` | Countable exhaustion now proves that Q is almost surely Morse with distinct critical values. | that sample paths are almost surely Morse (Section 8 of the source). | +| `SuperlevelBarInterface` | structure | `lifetime-parent` | On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint. | Borel measurability of the elder event (Section 8 / the Section 9 replacement), identification with a persistence-module elder rule, or any behaviour off the Morse distinct-value locus. | +| `IsFiniteBar` | def_prop | `lifetime-parent` | each finite superlevel H0 bar has exactly one local-maximum birth and one index-(d-1) merging-saddle death. | the identification of these pairs with bars of a persistence module; that the count of such pairs equals the finite-bar count off the Morse locus. | +| `IsLifetimeDensityVersion` | def_prop | `lifetime-parent` | These densities admit versions with | existence of a version; the weighted Kac-Rice identity; the radial ledger (10.2). Measurability of the pair count as a function of the sample is REQUIRED by the predicate (through `Integrable`, hence `AEStronglyMeasurable`), not established here. | + +### `Specifications/Lifetime.lean` (24 registered declarations) + +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | +|---|---|---|---|---| +| `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. | +| `GaussianFieldInterface` | structure | `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 | existence or uniqueness in law of the field; the Fourier representation and positive spectrum of Section 2; almost-sure Morse genericity of Section 8. | +| `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. | +| `IsNondegMax` | def_prop | `lifetime-parent` | A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue. | that every local maximum of a sample path is nondegenerate (Morse genericity is Section 8 of the source). | +| `IsIndexSaddle` | def_prop | `lifetime-parent` | A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue. | that such a saddle is a merging saddle of any superlevel bar. | +| `IsCandidatePair` | def_prop | `lifetime-parent` | let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume. | positivity of the height gap (added separately where the source requires it). | +| `IsMorseDistinct` | def_prop | `lifetime-parent` | Countable exhaustion now proves that Q is almost surely Morse with distinct critical values. | that sample paths are almost surely Morse (Section 8 of the source). | +| `SuperlevelBarInterface` | structure | `lifetime-parent` | On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint. | Borel measurability of the elder event (Section 8 / the Section 9 replacement), identification with a persistence-module elder rule, or any behaviour off the Morse distinct-value locus. | +| `IsFiniteBar` | def_prop | `lifetime-parent` | each finite superlevel H0 bar has exactly one local-maximum birth and one index-(d-1) merging-saddle death. | the identification of these pairs with bars of a persistence module; that the count of such pairs equals the finite-bar count off the Morse locus. | +| `IsLifetimeDensityVersion` | def_prop | `lifetime-parent` | These densities admit versions with | existence of a version; the weighted Kac-Rice identity; the radial ledger (10.2). Measurability of the pair count as a function of the sample is REQUIRED by the predicate (through `Integrable`, hence `AEStronglyMeasurable`), not established here. | +| `IsWindowPair` | def_prop | `lifetime-parent` | Count ordered maximum/saddle pairs with 00, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume. | anything about the off-diagonal (far) population separately (Section 14 of the source). | +| `PairLawInterface` | structure | `lifetime-parent` | Let p_r(b,k,R) be the Q^W probability that the **global ordinary superlevel elder death partner** of M is S. | existence of the regression law, the typed weight, the full normalizer `Z_r > 0`, or that `p` is the probability of any event under any measure declared here; that `p` depends on `G` or `I` in any way (no field mentions them; they only record which field and elder relation the law is meant to refer to). | +| `ParentUniformSelectionA` | def_prop | `lifetime-parent` | There exist r_*>0 and C= c r^3, 0 6` (the source restricts to `q ≤ 6`); nothing here encloses the coefficient. | +| `DerivativeBoundAtLeastOne` | def_prop | `side24-proof` | At \|x\|>=1 this is at most76 \|x\|^6 e^(-\|x\|^2/2). | the bound (statement only); the coefficient arithmetic `1+15+45+15=76` and `76` bounding the smaller orders are kernel-checked on main (`UniversalLaw.Side24.pairing_sum_six`, `pairing_sum_le_six`), not here; nothing here encloses the coefficient. | +| `DerivativeBoundAtZero` | def_prop | `side24-proof` | contraction has absolute value at most15. | the bound (statement only); `5!! = 15` is kernel-checked on main (`UniversalLaw.Side24.sixth_moment_double_factorial`), not here. | +| `LatticeImageSumBound` | def_prop | `side24-proof` | <=1458 e^(-288). | the bound (statement only); the intermediate `729 sum_j j^9 e^(-288 j^2)` shell count is not specified; `27*27 = 729`, `729*2 = 1458` are kernel-checked on main (`UniversalLaw.Side24.lattice_shell_constant`), not here. | +| `SuccessiveTermRatio` | def_prop | `side24-proof` | successive terms have ratio at most512e^(-864)<1/2. | the bounds (statement only); `2^9 = 512`, `288*3 = 864` are kernel-checked on main (`UniversalLaw.Side24.geometric_ratio_constants`), not here. | +| `ExponentialBounds` | def_prop | `side24-proof` | hence e^(-288)<10^(-125) and the geometric-ratio inequality follows too. | the inequalities (statement only); the rational Taylor sum through degree 20 exceeding `10` is kernel-checked on main (the `UniversalLaw.Side24` Taylor-sum target in `Ledger.lean`), not here. | +| `ImageBound` | def_prop | `side24-proof` | \|D^q K_24(0)-D^q K_infty(0)\| | the bound (statement only); differentiability of `K_24` (if `K_24` is not `C^q` the derivative is `0` in Lean); nothing here encloses the coefficient; nonauthor review or acceptance. | +| `CovarianceOrdering` | def_prop | `side24-proof` | The inequalities are in the positive-semidefinite ordering and hold in every | the ordering (statement only); the passage to marginal and conditional covariances (not specified); `60E < epsilon` is kernel-checked on main (`UniversalLaw.Side24.covariance_relative_bound`), not here; nothing here encloses the coefficient; nonauthor review or acceptance. | +| `EntrywiseCovarianceBound` | def_prop | `side24-proof` | Each covariance entry differs from the reference by at most2E, by (2). | the bound (statement only); the spectral-norm bound `20E` (not specified); nonauthor review or acceptance. | +| `ReferenceCovarianceLowerBound` | def_prop | `side24-proof` | Thus C_ref >= I/3 and | the bound (statement only); the odd-block eigenvalues `8 ± sqrt 58` and the minors `2/3`, `7/9` are kernel-checked on main (`UniversalLaw.Side24.odd_block_eigenvalues`, `odd_block_shifted_minors`, `smaller_eigenvalue_exceeds_third`), not here. | +| `ReferenceOddAndPinBlocks` | def_prop | `side24-proof` | Var(t)=15, Cov(t,G)=(-3,0,...,0), | the identities (statement only); `15 - 9 = 6` is kernel-checked on main (`UniversalLaw.Side24.conditional_third_derivative_variance`), not here. | +| `DensityComparison` | def_prop | `side24-proof` | [(1-epsilon)/(1+epsilon)]^(n/2) phi_((1-epsilon)C0) | the comparison (statement only); the cone-moment consequence `(1 ± epsilon)^m D0` and the scaling `a^m` of `h(A) = det(A)^2 1{A<0}` (not specified); nothing here encloses the coefficient; nonauthor review or acceptance. | +| `IntegrandRatioBounds` | def_prop | `side24-proof` | ratio between1-13epsilon and | the bounds (statement only); that the integrand ratio of the source equals these power quotients (not specified); `13 ≤ 32`, `28 ≤ 32` and `32*10^(-108) < 10^(-106)` are kernel-checked on main (`UniversalLaw.Side24.ratio_constant_ordering`, `density_comparison_below_reported`), not here. | +| `ReferencePinDensityProduct` | def_prop | `side24-proof` | p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3). | the identity (statement only); `det diag(3,1,…,1) = 3` is kernel-checked on main (`UniversalLaw.Side24.pin_determinant_and_joint_dimension`), not here. | +| `ReferenceCoefficientIdentity` | def_prop | `side24-proof` | This eliminates BOTH the angular and cone integrals for the reference expression. | the identity (statement only); that the left side is the parent formula (15.2) evaluated at the reference law (the angular integrand is constant by rotational invariance of `K_infty`, a fact not specified here); `6^2*2 = 3*24` is kernel-checked on main (`UniversalLaw.Side24.cube_root_simplification`), not here; nonauthor review or acceptance. | +| `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. | +| `ConeMomentD2Identity` | def_prop | `side24-proof` | D_2 = (1/2)[25/3-20/3+8-8sqrt(3/8)] | the identity (statement only); the Rayleigh integration route; that `D_2` is the parent's `D_u` for the reference law; nothing here encloses the coefficient; nonauthor review or acceptance. | +| `ConeMomentD2Algebra` | def_prop | `side24-proof` | = 29/6-sqrt(6). | the identity (statement only); its rational part and `4 sqrt(3/8) = sqrt 6` are kernel-checked on main (`UniversalLaw.Side24.cone_moment_m2_algebra`), not here. | +| `TraceMoments` | def_prop | `side24-proof` | E s^2=5/3, E s^4=25/3 and | the identities (statement only); `3 (5/3)^2 = 25/3` and `1 + 5/3 = 8/3` are kernel-checked on main (`UniversalLaw.Side24.fourth_moment_of_trace`, `exponential_moment_of_trace`), not here. | +| `RayleighSquareLaw` | def_prop | `side24-proof` | The variable R^2 has density e^(-z/2)/2. | the law (statement only); measurability of the event (if the event is not measurable, `P` is its outer measure). | +| `ElementaryConeIntegral` | def_prop | `side24-proof` | integral_0^a (a-z)^2 e^(-z/2) dz/2 = a^2-4a+8-8e^(-a/2). | the identity (statement only). | +| `PeriodicCoefficientInterface` | structure | `side24-proof` | Let c_d,24 be precisely its equation (15.2), using ordinary sphere area, ordered | existence of an object satisfying this interface; that `c24` is the finite-bar asymptotic constant (that reading is conditional on the parent's results, which are not touched); that the parent's "ordered maximum/saddle roles and the full height/gradient pin Jacobian" are encoded beyond the prefactor of (15.2). | +| `CoefficientEnclosure` | def_prop | `side24-proof` | 0.07340691930603427103 < c_2,24 < 0.07340691930603427104, | the enclosures (statement only: nothing here encloses the coefficient); that an object satisfying the interface exists or is the coefficient of the source; the parent's results or the persistence interpretation; any finite-radius error estimate, unrestricted remainder or RN/24-jet statement; the arithmetic skeleton (endpoint adjacency, `d3 < d2`, endpoints in `(0, 1/10)`) is kernel-checked on main (31 `UniversalLaw.Side24` targets at origin/main f6deeba7), not here; nonauthor review or acceptance. | +| `ReferenceComparison` | def_prop | `side24-proof` | \|c_d,24/c_d,ref -1\| < 10^(-106), d=2,3. | the bound (statement only); equality of the periodic and reference constants (the source says it is NOT claimed); `32*10^(-108) < 10^(-106)` is kernel-checked on main (`UniversalLaw.Side24.density_comparison_below_reported`), not here; nothing here encloses the coefficient. | + +### `Specifications/RN.lean` (28 registered declarations) + +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | +|---|---|---|---|---| +| `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. | +| `HolderCountBoundN1` | def_prop | `rn-count-interface` | E[N_r 1_Er] <= (E N_r^p)^(1/p) P(E_r)^(1-1/p). | that any count of the Gaussian model satisfies the moment hypothesis; the result or its proof; any rate in `r`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). | +| `HolderConsequenceN2` | def_prop | `rn-count-interface` | E[N_r 1_Er] <= A^(1/p) C^(1-1/p) | that the exponent can be rounded up to `3`; that any Gaussian count satisfies the hypotheses; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). | +| `ConditionalMeanIdentityN3` | def_prop | `rn-count-interface` | E[N_r 1_Er]=q E[N_r \| E_r]. | the equivalence with a bounded conditional mean, the lower comparison `q ≥ c r^3`, or anything about the Gaussian model; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). | +| `UniformCounterexampleN4` | def_prop | `rn-count-interface` | E N_r/r^3=ceil(log(1/r)) -> infinity. | the explicit bound `sup_(t>=1) e^(-3t)(t+1)^p` (only existence of a `p`-dependent bound); the one-`p` sharpness construction `N_r=ceil(r^(-3/p))1_Er`; the exponential-tail remark after (N5); that the construction says anything about the Gaussian field; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). | +| `ExponentialTailLayerCakeN5` | def_prop | `rn-count-interface` | = B q[1+log(A/q)]. | the intermediate integral identity `integral_0^infinity min(q,Ae^(-t/B))dt`; that the logarithmic loss is attained (the source's remark about (N4)); anything about the Gaussian model; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). | +| `PinnedLawInterface` | structure | `rn-fixed-remote-window` | Q_r denotes continuous Gaussian regression on these 2(d+1) observations. | that `Q_r` is the Gaussian regression of any field; that `W_r` equals `F_d(H_M) F_(d-1)(H_S)`; the normalizer floor `Z_r >= z_* r^2`; that `Λ` is given by (13); existence or uniqueness of an object satisfying the interface (an inhabitant is not asserted; the consistency remark above is not stated); that `0 < r < L` is the sources' unspecified `small r` threshold. | +| `FixedRemoteMeanMeasureA2` | def_prop | `rn-fixed-remote-window` | \| E_(Q_r^W) N_(r,j)(E)/(k r^3) | the result or its proof (uniform remote nondegeneracy, coupling (5), weight expansion (10), normalizer (11)); identification of `I.Λ` with the kernel (13); the total-variation convergence remark; the multi-witness Section 6; any numerical `C` or `r_*`; uniformity as `ρ → 0`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN is transcribed by the register, not here). | +| `FixedRemoteCountBoundA3` | def_prop | `rn-fixed-remote-window` | E_(Q_r^W) N_(r,j)(E) <= C k r^3 \|E\|, | a matching lower bound on the event probability (the source says this is NOT one); the unnormalized numerator bound (14); the result or its proof; any numerical constant; nonauthor review or acceptance. | +| `FixedAnnulusWindowCandidateA2` | def_prop | `rn-fixed-annulus-window` | E_Qr^W N_j(r E0) <= C*k*r^3*area(E0), | the result or its proof (contact vector (A10), eigenfloor (A12), cutoff (A13), three-Hessian estimate (A18), inner-strip bound (A26)); the all-index sum or the Markov event bound; an all-height assertion; uniformity in `L`, as `k → 0`, as `B0 → ∞`, or for `d ≥ 3`; any numerical constant; nonauthor review or acceptance (the landing disposition REVIEWED_SCOPED is transcribed by the register, not here). | +| `AnnulusBridgeAllHeight2` | def_prop | `rn-annulus-bridge` | E_(Q_r^W) N_j(rE) <= C r^3 area(E), 0=0} dTV(Law N, Pois(lambda)) <= A r^3. | the result (Theorem O) or its proof; that `2a ≤ A` (derived in the source, not a hypothesis here); that H1 or H2 hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment matching or the independent-replica limit; the conditional-law liminf remark; that the upper half is the source's infimum bound verbatim: `∃ lam ≥ 0, dTV(Law N_r, Pois(lam)) ≤ A r^3` implies `inf_{lam ≥ 0} dTV ≤ A r^3` and is stronger unless the infimum is attained, and the existential form is used because the source's proof exhibits the witness `lam = 0` (the source: The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p); nonauthor review or acceptance. | +| `RemoteCollisionSecondFactorialD` | def_prop | `remote-collision` | E_(Q_r^W) N(E)(N(E)-1) <= C r^5 \|E\|, | Theorem C (the ordered near-pair count) or its proof (Lemmas 1-5); Corollaries E and F; a torus-wide second factorial moment or `math.rn-region.witness-collision` in its full sense; pairs with a witness near the pins; equality of `N(E)` with the all-critical-point count `windowCountAll` (they agree only off the degenerate locus); any numerical constant; nonauthor review or acceptance. | +| `WitnessCollisionNearPinPairsTarget` | def_prop | `d5-reconciliation` | A torus-wide second factorial moment `E_{Q_r^W} N(N−1)` for the window count, with pairs where at least | that this target holds, is attainable, or is implied by any reviewed source (the Palm route's Theorem Q is an author-side candidate whose `q = 2` case would imply it; that candidate is stated separately as `PalmRouteFactorialMomentsQ` and is not consumed here); that the lower obstruction `E N(N−1) ≥ 2c r³` is stated here; the optimal order; any status transition of the node. | + +### `Specifications/P15.lean` (58 registered declarations) + +| declaration | kind | source | exact coverage (verbatim anchor) | Not established (verbatim `Does not claim:`) | +|---|---|---|---|---| +| `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. | +| `IsGood` | def_prop | `p15-realized-covers` | D={U subset X: \|U intersect X_i\|<=a_i for every i, | that `D` is decreasing or contains the empty set (facts of the sources, not stated separately). | +| `IsKDecomposable` | def_prop | `p15-realized-covers` | write I_k(D) for sets partitionable into at most k members of D | equivalence with coverings by at most `k` good sets (true for a decreasing family, not stated). | +| `IsGeneratorCover` | def_prop | `p15-realized-covers` | A generator cover G of O_k(D) means every obstructed set contains some g in G. | anything. | +| `IsPaletteFeasible` | def_prop | `p15-realized-covers` | intersection_(i in e) P_i = empty, for every e in H. (P6) | upward closure in `K` (true, not stated). | +| `PaletteNumberAttained` | def_prop | `p15-realized-covers` | By discarding excess labels within blocks, one may take \|P_i\|=d_i. | the statement (not proved here; disjoint palettes of sizes `d_i` are a witness in the sources' argument). | +| `IsMinimalForbidden` | def_prop | `p15-realized-covers` | The inclusion-minimal forbidden ORIGINAL sets are exactly | anything. | +| `IsTransversal` | def_prop | `p15-realized-covers` | (b) for each e in H, EVERY transversal {x_i:i in e}, with x_i in X_i. | anything. | +| `MinimalForbiddenCharacterisation` | def_prop | `p15-realized-covers` | (a) every (a_i+1)-subset of a single X_i; | the characterisation (statement only); its completeness clause is what the source calls "completeness, not just existence of some witnesses". | +| `LocalRestriction` | def_prop | `p15-realized-covers` | D_i=D intersect 2^(X_i)={S subset X_i: \|S\|<=a_i}. (P2) | the identity (statement only). | +| `LocalDecomposability` | def_prop | `p15-realized-covers` | I_k(D_i)={S:\|S\|<=a_i k}. | the identity (statement only). | +| `LocalObstructionIsFullBlock` | def_prop | `p15-realized-covers` | G_i={X_i} (P3) | the statement (statement only). | +| `LocalPriceChain` | def_prop | `p15-realized-covers` | =P(U_i=X_i)<=q_i<=phi(q_i), (P4) | the chain (statement only); the identity `prod p_v = P(U_i = X_i)` is the definition of the weight. | +| `IndependentBlockHazard` | def_prop | `p15-realized-covers` | <=-log(product_i(1-q_i))<=-log mu_p(D). (P5) | the inequalities (statement only); independence of crossing events ("Crossing events are not declared independent"). | +| `FullBlockCoverIff` | def_prop | `p15-realized-covers` | K>=K_H(d). (P7) | the criterion (statement only); optimality of any other cover ("It does not prove that no different cover with the same budget works at fewer colors"); unrestricted prize optimality; disposition transcribed from LANDING_CLAIMS for p15-realized-covers: HOLD_WITH_DOMAIN; nonauthor review and acceptance are not claimed here. | +| `CoverCostHazardBound` | def_prop | `p15-realized-covers` | covercost_c(O_K(D)) <= min(1,-log mu_p(D)). (P8) | the bound (statement only); anything for `c_v > p_v`; the unrestricted P15 prize; original P15-B; disposition transcribed from LANDING_CLAIMS for p15-realized-covers: HOLD_WITH_DOMAIN; nonauthor review and acceptance are not claimed here. | +| `WholeGroundChromaticNumber` | def_prop | `p15-realized-covers` | the chromatic number of the ENTIRE ground set X is K_H(d+1), because ceil(n_i/a_i)=d_i+1. | the statement (statement only). | +| `ElementaryPaletteFormula` | def_prop | `p15-realized-covers` | K_H(d)=max(max_i d_i, ceil(sum_i d_i/s)). (P9) | the formula (statement only); the matroid palette formula of Section 5 (not specified). | +| `TriangleNeedsThree` | def_prop | `p15-realized-covers` | at unit demands needs3 colors, although a formula considering only size-two edge loads would give2. | the value (statement only; finite and decidable in principle). | +| `Benchmark816IsRealized` | def_prop | `p15-realized-covers` | Every D-color class contains at most one vertex per block and occupies at most three blocks. | the statement (finite, decidable in principle; not proved here). | +| `Benchmark816Palette` | def_prop | `p15-realized-covers` | K_H(408,...,408)=816. | the value (statement only); the arithmetic `max(408, ceil(2448/3)) = 816` is a core-Lean target on main, not here. | +| `Benchmark816Cover` | def_prop | `p15-realized-covers` | At816 colors it covers the obstruction; at815 it does not, because omitting one vertex from each block leaves2448 vertices, each color holds at most3, and815*3<2448. | the statement (finite over `2^2454` subsets, not decidable in practice); `815*3 < 2448` is a core-Lean target on main. | +| `Benchmark816WholeGround` | def_prop | `p15-realized-covers` | The whole ground set, unlike the cover-avoiding sets, needs818 colors, since2454/3=818. | the statement (statement only); `2454/3 = 818` is a core-Lean target on main. | +| `Benchmark816MinimalForbiddenCounts` | def_prop | `p15-realized-covers` | 15*409^4=419743994415 crossing transversal quadruples. (P10) | the counts (statement only); the evaluations `500616` and `419743994415` are core-Lean targets on main; the source stresses these are "not a claim that the program enumerated more than four hundred billion sets". | +| `Benchmark816Price` | def_prop | `p15-realized-covers` | 6/(40900^409)>0. | the identity (statement only; the six blocks are distinct members of the image). | +| `Benchmark816UnionBound` | def_prop | `p15-realized-covers` | =2449227/8180000000 < 3/10000. (P11) | the bound (statement only; "only a union bound, not an equality of the global bad probability"); the two rational identities are core-Lean targets on main. | +| `TwoCoordGeneratorPrices` | def_prop | `p15-price-boundary` | The possible generator costs are1,2/3,2/3,4/9 for the empty | the values (statement only; as integers over `9` they are core-Lean targets on main). | +| `DemandOneRefutation` | def_prop | `p15-price-boundary` | covercost_c(O_1(D))=4/9 > 1/3 > log(4/3). | the statement (statement only; the finite parts are decidable in principle, the two logarithm inequalities are analysis); that this contradicts original P15-B ("It does not contradict the original P15-B amalgamation"); anything about the `408`-scale or the unrestricted prize; the integral representation `log 2 = 2 integral_0^(1/3) 1/(1-t^2) dt` is not specified; disposition transcribed from LANDING_CLAIMS for p15-price-boundary: EXACT_COUNTEREXAMPLE; nonauthor review and acceptance are not claimed here. | +| `LogOnePlusBelowIdentity` | def_prop | `p15-price-boundary` | using log(1+x)0. | the inequality (statement only; the source proves it, this module does not). | +| `LogRatioBound` | def_prop | `p15-price-budget` | -log(1-p)=integral_0^p dt/(1-t) <= p/(1-p). (T2) | the inequality (statement only); the integral representation is not specified. | +| `RatioExponentBound` | def_prop | `p15-price-budget` | <= (4/3)*(4/9)^a <=16/27. (T3) | the inequalities (statement only); the preceding ratio steps of (T3) (price over `prod_{v in S} p_v`) are stated through their consequence `RestrictedLocalBudget`; the case `a = 1, n = 3` giving `16/27` is a core-Lean target on main. | +| `RestrictedLocalBudget` | def_prop | `p15-price-budget` | price({X}) <= (16/27)q <= phi(q). (T4) | the bound (statement only); the source notes "The extra demand hypothesis d>=2 is load-bearing" and that `p_v <= 1/4` "is a sufficient range, not an asserted sharp threshold". | +| `RestrictedTransformedPriceBudget` | def_prop | `p15-price-budget` | covercost_c(O_K(D)) <= min(1,(16/27)*[-log mu_p(D)]). (T1) | the bound (statement only); any retraction of the demand-one refutation; the unrestricted prize; new classical coloring theory; disposition transcribed from LANDING_CLAIMS for p15-price-budget-restricted: HOLD_WITH_DOMAIN; nonauthor review ("nonauthor review open") and acceptance are not claimed here. | +| `RhoTestSufficient` | def_prop | `p15-price-budget` | If rho<=1 the same argument proves price({X})<=q for that block. | the statement (statement only); anything when `rho > 1` (the source: a block with `rho > 1` "must not be labeled infeasible"). | +| `CapTestAtTwoThirds` | def_prop | `p15-price-budget` | or if q>=2/3 (then phi(q)=1, since log3>1) | the statement (statement only); the integral proof of `log 3 > 1` is not specified; the checker program of that section is not specified. | +| `Benchmark816TransformedRange` | def_prop | `p15-price-budget` | 0<=c_v<=log(40900/40899), | the statement (statement only); "No large witness enumeration" is involved, nor any claim about earlier P15 inputs. | +| `Benchmark816RationalPrice` | def_prop | `p15-price-budget` | 6*log(40900/40899)^409 <=6/(40899^409). (T7) | the statements (statement only); the rational identity `1/40900 + (1/40900)^2/2 = 81801/3345620000` is exact arithmetic not re-proved here; admissibility rests on `integral_0^p 1/(1-t) dt > integral_0^p (1+t) dt`, which is not specified. | +| `FullTransformedPriceBudget` | def_prop | `p15-full-price` | covercost_c(O_K(D)) <= min(1,rho_star[-log mu_p(D)]). (F2) | the bound (statement only); optimality of `rho_star` (that is `SharpnessOfRhoStar`); any unrestricted downset or prize statement (the source: "palette size remains K_H(d), which can depend on H and d"); the nonauthor review recorded in LANDING_CLAIMS (REVIEWED_SCOPED) is not reproduced or extended here. | +| `RhoStarEnclosure` | def_prop | `p15-full-price` | 0.84547981724898672067 < rho_star | the enclosure (statement only; the source's `full_price.py` Fraction procedure is not a kernel proof and is not specified); this is the next full-result target named in main's `docs/FORMAL_VERIFICATION_ROLLOUT_20260927.md`, still at `specified`. | +| `RhoStarBelowSixSevenths` | def_prop | `p15-full-price` | Thus 6/7 is a simpler valid rational factor. | the inequality (statement only); `16/27 < 6/7` is a core-Lean target on main. | +| `IsProperDownset` | def_prop | `p15-full-price` | Let A be any proper decreasing family on n>=1 coordinates containing the empty set. | anything. | +| `HazardStarPositive` | def_prop | `p15-full-price` | The probability is positive on this compact cube because the empty set is admitted and every p_i(t)<1. | the statement (statement only). | +| `CoordinatewiseConcavity` | def_prop | `p15-full-price` | Only SEPARATE coordinatewise concavity is asserted; joint concavity is not needed or claimed. | the statement (statement only); joint concavity; the derivative formulas (F4)-(F5) themselves (not specified; concavity is their consequence). | +| `HazardProductBound` | def_prop | `p15-full-price` | F_A(t) >= (product_i t_i) H_A. (F6) | the inequality (statement only). | +| `HazardPhiBound` | def_prop | `p15-full-price` | -log mu_p(A) >= H_A product_i phi(p_i). (F7) | the inequality (statement only). | +| `FullGroundPriceBound` | def_prop | `p15-full-price` | product_i c_i <= min(1, [-log mu_p(A)]/H_A). (F8) | the inequality (statement only); that the full-ground generator covers any obstruction ("It does not assert that this generator alone covers an arbitrary obstruction"). | +| `PhiAtPStar` | def_prop | `p15-full-price` | at every p_i=p_star and every c_i=1, the two sides of (F7) are equal | the identity (statement only). | +| `FullGroundBudgetIff` | def_prop | `p15-full-price` | a full-ground generator universally meets the capped hazard budget precisely when H_A>=1 | the equivalence (statement only); coverage of arbitrary `A` ("it does not infer coverage of arbitrary A"). | +| `CapacityHazardIsBinomial` | def_prop | `p15-full-price` | H_(n,a)=-log P(Bin(n,p_star)<=a). (F9) | the identity (statement only; a finite identity for each `n`, general in `n`). | +| `OddMajorityIdentity` | def_prop | `p15-full-price` | =p(1-2p)P(S=a)<0. (F10) | the identities (statement only; polynomial identities in `p`, a future kernel target for each fixed `a` and by induction in general). | +| `CapacityWorstCase` | def_prop | `p15-full-price` | =3-log(3e-2)=h_star>1. (F11) | the statements (statement only); the source stresses "it is not a finite-grid extrapolation". | +| `HStarExceedsOne` | def_prop | `p15-full-price` | e^2-(3e-2)=(e-1)(e-2)>0, | the statements (statement only). | +| `LocalPriceBoundFull` | def_prop | `p15-full-price` | price({X_i}) <= rho_star [-log mu_p(D_i)], (F12) | the bound (statement only); `price <= q_i` ("The inequality price<=q_i is NOT required and is in general stronger than what is proved here"). | +| `GlobalAssemblyHazard` | def_prop | `p15-full-price` | sum_i [-log mu_p(D_i)] <= -log mu_p(D). (F13) | the inequalities (statement only; "without independence of crossing failures"). | +| `SharpnessOfRhoStar` | def_prop | `p15-full-price` | Replacing rho_star by any smaller constant in (F2) would make its right side strictly less than one. | the statements (statement only; the finite parts are decidable in principle); a necessity claim for multi-block families with some demand one (the source: "other constraints may change its budget"). | +| `DemandOneBoundary` | def_prop | `p15-full-price` | mu_p(D)=1-p_star^(a+1)>1-p_star=exp(-1), | the statements (statement only); any retraction of original P15-B ("whose local-budget premise is missing in that case"). | +| `SixSeventhsCertificate` | def_prop | `p15-full-price` | Therefore 3e-2<197/327/6 and rho_star<6/7. | the statements (statement only); the two exact rational identities are exact arithmetic not re-proved here; the series and geometric-tail bounds behind `e < 31967/11760` are not specified. | + +## 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`); 31 statement entries are kernel-proved arithmetic theorems of main's `formal/` package (label `proved`: the 31 Side24 targets at origin/main `f6deeba7`; main's 28 P15 arithmetic targets are NOT listed because they exist only in this programme's uncommitted main worktree, on no commit), listed as the arithmetic skeleton of the claim they sit under and never as the claim. `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): `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` | 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. The persistence interpretation through the parent Theorem C (eq. 15.2) is conditional on the parent and untouched. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. | Formal lane (Math- Mathlib package, by the lane's admission rules): construct the conditional law A_u \| V_u = 0 and prove ConeMomentD1Identity / ConeMomentD2Identity for the reference law under ConeMomentInterface; then prove ReferenceCoefficientIdentity; the enclosure needs a proof-producing interval pipeline for Gamma(7/6), pi, exp and the theta sums (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md). Independent reviewer: alignment verdicts on the 28 Side24 statements. | +| `lifetime-remainder` | landing_claim | REVIEWED_SCOPED | `specified` | 4 specified: `RemainderBoundedR1`, `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `IsLifetimeDensityVersion` | RemainderBoundedR1 states the shape of (R1) for versions of the densities under the interface structures; the result is not proved. Not stated: (R19)-(R20), the observation transform and determinant lemmas R3.1/R3.2, the cap-loss bound, the parent interfaces D1-A-E it imports, the marked-cylinder-cap support. The coefficient c enters only through IsUnrestrictedLeadingCoefficient (c > 0 and the leading-term role), not through the representation (15.2). No numerical C or ell_*, no second coefficient, no convergence of the remainder, no compact-window O(ell^(2/3)) difference without cutoffs. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (R19) (cumulative form) as a further Prop; transcribe the parent interfaces consumed (section 8 comparison, section 9 as replaced by v1.1, section 10, (14.1), (15.2)) as explicit hypotheses; independent reviewer: alignment verdict on RemainderBoundedR1 against (R1), including the "versions" clause and the exclusion of the essential global maximum. | +| `rn-count-interface` | landing_claim | HOLD_WITH_DOMAIN | `none` | 5 specified: `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5` | No Prop states Theorem N as a whole: (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; (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". Nothing about the Gaussian model is stated or proved; no RN rate, no 24-jet enclosure, no ENV-RESCOV discharge. 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 (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`. (N1)-(N3), (N5) are candidates for proofs in the Math- Mathlib lane (Hoelder, layer cake). Independent reviewer: alignment of the five Props with (N1)-(N5). | +| `rn-fixed-remote-window` | landing_claim | HOLD_WITH_DOMAIN | `specified` | 5 specified: `PinnedLawInterface`, `IsCriticalPoint`, `HessianHasInertia`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | FixedRemoteMeanMeasureA2 and FixedRemoteCountBoundA3 state the shapes of displays (2) and (3) for any object satisfying PinnedLawInterface (the regression law Q_r, the weight W_r, the normalizer and the contact kernel Lambda_j are interface fields, gated on 0 < r < L); no inhabitant of the interface is constructed and the kernel is not identified with formula (13). Not stated: the observation transform U_r and target v_r, the coupling (5), the weight expansion (10)-(11), the Kac-Rice formula (12), the kernel formula (13), the unnormalized numerator bound (14), and the multi-witness Section 6 displays (15)-(16). Nothing is proved; no numerical C or r_*; no shrinking exclusion rho -> 0, no collisions, no global RN complement (the disposition reason). Disposition HOLD_WITH_DOMAIN is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Section 6 ((15)-(16), separated m-tuples) as Props; record an inhabitant construction of PinnedLawInterface as a proof-lane target (Dirac mass at one admissible smooth path); independent reviewer: alignment of A2/A3 with displays (2)-(3), in particular the uniformity in b, k, R, j, E and the fixed rho. | +| `rn-fixed-annulus-window` | landing_claim | REVIEWED_SCOPED | `specified` | 1 specified: `FixedAnnulusWindowCandidateA2` | FixedAnnulusWindowCandidateA2 states the shape of (A2) (d = 2, fixed L, fixed scaled annulus, compact positive gaps, between-pin height window) over PinnedLawInterface; the result is not proved. Not stated: the six-pin regression (A3)-(A5), normalizer (A8), contact vector (A10)-(A12), cutoff (A13)-(A14), density (A17), Hessian bounds (A18)-(A20), intensities (A21)-(A26), the TWO_SCALE_ADDENDUM S6-S21 support (not pinned here), the eight reviewed interfaces as separate statements. No all-height assertion, no positive leading coefficient, no probability lower bound, no uniformity as k -> 0, B0 -> infinity, in L or for d >= 3; no global D5/RN/JETMOD transfer. Disposition REVIEWED_SCOPED is transcribed, not re-derived. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin TWO_SCALE_ADDENDUM.md and state (S3)-(S5) and (S20)-(S21); independent reviewer: alignment of FixedAnnulusWindowCandidateA2 with (A2), checking that the height window is strict and the constants depend only on L, A0, B0 and the mark compacts. | +| `p15-realized-covers` | landing_claim | HOLD_WITH_DOMAIN | `specified` | 27 specified: `RealizedData`, `IsRealizedFamily`, `IsGood`, `IsKDecomposable`, `IsGeneratorCover`, `IsPaletteFeasible`, `PaletteNumberAttained`, `IsMinimalForbidden`, `IsTransversal`, `MinimalForbiddenCharacterisation`, `LocalRestriction`, `LocalDecomposability`, `LocalObstructionIsFullBlock`, `LocalPriceChain`, `IndependentBlockHazard`, `FullBlockCoverIff`, `CoverCostHazardBound`, `WholeGroundChromaticNumber`, `ElementaryPaletteFormula`, `TriangleNeedsThree`, `Benchmark816IsRealized`, `Benchmark816Palette`, `Benchmark816Cover`, `Benchmark816WholeGround`, `Benchmark816MinimalForbiddenCounts`, `Benchmark816Price`, `Benchmark816UnionBound` | MinimalForbiddenCharacterisation (Proposition P1), FullBlockCoverIff (Theorem P2, (P7)) and CoverCostHazardBound ((P8)) state the results for every realized family; none is proved here. No `proved` statement is listed: main's 12 P15 arithmetic theorems for this claim (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, and the elementary instances of (P9)) exist only in this programme's uncommitted main worktree, on no commit, and are not listed until a commit and a hosted formal-verification.yml run exist; even then they would not count the minimal forbidders, show the palette is 816 or evaluate 6/40900^409. The benchmark Props Benchmark816Cover/WholeGround/MinimalForbiddenCounts/Price/UnionBound quantify over 2^2454 subsets and are not decidable in practice; PaletteNumberAttained (nonemptiness of the feasible palette set) is stated, not proved. Nothing is asserted for prices p_v < c_v <= phi(p_v), 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; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Proof lane: TriangleNeedsThree, Benchmark816IsRealized, LocalRestriction and LocalDecomposability are finite or elementary and are the first kernel targets; PaletteNumberAttained via the disjoint-palette witness. Independent reviewer: alignment of MinimalForbiddenCharacterisation with Proposition P1 (completeness clause) and of FullBlockCoverIff with (P7). main: commit and push the P15 modules (hosted formal-verification.yml green), then list the P15 arithmetic theorems here as `proved` statements citing the 40-hex commit and the workflow run; until then no P15 theorem is listed. | +| `p15-price-boundary` | landing_claim | EXACT_COUNTEREXAMPLE | `specified` | 3 specified: `TwoCoordGeneratorPrices`, `DemandOneRefutation`, `LogOnePlusBelowIdentity` | 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)); it is not proved. No `proved` statement is listed: main's 6 P15 arithmetic theorems for this claim (the finite arithmetic: good sets, obstruction, 3/4 over denominator 4, prices 9,6,6,4 over 9, 3*4 > 9) exist only in this programme's uncommitted main worktree, on no commit, and are not listed until a commit and a hosted formal-verification.yml run exist; the two logarithm inequalities phi(1/2) = log 2 > 2/3 and log(4/3) < 1/3 are analysis and are not proved anywhere; log(1+x) < x (LogOnePlusBelowIdentity) is stated only. The counterexample does not refute d_i >= 2 successors or original P15-B (the disposition reason). Disposition EXACT_COUNTEREXAMPLE 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. | Math- Mathlib lane: prove the two logarithm inequalities (Real.log bounds) and the finite conjuncts of DemandOneRefutation; this is the smallest complete research statement in the packet that could move from `specified` to `proved`. Independent reviewer: alignment of DemandOneRefutation with the source paragraph. main: commit and push the P15 modules (hosted formal-verification.yml green), then list the P15 arithmetic theorems here as `proved` statements citing the 40-hex commit and the workflow run; until then no P15 theorem is listed. | +| `p15-price-budget-restricted` | landing_claim | HOLD_WITH_DOMAIN | `specified` | 8 specified: `LogRatioBound`, `RatioExponentBound`, `RestrictedLocalBudget`, `RestrictedTransformedPriceBudget`, `RhoTestSufficient`, `CapTestAtTwoThirds`, `Benchmark816TransformedRange`, `Benchmark816RationalPrice` | RestrictedTransformedPriceBudget states (T1) for every realized family with a_i >= 1, d_i >= 2, p_v <= 1/4, c_v <= phi(p_v); not proved. (T2), the exponent arithmetic of (T3), (T4), the per-block sufficient test (T5), the cap test and the Section 4 benchmark statements are stated only; the intermediate ratio steps of (T3), the integral representation behind (T2) and the admissibility integral are not specified. No `proved` statement is listed: main's 3 P15 arithmetic theorems for this claim (the a = 1, n = 3 instance 16/27, finite instances of the (T3) bound and the rational identity 81801/3345620000) exist only in this programme's uncommitted main worktree, on no commit, and are not listed until a commit and a hosted formal-verification.yml run exist; they are not (T1) in any case. The dependency on claim p15-realized-covers is not discharged here. 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; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Math- Mathlib lane: LogRatioBound (-log(1-p) <= p/(1-p)) and RatioExponentBound are provable from Mathlib Real.log facts and are the first targets; independent reviewer: alignment of RestrictedTransformedPriceBudget with (T1) including the per-coordinate price hypothesis and the K >= K_H(d) palette. main: commit and push the P15 modules (hosted formal-verification.yml green), then list the P15 arithmetic theorems here as `proved` statements citing the 40-hex commit and the workflow run; until then no P15 theorem is listed. | +| `p15-full-price` | landing_claim | REVIEWED_SCOPED | `specified` | 20 specified: `FullTransformedPriceBudget`, `RhoStarEnclosure`, `RhoStarBelowSixSevenths`, `IsProperDownset`, `HazardStarPositive`, `CoordinatewiseConcavity`, `HazardProductBound`, `HazardPhiBound`, `FullGroundPriceBound`, `PhiAtPStar`, `FullGroundBudgetIff`, `CapacityHazardIsBinomial`, `OddMajorityIdentity`, `CapacityWorstCase`, `HStarExceedsOne`, `LocalPriceBoundFull`, `GlobalAssemblyHazard`, `SharpnessOfRhoStar`, `DemandOneBoundary`, `SixSeventhsCertificate` | FullTransformedPriceBudget states (F2), RhoStarEnclosure states (F3), SharpnessOfRhoStar states the attainment clause; none is proved. The hazard interpolation lemma is stated as separate coordinatewise concavity (CoordinatewiseConcavity) and its consequences (F6)-(F8); the derivative formulas (F4)-(F5) are not specified. No `proved` statement is listed: main's 7 P15 arithmetic theorems for this claim (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) exist only in this programme's uncommitted main worktree, on no commit, and are not listed until a commit and a hosted formal-verification.yml run exist; they would not place rho_star = 1/(3 - log(3e - 2)) between the endpoints, prove e < 87/32 or 197/32 < exp(11/6), or be 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; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Math- Mathlib lane: RhoStarBelowSixSevenths / SixSeventhsCertificate (Real.exp and Real.log bounds e < 31967/11760 < 87/32, 197/32 < exp(11/6)) are the next full-theorem target named in main docs/FORMAL_VERIFICATION_ROLLOUT_20260927.md; then HStarExceedsOne and PhiAtPStar. Independent reviewer: alignment of FullTransformedPriceBudget with (F2), in particular the quantification over ALL p in [0,1]^X and the SAME palette K >= K_H(d). main: commit and push the P15 modules (hosted formal-verification.yml green), then list the P15 arithmetic theorems here as `proved` statements citing the 40-hex commit and the workflow run; until then no P15 theorem is listed. | +| `downstream-hard-gate` | landing_claim | ENGINEERING_HOLD | `none` | — | An engineering control, not a mathematical claim: there is no theorem to state in Lean and none is stated. The gate's terminal classifications and non-discharge tokens are transcribed in the graph_node rows of this register only as source identity. Disposition ENGINEERING_HOLD ("repository-wide external controlling-source binding remains unresolved") is transcribed, not re-derived. | No Lean target. The control's semantics live in Math- frontiers/downstream_gate_20260925 and its workflow; the register only records that every GRAPH math.* node has a row here. | +| `math.uniform-matrix-cap-lifetime` | graph_node | — | `specified` | 32 specified: `IsLPeriodic`, `IsSmooth`, `GaussianFieldInterface`, `IsCriticalPoint`, `HessianHasInertia`, `IsNondegMax`, `IsIndexSaddle`, `IsCandidatePair`, `IsMorseDistinct`, `SuperlevelBarInterface`, `IsFiniteBar`, `IsLifetimeDensityVersion`, `IsLPeriodic`, `IsSmooth`, `GaussianFieldInterface`, `IsCriticalPoint`, `HessianHasInertia`, `IsNondegMax`, `IsIndexSaddle`, `IsCandidatePair`, `IsMorseDistinct`, `SuperlevelBarInterface`, `IsFiniteBar`, `IsLifetimeDensityVersion`, `IsWindowPair`, `IsWindowElderPair`, `IsUnrestrictedPair`, `PairLawInterface`, `ParentUniformSelectionA`, `ParentCompactWindowDensityB`, `IsUnrestrictedLeadingCoefficient`, `ParentUnrestrictedLeadingDensityC` | ParentUniformSelectionA, ParentCompactWindowDensityB and ParentUnrestrictedLeadingDensityC state the shapes of (1.1), (1.2), (1.3) for every object satisfying the interfaces; nothing is proved. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. The reading rule that the GRAPH binds to this node (congruence erratum E1, Section 9 replacement v1.1, marked-cylinder cap sections 1-5, reconciliation record with W1 and the embedding radius r < L/(4 sqrt 2)) is NOT transcribed into any hypothesis; the Props are stated from the parent bytes alone. Not stated: the regression law Q_{r,b,k,R}, the typed weight W_r, the full normalizer Z_r and Q^W (p_r is a primitive interface field), the marked Kac-Rice representation (Sections 9-10, radial ledger (10.2)), the Gaussian/angular representation (15.2) of c_{d,L} and (11.3) of c_{B,K}, (12.1)-(12.3), (14.1), (5.5), Sections 2-8. No numerical r_*, C, z_*, c_{B,K}, c_{d,L}; no lower bound for d >= 3. The GRAPH classification PROVED_REVIEWED and the review disposition ACCEPT_AT_EXISTENTIAL_SCOPE are source identity, not re-derived; the five validity premises of the hardening-branch claim graph are not touched. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p or the field: the parent-shape Props are predicates on the interface objects, not closed statements; the universal closure of ParentUniformSelectionA over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed. | Pin ERRATUM_CONGRUENCE.md, REPAIR.md (Section 9 v1.1) and the reconciliation record, and add the reading rule as explicit hypotheses or separate Props; state (15.2) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of the three parent Props against (1.1)-(1.3) with the honesty checklist (quantifier order, existential constants outside the universal quantifiers, "versions", compact marks, all frames). | +| `math.d1-component.congruence-erratum` | graph_node | — | `none` | — | No Lean statement: the erratum's source bytes are not pinned in this packet and no Prop states the corrected congruence D_r H_i D_r or det(D_r H_i D_r) = det(H_i)/r. The parent Props of this register do not transcribe section 5 at all, so the amendment has nothing to attach to yet. | Pin ERRATUM_CONGRUENCE.md (blob 213594d6, 1782 B); state the corrected congruence as a matrix identity over Matrix (Fin d) (Fin d) R (a candidate kernel-proof target: block-matrix algebra). | +| `math.d1-component.section9-replacement-v1_1` | graph_node | — | `none` | — | No Lean statement: the repair's bytes are not pinned and no Prop states the weighted pair Kac-Rice identity (9.1) or its monotone-class extension to bounded Borel marks. In this register the marked Kac-Rice representation is deliberately NOT transcribed (densities are pinned only through IsLifetimeDensityVersion), so the Borel elder mark enters only as the ElderPartner field of SuperlevelBarInterface, whose measurability is disclaimed there. | Pin REPAIR.md (blob fe9b9ce4, 9062 B); state (9.1)-(9.4) as Props over MeasureTheory.Measure on the pair domain with the Armentano-Azais-Leon identity as an interface hypothesis. | +| `math.d1-component.marked-cylinder-cap` | graph_node | — | `none` | — | No Prop of this package states the deterministic cap theorem (unique transverse ridge, closed cap through S separating M from every older point, elder death partner of M is S under conditions (1)). The pinned bytes exist (sources/marked-cylinder-cap). Kernel evidence exists outside this package: Math- frontiers/cap_first_exit_lean_20261002 proves the topological global step and the analytic chain L1-L13, L16-L20 (torus_capHyp_blocks, torus_elder_death_level_blocks), but its 152 declarations are not registered here with anchors, its alignment review is NOT_CLAIMED, and it does not formalize that the field on G_r satisfies SP section 2, L14/L15, L23 or the persistence-pairing identification. The probabilistic corollary (2)-(3) and sections 6-8 are not consumed by D1 and not stated. | Register the CapFirstExit declarations here with their ALIGNMENT.md anchors against the pinned CAP bytes (statements of kind theorem, label proved, kernel_evidence cap-first-exit-lean.yml), or state the deterministic theorem as a def : Prop in Lifetime.lean so that the Math- lane's delta review has a target; independent (non-Claude) reviewer for the v1.5-v2.6 delta. | +| `math.d1-component.reconciliation-record` | graph_node | — | `none` | — | A review/reading-rule record, not a mathematical statement: nothing to state in Lean beyond what its components say. The embedding-radius condition r < L/(4 sqrt 2) and the W1 wording are not transcribed into any hypothesis of the parent Props. Its ACCEPT verdicts are not re-derived; the OpenAI author-provider check it records is non-discharge by its own terms. | Add the embedding radius as an explicit hypothesis to ParentUniformSelectionA / PairLawInterface once the reading rule is pinned; no proof target. | +| `math.lifetime-remainder` | graph_node | — | `specified` | 1 specified: `RemainderBoundedR1` | See row lifetime-remainder: (R1) is stated, not proved; the imports of parent interfaces D1-A-E are not transcribed; Theorem A's cubic bound (1.1) is not consumed by R and is not a hypothesis here. The GRAPH classification and review disposition are source identity only. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row lifetime-remainder. | +| `math.side24-coefficient` | graph_node | — | `specified` | 3 specified: `PeriodicCoefficientInterface`, `CoefficientEnclosure`, `ReferenceComparison` | See row side24-coefficient: the enclosure is stated for any object satisfying PeriodicCoefficientInterface and is not proved; the "evaluates_eq_15_2" edge to the parent is represented only by the interface field that takes (15.2) as a hypothesis. No finite-radius band. The 27 arithmetic theorems of main are listed on the landing row. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine c_{d,24}: CoefficientEnclosure and ReferenceComparison are predicates on the interface object, not closed statements; their universal closure over all inhabitants is false whenever the interface is inhabited (scaling the given cone moment by a positive constant scales c24 through eq_15_2 and gives another inhabitant), and `specified` refers to the statement shape at the source's object, which is not constructed. | As for row side24-coefficient. | +| `math.rn-count-interface` | graph_node | — | `none` | 5 specified: `HolderCountBoundN1`, `HolderConsequenceN2`, `ConditionalMeanIdentityN3`, `UniformCounterexampleN4`, `ExponentialTailLayerCakeN5` | See row rn-count-interface: (N1)-(N5) stated, (N6)-(N7) not stated, so no Prop states the interface as a whole; nothing proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-count-interface. | +| `math.rn-fixed-remote-window` | graph_node | — | `specified` | 3 specified: `PinnedLawInterface`, `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | See row rn-fixed-remote-window: displays (2)-(3) stated over PinnedLawInterface, not proved; Section 6 not stated; no inhabitant of the interface. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window. | +| `math.rn-region.fixed-remote` | graph_node | — | `specified` | 2 specified: `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | The region's mathematical content is the fixed-remote Theorem A at exactly this scope, stated by A2/A3 and not proved. The coverage classification itself (COVERED_BY_CANDIDATE, coverage_source) is graph bookkeeping and is not a Lean statement. No all-height bound on the remote region, no rho -> 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window; nothing region-specific to add. | +| `math.rn-region.mesoscopic-scaled-annulus` | graph_node | — | `specified` | 2 specified: `D5dSummedPinNeighbourhoodC2`, `AnnulusBridgeAllHeight2` | D5dSummedPinNeighbourhoodC2 states the summed pin-neighbourhood bound (all heights, scaled ball of radius R minus the pins) and AnnulusBridgeAllHeight2 the all-height fixed-annulus bridge; neither is proved, and the "covers the reviewed fixed annulus" reading (R >= B) is not a Lean statement. No d > 2, no numerical constants, no elder selection, no JETMOD or witness-collision closure. The eight reading-rule component nodes are source identity, not transcribed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Independent reviewer: alignment of D5dSummedPinNeighbourhoodC2 with row D5-d; proof lane: none yet (the composition D5-a + D5-c at eta = 1/4 is a proof, not a statement). | +| `math.rn-region.pin-collision` | graph_node | — | `specified` | 3 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk`, `D5dSummedPinNeighbourhoodC2` | The three row shapes are stated for any object satisfying PinnedLawInterface and are not proved; the P2 source (PUNCTURED_PIN_PROOF.md) is not pinned here, so the Props are anchored to the reconciliation's transcription of the rows, not to the proof's own bytes. Including the deterministic pin atoms would invalidate the vanishing bounds (the reconciliation's caveat); the Props puncture the disks accordingly. No d > 2, no numerical constants, no uniformity in kappa. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin PUNCTURED_PIN_PROOF.md and COLLAR_PROOF.md (reviews/d5_local_collar_20260928) and re-anchor D5a/D5b/D5c to their own displays; independent reviewer: alignment of the three Props with rows D5-a, D5-b, D5-d. | +| `math.rn-region.intermediate-r-to-rho` | graph_node | — | `specified` | 1 specified: `D5eIntermediateShellsI3I4` | D5eIntermediateShellsI3I4 states the window-only bound E N <= C r^3 (A_0^(-2) + rho^2) on {A_0 r <= \|X\| <= rho} with side conditions taken from dimension-lift (1.6); not proved. The I3/I4 source (frontiers/intermediate_window_20260928) is not pinned here; no all-height intermediate count exists or is claimed; no numerical constants. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin intermediate_window_20260928/PROOF.md and anchor the single-shell display there; independent reviewer: alignment of D5eIntermediateShellsI3I4 with row D5-e and with dimension-lift (1.6). | +| `math.rn-region.witness-collision` | open_obligation | — | `specified` | 2 specified: `WitnessCollisionNearPinPairsTarget`, `RemoteCollisionSecondFactorialD` | WitnessCollisionNearPinPairsTarget states the OPEN obligation as a target Prop, not as a result: nothing asserts that the target holds, is attainable, or follows from any reviewed source. RemoteCollisionSecondFactorialD states only the fixed-remote, fixed-eta part (Corollary D of remote_collision), which the GRAPH records as reviewed evidence and NOT as a proof of this node. The Palm route's Theorem Q (q = 2) would imply the target but is an author-side candidate stated separately (PalmRouteFactorialMomentsQ) and not consumed here; the c6_witness_collision reconciliation's proposed discharge is declarative with execution pending. The node stays OPEN_ACTIVE; this register cannot and does not change that. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | None for Lean beyond the target statement: the obligation needs a source-bound proof and nonauthor review in Math-; if Theorem Q is folded, re-anchor the target to the fold record. | +| `math.rn-mesoscopic-reduction` | graph_node | — | `none` | — | No Lean statement: the PR7 route's bytes (frontiers/rn_mesoscopic_20260925/PROOF.md) are not pinned here and its Lemma A (positive-definite covariance of the rows of (U_0, J_0(y)) on K_AB) is not stated. The supersession is a graph classification, not mathematics. | Pin rn_mesoscopic_20260925/PROOF.md and state Lemma A (Matrix.PosDef lower bound on a compact annulus) if the route is ever revived; otherwise none. | +| `math.p15-full-price` | graph_node | — | `specified` | 3 specified: `FullTransformedPriceBudget`, `RhoStarEnclosure`, `SharpnessOfRhoStar` | See row p15-full-price: (F2), (F3) and sharpness stated, not proved; 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; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row p15-full-price. | +| `math.p15-price-boundary` | graph_node | — | `specified` | 2 specified: `TwoCoordGeneratorPrices`, `DemandOneRefutation` | See row p15-price-boundary: the refutation is stated, its two logarithm inequalities are not proved anywhere; REFUTED is the graph classification of the extension, transcribed as source identity. 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 p15-price-boundary. | +| `math.rn-selector-region-crosswalk` | graph_node | — | `none` | — | An engineering inventory, not a mathematical claim; nothing to state in Lean and nothing stated. | No Lean target. | +| `math.rn-fixed-annulus-window` | graph_node | — | `specified` | 1 specified: `FixedAnnulusWindowCandidateA2` | See row rn-fixed-annulus-window: (A2) stated, not proved; no global D5/RN/JETMOD, k -> 0, d >= 3, pin-neighbourhood, intermediate-scale, collision or elder-selection closure. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-annulus-window. | +| `math.rn-region.fixed-annulus-window` | graph_node | — | `specified` | 1 specified: `FixedAnnulusWindowCandidateA2` | The region's content is (A2) at its exact scope, stated and not proved; the coverage classification is graph bookkeeping, not a Lean statement; the superseded high-jet route (regional.fixed-annulus.high-jet-route, SUPERSEDED_NONBLOCKING) is not represented. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-annulus-window. | +| `math.d5-pin-neighborhood-first-moment` | graph_node | — | `specified` | 6 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk`, `D5cCollarC1`, `D5dSummedPinNeighbourhoodC2`, `D5eIntermediateShellsI3I4`, `D5fGlobalWindowFirstMomentI5` | The six row shapes are stated for any object satisfying PinnedLawInterface (sample space Omega, gated pin regime 0 < r < L) and are not proved. The consumption graph (D5-d from D5-a + D5-c, D5-f from D5-d + I4 + remote Theorem A) is a proof structure and is not stated; the component sources PUNCTURED_PIN_PROOF.md, COLLAR_PROOF.md and intermediate_window PROOF.md are not pinned here, so D5a/b/c/e/f are anchored to the reconciliation's row text. The lower bound c r^3 <= P(A_r) of D5-f has its source in remote_collision Corollary F-, not pinned. No d > 2, no numerical constants, no uniformity in T, R, eta, k -> 0; no elder pairing; no RN/24-jet/JETMOD; the witness-collision region stays OPEN_ACTIVE. The STATUS.md note that D5 GRAPH transitions were not executed by that summary is a recorded discrepancy, not resolved here. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin the three component proofs and re-anchor; independent reviewer: alignment of the six Props with the section 2 table (all-height versus window I_r per row, exclusion of M and S). | +| `math.d5-component.punctured-pin-proof` | graph_node | — | `specified` | 2 specified: `D5aPuncturedPinDisks`, `D5bMicrodisk` | The Props state rows D5-a and D5-b as the reconciliation transcribes P2, not P2 from its own bytes (not pinned); the slices P4, P9, P17, P20 and the continuum argument reviewed are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin PUNCTURED_PIN_PROOF.md (blob f972e50b per GRAPH fingerprint) and anchor D5a/D5b to its own displays. | +| `math.d5-component.punctured-pin-review-111` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | +| `math.d5-component.punctured-pin-continuum-review` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | +| `math.d5-component.punctured-pin-continuum-check` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | +| `math.d5-component.collar-review` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | +| `math.d5-component.annulus-review` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | +| `math.d5-component.i5-review-xai` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | +| `math.d5-component.i5-review-claude` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | +| `math.d5-component.offpin-second-moment-review` | graph_node | — | `none` | — | A review record (or its checker) is not a mathematical statement; nothing to state in Lean and nothing stated. Its verdict is source identity in the GRAPH and is not re-derived or re-graded here; all lanes share one GitHub account, so no record carries organizational-independence credit. | No Lean target. The proof it reviews is specified under its proof-component row. | +| `math.d5-component.collar-proof` | graph_node | — | `specified` | 1 specified: `D5cCollarC1` | D5cCollarC1 states row D5-c as the reconciliation transcribes C1, not C1 from its own bytes (not pinned); the normalizer (P14) it imports and the corollary C2 are not stated separately; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin COLLAR_PROOF.md (blob 794babe0 per GRAPH fingerprint) and anchor D5cCollarC1 to display (C1). | +| `math.d5-component.annulus-proof` | graph_node | — | `specified` | 1 specified: `AnnulusBridgeAllHeight2` | AnnulusBridgeAllHeight2 states display (2) (d = 2, fixed L, compact birth, k_- > 0, 1 < A < B, all heights) over PinnedLawInterface; not proved. Not stated: displays (4)-(20), the R1-R7 review interfaces as separate statements, the deterministic three-Hessian lemma, the crossover factor min(1,(r/\|v\|)^6). No matching lower bound, no numerical constant or radius, no pin neighbourhoods, no k -> 0 uniformity, no d >= 3, no fixed-remote all-height bound, no all-scales RN theorem. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State the R1-R7 interfaces as Props; independent reviewer: alignment of AnnulusBridgeAllHeight2 with display (2), checking "all heights" and the uniformity over E and j. | +| `math.d5-component.intermediate-window-proof` | graph_node | — | `specified` | 2 specified: `D5eIntermediateShellsI3I4`, `D5fGlobalWindowFirstMomentI5` | The Props state rows D5-e and D5-f as the reconciliation transcribes I3/I4/I5, not from the proof's own bytes (not pinned); the tiling N1 of the Claude review and the planar F+ lower bound source are not stated; nothing is proved. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Pin intermediate_window_20260928/PROOF.md (blob b3eb9456 per GRAPH fingerprint) and anchor the shell displays there. | +| `math.d5-component.remote-window-proof` | graph_node | — | `specified` | 2 specified: `FixedRemoteMeanMeasureA2`, `FixedRemoteCountBoundA3` | See rows rn-fixed-remote-window and math.rn-fixed-remote-window; the D5 consumption at rho = s_0 is a proof step, not stated. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | As for row rn-fixed-remote-window. | +| `D1-THM-A` | proof_index_result | — | `specified` | 2 specified: `PairLawInterface`, `ParentUniformSelectionA` | ParentUniformSelectionA states the shape of (1.1): existential r_* > 0, C < infinity outside the universal quantifiers over 0 < r <= r_*, b in [bLo,bHi], k in [kLo,kHi] and all frames, for the interface field p of PairLawInterface. Not stated or proved: the full-normalizer floor Z_r >= z_* r^2 (5.5), the transverse-matrix boundary layer, the regression law and typed weight, the reading-rule amendments (E1, Section 9 v1.1, W1, r < L/(4 sqrt 2)), any numerical r_*, C, z_*. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. No globally uniform constant over unbounded b, k; no lower bound. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: ParentUniformSelectionA is a predicate on the interface objects, not a closed statement; its universal closure over all inhabitants is false whenever the interfaces are inhabited (take p identically 0), and `specified` refers to the statement shape at the source's object, which is not constructed. | Independent reviewer: alignment of ParentUniformSelectionA with (1.1) under the honesty checklist; formal lane: none until the regression law is constructed. | +| `D1-THM-B` | proof_index_result | — | `specified` | 4 specified: `IsWindowPair`, `IsWindowElderPair`, `IsLifetimeDensityVersion`, `ParentCompactWindowDensityB` | ParentCompactWindowDensityB states the shape of (1.2) (asymptotics of versions of nu_cand and nu_eld, c_{B,K} > 0, the O(ell^(2/3)) difference) for compact B, K of positive length; not proved. Not stated: the exact coefficient formula (11.3), the radial ledger (10.2), the pushforward r = (ell/k)^(1/3); the distance cutoff is quantified existentially and not identified with the r_* of (1.1). The difference is NOT asserted for unrestricted densities; no numerical c_{B,K}, C_{B,K}; the historical H3 finite-band floor is not re-certified. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (11.3) as a Prop over an explicit Gaussian conditional law; independent reviewer: alignment of ParentCompactWindowDensityB with (1.2), including the positive-length requirement. | +| `D1-THM-C` | proof_index_result | — | `specified` | 3 specified: `IsUnrestrictedPair`, `IsUnrestrictedLeadingCoefficient`, `ParentUnrestrictedLeadingDensityC` | ParentUnrestrictedLeadingDensityC states the shape of (1.3) (leading term only) with the coefficient as an abstract positive constant c in IsUnrestrictedLeadingCoefficient; not proved. The Gaussian/angular representation (15.2), the target-growth majorant (13.3)-(13.6), the integrability (13.5) and the off-diagonal bound (14.1) are not stated; the SIDE24 enclosure of c_{d,24} is a separate row. No unrestricted remainder or difference rate (that is Theorem R), no elementary closed form. The identification of the interface structures (GaussianFieldInterface, SuperlevelBarInterface, PairLawInterface) with the source's field, elder rule and weighted regression law is not established: the Props hold for every object satisfying the interface, and no inhabitant is constructed. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (15.2) and (14.1) as Props; independent reviewer: alignment of ParentUnrestrictedLeadingDensityC with (1.3), in particular the exclusion of the essential global-maximum class. | +| `OA-ELDER-LOWER-DENSITY-GAP-20260928-v1` | proof_index_result | — | `specified` | 3 specified: `PlanarElderLowerE2`, `PlanarTwoSidedRateE4`, `PlanarDensityGapE11` | PlanarElderLowerE2 states (E2) for d = 2 over PairLawInterface; PlanarTwoSidedRateE4 states (E4) CONDITIONALLY on the parent (1.1) as the source's Corollary E2 does; PlanarDensityGapE11 states (E11) under the same (1.1) hypothesis, which the source needs only for the upper half (the lower half is thereby stated under a hypothesis the source does not need, disclosed in the docstring). None is proved. Not stated: (E3), (E7)-(E10), the polygonal-path and tilted jet-box argument. No lower bound for d >= 3 (separate packets CL-ELDER-LOWER-ALL-D / OA-ELDER-DIMENSION-LIFT, not extracted here); no numerical c or r_*. The review verdict of reviews/d1_elder_lower_claude_20260928 is not transcribed here; the pinned bytes carry their own disposition line. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. The interface does not determine p: PlanarElderLowerE2, PlanarTwoSidedRateE4 and PlanarDensityGapE11 are predicates on the interface objects, not closed statements; the universal closure of E2 and E4 over all inhabitants is false whenever the interfaces are inhabited (take p identically 1), and `specified` refers to the statement shape at the source's object, which is not constructed. | Independent reviewer: record whether (E11)'s lower half should be stated without the (1.1) hypothesis (a narrower-scope verdict is the reviewer's to write); transcribe the review verdict once pinned. | +| `two-scale-s6-s21` | proof_index_result | — | `none` | — | No Lean statement: the addendum is not pinned here and none of (S3)-(S5), (S20)-(S21) is stated; it enters the register only as the named support of FixedAnnulusWindowCandidateA2. Does not establish uniform O(r^3) on a fixed scaled-width annulus, anything at pin sites, near u = 0, r << \|x\| << rho, witness collisions, d >= 3, k_- = 0, uniform L. | Pin TWO_SCALE_ADDENDUM.md (blob 89cae3a9, 15902 B) and state (S3)-(S5) over PinnedLawInterface with the Stecconi weighted Kac-Rice input as an interface field. | +| `axial-density` | proof_index_result | — | `none` | — | No Lean statement: not pinned, not stated. The bound contains no pin-density factor, spatial Jacobian or Palm weight; nothing about the height-conditioned density, Hessian moments, the endpoint-weighted count, the full normalizer, the remaining thin belt, collisions, d = 3 or numerical constants. | Pin axial_density_20260925/PROOF.md (blob a72418d7) and state (1) as a density bound over PinnedLawInterface with the conditional gradient density as an interface field. | +| `fixed-transverse-chart` | proof_index_result | — | `none` | — | No Lean statement: not pinned, not stated. Constants are qualitative and not uniform as eta -> 0, B -> infinity or the gap interval approaches zero; no pin collisions, no intermediate distances, no d = 3, no positive leading coefficient, no probability lower bound. | Pin TRANSVERSE_BOUND_CANDIDATE.md (blob 024d9277) and state the chart bound as a Prop parallel to FixedAnnulusWindowCandidateA2 with the extra \|v\| >= eta restriction. | +| `CL-D5-DIMENSION-LIFT-20260929-v1` | proof_index_result | — | `specified` | 4 specified: `DimensionLiftPuncturedPinBallP`, `DimensionLiftCompactCollarC`, `DimensionLiftIntermediateShellsI`, `DimensionLiftGlobalWindowG` | The four theorem shapes (1.2), (1.3), (1.5)-(1.6), (1.7) are stated over PinnedLawInterface for fixed d >= 2 and not proved. Not stated: display (1.4), Corollaries F_d (1.8) and M_d (1.9), the five devices D1-D5, the consumed sources ([LP] sections 2-5, (5.5), the planar templates, [RM], [RC], [EDL], [LOCAL]). No all-height intermediate or global O(r^3) count; no torus-wide second factorial moment; no elder selection; no uniformity in k, L, d; no numerical C, c, r_*, s_0; no recovery of historical RN/JETMOD/24-jet carriers. The reviews (Math-#141 OpenAI, source-exposed) are source identity; the 2026-10-05 erratum may amend these statements and was not read. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | Read reviews/d5_dimension_lift_erratum_20261005 and amend the four Props if the displays changed; state Corollaries F_d and M_d; independent reviewer: alignment of the four Props with (1.2)-(1.7). | +| `OA-C6-FOURIER-20260929-v1` | proof_index_result | — | `none` | 1 specified: `FourierCutoffPlanarSecondFactorialF4` | No Prop states Theorem F (the cap functional Psi_F and its stretched-exponential tail (F1)-(F2)), Theorem W in general d and p (F3), or Proposition B; only the planar consequence (F4) is stated, GIVEN the hypothesis G_2 (E N <= C r^3), so the claim as a whole is not specified. Nothing is proved; the logarithm is neither shown sharp nor removable here; no numerical cutoff or parameter-uniform constants; the abstract barrier is not a Gaussian counterexample. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State (F1)-(F2) with Psi_F as an interface field (a measurable functional dominating the total critical count) and (F3) for general d, p; then the claim would be `specified`. | +| `CL-C6-PALM-20260929-v1` | proof_index_result | — | `specified` | 2 specified: `PalmRouteFactorialMomentsQ`, `PalmRouteSecondFactorialTheta` | PalmRouteFactorialMomentsQ states (1.1) (with r_* allowed to depend on q) and PalmRouteSecondFactorialTheta states (1.2) over PinnedLawInterface; not proved. Corollary P (size-biased law) and Sections 3-7 (the cap Psi, the parametric Rouche tail, the marked Kac-Rice formula with Hoelder insertion, regimes R1-R4) are not stated; the dependence on [C6L] and [DL] at their bytes is not transcribed; the lower bounds' source [EDL] (A4) is not stated. No lower bound for q >= 3; no distribution of N beyond factorial moments; no elder selection; no numerical constants; for d >= 3 conditional on [DL]. The GRAPH records that this global result is not a proof of the regional witness-collision node. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Corollary P; independent reviewer: alignment of the two Props with (1.1)-(1.2), including whether r_* may depend on q. | +| `OA-C6-RARE-CLUSTER-20260929-v1` | proof_index_result | — | `none` | 1 specified: `RareClusterPoissonObstructionO` | Only Theorem O (3.1) is stated, for a family of laws with hypotheses (pmf, summable mean, H1, H2) on 0 < r <= r0; Theorems C (compound Poisson (4.1)), S (6.1), the replica limit (7.1), display (5.2) and the addendum (A1)-(A2) are not stated, so the packet as a whole is not specified. Nothing is proved; H1/H2/Hq are not revalidated and are not asserted for the Gaussian window count; for A < 2a the hypotheses are unsatisfiable (the source's own remark), so the Prop is vacuous there by the mathematics. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Theorems C and S and (7.1) over pmfs on N (Mathlib PMF or explicit functions N -> R); Theorem O is a candidate kernel target (finite total-variation arithmetic). | +| `CL-D5-REMOTE-COLLISION-20260928-v1` | proof_index_result | — | `none` | 1 specified: `RemoteCollisionSecondFactorialD` | Only Corollary D (second factorial moment <= C r^5 \|E\| in the fixed remote region) is stated; Theorem C (ordered near-pair count), Corollaries E and F (event asymptotics, torus-wide lower bound) are not, so the packet is not specified as a whole. Nothing proved; pairs with a witness near the pins are excluded; no torus-wide second factorial moment; this is explicitly NOT the open witness-collision mechanism. Author Anthropic/Claude: same provider as this packet, independence credit 0. Nonauthor review and acceptance are not claimed by any statement; alignment of every statement with its anchor is PENDING_INDEPENDENT_REVIEW; organizational-independence credit of this packet is 0. | State Theorem C and Corollaries E, F; independent (non-Anthropic) reviewer for both the source and the Props. | +| `cumulative-transfer-correction` | proof_index_result | — | `none` | — | No Prop states the corrected cumulative theorem (sigma-finite (Z, lambda), alpha > -1, m > 0, ratio hypothesis h/(kappa r^m) -> 1, limit N(ell)/ell^beta -> (1/(alpha+1)) int B0 kappa^(-beta)), although its bytes are pinned. Nothing is proved; the correction does not license inferring a density asymptotic from a cumulative one; Section A of NOTE.md is outside both acceptances. | State the corrected theorem as a def : Prop over MeasureTheory.Measure Z (an abstract measure-space statement with no Gaussian input; a natural Mathlib-lane proof target afterwards). | +| `hist.D0-RN-UNIF-historical-predicates` | open_obligation | — | `none` | — | No Lean statement: the historical numerical RN all-cell / 24-jet certification predicates have no in-tree source text to anchor (their carriers are ABSENT by fingerprint) and no Prop states any of them. The GRAPH records no complete proof where not explicitly superseded, and hist.lemma_closed is FALSE; this register changes none of that. The regional bypass (regional.fixed-annulus.high-jet-route, SUPERSEDED_NONBLOCKING) does not reopen or close these predicates in their original replay scopes; no numerical certificate exists in tree. | None for Lean until a source text exists; recovery of the absent carriers is a custody question (reviews/d0_custody_audit_20260926), not a formalization one. | +| `lean:main-formal-UniversalLaw` | existing_lean_package | — | `none` | 4 proved (main): `d2_endpoints_adjacent`, `d3_endpoints_adjacent`, `d3_below_d2`, `endpoints_in_unit_tenth` | The package states no research claim as a Prop, so the claim-level label is none; its 31 theorems at origin/main f6deeba7 are finite integer/cross-multiplied rational facts proved by decide (label `proved` in the manifest, `kernel-checked` only in a run receipt). The four Endpoints theorems listed here show the published 20-digit endpoints are adjacent, ordered and below 1/10 as Nat; they do not place c_{2,24} or c_{3,24} between them and say nothing about the true coefficients (the module's own note). The 28 P15 arithmetic targets of this programme's main worktree are on no commit, are not listed in this register, and have only a local untrusted --run-lean receipt. main has NO mechanism for `specified`: its gate refuses any manifest label but `proved`, so statement-only work cannot live in this package. Alignment review by a non-Anthropic lane is an open ask (formal/reviews holds only the template). | Commit and push the P15 modules so the hosted formal-verification.yml run produces a trusted receipt, then list them here as `proved` statements citing the 40-hex commit and the run; obtain a non-Anthropic alignment review of the 31 committed targets (59 once the P15 modules are committed); keep real-number statements in the Math- Mathlib lane. | +| `lean:math-formal-ResearchFormalCoreR1` | existing_lean_package | — | `none` | — | The package states no research claim of this register as a Prop and none of its statements can be listed here: its bound sources (P02-LM-008 proof.md.export.txt at 22876eda, MOMENT_TAIL.md, WEIGHT_PERTURBATION.md, the D2 small-L NOTE.md) are not among this packet's pinned sources, so no verbatim anchor is available. Not established by the package (its own does_not_establish): any concrete Gaussian or typed Palm law, moment bounds M, cW, cZ, lower normalizers, event tails, constant uniformity, GT5, P0.2, the parent persistence theorem, identification of weightedLaw with the program's conditioned law, D2 Schur in d >= 3, the coefficient formula (15.2). Alignment: ACCEPTED for 20 targets at an older head only, STALE for the current 63, credit 0. | Pin the Math- formal sources in a later packet revision and register the 63 targets as `proved` statements with anchors; obtain a fresh alignment record for the 63-target manifest from a lane differing in provider, family and agent. | +| `lean:math-companion-d2_moment_bridge_20261006` | existing_lean_package | — | `none` | — | States no research claim of this register; its sources (Math-#302 NOTE.md section 4 / C7, #326, #328, #343 comments) are not pinned here, so no statement can be anchored and listed. Does not establish: no uniform spectral floor, no concrete periodic probability law, no moment summability for that law, no trigonometric q map, no Gaussian/Palm identification, no interval quadrature, no numerical coefficient or persistence/cap theorem; the finite countermodel law is tested in Python, not constructed in Lean. Same GitHub account as every lane: organizational-independence credit 0. | Pin the companion sources and register its theorems as `proved` statements in a later packet revision; a reviewer differing in provider, family and agent for alignment. | +| `lean:math-companion-d2_square_schur_20261006` | existing_lean_package | — | `none` | — | States no research claim of this register; its sources (Math-#302 NOTE.md section 4 / C7, #326, #328, #343 comments) are not pinned here, so no statement can be anchored and listed. Does not establish: no atom-measure integral lower bound, moment realizability, periodic spectral law, trigonometric parameterization, Gaussian/Palm covariance identification, quadrature or persistence theorem; the #328 Part B stronger proposal is not incorporated. Same GitHub account as every lane: organizational-independence credit 0. | Pin the companion sources and register its theorems as `proved` statements in a later packet revision; a reviewer differing in provider, family and agent for alignment. | +| `lean:math-companion-d2_atom_integral_20261006` | existing_lean_package | — | `none` | — | States no research claim of this register; its sources (Math-#302 NOTE.md section 4 / C7, #326, #328, #343 comments) are not pinned here, so no statement can be anchored and listed. Does not establish: no formal sharpness, no actual periodic spectral measure or its summability, no numerical values or radii, no uniform-L or family floors, no Gaussian/Palm conditioning, no coefficient integral, no persistence matching. Same GitHub account as every lane: organizational-independence credit 0. | Pin the companion sources and register its theorems as `proved` statements in a later packet revision; a reviewer differing in provider, family and agent for alignment. | +| `lean:math-frontier-cap_first_exit_lean_20261002` | existing_lean_package | — | `none` | — | The frontier states and kernel-checks the topological global step of D1 component III (maximin level, elder merge, elder death level, saddle cut, older peak on compact locally connected spaces, torus transport) and the analytic chain L1-L13, L16-L20 from f in C^4 and CAP (1), ending in torus_capHyp_blocks and torus_elder_death_level_blocks. None of its 152 declarations is listed here as a statement because their per-declaration anchors (ALIGNMENT.md sections 1-3) were not transcribed in this programme. Not established (its own does_not_establish): that the local field on G_r satisfies SP section 2; the identification of SP's coordinates/contact frame and P's field with the Lean objects; L14 (critical-point count), L15 (Morse indices); the definition of the H0 superlevel persistence pairing (only the level s is proved); L23, P section 8, Theorem A, measurability and Q^W(G_r^c) <= C r^3 (every probabilistic statement). Math-#193 D1 kernel formalization remains partially unsupplied. Same provider as this packet: independence credit 0; the author will not merge. | Transcribe ALIGNMENT.md row by row and register the 152 declarations as `proved` statements anchored in the pinned CAP bytes (and a pinned SP review); non-Claude delta review of v1.5-v2.6. | +| `lean:math-w3scratch-side24-proposal` | existing_lean_package | — | `none` | — | No kernel evidence exists for this package in any workflow or committed receipt: builds and axiom reports are author-reported, box-local, in a scratch copy of formal/ with an appended lean_lib, and gate.py --execute was never satisfied (namespace mismatch; a 140-target option-A variant hit the 900 s build timeout). Nothing here is reviewed, aligned or a discharge; the identification of the Poisson-dual law with the NOTE's spectral law (its question (a)) is open; the all-L positivity statement is not the source's six-L interval certificate; no status field changes. Its declarations are not listed here (unpinned sources, no workflow). | If the proposal is adopted, register it under a lakefile and workflow in Math- first; only then could its statements enter a register with anchors and `proved` labels. | +| `main:experiments/universality/finite_gaussian_contact` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/finite_gaussian_contact/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/universality/finite_candidate_lifetime` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/finite_candidate_lifetime/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/universality/finite_fold_chart` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/finite_fold_chart/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/universality/finite_weighted_fold_window` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/finite_weighted_fold_window/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/universality/finite_contact_genericity` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/finite_contact_genericity/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/universality/finite_elder_transfer` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/finite_elder_transfer/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/universality/finite_h0_lifetime` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/finite_h0_lifetime/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/universality/fold_structural_universality` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/fold_structural_universality/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/universality/periodized_gaussian_h0` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/periodized_gaussian_h0/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/universality/iid_short_bar_process` | main_experiment_proof | — | `none` | — | A prose proof on main, indexed only in experiments/universality/README.md (not in PROOF_INDEX, LANDING_CLAIMS or STATUS.md, which are Math- registers); no Lean statement exists and none is stated here; its bytes are not pinned in this packet and its theorem text was read only at the header. The REVIEW.md PASS verdicts are by the same provider (OpenAI) under one GitHub account: credit 0, no human, blind or provider-distinct acceptance. Nothing is proved in Lean; no catalogue model is admitted to the full persistence theorem by this row. | Pin experiments/universality/iid_short_bar_process/PROOF.md by SHA-256, extract the theorem and its interfaces (the five field-admission interfaces as structure fields), and state it as a def : Prop in a main-side or Math- Mathlib statement module; then an independent (non-OpenAI) review. | +| `main:experiments/periodic_h0/spectral_shrinking_bins` | main_experiment_proof | — | `none` | — | A conditional prose theorem under explicit input interfaces (spectral error, spatial interpolation, exceptional expected counts); not pinned here, not stated in Lean, review pending by its own header. It does not prove its imported interfaces anew or change their status; the C131 exact-sample shrinking-bin reading record (EXACT_SAMPLE_SHRINKING_BIN.md) is a reading record, not a claim, and has no row. | Pin the bytes, extract the conditional theorem with its three input interfaces as structure fields, state as def : Prop; nonauthor review first. | +| `main:unmerged-codex-prs-330-350` | main_experiment_proof | — | `none` | — | Nothing on origin/main: these are open pull requests whose PROOF.md, guard and certificate files are not merged, not pinned and not stated; their RESULTS.md scope statuses record lean_kernel NOT_RUN and scientific_promotion NOT_RUN. Listing them here is inventory only; no verdict, no landing and no statement is implied. | Revisit after merge: pin, extract, state. The exact Fraction certificates of #343/#349 (which import periodic_h0 sqrt/exp/log bound helpers) would be exact_certificate rows then. | +| `cert:main-periodic_h0-exact_h0_1` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet. | Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses. | +| `cert:main-periodic_h0-certificate8` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet. | Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses. | +| `cert:main-periodic_h0-nodal1` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet. | Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses. | +| `cert:main-periodic_h0-hessian1` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet. | Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses. | +| `cert:main-periodic_h0-gaussian_tail1` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet. | Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses. | +| `cert:main-periodic_h0-gaussian_coupling1` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet. | Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses. | +| `cert:main-periodic_h0-word_polynomial1` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet. | Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses. | +| `cert:main-periodic_h0-finite_count_loss1` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet. | Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses. | +| `cert:main-periodic_h0-count_confidence1` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (Python stdlib Fraction / integer) about one finite rounded object: a Python certificate is not a kernel proof and is not Lean. The CERTIFICATE.json scope flags are all false for what is not certified (gaussian_law, infinite_field, lifetime_law, historical_fft / historical_gaussian_draw, iid_input_law where present; input_law_is_hypothesis true where present): no Gaussian law, infinite field, lifetime law or held-out confirmation is certified. No Lean statement exists or is stated here; the CERTIFICATE.json bytes are not pinned in this packet. | Restate the finite inequality that the Python certificate checks in exact rational arithmetic (the bounds for the fixed object) as a Lean Prop over ℚ / ℕ and prove it in a proof package (norm_num / decide, or interval arithmetic in Mathlib for the pi / sqrt / exp / log bound helpers); the input-law hypotheses stay hypotheses. | +| `cert:main-universality-fold_controls` | exact_certificate | — | `none` | — | Exact Fraction toy control (L2 evidence), by its own words "no random-field admissibility or Lean claim": not a stationary random-field counterexample, not a formal kernel proof, not stated in Lean. | The exact pushforward exponent identities are elementary and could be stated and proved over ℚ in a proof package; low priority. | +| `cert:math-side24_v1-coefficient.py` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence: a Python outward-rounding procedure is not a kernel proof, and main's 31 Side24 theorems formalize only its integer arithmetic skeleton (not the interval arithmetic, not the Gamma / pi / exp enclosures, not the theta sums). The enclosure statement itself is CoefficientEnclosure (specified, not proved) on row side24-coefficient; ENCLOSURE.json says scientific_acceptance false. | A proof-producing enclosure pipeline (the numerical formalizer work offer of main docs/FORMAL_VERIFICATION.md): proof-producing Gamma(7/6), pi, exp and theta-sum enclosures in Mathlib, then CoefficientEnclosure under ConeMomentInterface. | +| `cert:math-full_price-full_price.py` | exact_certificate | — | `none` | — | L2 exact-arithmetic evidence (stdlib Fraction) for finite rational facts; not a kernel proof. The FullPrice.lean module staged in this programme's uncommitted main worktree (on no commit; not listed in this register) targets only the adjacency of the endpoints as Nat, 16/27 < 6/7 and the rational orderings / Taylor partial sum of Section 6, not that rho_star = 1/(3 - log(3e - 2)) lies between the endpoints (RhoStarEnclosure, specified only). The finite suite "supports algebra only" (the source's own words); no prize status (original_prize_closed stays false). | Prove RhoStarBelowSixSevenths / SixSeventhsCertificate in the Math- Mathlib lane (Real.exp / Real.log bounds); then RhoStarEnclosure via proof-producing exp/log enclosures. | + +### Packets listed by id only (82 rows) + +Kind `proof_index_result`, formal progress `none`, no statements, no kernel evidence, layer 0 source null (bytes not pinned here). Shared `not_established`: Statement not extracted in this programme: the packet was inventoried by id only, its bytes are not pinned here and nothing about it is stated in Lean. Its own disposition and review records govern and are not transcribed here; no verdict is implied by this row. Nothing is proved. Shared `next_step` (per packet path): Read Math-:frontiers/side24_periodization_20260929 at 6c020d6a, pin the bytes under sources/, extract the headline statement and its review disposition, and decide whether a def : Prop is warranted. + +| claim_id | title | +|---|---| +| `CL-SIDE24-PFV-20260929-v1` | SIDE24 periodization first variation, P_3(L) = -L^2(10L^4-147L^2+315)/105, P_3(24) = -620813376/35 (#144): Math- frontiers/side24_periodization_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-SIDE24-REMAINDER-20260929-v1` | SIDE24 remainder \|c_{d,24}/c_{d,ref} - 1 - P_d(24) e^{-288}\| < 10^{-216} (#154): Math- coefficients/side24_remainder_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `CL-SIDE24-PERIODIZATION-REMAINDER-20260929-v1` | SIDE24 periodization remainder (#156): Math- frontiers/side24_periodization_remainder_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-CONTACT-ANGULAR-TAIL-20260925-v1` | contact angular tail (#35): Math- reviews/contact_angular_tail_20260925 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-PIN-MICRO-COV-20260926-v1` | pin micro covariance (#60): Math- reviews/pin_micro_covariance_20260926 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-D5-MICRODISK-20260926-v1` | D5 microdisk (#82): Math- reviews/d5_microdisk_20260926 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `D5-PIN-MICRODISK-20260927-v1` | D5 pin microdisk (#90): Math- reviews/d5_pin_microdisk_20260927 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-BF-SIX-PIN-HESSIAN-20260926-v1` | Bargmann-Fock six-pin Hessian certificate (+SCALAR) (#89): Math- certificates/bf_six_pin_hessian_20260926 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-CONTACT-KERNEL-SUBSTITUTE-20260926-v1` | contact kernel substitute (#80): Math- reviews/contact_kernel_substitute_20260926 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-D5-SHORT-EDGE-20260928-v1` | D5 short edge (#103): Math- reviews/d5_short_edge_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `CL-D5-PIN-DISK-20260928-v1` | D5 pin disk, all directions (#104): Math- reviews/d5_pin_disk_all_directions_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-D5-REMOTE-SINGLETON-LAW-20260928-v1` | D5 remote singleton law (#115): Math- frontiers/remote_singleton_law_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-REMOTE-HEIGHT-DECOUPLING-20260928-v1` | remote height decoupling (#120): Math- frontiers/remote_height_decoupling_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-WINDOW-MULTIPLICITY-LOCAL` | window multiplicity law, local (#116 C2; id abbreviated as listed by the navigation refresh): Math- frontiers/window_multiplicity_laws_20260928 (C2) at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-WINDOW-MULTIPLICITY-REMOTE` | window multiplicity law, remote (#116 C3; id abbreviated as listed): Math- frontiers/window_multiplicity_laws_20260928 (C3) at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-WINDOW-MULTIPLICITY-DISTANCE` | window multiplicity law, distance (#116 C4; id abbreviated as listed): Math- frontiers/window_multiplicity_laws_20260928 (C4) at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-WINDOW-MULTIPLICITY-HEIGHT` | window multiplicity law, height (#116 C5; id abbreviated as listed): Math- frontiers/window_multiplicity_laws_20260928 (C5) at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-REMOTE-INVERSE-SEPARATION-20260928-v1` | remote inverse separation (#128): Math- frontiers/remote_inverse_separation_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-ELDER-DIMENSION-LIFT-20260928-v1` | elder lower bound 1 - p_r >= c r^3 in every fixed d (#125): Math- frontiers/elder_dimension_lift_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `CL-ELDER-LOWER-ALL-D-20260929-v1.1` | elder lower bound, all d (#149): Math- frontiers/elder_lower_all_d_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `CL-C6-FACTORIAL-20260929-v1` | C6 factorial moment, Lemma M and cap Psi (#140, v1.3): Math- frontiers/c6_factorial_moment_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `CL-C6-SHARPENED-20260929-v1` | C6 sharpened (#142, v1.2): Math- frontiers/c6_sharpened_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-C6-COUNT-CAP-BOUNDARY-20260929-v1` | C6 count cap boundary (#146): Math- frontiers/c6_count_cap_boundary_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-LOCAL-CLUSTER-TIGHTNESS-20260929-v1` | local cluster tightness (#161): Math- frontiers/local_cluster_tightness_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-REMOTE-MIXED-INVERSE-20260929-v1` | remote mixed inverse (#132): Math- frontiers/remote_mixed_inverse_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `CL-D1-UNRESTRICTED-DIFFERENCE-20260929-v3` | Theorem U: the C7 unrestricted O(ell^{2/3}) rate is false (#133): Math- frontiers/unrestricted_selection_difference_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `CL-C7-TOTAL-BOUNDED-20260929-v1.2` | C7 total bounded, Theorem K; read with ERRATUM_NORMALIZED_FORM.md and RATE_READING.md per the 2026-10-07 guidance (#150): Math- frontiers/c7_total_bounded_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-C7-ZERO-GAP-LIMIT-20260929-v1` | C7 zero-gap limit, Theorem Z (#155): Math- frontiers/c7_zero_gap_limit_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-SARD-ROBUST-CHARTS-20260929-v1` | SARD-G robust charts (#135; STATUS.md ACCEPT, scoped, qualitative successor; A1/A6 original formulation remains AMEND): Math- frontiers/sard_g_robust_charts_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-SARD-A2-FIXED-FRAME-20260929-v2` | SARD-G A2 fixed frame (#137): Math- frontiers/sard_g_a2_parameter_20260929/v2/A2_FIXED_FRAME.md at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:sard_g_a2_regularity_20260929` | SARD-G A2 regularity (reading rule #139 e947ade5): Math- frontiers/sard_g_a2_regularity_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:sard_g_pinned_transfer_20260929` | SARD-G pinned transfer (reading rule #139 e947ade5): Math- frontiers/sard_g_pinned_transfer_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-P15-BIPARTITE-OVERLAP-20260928-v1` | P15 bipartite overlap (#118; prize reconnaissance, HOLD): Math- frontiers/p15_bipartite_overlap_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `OA-P15-TAIL-LOAD-20260928-v1` | P15 tail load (#122; prize reconnaissance, HOLD): Math- frontiers/p15_tail_load_20260928 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_cubic_cluster_20260929` | planar cubic cluster (#158 aa288385): Math- frontiers/planar_cubic_cluster_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:c6_marked_fourier_20260929` | C6 marked Fourier (#157 2479a7c2): Math- frontiers/c6_marked_fourier_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:c6_cluster_law_20260929` | C6 cluster law (#159 98fdb549): Math- frontiers/c6_cluster_law_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:cluster_exponential_weight_20260930` | cluster exponential weight (#164 7d0c89a5): Math- frontiers/cluster_exponential_weight_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:spectral_cluster_closure_20260929` | spectral cluster closure (#162 358f2562): Math- frontiers/spectral_cluster_closure_20260929 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:lifetime_moment_boundary_20260930` | lifetime moment boundary (#177 dda8991f): Math- frontiers/lifetime_moment_boundary_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:radial_rate_20260930` | radial rate (#176 7ca3eb00): Math- frontiers/radial_rate_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:extreme_cluster_sampling_20260930` | extreme cluster sampling (#169 f430fdec): Math- frontiers/extreme_cluster_sampling_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:fixed_r_inverse_lifetime_20260930` | fixed-r inverse lifetime (#182 2a10ed34): Math- frontiers/fixed_r_inverse_lifetime_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:local_elder_geometry_20260930` | local elder geometry (#170 fa2e9909): Math- frontiers/local_elder_geometry_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:concave_fibre_elder_20260930` | concave fibre elder (#175 eb659bd3): Math- frontiers/concave_fibre_elder_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:elder_partner_extremes_20260930` | elder partner extremes (#181 443212fb): Math- frontiers/elder_partner_extremes_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:remainder_vanishing_20260930` | remainder vanishing (#191 95565474): Math- frontiers/remainder_vanishing_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:elder_selected_extremes_20260930` | elder selected extremes (#179 6e5f36d7): Math- frontiers/elder_selected_extremes_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:remainder_rate_20260930` | remainder rate (#198 124c37d9): Math- frontiers/remainder_rate_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:far_elder_rate_20260930` | far elder rate (#187 c2f1270d): Math- frontiers/far_elder_rate_20260930 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:short_bar_endpoint_law_20261001` | short bar endpoint law (#221 f166ef52): Math- frontiers/short_bar_endpoint_law_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:c2_finite_jet_transfer_20261001` | C2 finite jet transfer (#232 7fe06b01): Math- frontiers/c2_finite_jet_transfer_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:cusp_second_order_20261001` | cusp second order (#207 566b1a14): Math- frontiers/cusp_second_order_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:candidate_third_order_20261001` | candidate third order (#218 fb6ee97b): Math- frontiers/candidate_third_order_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:elder_third_order_20261001` | elder third order (#220 0d797780): Math- frontiers/elder_third_order_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:third_order_rate_20261001` | third order rate (#229 6f74f7a7): Math- frontiers/third_order_rate_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:elder_cusp_parity_20261002` | elder cusp parity (#240 7636cae5): Math- frontiers/elder_cusp_parity_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:separated_fold_multiplicity_20261002` | separated fold multiplicity (#239 240a9bfa): Math- frontiers/separated_fold_multiplicity_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:subpin_collision_20261002` | subpin collision (#245 2d8616ef): Math- frontiers/subpin_collision_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:soft_rejected_pairs_20261002` | soft rejected pairs (#242 54c6759e): Math- frontiers/soft_rejected_pairs_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:soft_fold_limit_20261002` | soft fold limit (#243 fa8b0d21): Math- frontiers/soft_fold_limit_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:soft_closed_form_20261002` | soft closed form (#244 07320089): Math- frontiers/soft_closed_form_20261002 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:cusp_torus_transfer_20261001` | cusp torus transfer (3 Oct supplement): Math- frontiers/cusp_torus_transfer_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:moving_remote_collision_20261003` | moving remote collision (3 Oct supplement): Math- frontiers/moving_remote_collision_20261003 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:ordinary_short_bar_occurrence_20261001` | ordinary short bar occurrence (3 Oct supplement): Math- frontiers/ordinary_short_bar_occurrence_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:replacement_bar_occurrence_20261001` | replacement bar occurrence (3 Oct supplement): Math- frontiers/replacement_bar_occurrence_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:strict_unique_replacement_coefficient_20261003` | strict unique replacement coefficient (3 Oct supplement): Math- frontiers/strict_unique_replacement_coefficient_20261003 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:unique_replacement_bar_intensity_20261001` | unique replacement bar intensity (3 Oct supplement): Math- frontiers/unique_replacement_bar_intensity_20261001 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_rejected_endpoint_margin_20261003/A4` | planar rejected endpoint margin A4 (3 Oct supplement, nested): Math- frontiers/planar_rejected_endpoint_margin_20261003/A4 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C91` | planar soft layer chain C91 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C91 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C92` | planar soft layer chain C92 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C92 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C93` | planar soft layer chain C93 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C93 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C94` | planar soft layer chain C94 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C94 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C95` | planar soft layer chain C95 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C95 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C96` | planar soft layer chain C96 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C96 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C97` | planar soft layer chain C97 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C97 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C98` | planar soft layer chain C98 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C98 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C99` | planar soft layer chain C99 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C99 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C101` | planar soft layer chain C101 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C101 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C102` | planar soft layer chain C102 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C102 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C103` | planar soft layer chain C103 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C103 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | +| `packet:planar_soft_layer_chain_20261003/C124` | planar soft layer chain C124 (3 Oct supplement, nested): Math- frontiers/planar_soft_layer_chain_20261003/C124 at 6c020d6a; listed by PROOF_INDEX (navigation refresh 26-29 Sep 2026 / 2 Oct reachability inventory / 3 Oct supplement); statement not extracted | + +## Not established (by construction) + +| the package does | the package does not | +|---|---| +| state each registered result as a Prop with explicit hypotheses and interface fields | prove any of them; `theorem` is refused | +| elaborate against Mathlib `d13f23b7` under Lean `v4.34.1` in `--execute` | establish that any statement is true | +| pin the Layer 0 bytes each statement is anchored to | establish that a statement says what its anchor says (alignment is PENDING_INDEPENDENT_REVIEW) | +| transcribe LANDING_CLAIMS dispositions verbatim | decide, re-derive or change any disposition, premise or obligation | +| report `#print axioms` for every registered declaration (the three standard axioms `propext`, `Classical.choice`, `Quot.sound` are allowed and expected: real-analytic definitions depend on them; `sorryAx`, `Lean.ofReduceBool` and any custom axiom are refused) | turn an axiom report into a proof receipt; the receipt says `specified` | +| audit every constant the modules add by Lean's `ConstantInfo` (no theorem, axiom, opaque constant, instance or proof-typed definition beyond what `structure`/`def` generate; inventory equal to the register) | turn the absence of proofs into evidence about any claim; a module that passes the audit states, it does not prove | +| record the checked commit and worktree state in the receipt | make a `--allow-dirty` receipt evidence about any commit | + +The identification of any interface structure with the object of the source texts, the parent +theorems themselves, the review of the statements, and acceptance of any claim remain open. +Nothing under this directory may be cited as evidence that a research claim is proved, verified, +closed or certified. diff --git a/specifications/lean_statements_20261010/Specifications.lean b/specifications/lean_statements_20261010/Specifications.lean new file mode 100644 index 000000000..e5b750c01 --- /dev/null +++ b/specifications/lean_statements_20261010/Specifications.lean @@ -0,0 +1,5 @@ +import Specifications.Field +import Specifications.Lifetime +import Specifications.Side24 +import Specifications.RN +import Specifications.P15 diff --git a/specifications/lean_statements_20261010/Specifications/Field.lean b/specifications/lean_statements_20261010/Specifications/Field.lean new file mode 100644 index 000000000..cc40e9d19 --- /dev/null +++ b/specifications/lean_statements_20261010/Specifications/Field.lean @@ -0,0 +1,380 @@ +import Mathlib + +/-! +# Shared objects of the periodized Gaussian lifetime track (statements only) + +Scientific effect: NONE. Formal progress of every declaration here: `specified`. +Organizational-independence credit of this transcription: 0 (Anthropic / Claude). + +This module declares, as definitions and interface structures, the objects that the +Layer 0 texts of the lifetime track share: the periodized Gaussian covariance `K_L` +(as a quotient of theta sums written with `tsum`), the centred stationary Gaussian field +on the torus `R^d/(L Z^d)` (an interface: probability space, smooth `L`-periodic sample +paths, Gaussian law of every finite real linear combination of point evaluations, and the +covariance identity), critical points, the Hessian form through `iteratedFDeriv`, the +inertia predicates for nondegenerate maxima and index-(d-1) saddles, ordered +maximum/saddle pairs with their height gap and scaled gap, the superlevel H0 bar and +elder-partner interface, and the per-unit-volume expected lifetime density "versions". + +What this module does not establish. It consists only of definitions and structures; +it states nothing as a result and proves nothing. It does not construct the Gaussian +field (the existence of a process with these finite-dimensional laws and smooth periodic +sample paths is a hypothesis field), does not prove summability of the theta sums (a +hypothesis field), does not identify the elder-partner relation with any Kac-Rice or +persistence-module construction, does not show that lifetime density versions exist, +does not assert almost-sure Morse genericity, and does not assert that an object +satisfying an interface here coincides with the object of the source texts. Nothing here +promotes, reclassifies or discharges any claim, premise or obligation of the registers; +alignment of these statements with the sources is PENDING_INDEPENDENT_REVIEW. + +Primary source (Layer 0, byte-pinned): `d6g8k5htny-coder/Math-` commit +`c39164254ad1529ed49543743598d5dcb24bab98`, path +`imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, +sha256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7` +(source id `lifetime-parent`). +-/ + +namespace UniversalLaw.Spec.Field + +open MeasureTheory + +/-- The ambient Euclidean space `R^d` in which the torus `X = R^d/(L Z^d)` is realised +by `L`-periodic functions. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance" +Interface: none; this is the ambient space `EuclideanSpace ℝ (Fin d)`. +Does not claim: that the torus is represented by any quotient type; periodicity is a hypothesis on functions. -/ +abbrev E (d : ℕ) : Type := EuclideanSpace ℝ (Fin d) + +/-- The lattice vector `L n` for `n ∈ Z^d`, written in the standard coordinate vectors. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2)" +Interface: none. +Does not claim: anything beyond the definition of the vector `L n`. -/ +noncomputable def latticeVec (d : ℕ) (L : ℝ) (n : Fin d → ℤ) : E d := + ∑ i, EuclideanSpace.single i (L * (n i : ℝ)) + +/-- The unnormalised periodized Gaussian (theta) sum `sum_{n in Z^d} exp(-|z+Ln|^2/2)`, +written as a `tsum` over `Z^d`. If the family were not summable the `tsum` would be `0`; +summability is a hypothesis field of `GaussianFieldInterface`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2)" +Interface: summability of the family is the field `GaussianFieldInterface.summable`, not derived here. +Does not claim: convergence of the sum. -/ +noncomputable def thetaSum (d : ℕ) (L : ℝ) (z : E d) : ℝ := + ∑' n : Fin d → ℤ, Real.exp (-(‖z + latticeVec d L n‖ ^ 2) / 2) + +/-- The periodized Gaussian covariance +`K_L(z) = sum_n exp(-|z+Ln|^2/2) / sum_n exp(-|Ln|^2/2)`; the denominator is the theta +sum at `z = 0`, so `K_L(0) = 1` (variance one) whenever that denominator is nonzero (the +summability field is meant to provide this; not shown here). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2)" +Interface: summability of both theta sums is the field `GaussianFieldInterface.summable`. +Does not claim: positivity of the Fourier coefficients (Section 2 of the source) or any property of `K_L` beyond its definition. -/ +noncomputable def periodicCovariance (d : ℕ) (L : ℝ) (z : E d) : ℝ := + thetaSum d L z / thetaSum d L 0 + +/-- The `K_L`-quadratic form `sum_{i,j} a_i a_j K_L(x_i - x_j)` of a finite real linear +combination `sum_i a_i f(x_i)` of point evaluations: its variance under a centred +stationary field with covariance `K_L`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "let f be the centered stationary Gaussian field of variance one with covariance" +Interface: nonnegativity of this form is the field `GaussianFieldInterface.quad_nonneg`. +Does not claim: positive definiteness (Section 2 of the source). -/ +noncomputable def covQuadForm (d : ℕ) (L : ℝ) {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ) : ℝ := + ∑ i, ∑ j, a i * a j * periodicCovariance d L (x i - x j) + +/-- `L`-periodicity of a function on `R^d`: invariance under every lattice translation. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance" +Interface: none; this is the defining predicate of a function on the torus. +Does not claim: anything about the field. -/ +def IsLPeriodic (d : ℕ) (L : ℝ) (g : E d → ℝ) : Prop := + ∀ (x : E d) (n : Fin d → ℤ), g (x + latticeVec d L n) = g x + +/-- Smoothness of a sample path: `C^n` for every finite `n` (written without `⊤`). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "In particular the field has a smooth version and all finite moments of its global derivative suprema exist." +Interface: smoothness of every sample path is the field `GaussianFieldInterface.smooth`. +Does not claim: existence of a smooth version, or moment bounds on derivative suprema. -/ +def IsSmooth (d : ℕ) (g : E d → ℝ) : Prop := + ∀ n : ℕ, ContDiff ℝ n g + +/-- Interface for the centred stationary Gaussian field of variance one on the torus +`R^d/(L Z^d)` with covariance `K_L`. The field is GIVEN: its probability space, its sample +paths, their smoothness and periodicity, the Gaussian law of every finite real linear +combination of point evaluations (law `gaussianReal 0 v` with `v` the `K_L`-quadratic +form) and the covariance identity are hypothesis fields, as is summability of the theta +sums. Existence of such an object is not asserted. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance" +Interface: every field of this structure is a given property of the field, never derived. +Does not claim: existence or uniqueness in law of the field; the Fourier representation and positive spectrum of Section 2; almost-sure Morse genericity of Section 8. -/ +structure GaussianFieldInterface (d : ℕ) (L : ℝ) where + /-- The underlying probability space (sample space). -/ + Ω : Type + -- Its sigma-algebra (square-bracket field, as in Mathlib's bundled categories). It is not + -- registered for typeclass resolution after this declaration; consumers bring it into + -- scope with `haveI := G.mΩ`. + [mΩ : MeasurableSpace Ω] + /-- The probability measure `P`. -/ + P : Measure Ω + -- `P` is a probability measure (square-bracket field). + [isProb : IsProbabilityMeasure P] + /-- The sample paths `ω ↦ f_ω : R^d → ℝ`. -/ + F : Ω → E d → ℝ + /-- Section 2 clause: "the field has a smooth version"; every sample path is smooth. -/ + smooth : ∀ ω, IsSmooth d (F ω) + /-- Section 1 clause: the field lives "On the flat torus X = R^d/(L Z^d)"; every sample path + is `L`-periodic. -/ + periodic : ∀ ω, IsLPeriodic d L (F ω) + /-- Section 1 clause: the theta sums defining `K_L` converge (taken as a hypothesis). -/ + summable : ∀ z : E d, Summable (fun n : Fin d → ℤ => Real.exp (-(‖z + latticeVec d L n‖ ^ 2) / 2)) + /-- The `K_L`-quadratic form is nonnegative (so it is a variance). -/ + quad_nonneg : ∀ {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ), 0 ≤ covQuadForm d L x a + /-- Section 1 clause: "the centered stationary Gaussian field of variance one with covariance" + `K_L`; every finite real linear combination of point evaluations has the centred Gaussian + law whose variance is the `K_L`-quadratic form. -/ + gaussian : ∀ {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ), + Measure.map (fun ω => ∑ i, a i * F ω (x i)) P = + ProbabilityTheory.gaussianReal 0 ⟨covQuadForm d L x a, quad_nonneg x a⟩ + /-- Covariance identity `E[f(x) f(y)] = K_L(x - y)` (centred, stationary). -/ + covariance : ∀ x y : E d, ∫ ω, F ω x * F ω y ∂P = periodicCovariance d L (x - y) + +/-- The Hessian of `g` at `x` as a bilinear form, through the second iterated Fréchet +derivative: `hessian d g x v w = D^2 g(x)[v, w]`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue." +Interface: none. +Does not claim: symmetry of the form (true for `C^2` functions, not stated here). -/ +noncomputable def hessian (d : ℕ) (g : E d → ℝ) (x v w : E d) : ℝ := + iteratedFDeriv ℝ 2 g x ![v, w] + +/-- Critical point: vanishing Fréchet derivative (`grad f = 0`). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "grad f(M)=grad f(S)=0." +Interface: none. +Does not claim: differentiability of `g` at `x` (if `g` is not differentiable, `fderiv` is `0`). -/ +def IsCriticalPoint (d : ℕ) (g : E d → ℝ) (x : E d) : Prop := + fderiv ℝ g x = 0 + +/-- Sylvester-inertia encoding of "the Hessian at `x` is nondegenerate with exactly `j` +negative and `d - j` positive eigenvalues": there are complementary subspaces `V`, `W` of +dimensions `j` and `d - j` on which the Hessian form is negative definite, respectively +positive definite. Intended to agree with the eigenvalue count for symmetric (`C^2`) +Hessians; not shown here. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "The index counts negative eigenvalues." +Interface: none. +Does not claim: equivalence with an eigenvalue count for a non-symmetric form. -/ +def HessianHasInertia (d : ℕ) (g : E d → ℝ) (x : E d) (j : ℕ) : Prop := + ∃ V W : Submodule ℝ (E d), IsCompl V W ∧ + Module.finrank ℝ V = j ∧ Module.finrank ℝ W = d - j ∧ + (∀ v ∈ V, v ≠ 0 → hessian d g x v v < 0) ∧ + (∀ w ∈ W, w ≠ 0 → 0 < hessian d g x w w) + +/-- Nondegenerate local maximum: a critical point whose Hessian has `d` negative +eigenvalues (negative definite). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue." +Interface: none. +Does not claim: that every local maximum of a sample path is nondegenerate (Morse genericity is Section 8 of the source). -/ +def IsNondegMax (d : ℕ) (g : E d → ℝ) (x : E d) : Prop := + IsCriticalPoint d g x ∧ HessianHasInertia d g x d + +/-- Index-(d-1) saddle: a critical point whose Hessian has `d - 1` negative and one +positive eigenvalue. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue." +Interface: none. +Does not claim: that such a saddle is a merging saddle of any superlevel bar. -/ +def IsIndexSaddle (d : ℕ) (g : E d → ℝ) (x : E d) : Prop := + IsCriticalPoint d g x ∧ HessianHasInertia d g x (d - 1) + +/-- The pin `M = -(r/2) u` of the embedded local cylinder with axial unit vector `u`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "M=-(r/2)u, S=(r/2)u," +Interface: none. +Does not claim: that `u` is a unit vector; this pin is an auxiliary definition consumed by no statement of this packet (the typed pair law `p` is a primitive), so it is attached to no claim. -/ +noncomputable def pinM (d : ℕ) (r : ℝ) (u : E d) : E d := + -((r / 2) • u) + +/-- The pin `S = (r/2) u` of the embedded local cylinder with axial unit vector `u`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "M=-(r/2)u, S=(r/2)u," +Interface: none. +Does not claim: that `u` is a unit vector; this pin is an auxiliary definition consumed by no statement of this packet (the typed pair law `p` is a primitive), so it is attached to no claim. -/ +noncomputable def pinS (d : ℕ) (r : ℝ) (u : E d) : E d := + (r / 2) • u + +/-- Torus distance between two points of `R^d`: the infimum over the lattice of +`|x - y + L n|`. For `L > 0` it is intended to vanish exactly when `x` and `y` represent the +same torus point; not shown here. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Count ordered maximum/saddle pairs with 00, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume." +Interface: none. +Does not claim: that the height gap of an elder pair is the persistence of a bar in any persistence module. -/ +def heightGap (d : ℕ) (g : E d → ℝ) (M S : E d) : ℝ := + g M - g S + +/-- Scaled gap `(f(M) - f(S)) / distance^3` of an ordered pair, with the torus distance. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Count ordered maximum/saddle pairs with 00, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume." +Interface: none. +Does not claim: positivity of the height gap (added separately where the source requires it). -/ +def IsCandidatePair (d : ℕ) (L : ℝ) (g : E d → ℝ) (M S : E d) : Prop := + IsNondegMax d g M ∧ IsIndexSaddle d g S ∧ 0 < torusDist d L M S + +/-- Morse genericity with distinct critical values on the torus: every critical point is +nondegenerate (has some inertia) and critical points at distinct torus positions have +distinct critical values. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Countable exhaustion now proves that Q is almost surely Morse with distinct critical values." +Interface: none; used only as a hypothesis of the elder-partner interface clause. +Does not claim: that sample paths are almost surely Morse (Section 8 of the source). -/ +def IsMorseDistinct (d : ℕ) (L : ℝ) (g : E d → ℝ) : Prop := + (∀ x : E d, IsCriticalPoint d g x → ∃ j : ℕ, HessianHasInertia d g x j) ∧ + (∀ x y : E d, IsCriticalPoint d g x → IsCriticalPoint d g y → + 0 < torusDist d L x y → g x ≠ g y) + +/-- Interface for the superlevel-set H0 persistence objects used by the lifetime track: +the maximin death height `d_f(M)`, the essential (global-maximum) class, and the ordinary +elder death partner relation. These are GIVEN relations; the fields transcribe the source +clauses that characterise them on smooth periodic functions, and the elder relation is +pinned only on the Morse distinct-value locus, exactly as the source does. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint." +Interface: `deathHeight`, `IsEssential`, `ElderPartner` are given; `deathHeight_maximin`, `essential_iff`, `elderPartner_iff` are their hypothesis clauses. +Does not claim: Borel measurability of the elder event (Section 8 / the Section 9 replacement), identification with a persistence-module elder rule, or any behaviour off the Morse distinct-value locus. -/ +structure SuperlevelBarInterface (d : ℕ) (L : ℝ) where + /-- Section 8 clause: the maximin connection level + `d_f(M)=sup_{gamma(0)=M, f(gamma(1))>f(M)} min_t f(gamma(t))`. -/ + deathHeight : (E d → ℝ) → E d → ℝ + /-- The essential class: the global maximum, which "has no older endpoint" and whose + superlevel H0 bar is not finite. -/ + IsEssential : (E d → ℝ) → E d → Prop + /-- Section 1 clause: "the global ordinary superlevel elder death partner of M is S". -/ + ElderPartner : (E d → ℝ) → E d → E d → Prop + /-- `M` is essential exactly when no point is strictly higher (so the supremum in the + maximin is over an empty set, "equal to -infinity"). -/ + essential_iff : ∀ (g : E d → ℝ), IsSmooth d g → IsLPeriodic d L g → + ∀ M : E d, IsEssential g M ↔ ∀ y : E d, g y ≤ g M + /-- Section 8 maximin clause, in the form `h < d_f(M)` iff some path from `M` to a strictly + higher point stays strictly above `h`. -/ + deathHeight_maximin : ∀ (g : E d → ℝ), IsSmooth d g → IsLPeriodic d L g → + ∀ M : E d, ¬ IsEssential g M → ∀ h : ℝ, + h < deathHeight g M ↔ + ∃ y : E d, g M < g y ∧ ∃ γ : Path M y, ∀ t : unitInterval, h < g (γ t) + /-- Section 8 / Section 14 clause: on the Morse distinct-value locus, "equality + d_f(M)=f(S)" with `M` a nondegenerate maximum and `S` an index-(d-1) saddle "expresses + ordinary elder death at S". -/ + elderPartner_iff : ∀ (g : E d → ℝ), IsSmooth d g → IsLPeriodic d L g → IsMorseDistinct d L g → + ∀ M S : E d, ElderPartner g M S ↔ + (IsNondegMax d g M ∧ IsIndexSaddle d g S ∧ ¬ IsEssential g M ∧ deathHeight g M = g S) + +/-- Finite ordinary superlevel H0 bar, as an ordered pair: a nondegenerate maximum `M` +(birth) with its ordinary elder death partner `S`, an index-(d-1) saddle at positive torus +distance. The essential global-maximum class is excluded because `ElderPartner` requires +`¬ IsEssential` on the Morse locus. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "each finite superlevel H0 bar has exactly one local-maximum birth and one index-(d-1) merging-saddle death." +Interface: `I : SuperlevelBarInterface d L` is given. +Does not claim: the identification of these pairs with bars of a persistence module; that the count of such pairs equals the finite-bar count off the Morse locus. -/ +def IsFiniteBar (d : ℕ) (L : ℝ) (I : SuperlevelBarInterface d L) (g : E d → ℝ) (M S : E d) : Prop := + IsCandidatePair d L g M S ∧ 0 < heightGap d g M S ∧ I.ElderPartner g M S + +/-- The set of pairs of a population `Pair` with both points in the fundamental domain and +lifetime (height gap) in `A`; `Pair g M S` is the membership predicate of the population. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Let nu_cand(ell) be their expected lifetime density per unit volume; let nu_eld(ell) count only pairs which are actual ordinary elder partners." +Interface: the population predicate `Pair` is a parameter. +Does not claim: finiteness of this set (a field of the density-version predicate). -/ +def pairSet (d : ℕ) (L : ℝ) (Pair : (E d → ℝ) → E d → E d → Prop) (g : E d → ℝ) (A : Set ℝ) : + Set (E d × E d) := + {p | p.1 ∈ FundDomain d L ∧ p.2 ∈ FundDomain d L ∧ Pair g p.1 p.2 ∧ heightGap d g p.1 p.2 ∈ A} + +/-- Number of pairs of the population with lifetime in `A` and base points in the +fundamental domain (`Set.ncard`, which is `0` for an infinite set). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Let nu_cand(ell) be their expected lifetime density per unit volume; let nu_eld(ell) count only pairs which are actual ordinary elder partners." +Interface: the population predicate `Pair` is a parameter. +Does not claim: finiteness of the counted set. -/ +noncomputable def pairCount (d : ℕ) (L : ℝ) (Pair : (E d → ℝ) → E d → E d → Prop) + (g : E d → ℝ) (A : Set ℝ) : ℕ := + Set.ncard (pairSet d L Pair g A) + +/-- `ν` is a VERSION of the expected per-unit-volume lifetime density of the population +`Pair` under the field `G`: it is nonnegative and measurable, the pair sets are almost +surely finite with integrable counts, `ν` is integrable on every Borel `A ⊆ (0, ∞)`, and +`∫_A ν = E[#pairs with lifetime in A and base point in [0,L)^d] / L^d` for every such `A`. +The source's densities are only defined up to such versions. The sigma-algebra of the sample +space is brought into scope by the inlined term-level `haveI := G.mΩ` (a square-bracket +field of a bundled record is not registered for typeclass resolution after the declaration). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "These densities admit versions with" +Interface: the field `G` and the population predicate `Pair` are given; the Kac-Rice representation of the expectation (Sections 9-10 of the source) is NOT used; the density is characterised only by the integral identity. +Does not claim: existence of a version; the weighted Kac-Rice identity; the radial ledger (10.2). Measurability of the pair count as a function of the sample is REQUIRED by the predicate (through `Integrable`, hence `AEStronglyMeasurable`), not established here. -/ +def IsLifetimeDensityVersion (d : ℕ) (L : ℝ) (G : GaussianFieldInterface d L) + (Pair : (E d → ℝ) → E d → E d → Prop) (ν : ℝ → ℝ) : Prop := + haveI := G.mΩ + (∀ ℓ : ℝ, 0 ≤ ν ℓ) ∧ Measurable ν ∧ + (∀ A : Set ℝ, MeasurableSet A → A ⊆ Set.Ioi 0 → + (∀ᵐ ω ∂G.P, (pairSet d L Pair (G.F ω) A).Finite) ∧ + Integrable (fun ω => ((pairCount d L Pair (G.F ω) A : ℕ) : ℝ)) G.P ∧ + IntegrableOn ν A ∧ + ∫ ℓ in A, ν ℓ = (∫ ω, ((pairCount d L Pair (G.F ω) A : ℕ) : ℝ) ∂G.P) / L ^ d) + +end UniversalLaw.Spec.Field diff --git a/specifications/lean_statements_20261010/Specifications/Lifetime.lean b/specifications/lean_statements_20261010/Specifications/Lifetime.lean new file mode 100644 index 000000000..f2508f7f2 --- /dev/null +++ b/specifications/lean_statements_20261010/Specifications/Lifetime.lean @@ -0,0 +1,634 @@ +import Mathlib + +/-! +# Lifetime-track statements: parent Theorems A, B, C, the bounded remainder R and the +# planar elder lower bound / density gap (statements only) + +Scientific effect: NONE. Formal progress of every declaration here: `specified`. +Organizational-independence credit of this transcription: 0 (Anthropic / Claude). + +This module states, as `Prop`-valued definitions, the shapes of the parent lifetime +results (1.1), (1.2), (1.3) of UNIFORM_MATRIX_CAP_AND_LIFETIME.md, of the bounded +remainder (R1) of LIFETIME_REMAINDER.md, and of the planar lower bound (E2), the +two-sided rate (E4) and the compact-window density gap (E11) of +ELDER_LOWER_AND_DENSITY_GAP.md. Every statement is parametrised by interface objects +(the Gaussian field, the superlevel-bar / elder-partner relation, and the typed pair law +probability `p_r(b,k,R)`); it is a statement about whatever objects satisfy those +interfaces, and the identification of those objects with the objects of the source texts +is not part of the statement. + +This module is self-contained: it imports only Mathlib and re-declares, verbatim up to +the namespace, the shared objects of `Specifications/Field.lean` (covariance, field +interface, critical points, Hessian inertia, pairs, superlevel bars, density versions). +The duplication is deliberate at this layer and is recorded as such. + +What this module does not establish. It consists only of definitions and structures; +it states nothing as a result and proves nothing: none of the parent results, the +remainder bound or the planar bounds is shown here, and no kernel receipt about them +exists. It does not transcribe the explicit Gaussian/angular integral (15.2) for the +leading coefficient `c_{d,L}` (the coefficient is characterised only through the +leading-term asymptotics of (1.3)); it does not transcribe the regression law +`Q_{r,b,k,R}`, the typed weight `W_r`, the full normalizer `Z_r` or the marked Kac-Rice +representation (all informal in the source and taken as the interface field `p`); it +does not transcribe the reading-rule amendments bound to the parent (congruence erratum, +Section 9 replacement v1.1, wording W1, embedding radius `r < L/(4 sqrt 2)`), nor the +marked-cylinder cap support; it asserts no numerical `C`, `r_*`, `z_*`, `ell_*` or +coefficient value; it does not state Theorem A as consumed by Theorem R; and it does +not state the compact-window `O(ell^(2/3))` difference for the unrestricted densities. +The planar statements (E4) and (E11) carry the parent upper bound (1.1), for the same +typed pair law at `d = 2`, as an explicit hypothesis, as the source does ("If the precise +Theorem A upper bound in [LP] is consumed"; its Section 4 says the upper half of (E11) +consumes the parent upper bound already identified there); the direct lower bound (E2) is +unconditional, as in the source. +Nothing here promotes, reclassifies or discharges any claim, premise or obligation of +the registers; the landing disposition `REVIEWED_SCOPED` (lifetime-remainder) and the +graph classification and review disposition recorded for +`math.uniform-matrix-cap-lifetime` are transcribed from the registers by the statement +register, never re-derived here; alignment of these statements with the sources is +PENDING_INDEPENDENT_REVIEW. + +Sources (Layer 0, byte-pinned, `d6g8k5htny-coder/Math-`): +* `lifetime-parent`: commit `c39164254ad1529ed49543743598d5dcb24bab98`, path + `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, + sha256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. +* `lifetime-remainder`: commit `760340e921ac4ceda296b8118da936f1133e956e`, path + `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md`, + sha256 `380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a`. +* `elder-lower-density-gap`: commit `6c020d6a72936632f2055122e71a7818a4ff497e`, path + `frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md`, + sha256 `8ce05a43bde8e0d718f09e2da1ffe75d3ec5893489254afdbf5b58b602cdba0c`. +-/ + +namespace UniversalLaw.Spec.Lifetime + +open MeasureTheory + +/-! ### Shared objects, re-declared from `Specifications/Field.lean` (same text, this +namespace). The duplication is deliberate: modules of this packet import only Mathlib. -/ + +/-- The ambient Euclidean space `R^d` in which the torus `X = R^d/(L Z^d)` is realised +by `L`-periodic functions. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance" +Interface: none; this is the ambient space `EuclideanSpace ℝ (Fin d)`. +Does not claim: that the torus is represented by any quotient type; periodicity is a hypothesis on functions. -/ +abbrev E (d : ℕ) : Type := EuclideanSpace ℝ (Fin d) + +/-- The lattice vector `L n` for `n ∈ Z^d`, written in the standard coordinate vectors. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2)" +Interface: none. +Does not claim: anything beyond the definition of the vector `L n`. -/ +noncomputable def latticeVec (d : ℕ) (L : ℝ) (n : Fin d → ℤ) : E d := + ∑ i, EuclideanSpace.single i (L * (n i : ℝ)) + +/-- The unnormalised periodized Gaussian (theta) sum `sum_{n in Z^d} exp(-|z+Ln|^2/2)`, +written as a `tsum` over `Z^d`. If the family were not summable the `tsum` would be `0`; +summability is a hypothesis field of `GaussianFieldInterface`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2)" +Interface: summability of the family is the field `GaussianFieldInterface.summable`, not derived here. +Does not claim: convergence of the sum. -/ +noncomputable def thetaSum (d : ℕ) (L : ℝ) (z : E d) : ℝ := + ∑' n : Fin d → ℤ, Real.exp (-(‖z + latticeVec d L n‖ ^ 2) / 2) + +/-- The periodized Gaussian covariance +`K_L(z) = sum_n exp(-|z+Ln|^2/2) / sum_n exp(-|Ln|^2/2)`; the denominator is the theta +sum at `z = 0`, so `K_L(0) = 1` (variance one) whenever that denominator is nonzero (the +summability field is meant to provide this; not shown here). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2)" +Interface: summability of both theta sums is the field `GaussianFieldInterface.summable`. +Does not claim: positivity of the Fourier coefficients (Section 2 of the source) or any property of `K_L` beyond its definition. -/ +noncomputable def periodicCovariance (d : ℕ) (L : ℝ) (z : E d) : ℝ := + thetaSum d L z / thetaSum d L 0 + +/-- The `K_L`-quadratic form `sum_{i,j} a_i a_j K_L(x_i - x_j)` of a finite real linear +combination `sum_i a_i f(x_i)` of point evaluations: its variance under a centred +stationary field with covariance `K_L`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "let f be the centered stationary Gaussian field of variance one with covariance" +Interface: nonnegativity of this form is the field `GaussianFieldInterface.quad_nonneg`. +Does not claim: positive definiteness (Section 2 of the source). -/ +noncomputable def covQuadForm (d : ℕ) (L : ℝ) {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ) : ℝ := + ∑ i, ∑ j, a i * a j * periodicCovariance d L (x i - x j) + +/-- `L`-periodicity of a function on `R^d`: invariance under every lattice translation. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance" +Interface: none; this is the defining predicate of a function on the torus. +Does not claim: anything about the field. -/ +def IsLPeriodic (d : ℕ) (L : ℝ) (g : E d → ℝ) : Prop := + ∀ (x : E d) (n : Fin d → ℤ), g (x + latticeVec d L n) = g x + +/-- Smoothness of a sample path: `C^n` for every finite `n` (written without `⊤`). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "In particular the field has a smooth version and all finite moments of its global derivative suprema exist." +Interface: smoothness of every sample path is the field `GaussianFieldInterface.smooth`. +Does not claim: existence of a smooth version, or moment bounds on derivative suprema. -/ +def IsSmooth (d : ℕ) (g : E d → ℝ) : Prop := + ∀ n : ℕ, ContDiff ℝ n g + +/-- Interface for the centred stationary Gaussian field of variance one on the torus +`R^d/(L Z^d)` with covariance `K_L`. The field is GIVEN: its probability space, its sample +paths, their smoothness and periodicity, the Gaussian law of every finite real linear +combination of point evaluations (law `gaussianReal 0 v` with `v` the `K_L`-quadratic +form) and the covariance identity are hypothesis fields, as is summability of the theta +sums. Existence of such an object is not asserted. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance" +Interface: every field of this structure is a given property of the field, never derived. +Does not claim: existence or uniqueness in law of the field; the Fourier representation and positive spectrum of Section 2; almost-sure Morse genericity of Section 8. -/ +structure GaussianFieldInterface (d : ℕ) (L : ℝ) where + /-- The underlying probability space (sample space). -/ + Ω : Type + -- Its sigma-algebra (square-bracket field, as in Mathlib's bundled categories). It is not + -- registered for typeclass resolution after this declaration; consumers bring it into + -- scope with `haveI := G.mΩ`. + [mΩ : MeasurableSpace Ω] + /-- The probability measure `P`. -/ + P : Measure Ω + -- `P` is a probability measure (square-bracket field). + [isProb : IsProbabilityMeasure P] + /-- The sample paths `ω ↦ f_ω : R^d → ℝ`. -/ + F : Ω → E d → ℝ + /-- Section 2 clause: "the field has a smooth version"; every sample path is smooth. -/ + smooth : ∀ ω, IsSmooth d (F ω) + /-- Section 1 clause: the field lives "On the flat torus X = R^d/(L Z^d)"; every sample path + is `L`-periodic. -/ + periodic : ∀ ω, IsLPeriodic d L (F ω) + /-- Section 1 clause: the theta sums defining `K_L` converge (taken as a hypothesis). -/ + summable : ∀ z : E d, Summable (fun n : Fin d → ℤ => Real.exp (-(‖z + latticeVec d L n‖ ^ 2) / 2)) + /-- The `K_L`-quadratic form is nonnegative (so it is a variance). -/ + quad_nonneg : ∀ {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ), 0 ≤ covQuadForm d L x a + /-- Section 1 clause: "the centered stationary Gaussian field of variance one with covariance" + `K_L`; every finite real linear combination of point evaluations has the centred Gaussian + law whose variance is the `K_L`-quadratic form. -/ + gaussian : ∀ {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ), + Measure.map (fun ω => ∑ i, a i * F ω (x i)) P = + ProbabilityTheory.gaussianReal 0 ⟨covQuadForm d L x a, quad_nonneg x a⟩ + /-- Covariance identity `E[f(x) f(y)] = K_L(x - y)` (centred, stationary). -/ + covariance : ∀ x y : E d, ∫ ω, F ω x * F ω y ∂P = periodicCovariance d L (x - y) + +/-- The Hessian of `g` at `x` as a bilinear form, through the second iterated Fréchet +derivative: `hessian d g x v w = D^2 g(x)[v, w]`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue." +Interface: none. +Does not claim: symmetry of the form (true for `C^2` functions, not stated here). -/ +noncomputable def hessian (d : ℕ) (g : E d → ℝ) (x v w : E d) : ℝ := + iteratedFDeriv ℝ 2 g x ![v, w] + +/-- Critical point: vanishing Fréchet derivative (`grad f = 0`). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "grad f(M)=grad f(S)=0." +Interface: none. +Does not claim: differentiability of `g` at `x` (if `g` is not differentiable, `fderiv` is `0`). -/ +def IsCriticalPoint (d : ℕ) (g : E d → ℝ) (x : E d) : Prop := + fderiv ℝ g x = 0 + +/-- Sylvester-inertia encoding of "the Hessian at `x` is nondegenerate with exactly `j` +negative and `d - j` positive eigenvalues": there are complementary subspaces `V`, `W` of +dimensions `j` and `d - j` on which the Hessian form is negative definite, respectively +positive definite. Intended to agree with the eigenvalue count for symmetric (`C^2`) +Hessians; not shown here. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "The index counts negative eigenvalues." +Interface: none. +Does not claim: equivalence with an eigenvalue count for a non-symmetric form. -/ +def HessianHasInertia (d : ℕ) (g : E d → ℝ) (x : E d) (j : ℕ) : Prop := + ∃ V W : Submodule ℝ (E d), IsCompl V W ∧ + Module.finrank ℝ V = j ∧ Module.finrank ℝ W = d - j ∧ + (∀ v ∈ V, v ≠ 0 → hessian d g x v v < 0) ∧ + (∀ w ∈ W, w ≠ 0 → 0 < hessian d g x w w) + +/-- Nondegenerate local maximum: a critical point whose Hessian has `d` negative +eigenvalues (negative definite). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue." +Interface: none. +Does not claim: that every local maximum of a sample path is nondegenerate (Morse genericity is Section 8 of the source). -/ +def IsNondegMax (d : ℕ) (g : E d → ℝ) (x : E d) : Prop := + IsCriticalPoint d g x ∧ HessianHasInertia d g x d + +/-- Index-(d-1) saddle: a critical point whose Hessian has `d - 1` negative and one +positive eigenvalue. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue." +Interface: none. +Does not claim: that such a saddle is a merging saddle of any superlevel bar. -/ +def IsIndexSaddle (d : ℕ) (g : E d → ℝ) (x : E d) : Prop := + IsCriticalPoint d g x ∧ HessianHasInertia d g x (d - 1) + +/-- The pin `M = -(r/2) u` of the embedded local cylinder with axial unit vector `u`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "M=-(r/2)u, S=(r/2)u," +Interface: none. +Does not claim: that `u` is a unit vector; this pin is an auxiliary definition consumed by no statement of this packet (the typed pair law `p` is a primitive), so it is attached to no claim. -/ +noncomputable def pinM (d : ℕ) (r : ℝ) (u : E d) : E d := + -((r / 2) • u) + +/-- The pin `S = (r/2) u` of the embedded local cylinder with axial unit vector `u`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "M=-(r/2)u, S=(r/2)u," +Interface: none. +Does not claim: that `u` is a unit vector; this pin is an auxiliary definition consumed by no statement of this packet (the typed pair law `p` is a primitive), so it is attached to no claim. -/ +noncomputable def pinS (d : ℕ) (r : ℝ) (u : E d) : E d := + (r / 2) • u + +/-- Torus distance between two points of `R^d`: the infimum over the lattice of +`|x - y + L n|`. For `L > 0` it is intended to vanish exactly when `x` and `y` represent the +same torus point; not shown here. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Count ordered maximum/saddle pairs with 00, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume." +Interface: none. +Does not claim: that the height gap of an elder pair is the persistence of a bar in any persistence module. -/ +def heightGap (d : ℕ) (g : E d → ℝ) (M S : E d) : ℝ := + g M - g S + +/-- Scaled gap `(f(M) - f(S)) / distance^3` of an ordered pair, with the torus distance. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Count ordered maximum/saddle pairs with 00, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume." +Interface: none. +Does not claim: positivity of the height gap (added separately where the source requires it). -/ +def IsCandidatePair (d : ℕ) (L : ℝ) (g : E d → ℝ) (M S : E d) : Prop := + IsNondegMax d g M ∧ IsIndexSaddle d g S ∧ 0 < torusDist d L M S + +/-- Morse genericity with distinct critical values on the torus: every critical point is +nondegenerate (has some inertia) and critical points at distinct torus positions have +distinct critical values. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Countable exhaustion now proves that Q is almost surely Morse with distinct critical values." +Interface: none; used only as a hypothesis of the elder-partner interface clause. +Does not claim: that sample paths are almost surely Morse (Section 8 of the source). -/ +def IsMorseDistinct (d : ℕ) (L : ℝ) (g : E d → ℝ) : Prop := + (∀ x : E d, IsCriticalPoint d g x → ∃ j : ℕ, HessianHasInertia d g x j) ∧ + (∀ x y : E d, IsCriticalPoint d g x → IsCriticalPoint d g y → + 0 < torusDist d L x y → g x ≠ g y) + +/-- Interface for the superlevel-set H0 persistence objects used by the lifetime track: +the maximin death height `d_f(M)`, the essential (global-maximum) class, and the ordinary +elder death partner relation. These are GIVEN relations; the fields transcribe the source +clauses that characterise them on smooth periodic functions, and the elder relation is +pinned only on the Morse distinct-value locus, exactly as the source does. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint." +Interface: `deathHeight`, `IsEssential`, `ElderPartner` are given; `deathHeight_maximin`, `essential_iff`, `elderPartner_iff` are their hypothesis clauses. +Does not claim: Borel measurability of the elder event (Section 8 / the Section 9 replacement), identification with a persistence-module elder rule, or any behaviour off the Morse distinct-value locus. -/ +structure SuperlevelBarInterface (d : ℕ) (L : ℝ) where + /-- Section 8 clause: the maximin connection level + `d_f(M)=sup_{gamma(0)=M, f(gamma(1))>f(M)} min_t f(gamma(t))`. -/ + deathHeight : (E d → ℝ) → E d → ℝ + /-- The essential class: the global maximum, which "has no older endpoint" and whose + superlevel H0 bar is not finite. -/ + IsEssential : (E d → ℝ) → E d → Prop + /-- Section 1 clause: "the global ordinary superlevel elder death partner of M is S". -/ + ElderPartner : (E d → ℝ) → E d → E d → Prop + /-- `M` is essential exactly when no point is strictly higher (so the supremum in the + maximin is over an empty set, "equal to -infinity"). -/ + essential_iff : ∀ (g : E d → ℝ), IsSmooth d g → IsLPeriodic d L g → + ∀ M : E d, IsEssential g M ↔ ∀ y : E d, g y ≤ g M + /-- Section 8 maximin clause, in the form `h < d_f(M)` iff some path from `M` to a strictly + higher point stays strictly above `h`. -/ + deathHeight_maximin : ∀ (g : E d → ℝ), IsSmooth d g → IsLPeriodic d L g → + ∀ M : E d, ¬ IsEssential g M → ∀ h : ℝ, + h < deathHeight g M ↔ + ∃ y : E d, g M < g y ∧ ∃ γ : Path M y, ∀ t : unitInterval, h < g (γ t) + /-- Section 8 / Section 14 clause: on the Morse distinct-value locus, "equality + d_f(M)=f(S)" with `M` a nondegenerate maximum and `S` an index-(d-1) saddle "expresses + ordinary elder death at S". -/ + elderPartner_iff : ∀ (g : E d → ℝ), IsSmooth d g → IsLPeriodic d L g → IsMorseDistinct d L g → + ∀ M S : E d, ElderPartner g M S ↔ + (IsNondegMax d g M ∧ IsIndexSaddle d g S ∧ ¬ IsEssential g M ∧ deathHeight g M = g S) + +/-- Finite ordinary superlevel H0 bar, as an ordered pair: a nondegenerate maximum `M` +(birth) with its ordinary elder death partner `S`, an index-(d-1) saddle at positive torus +distance. The essential global-maximum class is excluded because `ElderPartner` requires +`¬ IsEssential` on the Morse locus. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "each finite superlevel H0 bar has exactly one local-maximum birth and one index-(d-1) merging-saddle death." +Interface: `I : SuperlevelBarInterface d L` is given. +Does not claim: the identification of these pairs with bars of a persistence module; that the count of such pairs equals the finite-bar count off the Morse locus. -/ +def IsFiniteBar (d : ℕ) (L : ℝ) (I : SuperlevelBarInterface d L) (g : E d → ℝ) (M S : E d) : Prop := + IsCandidatePair d L g M S ∧ 0 < heightGap d g M S ∧ I.ElderPartner g M S + +/-- The set of pairs of a population `Pair` with both points in the fundamental domain and +lifetime (height gap) in `A`; `Pair g M S` is the membership predicate of the population. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Let nu_cand(ell) be their expected lifetime density per unit volume; let nu_eld(ell) count only pairs which are actual ordinary elder partners." +Interface: the population predicate `Pair` is a parameter. +Does not claim: finiteness of this set (a field of the density-version predicate). -/ +def pairSet (d : ℕ) (L : ℝ) (Pair : (E d → ℝ) → E d → E d → Prop) (g : E d → ℝ) (A : Set ℝ) : + Set (E d × E d) := + {p | p.1 ∈ FundDomain d L ∧ p.2 ∈ FundDomain d L ∧ Pair g p.1 p.2 ∧ heightGap d g p.1 p.2 ∈ A} + +/-- Number of pairs of the population with lifetime in `A` and base points in the +fundamental domain (`Set.ncard`, which is `0` for an infinite set). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Let nu_cand(ell) be their expected lifetime density per unit volume; let nu_eld(ell) count only pairs which are actual ordinary elder partners." +Interface: the population predicate `Pair` is a parameter. +Does not claim: finiteness of the counted set. -/ +noncomputable def pairCount (d : ℕ) (L : ℝ) (Pair : (E d → ℝ) → E d → E d → Prop) + (g : E d → ℝ) (A : Set ℝ) : ℕ := + Set.ncard (pairSet d L Pair g A) + +/-- `ν` is a VERSION of the expected per-unit-volume lifetime density of the population +`Pair` under the field `G`: it is nonnegative and measurable, the pair sets are almost +surely finite with integrable counts, `ν` is integrable on every Borel `A ⊆ (0, ∞)`, and +`∫_A ν = E[#pairs with lifetime in A and base point in [0,L)^d] / L^d` for every such `A`. +The source's densities are only defined up to such versions. The sigma-algebra of the sample +space is brought into scope by the inlined term-level `haveI := G.mΩ` (a square-bracket +field of a bundled record is not registered for typeclass resolution after the declaration). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "These densities admit versions with" +Interface: the field `G` and the population predicate `Pair` are given; the Kac-Rice representation of the expectation (Sections 9-10 of the source) is NOT used; the density is characterised only by the integral identity. +Does not claim: existence of a version; the weighted Kac-Rice identity; the radial ledger (10.2). Measurability of the pair count as a function of the sample is REQUIRED by the predicate (through `Integrable`, hence `AEStronglyMeasurable`), not established here. -/ +def IsLifetimeDensityVersion (d : ℕ) (L : ℝ) (G : GaussianFieldInterface d L) + (Pair : (E d → ℝ) → E d → E d → Prop) (ν : ℝ → ℝ) : Prop := + haveI := G.mΩ + (∀ ℓ : ℝ, 0 ≤ ν ℓ) ∧ Measurable ν ∧ + (∀ A : Set ℝ, MeasurableSet A → A ⊆ Set.Ioi 0 → + (∀ᵐ ω ∂G.P, (pairSet d L Pair (G.F ω) A).Finite) ∧ + Integrable (fun ω => ((pairCount d L Pair (G.F ω) A : ℕ) : ℝ)) G.P ∧ + IntegrableOn ν A ∧ + ∫ ℓ in A, ν ℓ = (∫ ω, ((pairCount d L Pair (G.F ω) A : ℕ) : ℝ) ∂G.P) / L ^ d) + +/-! ### Populations of pairs -/ + +/-- Compact-window population of Theorem B: ordered maximum / index-(d-1)-saddle pairs with +`0 < distance <= ρ`, birth `f(M)` in `B = [bLo, bHi]` and scaled gap in `K = [kLo, kHi]`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Count ordered maximum/saddle pairs with 00, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume." +Interface: none beyond the shared predicates. +Does not claim: anything about the off-diagonal (far) population separately (Section 14 of the source). -/ +def IsUnrestrictedPair (d : ℕ) (L : ℝ) (g : E d → ℝ) (M S : E d) : Prop := + IsCandidatePair d L g M S ∧ 0 < heightGap d g M S + +/-! ### The typed pair law as an interface -/ + +/-- Interface for the weighted regression (typed pair) law: the only object consumed by the +statements is the probability `p_r(b,k,R)` that the global ordinary superlevel elder death +partner of the pinned maximum `M = -(r/2)u` is the pinned saddle `S = (r/2)u`, under the +law `Q^W = (W_r/Z_r) Q_{r,b,k,R}` with `W_r = |det H_M det H_S| 1{H_M<0, index(H_S)=d-1}`. +That law, its weight, its normalizer and the regression construction are informal in the +source and are NOT transcribed; `p` is a GIVEN function of radius `r`, birth `b`, gap mark +`k` and orthonormal frame `R` with values in `[0, 1]`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Let p_r(b,k,R) be the Q^W probability that the **global ordinary superlevel elder death partner** of M is S." +Interface: `p` is given; its only hypothesis fields are the bounds `0 ≤ p ≤ 1`. The parameters `G` and `I` record which field and which elder relation the law refers to; no field ties `p` to them. +Does not claim: existence of the regression law, the typed weight, the full normalizer `Z_r > 0`, or that `p` is the probability of any event under any measure declared here; that `p` depends on `G` or `I` in any way (no field mentions them; they only record which field and elder relation the law is meant to refer to). -/ +structure PairLawInterface (d : ℕ) (L : ℝ) (G : GaussianFieldInterface d L) + (I : SuperlevelBarInterface d L) where + /-- Section 1 clause: "Let p_r(b,k,R) be the Q^W probability that the global ordinary + superlevel elder death partner of M is S." -/ + p : ℝ → ℝ → ℝ → OrthonormalBasis (Fin d) ℝ (E d) → ℝ + /-- `p` is a probability: nonnegative. -/ + p_nonneg : ∀ (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)), 0 ≤ p r b k R + /-- `p` is a probability: at most one. -/ + p_le_one : ∀ (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)), p r b k R ≤ 1 + +/-! ### The parent statements (1.1), (1.2), (1.3) -/ + +/-- Shape of Theorem A (1.1), uniform selection candidate: for fixed `d ≥ 2`, `L > 0`, +compact `B = [bLo, bHi]` and `K = [kLo, kHi]` with `0 < kLo ≤ kHi`, there exist `r_* > 0` +and `C` (quantified OUTSIDE the universal quantifiers over `r`, `b`, `k` and the frame) +such that `0 <= 1 - p_r(b,k,R) <= C r^3` for all `0 < r ≤ r_*`, `b ∈ B`, `k ∈ K` and all +orthonormal frames `R`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "There exist r_*>0 and C 0` and +windows `B`, `K` of positive length with `kLo > 0`, there are a distance cutoff `r_* > 0`, a +constant `c_{B,K} > 0` and a constant `C_{B,K}` such that the compact-window candidate and +elder lifetime densities admit versions `νc`, `νe` with `νc(ℓ) ~ c_{B,K} ℓ^(-1/3)`, +`νe(ℓ) ~ c_{B,K} ℓ^(-1/3)` as `ℓ ↓ 0` (filter `nhdsWithin 0 (Set.Ioi 0)`) and +`0 ≤ νc(ℓ) - νe(ℓ) ≤ C_{B,K} ℓ^(2/3)` for all sufficiently small `ℓ > 0`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "0 <= nu_cand(ell)-nu_eld(ell) <= C_{B,K} ell^(2/3)," +Interface: `G` and `I` are given; "versions" are any `νc`, `νe` satisfying `IsLifetimeDensityVersion` (integral identity), not a Kac-Rice representation; the coefficient `c_{B,K}` is existential, not the Gaussian integral (11.3). +Does not claim: the parent result or its proof; the exact coefficient formula (11.3); existence of the versions; that the cutoff `r_*` is the one of (1.1); any numerical constant; nonauthor review or acceptance. -/ +def ParentCompactWindowDensityB (d : ℕ) (L : ℝ) (G : GaussianFieldInterface d L) + (I : SuperlevelBarInterface d L) (bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → bLo < bHi → 0 < kLo → kLo < kHi → + ∃ rStar : ℝ, 0 < rStar ∧ ∃ c : ℝ, 0 < c ∧ ∃ C : ℝ, ∃ νc νe : ℝ → ℝ, + IsLifetimeDensityVersion d L G (IsWindowPair d L rStar bLo bHi kLo kHi) νc ∧ + IsLifetimeDensityVersion d L G (IsWindowElderPair d L rStar bLo bHi kLo kHi I) νe ∧ + Asymptotics.IsEquivalent (nhdsWithin (0 : ℝ) (Set.Ioi 0)) νc + (fun ℓ : ℝ => c * ℓ ^ ((-1 : ℝ) / 3)) ∧ + Asymptotics.IsEquivalent (nhdsWithin (0 : ℝ) (Set.Ioi 0)) νe + (fun ℓ : ℝ => c * ℓ ^ ((-1 : ℝ) / 3)) ∧ + ∃ ℓStar : ℝ, 0 < ℓStar ∧ ∀ ℓ : ℝ, 0 < ℓ → ℓ ≤ ℓStar → + 0 ≤ νc ℓ - νe ℓ ∧ νc ℓ - νe ℓ ≤ C * ℓ ^ ((2 : ℝ) / 3) + +/-- `c` is a leading coefficient of the unrestricted finite-lifetime densities: `c > 0` and +versions `νc`, `νe` of the unrestricted candidate density and of the finite ordinary +superlevel H0 bar density satisfy `νc(ℓ) ~ c ℓ^(-1/3)` and `νe(ℓ) ~ c ℓ^(-1/3)` as `ℓ ↓ 0`. +The essential global-maximum class is excluded from `νe` through `IsFiniteBar`. This is the +conclusion of (1.3) for a given `c`; it characterises `c_{d,L}` by its leading-term role, +not by the integral (13.6)/(15.2). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "There is a positive finite constant c_{d,L} such that" +Interface: `G` and `I` are given; versions are characterised by the integral identity only. +Does not claim: the explicit Gaussian/angular representation (15.2) of `c_{d,L}`; uniqueness of `c` (which follows from the asymptotics but is not stated); existence of versions. -/ +def IsUnrestrictedLeadingCoefficient (d : ℕ) (L : ℝ) (G : GaussianFieldInterface d L) + (I : SuperlevelBarInterface d L) (c : ℝ) : Prop := + 0 < c ∧ ∃ νc νe : ℝ → ℝ, + IsLifetimeDensityVersion d L G (IsUnrestrictedPair d L) νc ∧ + IsLifetimeDensityVersion d L G (IsFiniteBar d L I) νe ∧ + Asymptotics.IsEquivalent (nhdsWithin (0 : ℝ) (Set.Ioi 0)) νc + (fun ℓ : ℝ => c * ℓ ^ ((-1 : ℝ) / 3)) ∧ + Asymptotics.IsEquivalent (nhdsWithin (0 : ℝ) (Set.Ioi 0)) νe + (fun ℓ : ℝ => c * ℓ ^ ((-1 : ℝ) / 3)) + +/-- Shape of Theorem C (1.3), unrestricted finite-lifetime leading density candidate: for +fixed `d ≥ 2` and `L > 0` there is a positive finite constant `c_{d,L}` such that the +unrestricted candidate density and the finite ordinary superlevel H0 bar density (all birth +heights, all positive gap marks, all separations; essential global maximum excluded) admit +versions with `ν(ℓ) ~ c_{d,L} ℓ^(-1/3)` as `ℓ ↓ 0`. Leading term only. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "nu_cand^all(ell) ~ c_{d,L} ell^(-1/3)," +Interface: `G` and `I` are given; versions are characterised by the integral identity only. +Does not claim: the parent result or its proof (dominated convergence of Section 13, off-diagonal bound (14.1)); the representation (15.2); the compact-window `O(ell^(2/3))` difference for the unrestricted densities; any numerical enclosure of `c_{d,L}` (the SIDE24 coefficient track is separate); nonauthor review or acceptance. -/ +def ParentUnrestrictedLeadingDensityC (d : ℕ) (L : ℝ) (G : GaussianFieldInterface d L) + (I : SuperlevelBarInterface d L) : Prop := + 2 ≤ d → 0 < L → ∃ c : ℝ, IsUnrestrictedLeadingCoefficient d L G I c + +/-! ### The bounded remainder (R1) -/ + +/-- Shape of Theorem R (R1), bounded unrestricted short-lifetime remainder: for fixed +`d ≥ 2`, `L > 0` and with `c = c_{d,L}` the parent leading coefficient (characterised here +by `IsUnrestrictedLeadingCoefficient`), there are `ell_* > 0` and `C` such that versions of +the unrestricted candidate and finite-bar densities satisfy, for `0 < ℓ ≤ ell_*`, +`|νc(ℓ) - c ℓ^(-1/3)| ≤ C`, `0 ≤ νc(ℓ) - νe(ℓ) ≤ C` and `|νe(ℓ) - c ℓ^(-1/3)| ≤ C`. +The `O(1)` remainder is stated with an explicit absolute-value bound. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md sha256 380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a +Anchor: "|nu_cand(ell)-c ell^(-1/3)| <= C," +Interface: `G` and `I` are given; `c` is any real satisfying the parent leading-coefficient property (the source says the coefficient "is exactly the parent's equation (15.2)"; that integral is not transcribed, so `c` is pinned through (1.3) instead); versions are characterised by the integral identity only. The statement does not mention the typed pair law: Theorem R does not consume (1.1). +Does not claim: the result or its proof (quadratic determinant cancellation (R7), coupling (R4), cap-loss bound (R12)); convergence of the remainder; a second coefficient; numerical `C` or `ell_*`; the compact-window `O(ell^(2/3))` difference without cutoffs; the parent interfaces it imports (D1-A-E); nonauthor review or acceptance. If no `c` satisfies the parent property the statement holds vacuously; that is a transcription of its dependency on (1.3), not a claim. -/ +def RemainderBoundedR1 (d : ℕ) (L : ℝ) (G : GaussianFieldInterface d L) + (I : SuperlevelBarInterface d L) : Prop := + 2 ≤ d → 0 < L → ∀ c : ℝ, IsUnrestrictedLeadingCoefficient d L G I c → + ∃ ℓStar : ℝ, 0 < ℓStar ∧ ∃ C : ℝ, ∃ νc νe : ℝ → ℝ, + IsLifetimeDensityVersion d L G (IsUnrestrictedPair d L) νc ∧ + IsLifetimeDensityVersion d L G (IsFiniteBar d L I) νe ∧ + ∀ ℓ : ℝ, 0 < ℓ → ℓ ≤ ℓStar → + |νc ℓ - c * ℓ ^ ((-1 : ℝ) / 3)| ≤ C ∧ + 0 ≤ νc ℓ - νe ℓ ∧ νc ℓ - νe ℓ ≤ C ∧ + |νe ℓ - c * ℓ ^ ((-1 : ℝ) / 3)| ≤ C + +/-! ### Planar elder lower bound and density gap (d = 2) -/ + +/-- Shape of the planar direct lower bound (E2): on the fixed planar torus (`d = 2`), with +birth `b` in a fixed compact interval, `k ∈ [kLo, kHi]` with `kLo > 0` and frames ranging +over `O(2)`, there are `c > 0` and `r_* > 0` such that `1 - p_r(b,k,R) ≥ c r^3` for all +`0 < r ≤ r_*`, uniformly in `b`, `k` and the frame. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md sha256 8ce05a43bde8e0d718f09e2da1ffe75d3ec5893489254afdbf5b58b602cdba0c +Anchor: "1-p_r(b,k,R) >= c r^3, 0 0` there are a distance cutoff +`r_* > 0`, constants `c_{B,K} > 0`, `C_{B,K}` and `ell_* > 0` such that versions of the +compact-window candidate and ordinary elder densities satisfy +`c_{B,K} ℓ^(2/3) ≤ νc(ℓ) - νe(ℓ) ≤ C_{B,K} ℓ^(2/3)` for all `0 < ℓ ≤ ell_*`. The parent upper +input is an explicit hypothesis, not folded into the conclusion. The lower half, which the +source derives from (E2) alone, is stated under the same hypothesis because the source displays +(E11) as one two-sided inequality; that is a hypothesis the source does not need for the lower +half, recorded here as such. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md sha256 8ce05a43bde8e0d718f09e2da1ffe75d3ec5893489254afdbf5b58b602cdba0c +Anchor: " <= C_{B,K} ell^(2/3). (E11)" +Interface: `G`, `I` and `PL` are given; the hypothesis `ParentUniformSelectionA 2 L G I PL bLo bHi kLo kHi` is the consumed parent upper input (a statement shape, not the parent result); versions are characterised by the integral identity only; the weighted Kac-Rice / radial ledger of [LP, Sections 9-11] that the source consumes is not transcribed. The interface does not determine `p` or the field: the Prop is a predicate on the interface objects, not a closed statement about the source's objects; nothing is asserted about its universal closure over all inhabitants, and `specified` refers to the statement shape at the source's object, which is not constructed. +Does not claim: the result or its proof; the upper half independently of the parent (1.1); the parent (1.1) itself; the lower half without the hypothesis (the source has it from (E2) alone); sharpness of the `O(1)` remainder of (R1); removal of the compact-mark restriction; that the cutoff `r_*` is the one of (1.1); any numerical constant; nonauthor review or acceptance. -/ +def PlanarDensityGapE11 (L : ℝ) (G : GaussianFieldInterface 2 L) (I : SuperlevelBarInterface 2 L) + (PL : PairLawInterface 2 L G I) (bLo bHi kLo kHi : ℝ) : Prop := + ParentUniformSelectionA 2 L G I PL bLo bHi kLo kHi → + (0 < L → bLo < bHi → 0 < kLo → kLo < kHi → + ∃ rStar : ℝ, 0 < rStar ∧ ∃ c : ℝ, 0 < c ∧ ∃ C : ℝ, ∃ ℓStar : ℝ, 0 < ℓStar ∧ + ∃ νc νe : ℝ → ℝ, + IsLifetimeDensityVersion 2 L G (IsWindowPair 2 L rStar bLo bHi kLo kHi) νc ∧ + IsLifetimeDensityVersion 2 L G (IsWindowElderPair 2 L rStar bLo bHi kLo kHi I) νe ∧ + ∀ ℓ : ℝ, 0 < ℓ → ℓ ≤ ℓStar → + c * ℓ ^ ((2 : ℝ) / 3) ≤ νc ℓ - νe ℓ ∧ νc ℓ - νe ℓ ≤ C * ℓ ^ ((2 : ℝ) / 3)) + +end UniversalLaw.Spec.Lifetime diff --git a/specifications/lean_statements_20261010/Specifications/P15.lean b/specifications/lean_statements_20261010/Specifications/P15.lean new file mode 100644 index 000000000..3544d4306 --- /dev/null +++ b/specifications/lean_statements_20261010/Specifications/P15.lean @@ -0,0 +1,1161 @@ +import Mathlib + +/-! +# P15 realized covers, price boundary, restricted and full transformed-price budgets (statements only) + +Scientific effect: NONE. Formal progress of every declaration here: `specified`. +Organizational-independence credit of this transcription: 0 (Anthropic / Claude). + +This module transcribes, as definitions and `Prop`-valued specifications, the finite +combinatorics of four Layer 0 notes of `d6g8k5htny-coder/Math-` at commit +`760340e921ac4ceda296b8118da936f1133e956e`: + +* `frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md` (P15-REALIZED-COVERS-20260924-v1, + source id `p15-realized-covers`): the realized capacity/clutter family, its downset `D` (P1), + the actual local restrictions (P2), the local covers (P3)-(P5), the palette demand `K_H(d)` + (P6), the full-block cover criterion (P7), the capped hazard bound (P8), the elementary + palette formula (P9), and the six-block 816-label realization with its counts (P10) and its + union bound (P11); +* `frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md` (P15-PRICE-BOUNDARY-20260924-v1, + source id `p15-price-boundary`): the exact two-coordinate refutation of the same-palette + transformed-price extension at demand one; +* `frontiers/price_budget_20260924/PROOF.md` (P15-TRANSFORMED-PRICE-BUDGET-20260924-v1, source + id `p15-price-budget`): the restricted budget (T1) with factor `16/27` for `p_v <= 1/4` and + `d_i >= 2`, its local displays (T2)-(T5), and the 816-label consequence (T7); +* `frontiers/full_price_20260924/PROOF.md` (P15-FULL-TRANSFORMED-PRICE-20260924-v1, source id + `p15-full-price`): the full-cube budget (F2) with the constant `rho_star = 1/(3 - log(3e - 2))`, + its enclosure (F3), the hazard interpolation bounds (F6)-(F8), the capacity worst case + (F9)-(F11), the assembly (F12)-(F13), the sharpness and demand-one boundary of Section 5 and + the rational certificates of Section 6. + +Because the domain is finite combinatorics, everything is DEFINED rather than given by +interface fields: a realized family is a data record `RealizedData` (finite ground type, +block map, capacities, demands, clutter) and its side conditions are one `Prop` +(`IsRealizedFamily`), so that the concrete realizations of the sources (`benchmark816`, +`twoCoord`, `tripleBlock`, `demandOneBlock`) are plain definitions without embedded proofs. +Two conventions are made explicit where the sources use extended arithmetic: the hazard +`-log 0 = +infinity` is rendered by `cappedHazard`, which returns the cap `1` when the +probability is zero (the sources: "the trivial price-one cover handles the endpoint"), and +`phi(1) = 1` is rendered by a guard in `phi`. The palette demand `K_H(d)` is `sInf` of the +feasible palette sizes; the sources call it "the minimum size K", and that the feasible set is +nonempty (so that `sInf` is a minimum) is NOT proved here but stated as `PaletteNumberAttained`. +The cover cost is `sInf` of the prices of all generator covers, a finite nonempty set of reals. + +What this module does not establish. It consists only of definitions and `Prop`s; it states +nothing as a result and proves nothing. It does not establish (P7), (P8), (T1), (F2), the +enclosure (F3), `rho_star < 6/7`, any count of (P10), the union bound (P11), the refutation of +the demand-one extension, or any of the displayed inequalities; it does not identify these +realized families with any earlier abstract P15 benchmark; it does not touch original P15-B +or the unrestricted P15 prize. The arithmetic skeleton of several displays (the two-coordinate +prices `9, 6, 6, 4` over `9`, `3/4` over `4`, `1/3 < 4/9`; the counts `500616` and +`419743994415`; `2449227/8180000000 < 3/10000`; `16/27 < 6/7`; the adjacency of the (F3) +endpoints and `upper * 7 < 6 * 10^20`) is the target of the core-Lean package +`formal/UniversalLaw/P15/*` on main and is not re-stated as a result here. Nothing here +promotes, reclassifies or discharges any claim, premise or obligation of the registers; +alignment of these statements with the sources is PENDING_INDEPENDENT_REVIEW. Finite +`Prop`s about the concrete realizations (`twoCoord`, `tripleBlock`, the palette numbers of small +clutters) are decidable in principle and are future kernel-proof targets; the `Prop`s about +`benchmark816` quantify over `2^2454` subsets and are finite but not decidable in practice. + +Primary sources (Layer 0, byte-pinned): `d6g8k5htny-coder/Math-` commit +`760340e921ac4ceda296b8118da936f1133e956e`; sha256 of the four paths: +`c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9` (realized covers), +`498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b` (price boundary), +`3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535` (price budget), +`87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9` (full price). +-/ + +namespace UniversalLaw.Spec.P15 + +/-! ## Product measure, prices, the transformed price `phi` and the capped hazard -/ + +/-- Independent-coordinate weight of a subset `U` of a finite ground type `Y` under the +probability vector `p`: `prod_{v in U} p_v * prod_{v notin U} (1 - p_v)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "Select the ORIGINAL coordinates independently with arbitrary probabilities p_v in [0,1], and let 0<=c_v<=p_v." +Interface: none; `p` is any real vector (the sources take `p_v in [0,1]`, imposed as a hypothesis where used). +Does not claim: that these weights sum to one over all subsets (not stated separately). -/ +noncomputable def weight {Y : Type} [Fintype Y] [DecidableEq Y] (p : Y → ℝ) (U : Finset Y) : ℝ := + (∏ v ∈ U, p v) * ∏ v ∈ Finset.univ \ U, (1 - p v) + +/-- Product-measure mass `mu_p(A)` of a finite family `A` of subsets of a finite ground type: +the sum of the weights of its members. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "F_A(t)=-log mu_(p(t))(A)," +Interface: none. +Does not claim: anything; this is the probability that the random set of independently selected coordinates lies in `A`. -/ +noncomputable def muOf {Y : Type} [Fintype Y] [DecidableEq Y] (A : Finset (Finset Y)) + (p : Y → ℝ) : ℝ := + ∑ U ∈ A, weight p U + +/-- Price of a generator `g`: `product_(v in g) c_v`; the empty generator has price `1`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "A generator's price is product_(v in g)c_v, and family price is the sum of generator prices." +Interface: none. +Does not claim: anything. -/ +noncomputable def genPrice {Y : Type} (c : Y → ℝ) (g : Finset Y) : ℝ := + ∏ v ∈ g, c v + +/-- Price of a generator family `G`: the sum of its generator prices; the empty family has +price `0`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "The empty generator has price1; the empty family has price0." +Interface: none. +Does not claim: anything. -/ +noncomputable def famPrice {Y : Type} (c : Y → ℝ) (G : Finset (Finset Y)) : ℝ := + ∑ g ∈ G, genPrice c g + +/-- The transformed price ceiling `phi(t) = min(1, -log(1 - t))` with `phi(1) = 1`. The guard +`t < 1` renders the source's convention at `t = 1` (Mathlib's `Real.log 0 = 0` would otherwise +give `0`); for `t > 1`, outside the source's domain, the value `1` is a placeholder. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "where phi(t)=min(1,-log(1-t)), phi(1)=1." +Interface: none. +Does not claim: any value outside `[0, 1]`. -/ +noncomputable def phi (t : ℝ) : ℝ := + if t < 1 then min 1 (-Real.log (1 - t)) else 1 + +/-- The capped, scaled hazard `min(1, rho * [-log m])` of a probability `m`, with the sources' +convention that the hazard of a zero probability is `+infinity`, so that the cap `1` applies: +`cappedHazard rho m = 1` when `m <= 0`. With `rho = 1` this is `min(1, -log mu_p(D))` of (P8); +with `rho = 16/27` the right side of (T1); with `rho = rho_star` the right side of (F2). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "If some q_i=1 the trivial price-one cover handles the endpoint." +Interface: none. +Does not claim: anything; the branch at `m <= 0` is the sources' endpoint convention, not a result. -/ +noncomputable def cappedHazard (ρ m : ℝ) : ℝ := + if 0 < m then min 1 (ρ * (-Real.log m)) else 1 + +/-! ## The realized capacity/clutter family -/ + +/-- Data of a realized family on a finite ground type `X`: `b` blocks, the block map `blk` +(the blocks `X_i` are its fibres, hence disjoint and covering), capacities `a_i`, demands `d_i` +and the clutter `H` of forbidden block supports. The side conditions of the sources are the +separate `Prop` `IsRealizedFamily`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "Choose positive integers a_i,d_i and disjoint original-coordinate blocks X_i with" +Interface: none; every field is a finite datum, nothing is a hypothesis field. +Does not claim: that the data satisfy the sources' side conditions (that is `IsRealizedFamily`). -/ +structure RealizedData (X : Type) where + /-- The number of blocks. -/ + b : ℕ + /-- The block of each original coordinate. -/ + blk : X → Fin b + /-- The capacities `a_i`. -/ + a : Fin b → ℕ + /-- The demands `d_i`. -/ + d : Fin b → ℕ + /-- The clutter `H` of forbidden block supports. -/ + H : Finset (Finset (Fin b)) + +/-- The original-coordinate block `X_i = {v : blk v = i}`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "disjoint original-coordinate blocks X_i" +Interface: none. +Does not claim: anything. -/ +def block {X : Type} [Fintype X] (F : RealizedData X) (i : Fin F.b) : Finset X := + Finset.univ.filter (fun v => F.blk v = i) + +/-- The side conditions of the realized family: `a_i >= 1`, `d_i >= 1` ("positive integers +a_i,d_i"), `|X_i| = a_i d_i + 1`, every edge of `H` has size at least `2`, and `H` is a clutter +(no edge properly contains another; edges are distinct as members of a `Finset`). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " n_i=|X_i|=a_i d_i+1." +Interface: none; this `Prop` IS the interface of the sources, stated for a data record. +Does not claim: that any particular data record satisfies it. -/ +def IsRealizedFamily {X : Type} [Fintype X] (F : RealizedData X) : Prop := + (∀ i, 1 ≤ F.a i) ∧ (∀ i, 1 ≤ F.d i) ∧ (∀ i, (block F i).card = F.a i * F.d i + 1) ∧ + (∀ e ∈ F.H, 2 ≤ e.card) ∧ (∀ e ∈ F.H, ∀ f ∈ F.H, e ⊆ f → e = f) + +/-- The block support `supp(U) = {i : U intersect X_i is nonempty}`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "supp(U)={i:U intersect X_i is nonempty}. (P1)" +Interface: none. +Does not claim: anything. -/ +def supp {X : Type} (F : RealizedData X) (U : Finset X) : Finset (Fin F.b) := + U.image F.blk + +/-- Membership in the downset `D` of display (P1): `|U intersect X_i| <= a_i` for every `i`, and +`supp(U)` contains no edge of `H`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "D={U subset X: |U intersect X_i|<=a_i for every i," +Interface: none. +Does not claim: that `D` is decreasing or contains the empty set (facts of the sources, not stated separately). -/ +def IsGood {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) (U : Finset X) : Prop := + (∀ i, (U ∩ block F i).card ≤ F.a i) ∧ ∀ e ∈ F.H, ¬ e ⊆ supp F U + +/-- The downset `D` as a set of subsets of `X`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " supp(U) contains no edge of H}," +Interface: none. +Does not claim: anything. -/ +def downset {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Set (Finset X) := + {U | IsGood F U} + +/-- `U in I_k(D)`: `U` is partitionable into at most `k` members of `D` (a family of `k` +pairwise disjoint good parts, possibly empty, with union `U`). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "write I_k(D) for sets partitionable into at most k members of D" +Interface: none. +Does not claim: equivalence with coverings by at most `k` good sets (true for a decreasing family, not stated). -/ +def IsKDecomposable {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) (k : ℕ) + (U : Finset X) : Prop := + ∃ parts : Fin k → Finset X, (∀ j, IsGood F (parts j)) ∧ + (∀ j j', j ≠ j' → Disjoint (parts j) (parts j')) ∧ + (Finset.univ : Finset (Fin k)).biUnion parts = U + +/-- The obstruction `O_K(D) = 2^X \ I_K(D)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "O_k(D)=2^X\I_k(D) for its obstruction" +Interface: none. +Does not claim: anything. -/ +def obstruction {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) (K : ℕ) : + Set (Finset X) := + {U | ¬ IsKDecomposable F K U} + +/-- `G` is a generator cover of `O_K(D)`: every obstructed set contains some `g in G`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "A generator cover G of O_k(D) means every obstructed set contains some g in G." +Interface: none. +Does not claim: anything. -/ +def IsGeneratorCover {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) (K : ℕ) + (G : Finset (Finset X)) : Prop := + ∀ U : Finset X, U ∈ obstruction F K → ∃ g ∈ G, g ⊆ U + +/-- The exact optimal cover price `covercost_c(O_K(D))`: the infimum of the family prices of all +generator covers of `O_K(D)`. The set of covers is finite and nonempty (`{empty set}` always +covers), so the infimum is a minimum; that fact is not proved here. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md sha256 498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b +Anchor: "Therefore the EXACT optimal cover price is" +Interface: none. +Does not claim: that the infimum is attained (finite, not proved here); for `c` with negative entries, outside the sources' domain, `sInf` of an unbounded set is Mathlib's default. -/ +noncomputable def coverCost {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) (K : ℕ) + (c : X → ℝ) : ℝ := + sInf {t : ℝ | ∃ G : Finset (Finset X), IsGeneratorCover F K G ∧ famPrice c G = t} + +/-- The product measure `mu_p(D)` of the downset: the sum over `U in D` of +`prod_{v in U} p_v prod_{v notin U} (1 - p_v)`, written as a sum over `2^X` of the indicator of +`D` times the weight. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "For EVERY independent original probability vector p in [0,1]^X," +Interface: none. +Does not claim: anything. -/ +noncomputable def mu {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) (p : X → ℝ) : ℝ := + ∑ U ∈ (Finset.univ : Finset X).powerset, Set.indicator (downset F) (weight p) U + +/-- The actual local good probability `mu_p(D_i) = P(U intersect X_i in D)`, with +`D_i = D intersect 2^(X_i)`: the sum over subsets `S` of the block of the indicator of `D` times +the block-restricted weight. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: "Write q_i=P(U intersect X_i notin D_i) for the ACTUAL capacity restriction." +Interface: none. +Does not claim: the identification `D_i = {S subset X_i : |S| <= a_i}` (that is `LocalRestriction`). -/ +noncomputable def localGood {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) + (p : X → ℝ) (i : Fin F.b) : ℝ := + ∑ S ∈ (block F i).powerset, + Set.indicator (downset F) (fun S => (∏ v ∈ S, p v) * ∏ v ∈ block F i \ S, (1 - p v)) S + +/-- The local failure probability `q_i = P(U_i notin D_i) = 1 - mu_p(D_i)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "With q_i=P(U_i notin D_i)," +Interface: none. +Does not claim: anything. -/ +noncomputable def localFailure {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) + (p : X → ℝ) (i : Fin F.b) : ℝ := + 1 - localGood F p i + +/-! ## Palette demand -/ + +/-- `K` is a feasible palette size for the clutter `H` and demands `d`: there are label sets +`P_i subset {0, ..., K-1}` with `|P_i| >= d_i` and empty intersection over every `H`-edge +(no label lies in `P_i` for all `i in e`). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " intersection_(i in e) P_i = empty, for every e in H. (P6)" +Interface: none. +Does not claim: upward closure in `K` (true, not stated). -/ +def IsPaletteFeasible (b : ℕ) (H : Finset (Finset (Fin b))) (d : Fin b → ℕ) (K : ℕ) : Prop := + ∃ P : Fin b → Finset (Fin K), (∀ i, d i ≤ (P i).card) ∧ + ∀ e ∈ H, ∀ ℓ : Fin K, ∃ i ∈ e, ℓ ∉ P i + +/-- The palette demand `K_H(d)`: the least feasible palette size, as `sInf` of the feasible set +(Mathlib's `sInf` on `ℕ` is the least element of a nonempty set and `0` for the empty set). The +sources define it as "the minimum size K"; nonemptiness of the feasible set is the separate +`Prop` `PaletteNumberAttained`, not proved here. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "Let K_H(d) be the minimum size K of an integer palette universe admitting subsets P_i with |P_i|>=d_i and" +Interface: none. +Does not claim: that the feasible set is nonempty; the source's remark that one may take `|P_i| = d_i`. -/ +noncomputable def paletteNumber (b : ℕ) (H : Finset (Finset (Fin b))) (d : Fin b → ℕ) : ℕ := + sInf {K : ℕ | IsPaletteFeasible b H d K} + +/-- The feasible set of palette sizes is nonempty, i.e. `K_H(d)` is itself feasible (the minimum +is attained), for every realized family. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "By discarding excess labels within blocks, one may take |P_i|=d_i." +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis. +Does not claim: the statement (not proved here; disjoint palettes of sizes `d_i` are a witness in the sources' argument). -/ +def PaletteNumberAttained {X : Type} [Fintype X] (F : RealizedData X) : Prop := + IsRealizedFamily F → IsPaletteFeasible F.b F.H F.d (paletteNumber F.b F.H F.d) + +/-- The specified cover: the family `G = {X_1, ..., X_b}` of full blocks. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "the fixed family G={X_1,...,X_b} covers O_K(D) if and only if" +Interface: none. +Does not claim: anything. -/ +def fullBlockFamily {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : + Finset (Finset X) := + (Finset.univ : Finset (Fin F.b)).image (block F) + +/-! ## Realized covers: Propositions P1, P2 and the local covers (P3)-(P5) -/ + +/-- `U` is an inclusion-minimal forbidden set: not good, and every proper subset obtained by +removing one element is good. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "The inclusion-minimal forbidden ORIGINAL sets are exactly" +Interface: none. +Does not claim: anything. -/ +def IsMinimalForbidden {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) + (U : Finset X) : Prop := + ¬ IsGood F U ∧ ∀ v ∈ U, IsGood F (U.erase v) + +/-- `U` is a transversal `{x_i : i in e}` of the edge `e`: its support is `e` and it meets each +block of `e` in exactly one coordinate. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "(b) for each e in H, EVERY transversal {x_i:i in e}, with x_i in X_i." +Interface: none. +Does not claim: anything. -/ +def IsTransversal {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) + (e : Finset (Fin F.b)) (U : Finset X) : Prop := + supp F U = e ∧ ∀ i ∈ e, (U ∩ block F i).card = 1 + +/-- Proposition P1: the inclusion-minimal forbidden sets are exactly (a) the `(a_i+1)`-subsets +of a single block and (b) the transversals of the edges of `H`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "(a) every (a_i+1)-subset of a single X_i;" +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis. +Does not claim: the characterisation (statement only); its completeness clause is what the source calls "completeness, not just existence of some witnesses". -/ +def MinimalForbiddenCharacterisation {X : Type} [Fintype X] [DecidableEq X] + (F : RealizedData X) : Prop := + IsRealizedFamily F → ∀ U : Finset X, IsMinimalForbidden F U ↔ + ((∃ i, U ⊆ block F i ∧ U.card = F.a i + 1) ∨ ∃ e ∈ F.H, IsTransversal F e U) + +/-- Display (P2): the actual local restriction is `D_i = D intersect 2^(X_i) = {S subset X_i : +|S| <= a_i}`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "D_i=D intersect 2^(X_i)={S subset X_i: |S|<=a_i}. (P2)" +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis. +Does not claim: the identity (statement only). -/ +def LocalRestriction {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → ∀ (i : Fin F.b) (S : Finset X), S ⊆ block F i → + (IsGood F S ↔ S.card ≤ F.a i) + +/-- Section 3: `I_k(D_i) = {S : |S| <= a_i k}` for subsets `S` of the block `X_i`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " I_k(D_i)={S:|S|<=a_i k}." +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis. +Does not claim: the identity (statement only). -/ +def LocalDecomposability {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → ∀ (i : Fin F.b) (k : ℕ) (S : Finset X), S ⊆ block F i → + (IsKDecomposable F k S ↔ S.card ≤ F.a i * k) + +/-- Display (P3): for `k = d_i` the only obstructed subset of `X_i` is `X_i` itself, so +`G_i = {X_i}` is an exact local obstruction cover. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " G_i={X_i} (P3)" +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis. +Does not claim: the statement (statement only). -/ +def LocalObstructionIsFullBlock {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : + Prop := + IsRealizedFamily F → ∀ (i : Fin F.b) (S : Finset X), S ⊆ block F i → + (¬ IsKDecomposable F (F.d i) S ↔ S = block F i) + +/-- Display (P4): for `0 <= c_v <= p_v <= 1`, `price(G_i) = prod_{v in X_i} c_v <= prod p_v +<= q_i <= phi(q_i)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " =P(U_i=X_i)<=q_i<=phi(q_i), (P4)" +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis; `p`, `c` with the displayed ranges. +Does not claim: the chain (statement only); the identity `prod p_v = P(U_i = X_i)` is the definition of the weight. -/ +def LocalPriceChain {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → ∀ (p c : X → ℝ), (∀ v, 0 ≤ p v ∧ p v ≤ 1) → (∀ v, 0 ≤ c v ∧ c v ≤ p v) → + ∀ i : Fin F.b, genPrice c (block F i) ≤ ∏ v ∈ block F i, p v ∧ + ∏ v ∈ block F i, p v ≤ localFailure F p i ∧ + localFailure F p i ≤ phi (localFailure F p i) + +/-- Display (P5) (also (T6)): `mu_p(D) <= prod_i (1 - q_i)` and, when `mu_p(D) > 0`, +`sum_i q_i <= -log prod_i (1 - q_i) <= -log mu_p(D)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " <=-log(product_i(1-q_i))<=-log mu_p(D). (P5)" +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis; `p in [0,1]^X`. +Does not claim: the inequalities (statement only); independence of crossing events ("Crossing events are not declared independent"). -/ +def IndependentBlockHazard {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → ∀ p : X → ℝ, (∀ v, 0 ≤ p v ∧ p v ≤ 1) → + mu F p ≤ ∏ i, (1 - localFailure F p i) ∧ + (0 < mu F p → + ∑ i, localFailure F p i ≤ -Real.log (∏ i, (1 - localFailure F p i)) ∧ + -Real.log (∏ i, (1 - localFailure F p i)) ≤ -Real.log (mu F p)) + +/-! ## Realized covers: the full-block cover criterion (P7) and the hazard bound (P8) -/ + +/-- Proposition P2, display (P7): the fixed family `G = {X_1, ..., X_b}` covers `O_K(D)` if and +only if `K >= K_H(d)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " K>=K_H(d). (P7)" +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis; `K_H(d)` is `paletteNumber`. +Does not claim: the criterion (statement only); optimality of any other cover ("It does not prove that no different cover with the same budget works at fewer colors"); unrestricted prize optimality; disposition transcribed from LANDING_CLAIMS for p15-realized-covers: HOLD_WITH_DOMAIN; nonauthor review and acceptance are not claimed here. -/ +def FullBlockCoverIff {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → ∀ K : ℕ, + IsGeneratorCover F K (fullBlockFamily F) ↔ paletteNumber F.b F.H F.d ≤ K + +/-- Display (P8): for `0 <= c_v <= p_v <= 1` and `K >= K_H(d)`, +`covercost_c(O_K(D)) <= min(1, -log mu_p(D))`, with the endpoint convention of `cappedHazard`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "covercost_c(O_K(D)) <= min(1,-log mu_p(D)). (P8)" +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis; `p in [0,1]^X`; prices in the range `0 <= c_v <= p_v` only ("No assertion is made here for the additional prices p_v p_v`; the unrestricted P15 prize; original P15-B; disposition transcribed from LANDING_CLAIMS for p15-realized-covers: HOLD_WITH_DOMAIN; nonauthor review and acceptance are not claimed here. -/ +def CoverCostHazardBound {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → ∀ (p c : X → ℝ), (∀ v, 0 ≤ p v ∧ p v ≤ 1) → (∀ v, 0 ≤ c v ∧ c v ≤ p v) → + ∀ K : ℕ, paletteNumber F.b F.H F.d ≤ K → coverCost F K c ≤ cappedHazard 1 (mu F p) + +/-- Section 4 comparison: the chromatic number of the ENTIRE ground set `X` (the least `K` with +`X in I_K(D)`) is `K_H(d+1)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "the chromatic number of the ENTIRE ground set X is K_H(d+1), because ceil(n_i/a_i)=d_i+1." +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis. +Does not claim: the statement (statement only). -/ +def WholeGroundChromaticNumber {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : + Prop := + IsRealizedFamily F → ∀ K : ℕ, + IsKDecomposable F K Finset.univ ↔ paletteNumber F.b F.H (fun i => F.d i + 1) ≤ K + +/-! ## Realized covers Section 5: the elementary palette formula (P9) -/ + +/-- The clutter of all `(s+1)`-subsets of `b` block indices. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "A fully elementary case is H consisting of all (s+1)-subsets of b indices, 1<=s<=b." +Interface: none. +Does not claim: anything; for `s = b` the family is empty ("Empty H is allowed"). -/ +def completeClutter (b s : ℕ) : Finset (Finset (Fin b)) := + (Finset.univ : Finset (Fin b)).powersetCard (s + 1) + +/-- Display (P9): for `H` all `(s+1)`-subsets of `b` indices, `1 <= s <= b`, and positive +demands, `K_H(d) = max(max_i d_i, ceil(sum_i d_i / s))`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " K_H(d)=max(max_i d_i, ceil(sum_i d_i/s)). (P9)" +Interface: none; the positivity of the demands is the sources' standing assumption ("positive integers a_i,d_i"), added as a hypothesis. +Does not claim: the formula (statement only); the matroid palette formula of Section 5 (not specified). -/ +def ElementaryPaletteFormula : Prop := + ∀ (b s : ℕ) (d : Fin b → ℕ), 1 ≤ s → s ≤ b → (∀ i, 1 ≤ d i) → + paletteNumber b (completeClutter b s) d = + max (Finset.univ.sup d) ⌈((∑ i, d i : ℕ) : ℚ) / (s : ℚ)⌉₊ + +/-- Section 5: the triangle on three block indices (all `2`-subsets) at unit demands needs `3` +colors. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "at unit demands needs3 colors, although a formula considering only size-two edge loads would give2." +Interface: none. +Does not claim: the value (statement only; finite and decidable in principle). -/ +def TriangleNeedsThree : Prop := + paletteNumber 3 (completeClutter 3 1) (fun _ => 1) = 3 + +/-! ## Realized covers Section 6: the six-block 816-label realization -/ + +/-- The six-block realization: ground type `Fin 6 × Fin 409` (six blocks of `409` coordinates, +`2454` in all), block map the first projection, `a_i = 1`, `d_i = 408`, and `H` all `15` +four-block subsets. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "Take six blocks, a_i=1 and d_i=408. Thus EACH original block has409 vertices and the original ground set has2454 vertices. H consists of all15 four-block subsets." +Interface: none; a concrete data record. +Does not claim: that it satisfies `IsRealizedFamily` (that is `Benchmark816IsRealized`); that any earlier six-block application had these coordinates ("it is not a statement that every earlier six-block application had409 vertices per block"). -/ +def benchmark816 : RealizedData (Fin 6 × Fin 409) where + b := 6 + blk := Prod.fst + a := fun _ => 1 + d := fun _ => 408 + H := completeClutter 6 3 + +/-- The six-block data satisfy the side conditions of a realized family. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "Every D-color class contains at most one vertex per block and occupies at most three blocks." +Interface: none. +Does not claim: the statement (finite, decidable in principle; not proved here). -/ +def Benchmark816IsRealized : Prop := + IsRealizedFamily benchmark816 + +/-- Section 6: `K_H(408, ..., 408) = 816` for `H` all four-block subsets of six. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " K_H(408,...,408)=816." +Interface: none. +Does not claim: the value (statement only); the arithmetic `max(408, ceil(2448/3)) = 816` is a core-Lean target on main, not here. -/ +def Benchmark816Palette : Prop := + paletteNumber 6 (completeClutter 6 3) (fun _ => 408) = 816 + +/-- Section 6: the six full blocks cover the obstruction at `816` colors and not at `815`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "At816 colors it covers the obstruction; at815 it does not, because omitting one vertex from each block leaves2448 vertices, each color holds at most3, and815*3<2448." +Interface: none. +Does not claim: the statement (finite over `2^2454` subsets, not decidable in practice); `815*3 < 2448` is a core-Lean target on main. -/ +def Benchmark816Cover : Prop := + IsGeneratorCover benchmark816 816 (fullBlockFamily benchmark816) ∧ + ¬ IsGeneratorCover benchmark816 815 (fullBlockFamily benchmark816) + +/-- Section 6: the whole ground set needs `818` colors (`2454/3 = 818`): it is `818`-decomposable +and not `817`-decomposable. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: "The whole ground set, unlike the cover-avoiding sets, needs818 colors, since2454/3=818." +Interface: none. +Does not claim: the statement (statement only); `2454/3 = 818` is a core-Lean target on main. -/ +def Benchmark816WholeGround : Prop := + IsKDecomposable benchmark816 818 Finset.univ ∧ ¬ IsKDecomposable benchmark816 817 Finset.univ + +/-- Display (P10): the minimal forbidden sets of the six-block realization are +`6 * binom(409, 2)` internal pairs (support of size one) and `15 * 409^4` crossing transversal +quadruples (support of size four), and nothing else. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " 15*409^4=419743994415 crossing transversal quadruples. (P10)" +Interface: none. +Does not claim: the counts (statement only); the evaluations `500616` and `419743994415` are core-Lean targets on main; the source stresses these are "not a claim that the program enumerated more than four hundred billion sets". -/ +def Benchmark816MinimalForbiddenCounts : Prop := + Set.ncard {U : Finset (Fin 6 × Fin 409) | + IsMinimalForbidden benchmark816 U ∧ (supp benchmark816 U).card = 1} = 6 * Nat.choose 409 2 ∧ + Set.ncard {U : Finset (Fin 6 × Fin 409) | + IsMinimalForbidden benchmark816 U ∧ (supp benchmark816 U).card = 4} = 15 * 409 ^ 4 ∧ + Set.ncard {U : Finset (Fin 6 × Fin 409) | IsMinimalForbidden benchmark816 U} = + 6 * Nat.choose 409 2 + 15 * 409 ^ 4 + +/-- Section 6: at `p_v = c_v = 1/40900` the cover price of the six full blocks is exactly +`6/(40900^409) > 0`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " 6/(40900^409)>0." +Interface: none. +Does not claim: the identity (statement only; the six blocks are distinct members of the image). -/ +def Benchmark816Price : Prop := + famPrice (fun _ => (1 : ℝ) / 40900) (fullBlockFamily benchmark816) = 6 / 40900 ^ 409 ∧ + (0 : ℝ) < 6 / 40900 ^ 409 + +/-- Display (P11): at `p_v = 1/40900`, `Q = 1 - mu_p(D) <= 500616/40900^2 + 15/100^4 = +2449227/8180000000 < 3/10000`, and the good probability exceeds `0.9997`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_REALIZED_COVERS.md sha256 c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9 +Anchor: " =2449227/8180000000 < 3/10000. (P11)" +Interface: none. +Does not claim: the bound (statement only; "only a union bound, not an equality of the global bad probability"); the two rational identities are core-Lean targets on main. -/ +def Benchmark816UnionBound : Prop := + 1 - mu benchmark816 (fun _ => (1 : ℝ) / 40900) ≤ 500616 / 40900 ^ 2 + 15 / 100 ^ 4 ∧ + (500616 / 40900 ^ 2 + 15 / 100 ^ 4 : ℝ) = 2449227 / 8180000000 ∧ + (2449227 / 8180000000 : ℝ) < 3 / 10000 ∧ + (9997 / 10000 : ℝ) < mu benchmark816 (fun _ => (1 : ℝ) / 40900) + +/-! ## Price boundary: the exact two-coordinate refutation at demand one -/ + +/-- The smallest one-block member: `X = {x, y}`, `a_1 = d_1 = 1`, no edges. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md sha256 498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b +Anchor: "The prescribed cover is {{x,y}}." +Interface: none; a concrete data record. +Does not claim: anything. -/ +def twoCoord : RealizedData (Fin 2) where + b := 1 + blk := fun _ => 0 + a := fun _ => 1 + d := fun _ => 1 + H := ∅ + +/-- The probabilities `p_x = p_y = 1/2` of the refutation. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md sha256 498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b +Anchor: "Take independent original probabilities p_x=p_y=1/2 and prices c_x=c_y=2/3." +Interface: none. +Does not claim: anything. -/ +noncomputable def twoCoordP : Fin 2 → ℝ := fun _ => 1 / 2 + +/-- The prices `c_x = c_y = 2/3` of the refutation. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md sha256 498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b +Anchor: "These are admissible TRANSFORMED prices: phi(1/2)=log 2>2/3." +Interface: none. +Does not claim: admissibility (that is part of `DemandOneRefutation`). -/ +noncomputable def twoCoordC : Fin 2 → ℝ := fun _ => 2 / 3 + +/-- The four generator costs `1, 2/3, 2/3, 4/9` of the empty set, the two singletons and `X`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md sha256 498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b +Anchor: "The possible generator costs are1,2/3,2/3,4/9 for the empty" +Interface: none. +Does not claim: the values (statement only; as integers over `9` they are core-Lean targets on main). -/ +def TwoCoordGeneratorPrices : Prop := + genPrice twoCoordC ∅ = 1 ∧ genPrice twoCoordC {0} = 2 / 3 ∧ genPrice twoCoordC {1} = 2 / 3 ∧ + genPrice twoCoordC Finset.univ = 4 / 9 + +/-- The exact refutation: the data are a realized family with `K_H(d) = 1` and one-piece +obstruction `{X}`; the prices are admissible transformed prices (`phi(1/2) = log 2 > 2/3`); +`mu_p(D) = 3/4`, `min(1, -log mu_p(D)) = log(4/3) < 1/3`; every cover costs at least `4/9`, +`covercost_c(O_1(D)) = 4/9 > 1/3 > log(4/3)`; hence the capped hazard is strictly below the +exact cover cost at the same palette. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md sha256 498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b +Anchor: " covercost_c(O_1(D))=4/9 > 1/3 > log(4/3)." +Interface: none; everything is concrete. +Does not claim: the statement (statement only; the finite parts are decidable in principle, the two logarithm inequalities are analysis); that this contradicts original P15-B ("It does not contradict the original P15-B amalgamation"); anything about the `408`-scale or the unrestricted prize; the integral representation `log 2 = 2 integral_0^(1/3) 1/(1-t^2) dt` is not specified; disposition transcribed from LANDING_CLAIMS for p15-price-boundary: EXACT_COUNTEREXAMPLE; nonauthor review and acceptance are not claimed here. -/ +def DemandOneRefutation : Prop := + IsRealizedFamily twoCoord ∧ + paletteNumber 1 ∅ (fun _ => 1) = 1 ∧ + (∀ U : Finset (Fin 2), U ∈ obstruction twoCoord 1 ↔ U = Finset.univ) ∧ + (∀ v, twoCoordC v ≤ phi (twoCoordP v)) ∧ + phi (1 / 2) = Real.log 2 ∧ (2 / 3 : ℝ) < Real.log 2 ∧ + mu twoCoord twoCoordP = 3 / 4 ∧ + cappedHazard 1 (mu twoCoord twoCoordP) = Real.log (4 / 3) ∧ Real.log (4 / 3) < (1 / 3 : ℝ) ∧ + (∀ G : Finset (Finset (Fin 2)), IsGeneratorCover twoCoord 1 G → 4 / 9 ≤ famPrice twoCoordC G) ∧ + coverCost twoCoord 1 twoCoordC = 4 / 9 ∧ (1 / 3 : ℝ) < 4 / 9 ∧ + cappedHazard 1 (mu twoCoord twoCoordP) < coverCost twoCoord 1 twoCoordC + +/-- The elementary inequality `log(1 + x) < x` for `x > 0` used by the refutation. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md sha256 498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b +Anchor: "using log(1+x)0." +Interface: none. +Does not claim: the inequality (statement only; the source proves it, this module does not). -/ +def LogOnePlusBelowIdentity : Prop := + ∀ x : ℝ, 0 < x → Real.log (1 + x) < x + +/-! ## Price budget: the restricted transformed-price extension (T1)-(T5), (T7) -/ + +/-- Display (T2): `-log(1 - p) <= p/(1 - p)` for `0 <= p < 1`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: " -log(1-p)=integral_0^p dt/(1-t) <= p/(1-p). (T2)" +Interface: none. +Does not claim: the inequality (statement only); the integral representation is not specified. -/ +def LogRatioBound : Prop := + ∀ p : ℝ, 0 ≤ p → p < 1 → -Real.log (1 - p) ≤ p / (1 - p) + +/-- The exponent arithmetic of display (T3): for `a >= 1` and `n >= 2a + 1`, +`4^(a+1)/3^n <= (4/3)(4/9)^a <= 16/27`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: " <= (4/3)*(4/9)^a <=16/27. (T3)" +Interface: none. +Does not claim: the inequalities (statement only); the preceding ratio steps of (T3) (price over `prod_{v in S} p_v`) are stated through their consequence `RestrictedLocalBudget`; the case `a = 1, n = 3` giving `16/27` is a core-Lean target on main. -/ +def RatioExponentBound : Prop := + ∀ a n : ℕ, 1 ≤ a → 2 * a + 1 ≤ n → + (4 : ℝ) ^ (a + 1) / 3 ^ n ≤ (4 / 3) * (4 / 9) ^ a ∧ (4 / 3 : ℝ) * (4 / 9) ^ a ≤ 16 / 27 + +/-- Display (T4): under `a_i >= 1`, `d_i >= 2`, `0 <= p_v <= 1/4` and `0 <= c_v <= phi(p_v)`, the +full-block generator satisfies `price({X_i}) <= (16/27) q_i <= phi(q_i)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: " price({X}) <= (16/27)q <= phi(q). (T4)" +Interface: the family data `F` with `IsRealizedFamily F` and `d_i >= 2` as hypotheses. +Does not claim: the bound (statement only); the source notes "The extra demand hypothesis d>=2 is load-bearing" and that `p_v <= 1/4` "is a sufficient range, not an asserted sharp threshold". -/ +def RestrictedLocalBudget {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → (∀ i, 2 ≤ F.d i) → + ∀ (p c : X → ℝ), (∀ v, 0 ≤ p v ∧ p v ≤ 1 / 4) → (∀ v, 0 ≤ c v ∧ c v ≤ phi (p v)) → + ∀ i : Fin F.b, genPrice c (block F i) ≤ (16 / 27) * localFailure F p i ∧ + (16 / 27) * localFailure F p i ≤ phi (localFailure F p i) + +/-- The restricted transformed-price extension, display (T1): for every realized family with +`a_i >= 1`, `d_i >= 2`, independent `0 <= p_v <= 1/4`, EVERY `0 <= c_v <= phi(p_v)` and the SAME +`K >= K_H(d)`, `covercost_c(O_K(D)) <= min(1, (16/27)[-log mu_p(D)])`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: " covercost_c(O_K(D)) <= min(1,(16/27)*[-log mu_p(D)]). (T1)" +Interface: the family data `F` with `IsRealizedFamily F` and `d_i >= 2` as hypotheses; `K_H(d)` is `paletteNumber`; the zero-probability endpoint follows the `cappedHazard` convention. +Does not claim: the bound (statement only); any retraction of the demand-one refutation; the unrestricted prize; new classical coloring theory; disposition transcribed from LANDING_CLAIMS for p15-price-budget-restricted: HOLD_WITH_DOMAIN; nonauthor review ("nonauthor review open") and acceptance are not claimed here. -/ +def RestrictedTransformedPriceBudget {X : Type} [Fintype X] [DecidableEq X] + (F : RealizedData X) : Prop := + IsRealizedFamily F → (∀ i, 2 ≤ F.d i) → + ∀ (p c : X → ℝ), (∀ v, 0 ≤ p v ∧ p v ≤ 1 / 4) → (∀ v, 0 ≤ c v ∧ c v ≤ phi (p v)) → + ∀ K : ℕ, paletteNumber F.b F.H F.d ≤ K → + coverCost F K c ≤ cappedHazard (16 / 27) (mu F p) + +/-- The sufficient ratio `rho(a, d, p_*) = p_*^(ad - a) / (1 - p_*)^(ad + 1)` of display (T5). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: " rho(a,d,p_*)=p_*^(ad-a)/(1-p_*)^(ad+1). (T5)" +Interface: none; natural-number subtraction `a*d - a` agrees with the source for `d >= 1`. +Does not claim: anything. -/ +noncomputable def rhoTest (a d : ℕ) (pBound : ℝ) : ℝ := + pBound ^ (a * d - a) / (1 - pBound) ^ (a * d + 1) + +/-- Display (T5) as a sufficient test, block by block: if a common upper bound `p_* < 1` has +`0 <= p_v <= p_*` for every coordinate `v` of the block `X_i` and `rho(a_i, d_i, p_*) <= 1`, then +for all `0 <= c_v <= phi(p_v)` the full-block price is at most `q_i`. The source: "This is only a +sufficient test; rho>1 is not a counterexample and must not be labeled infeasible." + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: "If rho<=1 the same argument proves price({X})<=q for that block." +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis; the bound `p_*` is imposed only on the coordinates of the block under test (the source: "for any common upper bound p_*<1", "for that block"), the price bound `0 <= c_v <= phi(p_v)` on every coordinate as in (T1). +Does not claim: the statement (statement only); anything when `rho > 1` (the source: a block with `rho > 1` "must not be labeled infeasible"). -/ +def RhoTestSufficient {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → ∀ (p c : X → ℝ) (pBound : ℝ), 0 ≤ pBound → pBound < 1 → + (∀ v, 0 ≤ c v ∧ c v ≤ phi (p v)) → + ∀ i : Fin F.b, (∀ v ∈ block F i, 0 ≤ p v ∧ p v ≤ pBound) → + rhoTest (F.a i) (F.d i) pBound ≤ 1 → + genPrice c (block F i) ≤ localFailure F p i + +/-- Section 5 of the price budget: the cap test `q >= 2/3` gives `phi(q) = 1`, since `log 3 > 1`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: "or if q>=2/3 (then phi(q)=1, since log3>1)" +Interface: none. +Does not claim: the statement (statement only); the integral proof of `log 3 > 1` is not specified; the checker program of that section is not specified. -/ +def CapTestAtTwoThirds : Prop := + 1 < Real.log 3 ∧ ∀ q : ℝ, 2 / 3 ≤ q → q ≤ 1 → phi q = 1 + +/-- Section 4 of the price budget: for the six-block realization at `p_v = 1/40900`, the SAME +`816`-label full-block cover is valid for the FULL transformed range +`0 <= c_v <= log(40900/40899)` (which is `phi(1/40900)`): it covers `O_816(D)` and its price meets +the (T1) budget `min(1, (16/27)[-log mu_p(D)])`; consequently so does the optimal cover cost. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: " 0<=c_v<=log(40900/40899)," +Interface: none; the specified cover is `fullBlockFamily benchmark816`; the last conjunct is the (T1) consequence for the infimum over all covers. +Does not claim: the statement (statement only); "No large witness enumeration" is involved, nor any claim about earlier P15 inputs. -/ +def Benchmark816TransformedRange : Prop := + phi (1 / 40900) = Real.log (40900 / 40899) ∧ + IsGeneratorCover benchmark816 816 (fullBlockFamily benchmark816) ∧ + ∀ c : Fin 6 × Fin 409 → ℝ, (∀ v, 0 ≤ c v ∧ c v ≤ Real.log (40900 / 40899)) → + famPrice c (fullBlockFamily benchmark816) ≤ + cappedHazard (16 / 27) (mu benchmark816 (fun _ => (1 : ℝ) / 40900)) ∧ + coverCost benchmark816 816 c ≤ + cappedHazard (16 / 27) (mu benchmark816 (fun _ => (1 : ℝ) / 40900)) + +/-- Display (T7) and the explicit rational price of Section 4: at maximal equal prices the union +price is `6 log(40900/40899)^409 <= 6/(40899^409)`; `c = p + p^2/2 = 81801/3345620000` for +`p = 1/40900` is admissible (`c < phi(p)`) and its cover price is exactly `6 c^409`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/price_budget_20260924/PROOF.md sha256 3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535 +Anchor: " 6*log(40900/40899)^409 <=6/(40899^409). (T7)" +Interface: none. +Does not claim: the statements (statement only); the rational identity `1/40900 + (1/40900)^2/2 = 81801/3345620000` is exact arithmetic not re-proved here; admissibility rests on `integral_0^p 1/(1-t) dt > integral_0^p (1+t) dt`, which is not specified. -/ +def Benchmark816RationalPrice : Prop := + 6 * Real.log (40900 / 40899) ^ 409 ≤ (6 : ℝ) / 40899 ^ 409 ∧ + (1 / 40900 : ℝ) + (1 / 40900) ^ 2 / 2 = 81801 / 3345620000 ∧ + (81801 / 3345620000 : ℝ) < phi (1 / 40900) ∧ + famPrice (fun _ => (81801 : ℝ) / 3345620000) (fullBlockFamily benchmark816) = + 6 * (81801 / 3345620000) ^ 409 + +/-! ## Full price: constants (F1), the result (F2) and the enclosure (F3) -/ + +/-- `p_star = 1 - exp(-1)`, display (F1). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " p_star=1-exp(-1)," +Interface: none. +Does not claim: anything. -/ +noncomputable def pStar : ℝ := 1 - Real.exp (-1) + +/-- `h_star = 3 - log(3e - 2)`, display (F1); ASCII `3e-2` of the source means `3*e - 2`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " h_star=3-log(3e-2)," +Interface: none. +Does not claim: `h_star > 1` (that is `HStarExceedsOne`). -/ +noncomputable def hStar : ℝ := 3 - Real.log (3 * Real.exp 1 - 2) + +/-- `rho_star = 1/h_star = 1/(3 - log(3e - 2))`, display (F1); written out so that it is +definitionally `1 / hStar`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " rho_star=1/h_star. (F1)" +Interface: none. +Does not claim: the enclosure (F3) or `rho_star < 6/7` (those are `RhoStarEnclosure`, `RhoStarBelowSixSevenths`). -/ +noncomputable def rhoStar : ℝ := 1 / (3 - Real.log (3 * Real.exp 1 - 2)) + +/-- Lower endpoint `0.84547981724898672067` of display (F3), as a rational over `10^20`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " 0.84547981724898672067 < rho_star" +Interface: none. +Does not claim: anything. -/ +noncomputable def rhoStarLower : ℝ := 84547981724898672067 / 10 ^ 20 + +/-- Upper endpoint `0.84547981724898672068` of display (F3), as a rational over `10^20`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " < 0.84547981724898672068 < 6/7. (F3)" +Interface: none. +Does not claim: anything; adjacency of the endpoints and `upper * 7 < 6 * 10^20` are core-Lean targets on main. -/ +noncomputable def rhoStarUpper : ℝ := 84547981724898672068 / 10 ^ 20 + +/-- The headline result F, display (F2): for every realized family with `a_i >= 1`, `d_i >= 2`, +EVERY independent `p in [0,1]^X`, every `0 <= c_v <= phi(p_v)` and the SAME `K >= K_H(d)`, +`covercost_c(O_K(D)) <= min(1, rho_star [-log mu_p(D)])`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " covercost_c(O_K(D)) <= min(1,rho_star[-log mu_p(D)]). (F2)" +Interface: the family data `F` with `IsRealizedFamily F` and `d_i >= 2` as hypotheses; `K_H(d)` is `paletteNumber`; the zero-probability endpoints follow the `cappedHazard` convention ("the price-one cap proves the assertion"). +Does not claim: the bound (statement only); optimality of `rho_star` (that is `SharpnessOfRhoStar`); any unrestricted downset or prize statement (the source: "palette size remains K_H(d), which can depend on H and d"); the nonauthor review recorded in LANDING_CLAIMS (REVIEWED_SCOPED) is not reproduced or extended here. -/ +def FullTransformedPriceBudget {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : + Prop := + IsRealizedFamily F → (∀ i, 2 ≤ F.d i) → + ∀ (p c : X → ℝ), (∀ v, 0 ≤ p v ∧ p v ≤ 1) → (∀ v, 0 ≤ c v ∧ c v ≤ phi (p v)) → + ∀ K : ℕ, paletteNumber F.b F.H F.d ≤ K → coverCost F K c ≤ cappedHazard rhoStar (mu F p) + +/-- Display (F3): `0.84547981724898672067 < rho_star < 0.84547981724898672068 < 6/7`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " 0.84547981724898672067 < rho_star" +Interface: none. +Does not claim: the enclosure (statement only; the source's `full_price.py` Fraction procedure is not a kernel proof and is not specified); this is the next full-result target named in main's `docs/FORMAL_VERIFICATION_ROLLOUT_20260927.md`, still at `specified`. -/ +def RhoStarEnclosure : Prop := + rhoStarLower < rhoStar ∧ rhoStar < rhoStarUpper ∧ rhoStarUpper < 6 / 7 + +/-- The simpler rational factor: `rho_star < 6/7`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "Thus 6/7 is a simpler valid rational factor." +Interface: none. +Does not claim: the inequality (statement only); `16/27 < 6/7` is a core-Lean target on main. -/ +def RhoStarBelowSixSevenths : Prop := + rhoStar < 6 / 7 + +/-! ## Full price Section 2: the hazard interpolation bounds (F4)-(F8) -/ + +/-- `A` is a proper decreasing family containing the empty set: `empty in A`, `X notin A`, and +`A` is closed under taking subsets. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "Let A be any proper decreasing family on n>=1 coordinates containing the empty set." +Interface: none. +Does not claim: anything. -/ +def IsProperDownset {Y : Type} [Fintype Y] (A : Finset (Finset Y)) : Prop := + ∅ ∈ A ∧ Finset.univ ∉ A ∧ ∀ U ∈ A, ∀ V : Finset Y, V ⊆ U → V ∈ A + +/-- `F_A(t) = -log mu_(p(t))(A)` with `p_i(t) = 1 - exp(-t_i)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " p_i(t)=1-exp(-t_i), t in [0,1]^n," +Interface: none. +Does not claim: anything. -/ +noncomputable def hazardCoord {Y : Type} [Fintype Y] [DecidableEq Y] (A : Finset (Finset Y)) + (t : Y → ℝ) : ℝ := + -Real.log (muOf A (fun i => 1 - Real.exp (-t i))) + +/-- `H_A = F_A(1, ..., 1) = -log mu_(p_star, ..., p_star)(A)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " H_A=F_A(1,...,1)>0." +Interface: none. +Does not claim: positivity (that is `HazardStarPositive`). -/ +noncomputable def hazardStar {Y : Type} [Fintype Y] [DecidableEq Y] (A : Finset (Finset Y)) : ℝ := + -Real.log (muOf A (fun _ => pStar)) + +/-- `H_A > 0` for every proper decreasing family on `n >= 1` coordinates containing the empty set. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "The probability is positive on this compact cube because the empty set is admitted and every p_i(t)<1." +Interface: `A` with `IsProperDownset A` as hypothesis. +Does not claim: the statement (statement only). -/ +def HazardStarPositive : Prop := + ∀ (n : ℕ) (A : Finset (Finset (Fin n))), 1 ≤ n → IsProperDownset A → 0 < hazardStar A + +/-- Display (F5) as separate coordinatewise concavity of `F_A` on `[0,1]` in each coordinate, +the other coordinates held fixed in `[0,1]`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "Only SEPARATE coordinatewise concavity is asserted; joint concavity is not needed or claimed." +Interface: `A` with `IsProperDownset A` as hypothesis. +Does not claim: the statement (statement only); joint concavity; the derivative formulas (F4)-(F5) themselves (not specified; concavity is their consequence). -/ +def CoordinatewiseConcavity : Prop := + ∀ (n : ℕ) (A : Finset (Finset (Fin n))), 1 ≤ n → IsProperDownset A → + ∀ (i : Fin n) (t : Fin n → ℝ), (∀ j, 0 ≤ t j ∧ t j ≤ 1) → + ConcaveOn ℝ (Set.Icc (0 : ℝ) 1) (fun s : ℝ => hazardCoord A (Function.update t i s)) + +/-- Display (F6): `F_A(t) >= (prod_i t_i) H_A` on the cube `[0,1]^n`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " F_A(t) >= (product_i t_i) H_A. (F6)" +Interface: `A` with `IsProperDownset A` as hypothesis. +Does not claim: the inequality (statement only). -/ +def HazardProductBound : Prop := + ∀ (n : ℕ) (A : Finset (Finset (Fin n))), 1 ≤ n → IsProperDownset A → + ∀ t : Fin n → ℝ, (∀ j, 0 ≤ t j ∧ t j ≤ 1) → (∏ i, t i) * hazardStar A ≤ hazardCoord A t + +/-- Display (F7): for `p in [0,1]^n` with `mu_p(A) > 0`, `-log mu_p(A) >= H_A prod_i phi(p_i)`; +when `mu_p(A) = 0` the source reads the inequality as an extended one and nothing is stated. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " -log mu_p(A) >= H_A product_i phi(p_i). (F7)" +Interface: `A` with `IsProperDownset A` as hypothesis. +Does not claim: the inequality (statement only). -/ +def HazardPhiBound : Prop := + ∀ (n : ℕ) (A : Finset (Finset (Fin n))), 1 ≤ n → IsProperDownset A → + ∀ p : Fin n → ℝ, (∀ j, 0 ≤ p j ∧ p j ≤ 1) → 0 < muOf A p → + hazardStar A * ∏ i, phi (p i) ≤ -Real.log (muOf A p) + +/-- Display (F8): for transformed prices `0 <= c_i <= phi(p_i)`, +`prod_i c_i <= min(1, [-log mu_p(A)]/H_A)`, with the `cappedHazard` endpoint convention. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " product_i c_i <= min(1, [-log mu_p(A)]/H_A). (F8)" +Interface: `A` with `IsProperDownset A` as hypothesis. +Does not claim: the inequality (statement only); that the full-ground generator covers any obstruction ("It does not assert that this generator alone covers an arbitrary obstruction"). -/ +def FullGroundPriceBound : Prop := + ∀ (n : ℕ) (A : Finset (Finset (Fin n))), 1 ≤ n → IsProperDownset A → + ∀ (p c : Fin n → ℝ), (∀ j, 0 ≤ p j ∧ p j ≤ 1) → (∀ j, 0 ≤ c j ∧ c j ≤ phi (p j)) → + ∏ i, c i ≤ cappedHazard (1 / hazardStar A) (muOf A p) + +/-- `phi(p_star) = 1`: at the reference vector `p_i = p_star`, `c_i = 1`, the two sides of (F7) +are equal. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "at every p_i=p_star and every c_i=1, the two sides of (F7) are equal" +Interface: none. +Does not claim: the identity (statement only). -/ +def PhiAtPStar : Prop := + phi pStar = 1 + +/-- Exactness of `1/H_A`: a full-ground generator universally meets the capped hazard budget +(for all `p in [0,1]^n` and all transformed prices) precisely when `H_A >= 1`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "a full-ground generator universally meets the capped hazard budget precisely when H_A>=1" +Interface: `A` with `IsProperDownset A` as hypothesis. +Does not claim: the equivalence (statement only); coverage of arbitrary `A` ("it does not infer coverage of arbitrary A"). -/ +def FullGroundBudgetIff : Prop := + ∀ (n : ℕ) (A : Finset (Finset (Fin n))), 1 ≤ n → IsProperDownset A → + ((∀ (p c : Fin n → ℝ), (∀ j, 0 ≤ p j ∧ p j ≤ 1) → (∀ j, 0 ≤ c j ∧ c j ≤ phi (p j)) → + ∏ i, c i ≤ cappedHazard 1 (muOf A p)) ↔ 1 ≤ hazardStar A) + +/-! ## Full price Section 3: capacity families and the sharp worst case (F9)-(F11) -/ + +/-- The capacity family `{S subset {0, ..., n-1} : |S| <= a}`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "For a capacity restriction A={S:|S|<=a} on n coordinates," +Interface: none. +Does not claim: anything. -/ +def capacityFamily (n a : ℕ) : Finset (Finset (Fin n)) := + (Finset.univ : Finset (Fin n)).powerset.filter (fun U => U.card ≤ a) + +/-- The binomial lower tail `P(Bin(n, p) <= a) = sum_{j <= a} C(n, j) p^j (1 - p)^(n - j)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " H_(n,a)=-log P(Bin(n,p_star)<=a). (F9)" +Interface: none. +Does not claim: anything. -/ +noncomputable def binomTail (n a : ℕ) (p : ℝ) : ℝ := + ∑ j ∈ Finset.range (a + 1), (Nat.choose n j : ℝ) * p ^ j * (1 - p) ^ (n - j) + +/-- Display (F9): `H_(n,a) = -log P(Bin(n, p_star) <= a)`, i.e. the product measure of the +capacity family at `p_star` is the binomial tail. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " H_(n,a)=-log P(Bin(n,p_star)<=a). (F9)" +Interface: none. +Does not claim: the identity (statement only; a finite identity for each `n`, general in `n`). -/ +def CapacityHazardIsBinomial : Prop := + ∀ n a : ℕ, hazardStar (capacityFamily n a) = -Real.log (binomTail n a pStar) + +/-- Display (F10): for `S ~ Bin(2a+1, p)`, +`P(Bin(2a+3, p) <= a+1) - P(S <= a) = (1-p)^2 P(S = a+1) - p^2 P(S = a) = p(1-2p) P(S = a)`, and +this is negative for `1/2 < p < 1` (the source writes `p > 1/2`; at `p = 1` the difference is +`0`, so the strict inequality is stated for `p < 1`, which covers `p_star`). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " =p(1-2p)P(S=a)<0. (F10)" +Interface: none. +Does not claim: the identities (statement only; polynomial identities in `p`, a future kernel target for each fixed `a` and by induction in general). -/ +def OddMajorityIdentity : Prop := + ∀ (a : ℕ) (p : ℝ), + binomTail (2 * a + 3) (a + 1) p - binomTail (2 * a + 1) a p = + (1 - p) ^ 2 * ((Nat.choose (2 * a + 1) (a + 1) : ℝ) * p ^ (a + 1) * (1 - p) ^ a) - + p ^ 2 * ((Nat.choose (2 * a + 1) a : ℝ) * p ^ a * (1 - p) ^ (a + 1)) ∧ + binomTail (2 * a + 3) (a + 1) p - binomTail (2 * a + 1) a p = + p * (1 - 2 * p) * ((Nat.choose (2 * a + 1) a : ℝ) * p ^ a * (1 - p) ^ (a + 1)) ∧ + (1 / 2 < p → p < 1 → binomTail (2 * a + 3) (a + 1) p - binomTail (2 * a + 1) a p < 0) + +/-- Display (F11): for `a >= 1` and `n >= 2a + 1`, `H_(n,a) >= H_(3,1) = -log[3exp(-2) - 2exp(-3)] += 3 - log(3e - 2) = h_star > 1`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " =3-log(3e-2)=h_star>1. (F11)" +Interface: none. +Does not claim: the statements (statement only); the source stresses "it is not a finite-grid extrapolation". -/ +def CapacityWorstCase : Prop := + (∀ a n : ℕ, 1 ≤ a → 2 * a + 1 ≤ n → + hazardStar (capacityFamily 3 1) ≤ hazardStar (capacityFamily n a)) ∧ + hazardStar (capacityFamily 3 1) = -Real.log (3 * Real.exp (-2) - 2 * Real.exp (-3)) ∧ + -Real.log (3 * Real.exp (-2) - 2 * Real.exp (-3)) = hStar ∧ + 1 < hStar + +/-- The last step of (F11): `e > 2`, `e^2 - (3e - 2) = (e - 1)(e - 2) > 0`, hence +`3exp(-2) - 2exp(-3) < exp(-1)` and `h_star > 1`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " e^2-(3e-2)=(e-1)(e-2)>0," +Interface: none. +Does not claim: the statements (statement only). -/ +def HStarExceedsOne : Prop := + 2 < Real.exp 1 ∧ + Real.exp 1 ^ 2 - (3 * Real.exp 1 - 2) = (Real.exp 1 - 1) * (Real.exp 1 - 2) ∧ + 0 < (Real.exp 1 - 1) * (Real.exp 1 - 2) ∧ + 3 * Real.exp (-2) - 2 * Real.exp (-3) < Real.exp (-1) ∧ + 1 < hStar + +/-! ## Full price Sections 3-4: local price bound (F12) and global assembly (F13) -/ + +/-- Display (F12): for each block with `d_i >= 2` and positive local good probability, under +`0 <= c_v <= phi(p_v)`, `price({X_i}) <= rho_star [-log mu_p(D_i)]`, and the price is at most +one. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " price({X_i}) <= rho_star [-log mu_p(D_i)], (F12)" +Interface: the family data `F` with `IsRealizedFamily F` and `d_i >= 2` as hypotheses. +Does not claim: the bound (statement only); `price <= q_i` ("The inequality price<=q_i is NOT required and is in general stronger than what is proved here"). -/ +def LocalPriceBoundFull {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → (∀ i, 2 ≤ F.d i) → + ∀ (p c : X → ℝ), (∀ v, 0 ≤ p v ∧ p v ≤ 1) → (∀ v, 0 ≤ c v ∧ c v ≤ phi (p v)) → + ∀ i : Fin F.b, 0 < localGood F p i → + genPrice c (block F i) ≤ rhoStar * (-Real.log (localGood F p i)) ∧ + genPrice c (block F i) ≤ 1 + +/-- Display (F13): `mu_p(D) <= prod_i mu_p(D_i)` and, when `mu_p(D) > 0`, +`sum_i [-log mu_p(D_i)] <= -log mu_p(D)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " sum_i [-log mu_p(D_i)] <= -log mu_p(D). (F13)" +Interface: the family data `F` with `IsRealizedFamily F` as hypothesis; `p in [0,1]^X`. +Does not claim: the inequalities (statement only; "without independence of crossing failures"). -/ +def GlobalAssemblyHazard {X : Type} [Fintype X] [DecidableEq X] (F : RealizedData X) : Prop := + IsRealizedFamily F → ∀ p : X → ℝ, (∀ v, 0 ≤ p v ∧ p v ≤ 1) → + mu F p ≤ ∏ i, localGood F p i ∧ + (0 < mu F p → ∑ i, -Real.log (localGood F p i) ≤ -Real.log (mu F p)) + +/-! ## Full price Section 5: sharpness and the exact demand boundary -/ + +/-- One block with `a = 1`, `d = 2`, `n = 3`, no edges. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "Every proper subset is two-decomposable and the full triple is not." +Interface: none; a concrete data record. +Does not claim: anything. -/ +def tripleBlock : RealizedData (Fin 3) where + b := 1 + blk := fun _ => 0 + a := fun _ => 1 + d := fun _ => 2 + H := ∅ + +/-- Sharpness of `rho_star` even among alternative covers: for the triple block at `K = 2`, the +obstruction is exactly `{X}`; at `p_i = p_star`, `c_i = 1` (admissible since `phi(p_star) = 1`) +every generator has price one, every cover of the obstruction costs at least one and the +one-generator full-block cover `{X}` attains it (so the exact cover cost is one), the hazard is +`h_star`, and replacing `rho_star` by any smaller constant makes the right side of (F2) strictly +less than one. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "Replacing rho_star by any smaller constant in (F2) would make its right side strictly less than one." +Interface: none; everything is concrete. +Does not claim: the statements (statement only; the finite parts are decidable in principle); a necessity claim for multi-block families with some demand one (the source: "other constraints may change its budget"). -/ +def SharpnessOfRhoStar : Prop := + IsRealizedFamily tripleBlock ∧ + paletteNumber 1 ∅ (fun _ => 2) = 2 ∧ + (∀ U : Finset (Fin 3), U ∈ obstruction tripleBlock 2 ↔ U = Finset.univ) ∧ + phi pStar = 1 ∧ + (∀ g : Finset (Fin 3), genPrice (fun _ => (1 : ℝ)) g = 1) ∧ + (∀ G : Finset (Finset (Fin 3)), IsGeneratorCover tripleBlock 2 G → + 1 ≤ famPrice (fun _ => (1 : ℝ)) G) ∧ + IsGeneratorCover tripleBlock 2 (fullBlockFamily tripleBlock) ∧ + famPrice (fun _ => (1 : ℝ)) (fullBlockFamily tripleBlock) = 1 ∧ + coverCost tripleBlock 2 (fun _ => (1 : ℝ)) = 1 ∧ + -Real.log (mu tripleBlock (fun _ => pStar)) = hStar ∧ + ∀ ρ : ℝ, ρ < rhoStar → cappedHazard ρ (mu tripleBlock (fun _ => pStar)) < 1 + +/-- One block with capacity `cap`, demand `1`, `n = cap + 1`, no edges. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "If d=1, n=a+1 and K=1, the sole obstruction is again the full block." +Interface: none; a concrete data record for each `cap`. +Does not claim: anything. -/ +def demandOneBlock (cap : ℕ) : RealizedData (Fin (cap + 1)) where + b := 1 + blk := fun _ => 0 + a := fun _ => cap + d := fun _ => 1 + H := ∅ + +/-- The exact demand boundary: for every `a >= 1`, in the demand-one block at `K = 1` the sole +obstruction is the full block; at `p_i = p_star`, `c_i = 1` every cover costs at least one, but +`mu_p(D) = 1 - p_star^(a+1) > 1 - p_star = exp(-1)`, so the hazard is less than one and no +constant `rho <= 1` makes the same-palette transformed-price guarantee hold. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: " mu_p(D)=1-p_star^(a+1)>1-p_star=exp(-1)," +Interface: none; concrete for each `cap`. +Does not claim: the statements (statement only); any retraction of original P15-B ("whose local-budget premise is missing in that case"). -/ +def DemandOneBoundary : Prop := + ∀ cap : ℕ, 1 ≤ cap → + IsRealizedFamily (demandOneBlock cap) ∧ + (∀ U : Finset (Fin (cap + 1)), U ∈ obstruction (demandOneBlock cap) 1 ↔ U = Finset.univ) ∧ + 1 ≤ coverCost (demandOneBlock cap) 1 (fun _ => (1 : ℝ)) ∧ + mu (demandOneBlock cap) (fun _ => pStar) = 1 - pStar ^ (cap + 1) ∧ + Real.exp (-1) < 1 - pStar ^ (cap + 1) ∧ + -Real.log (mu (demandOneBlock cap) (fun _ => pStar)) < 1 ∧ + ∀ ρ : ℝ, ρ ≤ 1 → + cappedHazard ρ (mu (demandOneBlock cap) (fun _ => pStar)) < + coverCost (demandOneBlock cap) 1 (fun _ => (1 : ℝ)) + +/-! ## Full price Section 6: rational certificates for `rho_star < 6/7` -/ + +/-- The short rational certificate chain of Section 6: `e < 31967/11760 < 87/32`, +`87/32 - 31967/11760 = 11/23520`, `sum_{j=0}^5 (11/6)^j/j! - 197/32 = 26081/933120 > 0`, +hence `3e - 2 < 197/32 < exp(11/6)`, `h_star > 7/6` and `rho_star < 6/7`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/full_price_20260924/PROOF.md sha256 87521901ca8e5405b4d1e47f1deb1cd0326affbd6f5967b53c4178590da993f9 +Anchor: "Therefore 3e-2<197/327/6 and rho_star<6/7." +Interface: none. +Does not claim: the statements (statement only); the two exact rational identities are exact arithmetic not re-proved here; the series and geometric-tail bounds behind `e < 31967/11760` are not specified. -/ +def SixSeventhsCertificate : Prop := + Real.exp 1 < 31967 / 11760 ∧ (31967 / 11760 : ℝ) < 87 / 32 ∧ + (87 / 32 - 31967 / 11760 : ℝ) = 11 / 23520 ∧ + (∑ j ∈ Finset.range 6, (11 / 6 : ℝ) ^ j / (Nat.factorial j : ℝ)) - 197 / 32 = 26081 / 933120 ∧ + (0 : ℝ) < 26081 / 933120 ∧ + 3 * Real.exp 1 - 2 < 197 / 32 ∧ (197 / 32 : ℝ) < Real.exp (11 / 6) ∧ + 7 / 6 < hStar ∧ rhoStar < 6 / 7 + +end UniversalLaw.Spec.P15 diff --git a/specifications/lean_statements_20261010/Specifications/RN.lean b/specifications/lean_statements_20261010/Specifications/RN.lean new file mode 100644 index 000000000..8af911ef6 --- /dev/null +++ b/specifications/lean_statements_20261010/Specifications/RN.lean @@ -0,0 +1,1009 @@ +import Mathlib + +/-! +# RN-track statements: count interface (Theorem N), fixed-remote and fixed-annulus +# height windows, the D5 reconciliation rows, the dimension lift, the C6 moment +# bounds, the rare-cluster Poisson obstruction and the open witness-collision target +# (statements only) + +Scientific effect: NONE. Formal progress of every declaration here: `specified`. +Organizational-independence credit of this transcription: 0 (Anthropic / Claude). + +This module states, as `Prop`-valued definitions, the shapes of the RN-track results: +the abstract probability-to-count interface (N1)-(N5) of RN_COUNT_INTERFACE.md, the +fixed-remote Theorem A displays (2) and (3) of remote_window_20260924/PROOF.md, the +fixed-annulus height-window candidate (A2) of FIXED_ANNULUS_CANDIDATE.md, the all-height +annulus bridge display (2) of rn_annulus_bridge_20260925/PROOF.md, the six reconciled +rows D5-a ... D5-f of RECONCILIATION.md, the headline displays of Theorems P_d, C_d, +I_d, G_d of d5_dimension_lift_20260929/PROOF.md, the planar (F4) display of +c6_fourier_cutoff_20260929/PROOF.md, Theorem Q and Corollary Theta of +c6_palm_route_20260929/PROOF.md, Theorem O of c6_rare_cluster_laws_20260929/PROOF.md, +Corollary D of remote_collision_20260928/PROOF.md, and, labelled as an OPEN obligation +whose Prop is a target and not a result, the regional near-pin witness-collision +second-moment target of GRAPH node `math.rn-region.witness-collision`. + +Every statement about the pinned Gaussian model is parametrised by an interface object +`PinnedLawInterface`: the sample space, the sample paths, the Gaussian regression law +`Q_r`, the typed endpoint weight `W_r`, the tilted law `Q_r^W = (W_r/Z_r) Q_r` and the +contact kernel `Lambda_j` are GIVEN; their Gaussian-regression construction, the +determinant form of the weight and the Kac-Rice representation of the kernel are informal +in the sources and are not transcribed. The pin equations, the positivity of the normalizer +and the exact tilt are required only in the pin regime `0 < r < L`, this module's reading of +the sources' "For small r>0 the pins are" (for `0 < r < L` the two pins are distinct modulo +`L Z^d`, so the pin equations are consistent with `L`-periodicity); outside that regime the +laws are unconstrained given probability measures. The statements are about whatever objects satisfy +the interface; the identification of those objects with the objects of the source texts is +not part of any statement. + +This module is self-contained: it imports only Mathlib and re-declares, up to the +namespace, the shared objects of `Specifications/Field.lean` that it needs (ambient space, +lattice vectors, Hessian form, critical points, inertia, pins, torus distance, fundamental +domain): the nine bodies are byte-identical to Field.lean's, and so are the docstrings except +the `Does not claim:` lines of `pinM` and `pinS`, which are module-specific (Field.lean +records that no statement there consumes the pins; here the interface and the window +statements consume them). The duplication is deliberate at this layer and is recorded as +such. + +What this module does not establish. It consists only of definitions and structures; it +states nothing as a result and proves nothing: none of Theorem N, Theorem A, the annulus +candidates, the D5 rows, the dimension lift, the C6 bounds, the rare-cluster Poisson +obstruction or the collision target is shown here, and no kernel receipt about them exists. It does not +construct the Gaussian field or the regression law `Q_r`; it does not transcribe the +determinant/inertia form of `W_r` (only its sign and typing), the normalizer floor +`Z_r >= z_* r^2`, the observation transform `U_r`, the coupling estimates, the weighted +Kac-Rice formulas (N6), (12), (A21), (16), (3.1) or the contact-kernel integral (13); it +asserts no numerical constant, radius or cutoff; it does not state the two-scale, thin-tube, +microdisk, collar, intermediate-window or elder sources that the D5 rows consume; it does +not state Theorem F (F1)-(F2) or Proposition B of the Fourier cutoff note, Corollary P of +the Palm route, Theorems C and S of the rare-cluster note, or Theorem C / Corollaries E, F +of the remote-collision note; and it does not assert that the open target is attainable. +Nothing here promotes, reclassifies or discharges any claim, premise or obligation of the +registers: the landing dispositions `HOLD_WITH_DOMAIN` (rn-count-interface, +rn-fixed-remote-window) and `REVIEWED_SCOPED` (rn-fixed-annulus-window), and the GRAPH +classifications of the `math.rn-region.*` nodes (including `OPEN_ACTIVE` for +`witness-collision`), are transcribed from the registers by the statement register, never +re-derived here; alignment of these statements with the sources is +PENDING_INDEPENDENT_REVIEW. + +Sources (Layer 0, byte-pinned, `d6g8k5htny-coder/Math-`): +* `lifetime-parent` (shared objects): commit `c39164254ad1529ed49543743598d5dcb24bab98`, + path `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, + sha256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. +* `rn-count-interface`: commit `760340e921ac4ceda296b8118da936f1133e956e`, path + `frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md`, + sha256 `aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab`. +* `rn-fixed-remote-window`: commit `760340e921ac4ceda296b8118da936f1133e956e`, path + `frontiers/remote_window_20260924/PROOF.md`, + sha256 `a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7`. +* `rn-fixed-annulus-window`: commit `760340e921ac4ceda296b8118da936f1133e956e`, path + `frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md`, + sha256 `1fd9fe7141e464fd1c09ebf0318d8a701729c9c61ea24f73f7e20a9cf10c552b`. +* `rn-annulus-bridge`: commit `760340e921ac4ceda296b8118da936f1133e956e`, path + `frontiers/rn_annulus_bridge_20260925/PROOF.md`, + sha256 `d55e2c03bb17e7977ff94130cc1ff21e54840cd1e20dc4e41ea9ad52228beb05`. +* `d5-reconciliation`: commit `6c020d6a72936632f2055122e71a7818a4ff497e`, path + `reviews/d5_reconciliation_20260929/RECONCILIATION.md`, + sha256 `14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432`. +* `d5-dimension-lift`: commit `6c020d6a72936632f2055122e71a7818a4ff497e`, path + `frontiers/d5_dimension_lift_20260929/PROOF.md`, + sha256 `6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80`. +* `c6-fourier-cutoff`: commit `6c020d6a72936632f2055122e71a7818a4ff497e`, path + `frontiers/c6_fourier_cutoff_20260929/PROOF.md`, + sha256 `c1692379a3a066589bd2522aaa5b4d480e736b737c8c39a474d1735185793733`. +* `c6-palm-route`: commit `6c020d6a72936632f2055122e71a7818a4ff497e`, path + `frontiers/c6_palm_route_20260929/PROOF.md`, + sha256 `aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b`. +* `c6-rare-cluster`: commit `6c020d6a72936632f2055122e71a7818a4ff497e`, path + `frontiers/c6_rare_cluster_laws_20260929/PROOF.md`, + sha256 `c52a3cf197b5cf26071a8cc951e15ef3ac564b3d7453d41647f91af540a1b8eb`. +* `remote-collision`: commit `6c020d6a72936632f2055122e71a7818a4ff497e`, path + `frontiers/remote_collision_20260928/PROOF.md`, + sha256 `b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2`. +-/ + +namespace UniversalLaw.Spec.RN + +open MeasureTheory + +/-! ### Shared objects, re-declared from `Specifications/Field.lean` (same bodies, this +namespace; see the module header for the two module-specific `Does not claim:` lines). The +duplication is deliberate: modules of this packet import only Mathlib. -/ + +/-- The ambient Euclidean space `R^d` in which the torus `X = R^d/(L Z^d)` is realised +by `L`-periodic functions. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance" +Interface: none; this is the ambient space `EuclideanSpace ℝ (Fin d)`. +Does not claim: that the torus is represented by any quotient type; periodicity is a hypothesis on functions. -/ +abbrev E (d : ℕ) : Type := EuclideanSpace ℝ (Fin d) + +/-- The lattice vector `L n` for `n ∈ Z^d`, written in the standard coordinate vectors. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2)" +Interface: none. +Does not claim: anything beyond the definition of the vector `L n`. -/ +noncomputable def latticeVec (d : ℕ) (L : ℝ) (n : Fin d → ℤ) : E d := + ∑ i, EuclideanSpace.single i (L * (n i : ℝ)) + +/-- The Hessian of `g` at `x` as a bilinear form, through the second iterated Fréchet +derivative: `hessian d g x v w = D^2 g(x)[v, w]`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue." +Interface: none. +Does not claim: symmetry of the form (true for `C^2` functions, not stated here). -/ +noncomputable def hessian (d : ℕ) (g : E d → ℝ) (x v w : E d) : ℝ := + iteratedFDeriv ℝ 2 g x ![v, w] + +/-- Critical point: vanishing Fréchet derivative (`grad f = 0`). + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "grad f(M)=grad f(S)=0." +Interface: none. +Does not claim: differentiability of `g` at `x` (if `g` is not differentiable, `fderiv` is `0`). -/ +def IsCriticalPoint (d : ℕ) (g : E d → ℝ) (x : E d) : Prop := + fderiv ℝ g x = 0 + +/-- Sylvester-inertia encoding of "the Hessian at `x` is nondegenerate with exactly `j` +negative and `d - j` positive eigenvalues": there are complementary subspaces `V`, `W` of +dimensions `j` and `d - j` on which the Hessian form is negative definite, respectively +positive definite. Intended to agree with the eigenvalue count for symmetric (`C^2`) +Hessians; not shown here. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "The index counts negative eigenvalues." +Interface: none. +Does not claim: equivalence with an eigenvalue count for a non-symmetric form. -/ +def HessianHasInertia (d : ℕ) (g : E d → ℝ) (x : E d) (j : ℕ) : Prop := + ∃ V W : Submodule ℝ (E d), IsCompl V W ∧ + Module.finrank ℝ V = j ∧ Module.finrank ℝ W = d - j ∧ + (∀ v ∈ V, v ≠ 0 → hessian d g x v v < 0) ∧ + (∀ w ∈ W, w ≠ 0 → 0 < hessian d g x w w) + +/-- The pin `M = -(r/2) u` of the embedded local cylinder with axial unit vector `u`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "M=-(r/2)u, S=(r/2)u," +Interface: none. +Does not claim: that `u` is a unit vector (a hypothesis where used). -/ +noncomputable def pinM (d : ℕ) (r : ℝ) (u : E d) : E d := + -((r / 2) • u) + +/-- The pin `S = (r/2) u` of the embedded local cylinder with axial unit vector `u`. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "M=-(r/2)u, S=(r/2)u," +Interface: none. +Does not claim: that `u` is a unit vector (a hypothesis where used). -/ +noncomputable def pinS (d : ℕ) (r : ℝ) (u : E d) : E d := + (r / 2) • u + +/-- Torus distance between two points of `R^d`: the infimum over the lattice of +`|x - y + L n|`. For `L > 0` it is intended to vanish exactly when `x` and `y` represent the +same torus point; not shown here. + +Source: d6g8k5htny-coder/Math- commit c39164254ad1529ed49543743598d5dcb24bab98 path imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md sha256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7 +Anchor: "Count ordered maximum/saddle pairs with 0 1`, +`E[N 1_E] ≤ (E N^p)^(1/p) P(E)^(1-1/p)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md sha256 aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab +Anchor: "E[N_r 1_Er] <= (E N_r^p)^(1/p) P(E_r)^(1-1/p)." +Interface: none; the probability space, event and count are universally quantified, integrability of `N^p` is a hypothesis. +Does not claim: that any count of the Gaussian model satisfies the moment hypothesis; the result or its proof; any rate in `r`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). -/ +def HolderCountBoundN1 : Prop := + ∀ (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (Ev : Set Ω) (N : Ω → ℕ) (p : ℝ), + MeasurableSet Ev → 1 < p → Integrable (fun ω => (N ω : ℝ) ^ p) P → + ∫ ω, (N ω : ℝ) * Ev.indicator (fun _ => (1 : ℝ)) ω ∂P ≤ + (∫ ω, (N ω : ℝ) ^ p ∂P) ^ (1 / p) * (P Ev).toReal ^ (1 - 1 / p) + +/-- Shape of (N2), the consequence of (N1) with a power-law moment bound: if +`E N^p ≤ A r^(-β)` and `P(E) ≤ C r^3` then +`E[N 1_E] ≤ A^(1/p) C^(1-1/p) r^((3(p-1)-β)/p)`. The powers refer to the `p`-th MOMENT +itself, as the source insists. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md sha256 aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab +Anchor: "E[N_r 1_Er] <= A^(1/p) C^(1-1/p)" +Interface: none; `A`, `β`, `C`, `p`, `r` and the probability space are universally quantified, the two bounds are hypotheses. +Does not claim: that the exponent can be rounded up to `3`; that any Gaussian count satisfies the hypotheses; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). -/ +def HolderConsequenceN2 : Prop := + ∀ (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (Ev : Set Ω) (N : Ω → ℕ) (p A β C r : ℝ), + MeasurableSet Ev → 1 < p → 0 ≤ A → 0 ≤ C → 0 < r → + Integrable (fun ω => (N ω : ℝ) ^ p) P → + ∫ ω, (N ω : ℝ) ^ p ∂P ≤ A * r ^ (-β) → + (P Ev).toReal ≤ C * r ^ 3 → + ∫ ω, (N ω : ℝ) * Ev.indicator (fun _ => (1 : ℝ)) ω ∂P ≤ + A ^ (1 / p) * C ^ (1 - 1 / p) * r ^ ((3 * (p - 1) - β) / p) + +/-- Shape of (N3), the exact identity `E[N 1_E] = q E[N | E]` with `q = P(E) > 0`, stated +with Mathlib's conditional measure `P[|E] = P(E)⁻¹ P|_E` for the conditional expectation. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md sha256 aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab +Anchor: "E[N_r 1_Er]=q E[N_r | E_r]." +Interface: none; the probability space, event and count are universally quantified, `P(E) > 0` and integrability of `N` are hypotheses. +Does not claim: the equivalence with a bounded conditional mean, the lower comparison `q ≥ c r^3`, or anything about the Gaussian model; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). -/ +def ConditionalMeanIdentityN3 : Prop := + ∀ (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (Ev : Set Ω) (N : Ω → ℕ), + MeasurableSet Ev → 0 < (P Ev).toReal → Integrable (fun ω => (N ω : ℝ)) P → + ∫ ω, (N ω : ℝ) * Ev.indicator (fun _ => (1 : ℝ)) ω ∂P = + (P Ev).toReal * ∫ ω, (N ω : ℝ) ∂(ProbabilityTheory.cond P Ev) + +/-- The law of `U` uniform on `[0,1]`: Lebesgue measure restricted to `[0, 1]`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md sha256 aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab +Anchor: "On one space take U uniform on [0,1]." +Interface: none; an explicit construction. +Does not claim: that this measure is a probability measure (true, not stated as a result here). -/ +noncomputable def uniformLaw : Measure ℝ := volume.restrict (Set.Icc (0 : ℝ) 1) + +/-- The counterexample count `N_r = ceil(log(1/r)) 1_{E_r}` with `E_r = {U ≤ r^3}`, as a +function of the uniform variable `u`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md sha256 aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab +Anchor: "N_r=ceil(log(1/r)) 1_Er." +Interface: none; an explicit construction. +Does not claim: that this count is a realization of any Gaussian field quantity (the source says it is NOT). -/ +noncomputable def counterCount (r : ℝ) (u : ℝ) : ℝ := + (⌈Real.log (1 / r)⌉₊ : ℝ) * (Set.Iic (r ^ 3)).indicator (fun _ => (1 : ℝ)) u + +/-- Shape of the sharp counterexample (N4): for `0 < r ≤ e^(-1)`, `P(E_r) = r^3` and +`E N_r / r^3 = ceil(log(1/r))`, which tends to infinity as `r ↓ 0`, while for EACH finite +`p ≥ 1` the `p`-th moments `E N_r^p` are bounded uniformly in `r` (by a constant that may +depend on `p`). So every finite moment does not preserve the cubic exponent. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md sha256 aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab +Anchor: "E N_r/r^3=ceil(log(1/r)) -> infinity." +Interface: none; the construction is explicit (`uniformLaw`, `counterCount`). +Does not claim: the explicit bound `sup_(t>=1) e^(-3t)(t+1)^p` (only existence of a `p`-dependent bound); the one-`p` sharpness construction `N_r=ceil(r^(-3/p))1_Er`; the exponential-tail remark after (N5); that the construction says anything about the Gaussian field; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). -/ +def UniformCounterexampleN4 : Prop := + (∀ r : ℝ, 0 < r → r ≤ Real.exp (-1) → + (uniformLaw (Set.Iic (r ^ 3))).toReal = r ^ 3 ∧ + (∫ u, counterCount r u ∂uniformLaw) / r ^ 3 = (⌈Real.log (1 / r)⌉₊ : ℝ)) ∧ + Filter.Tendsto (fun r : ℝ => (∫ u, counterCount r u ∂uniformLaw) / r ^ 3) + (nhdsWithin (0 : ℝ) (Set.Ioi 0)) Filter.atTop ∧ + (∀ p : ℝ, 1 ≤ p → ∃ A : ℝ, ∀ r : ℝ, 0 < r → r ≤ Real.exp (-1) → + ∫ u, counterCount r u ^ p ∂uniformLaw ≤ A) + +/-- Shape of (N5), the exponential-tail layer-cake bound: if `P(N > t) ≤ A exp(-t/B)` for +all `t ≥ 0` with `A ≥ 1`, `B > 0`, and `q = P(E) > 0`, then +`E[N 1_E] ≤ B q [1 + log(A/q)]`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md sha256 aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab +Anchor: "= B q[1+log(A/q)]." +Interface: none; the probability space, event, count and tail constants are universally quantified, the tail bound is a hypothesis. +Does not claim: the intermediate integral identity `integral_0^infinity min(q,Ae^(-t/B))dt`; that the logarithmic loss is attained (the source's remark about (N4)); anything about the Gaussian model; the result or its proof; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN of rn-count-interface is transcribed by the register, not here). -/ +def ExponentialTailLayerCakeN5 : Prop := + ∀ (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (Ev : Set Ω) (N : Ω → ℕ) (A B : ℝ), + MeasurableSet Ev → 1 ≤ A → 0 < B → 0 < (P Ev).toReal → + Integrable (fun ω => (N ω : ℝ)) P → + (∀ t : ℝ, 0 ≤ t → (P {ω | t < (N ω : ℝ)}).toReal ≤ A * Real.exp (-t / B)) → + ∫ ω, (N ω : ℝ) * Ev.indicator (fun _ => (1 : ℝ)) ω ∂P ≤ + B * (P Ev).toReal * (1 + Real.log (A / (P Ev).toReal)) + +/-! ### The pinned weighted law as an interface -/ + +/-- Interface for the pinned Gaussian regression law and its determinant tilt. The sample +space `Ω` and its sigma-algebra are parameters; the sample paths `F`, the regression law +`Q_r = Q r b k R` on the `2(d+1)` pin observations at `M = -(r/2)u`, `S = (r/2)u` +(`u = R e_1`), the typed endpoint weight `W_r`, the tilted law `Q_r^W` and the contact +kernel `Lambda_j` are GIVEN. The hypothesis fields transcribe: smooth `L`-periodic sample +paths; `Q_r` is a probability measure under which, in the pin regime `0 < r < L`, the +`2(d+1)` pin equations hold almost surely; `W_r ≥ 0` vanishes unless `H_M` is negative +definite and `H_S` has index `d-1`; in the same regime `Z_r = E_(Q_r) W_r > 0` and +`dQ_r^W = (W_r/Z_r) dQ_r` exactly (`Measure.withDensity`); and `Q_r^W` is a probability +measure. The regime `0 < r < L` is this module's reading of the source clause "For small r>0 +the pins are": for `0 < r < L` the pins `M`, `S` are distinct modulo `L Z^d` (`‖S - M‖ = r` +while every nonzero lattice vector has norm at least `L`), so the pin equations are +consistent with `L`-periodicity; without the gate the structure would have no inhabitant for +`L > 0` (at `r = L` along a coordinate frame the two pins coincide on the torus, so the pin +values `b` and `b - k r^3` would be prescribed at one point). The Gaussian-regression +construction of `Q_r`, the determinant factors `|det H_M| |det H_S|` of `W_r` and the +Kac-Rice contact representation (13) of `Lambda_j` are informal in the sources and are NOT +transcribed. Existence of such an object is not asserted (a Dirac mass at one smooth +`L`-periodic sample path with the prescribed pin values, a nondegenerate maximum at `M` and a +nondegenerate index-`(d-1)` saddle at `S`, with `W = 1` on that path, would satisfy the gated +fields; this consistency remark is not a statement of this module). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/remote_window_20260924/PROOF.md sha256 a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7 +Anchor: "Q_r denotes continuous Gaussian regression on these 2(d+1) observations." +Interface: every field is a given property of the law, never derived; in particular `Λ` is only a given kernel (its continuity and positivity are part of the Theorem A statement, not of this interface); the pin regime `0 < r < L` gating `Q_pinned`, `Z_pos` and `QW_eq` is part of the hypothesis fields, not a derived fact. +Does not claim: that `Q_r` is the Gaussian regression of any field; that `W_r` equals `F_d(H_M) F_(d-1)(H_S)`; the normalizer floor `Z_r >= z_* r^2`; that `Λ` is given by (13); existence or uniqueness of an object satisfying the interface (an inhabitant is not asserted; the consistency remark above is not stated); that `0 < r < L` is the sources' unspecified `small r` threshold. -/ +structure PinnedLawInterface (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] where + /-- The sample paths `ω ↦ f_ω : R^d → ℝ`. -/ + F : Ω → E d → ℝ + /-- Section 2 clause: "The real Fourier series therefore has smooth versions"; every + sample path is `C^n` for every finite `n`. -/ + smooth : ∀ (ω : Ω) (n : ℕ), ContDiff ℝ n (F ω) + /-- Section 1 clause: the field lives "On X=R^d/(L Z^d)"; every sample path is `L`-periodic. -/ + periodic : ∀ (ω : Ω) (x : E d) (n : Fin d → ℤ), F ω (x + latticeVec d L n) = F ω x + /-- Section 1 clause: "Q_r denotes continuous Gaussian regression on these 2(d+1) + observations", indexed by radius `r`, birth `b`, gap mark `k` and frame `R`. -/ + Q : ℝ → ℝ → ℝ → OrthonormalBasis (Fin d) ℝ (E d) → Measure Ω + /-- `Q_r` is a probability measure. -/ + Q_prob : ∀ (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)), IsProbabilityMeasure (Q r b k R) + /-- Section 1 clause: the pins "f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0" hold + `Q_r`-almost surely in the pin regime `0 < r < L` (source: "For small r>0 the pins are"). -/ + Q_pinned : ∀ (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)), 0 < r → r < L → + ∀ᵐ ω ∂(Q r b k R), + F ω (pinM d r (R 0)) = b ∧ F ω (pinS d r (R 0)) = b - k * r ^ 3 ∧ + IsCriticalPoint d (F ω) (pinM d r (R 0)) ∧ IsCriticalPoint d (F ω) (pinS d r (R 0)) + /-- Section 1 clause: the "actual endpoint weight" `W_r=F_d(H_M) F_(d-1)(H_S)`, as a + given functional of the sample. -/ + W : ℝ → ℝ → ℝ → OrthonormalBasis (Fin d) ℝ (E d) → Ω → ℝ + /-- `W_r ≥ 0` (a product of filtered absolute determinants). -/ + W_nonneg : ∀ (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)) (ω : Ω), 0 ≤ W r b k R ω + /-- Section 1 clause: `F_j(H)=|det H| 1{H has j negative eigenvalues}`, "with value zero + on singular matrices"; so `W_r ≠ 0` forces `H_M` negative definite and `H_S` of index + `d-1`. -/ + W_typed : ∀ (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)) (ω : Ω), W r b k R ω ≠ 0 → + HessianHasInertia d (F ω) (pinM d r (R 0)) d ∧ HessianHasInertia d (F ω) (pinS d r (R 0)) (d - 1) + /-- Section 1 clause: the "FULL normalizer" `Z_r=E_(Q_r) W_r` is positive in the pin + regime `0 < r < L`. -/ + Z_pos : ∀ (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)), 0 < r → r < L → + 0 < ∫ ω, W r b k R ω ∂(Q r b k R) + /-- The tilted law `Q_r^W`. -/ + QW : ℝ → ℝ → ℝ → OrthonormalBasis (Fin d) ℝ (E d) → Measure Ω + /-- Section 1 clause: `dQ_r^W=(W_r/Z_r)dQ_r`, exactly, in the pin regime `0 < r < L`. -/ + QW_eq : ∀ (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)), 0 < r → r < L → + QW r b k R = (Q r b k R).withDensity + (fun ω => ENNReal.ofReal (W r b k R ω / ∫ ω', W r b k R ω' ∂(Q r b k R))) + /-- `Q_r^W` is a probability measure. -/ + QW_prob : ∀ (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)), IsProbabilityMeasure (QW r b k R) + /-- Section 5 clause (13): "The contact kernel is Lambda_j(x;b,k,u)", a given function of + the index `j`, birth `b`, gap mark `k`, axial direction `u` and position `x`. -/ + Λ : ℕ → ℝ → ℝ → E d → E d → ℝ + +/-! ### Counts and regions -/ + +/-- `N_(r,j)(E)`: the number of index-`j` critical points of `g` in `A` with height STRICTLY +between `b - k r^3` and `b` (`Set.ncard`, which is `0` for an infinite set). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/remote_window_20260924/PROOF.md sha256 a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7 +Anchor: "N_(r,j)(E)=#{x in E: grad f(x)=0, index H_x=j," +Interface: none. +Does not claim: finiteness of the counted set; that boundary heights may be included (the source's remark about zero expected boundary counts is not stated). -/ +noncomputable def windowCount (d : ℕ) (g : E d → ℝ) (b k r : ℝ) (j : ℕ) (A : Set (E d)) : ℕ := + Set.ncard {x : E d | x ∈ A ∧ IsCriticalPoint d g x ∧ HessianHasInertia d g x j ∧ + b - k * r ^ 3 < g x ∧ g x < b} + +/-- All-index window count: the number of critical points of `g` in `A` (degenerate ones +included) with height strictly between `b - k r^3` and `b`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/c6_palm_route_20260929/PROOF.md sha256 aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b +Anchor: "`N` is the number of critical points of `f` in `X minus {M, S}` with height in `I_r`, all" +Interface: none. +Does not claim: finiteness of the counted set; equality with the sum of the index counts (true off the degenerate locus, not stated). -/ +noncomputable def windowCountAll (d : ℕ) (g : E d → ℝ) (b k r : ℝ) (A : Set (E d)) : ℕ := + Set.ncard {x : E d | x ∈ A ∧ IsCriticalPoint d g x ∧ b - k * r ^ 3 < g x ∧ g x < b} + +/-- `N(E) = sum_j N_j(E)`: the all-index window count of the remote-collision note, the sum +over `j = 0, ..., d` of the index-`j` window counts (critical points with nonsingular Hessian +of index `j`; an index above `d` does not occur in `R^d`). + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/remote_collision_20260928/PROOF.md sha256 b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2 +Anchor: "N(E) = sum_j N_j(E)." +Interface: none. +Does not claim: finiteness of the counted sets; equality with `windowCountAll` (the two agree only off the degenerate locus). -/ +noncomputable def windowCountIndexSum (d : ℕ) (g : E d → ℝ) (b k r : ℝ) (A : Set (E d)) : ℕ := + ∑ j ∈ Finset.range (d + 1), windowCount d g b k r j A + +/-- `N_j(E)`: the number of index-`j` critical points of `g` in `A` at all heights. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/rn_annulus_bridge_20260925/PROOF.md sha256 d55e2c03bb17e7977ff94130cc1ff21e54840cd1e20dc4e41ea9ad52228beb05 +Anchor: "let N_j(rE) count index-j critical points in rE at **all heights**." +Interface: none. +Does not claim: finiteness of the counted set. -/ +noncomputable def allHeightCount (d : ℕ) (g : E d → ℝ) (j : ℕ) (A : Set (E d)) : ℕ := + Set.ncard {x : E d | x ∈ A ∧ IsCriticalPoint d g x ∧ HessianHasInertia d g x j} + +/-- The remote region `D_rho = {x : dist_X(x,0) >= rho}`, represented in the fundamental +domain so that a Borel subset of it carries one representative per torus point. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/remote_window_20260924/PROOF.md sha256 a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7 +Anchor: "Fix 0=rho}." +Interface: none. +Does not claim: anything about the representation of torus Borel sets beyond this choice. -/ +def remoteRegion (d : ℕ) (L ρ : ℝ) : Set (E d) := + {x : E d | x ∈ FundDomain d L ∧ ρ ≤ torusDist d L x 0} + +/-- The torus with the two pins removed, `X minus {M, S}`, represented in the fundamental +domain: points at positive torus distance from both `M = -(r/2)u` and `S = (r/2)u`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/c6_palm_route_20260929/PROOF.md sha256 aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b +Anchor: "critical points of `f` in `X minus {M, S}`" +Interface: none. +Does not claim: anything beyond the definition. -/ +def torusMinusPins (d : ℕ) (L r : ℝ) (u : E d) : Set (E d) := + {x : E d | x ∈ FundDomain d L ∧ 0 < torusDist d L x (pinM d r u) ∧ 0 < torusDist d L x (pinS d r u)} + +/-- The torus-wide all-index window count `N` of the C6 notes: critical points other than +`M, S` with height in `I_r = (b - k r^3, b)`, under the frame `R` (axial direction `R 0`). + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/c6_palm_route_20260929/PROOF.md sha256 aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b +Anchor: "`N` is the number of critical points of `f` in `X minus {M, S}` with height in `I_r`, all" +Interface: `I : PinnedLawInterface` supplies the sample paths. +Does not claim: finiteness or measurability of the count as a function of the sample. -/ +noncomputable def torusWindowCount (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)) (ω : Ω) : ℕ := + windowCountAll d (I.F ω) b k r (torusMinusPins d L r (R 0)) + +/-- Ordered pairs of distinct window critical points (both in `X minus {M, S}`, both with +height in `I_r`, at positive torus distance from each other) such that the FIRST witness is +within torus distance `η` of a pin. All mutual separations are included, down to collision. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path reviews/d5_reconciliation_20260929/RECONCILIATION.md sha256 14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432 +Anchor: "one witness is within fixed `η` of a pin." +Interface: none. +Does not claim: finiteness or measurability of the counted set (`Set.ncard` is `0` on an infinite set); +that an ordered pair whose only near-pin witness is in second position is counted (it is not: it is +the swap of a counted pair, so in the finite case the source's "at least one witness near a pin" +count lies between this count and twice it, and a bound `C r^3` on this count is a bound `2 C r^3` +on the source's; the two functionals are not literally the same). -/ +noncomputable def nearPinPairCount (d : ℕ) (L : ℝ) (g : E d → ℝ) (b k r η : ℝ) (u : E d) : ℕ := + Set.ncard {q : E d × E d | + q.1 ∈ torusMinusPins d L r u ∧ q.2 ∈ torusMinusPins d L r u ∧ 0 < torusDist d L q.1 q.2 ∧ + IsCriticalPoint d g q.1 ∧ IsCriticalPoint d g q.2 ∧ + b - k * r ^ 3 < g q.1 ∧ g q.1 < b ∧ b - k * r ^ 3 < g q.2 ∧ g q.2 < b ∧ + (torusDist d L q.1 (pinM d r u) < η ∨ torusDist d L q.1 (pinS d r u) < η)} + +/-- The scaled chart `x0 + r E0` read through the frame `R`: the image of a coordinate set +`E0` (coordinates `p` with `p 0` axial) under `p ↦ x0 + r ∑ p_i R_i`. With `x0 = 0` these +are the midpoint scaled coordinates `X = r(u, v)`, in which the pins are `(±1/2, 0)`; with +`x0 = M` they are the endpoint chart `M + r(p, q)`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path reviews/d5_reconciliation_20260929/RECONCILIATION.md sha256 14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432 +Anchor: "Scaled coordinates are `X = r(u, v)` from the midpoint, so the pins are `(±1/2, 0)`." +Interface: none. +Does not claim: that the image lies in one fundamental domain (the sources take `r` small enough for the chart to embed in the torus). -/ +noncomputable def scaledAt (d : ℕ) (R : OrthonormalBasis (Fin d) ℝ (E d)) (r : ℝ) (x0 : E d) + (E0 : Set (E d)) : Set (E d) := + (fun p : E d => x0 + r • ∑ i, p i • R i) '' E0 + +/-- The compact collar `C(eta, R)` in midpoint scaled coordinates: the ball of radius `R` +(boundary included) with the two open balls of radius `eta` about the scaled pins +`(-1/2, 0)` and `(1/2, 0)` removed. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/d5_dimension_lift_20260929/PROOF.md sha256 6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80 +Anchor: "with the two open balls of radius `eta` about `(-1/2, 0)` and `(1/2, 0)` removed" +Interface: none. +Does not claim: anything beyond the definition; RECONCILIATION.md writes the same set `C_{η,R}` without defining it, and this definition is taken from the dimension-lift source. -/ +def collar (d : ℕ) [NeZero d] (η R0 : ℝ) : Set (E d) := + {p : E d | ‖p‖ ≤ R0 ∧ η ≤ ‖p - EuclideanSpace.single (0 : Fin d) (-(1 : ℝ) / 2)‖ ∧ + η ≤ ‖p - EuclideanSpace.single (0 : Fin d) ((1 : ℝ) / 2)‖} + +/-- The weight `omega_d(p, q) = max(|q|, r |p|)^(2-d)` of the endpoint chart, where `p` is +the axial coordinate (`p 0`) and `q` the transverse part; `omega_2 = 1`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/d5_dimension_lift_20260929/PROOF.md sha256 6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80 +Anchor: "omega_d(p, q) = max(|q|, r |p|)^(2-d)" +Interface: none. +Does not claim: integrability of the weight over the punctured ball (part of the Theorem P_d statement). -/ +noncomputable def omegaD (d : ℕ) [NeZero d] (r : ℝ) (p : E d) : ℝ := + (max ‖p - EuclideanSpace.single (0 : Fin d) (p 0)‖ (r * |p 0|)) ^ ((2 : ℝ) - d) + +/-- `E_(Q_r^W) N_(r,j)(A)`: the expected index-`j` window count in `A` under the tilted law. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/remote_window_20260924/PROOF.md sha256 a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7 +Anchor: "E_(Q_r^W) N_(r,j)(E)" +Interface: `I : PinnedLawInterface` supplies the sample paths and the tilted law; the Kac-Rice representation (12) of this expectation is NOT used. +Does not claim: integrability of the count (if it is not integrable the Bochner integral is `0`). -/ +noncomputable def meanWindowCount (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)) (j : ℕ) + (A : Set (E d)) : ℝ := + ∫ ω, (windowCount d (I.F ω) b k r j A : ℝ) ∂(I.QW r b k R) + +/-- `E_(Q_r^W) N_j(A)`: the expected index-`j` all-height count in `A` under the tilted law. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/rn_annulus_bridge_20260925/PROOF.md sha256 d55e2c03bb17e7977ff94130cc1ff21e54840cd1e20dc4e41ea9ad52228beb05 +Anchor: "E_(Q_r^W) N_j(rE)" +Interface: `I : PinnedLawInterface` supplies the sample paths and the tilted law; the weighted Kac-Rice intensity (16) is NOT used. +Does not claim: integrability of the count. -/ +noncomputable def meanAllHeightCount (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (r b k : ℝ) (R : OrthonormalBasis (Fin d) ℝ (E d)) (j : ℕ) + (A : Set (E d)) : ℝ := + ∫ ω, (allHeightCount d (I.F ω) j A : ℝ) ∂(I.QW r b k R) + +/-! ### Fixed-remote Theorem A, displays (2) and (3) -/ + +/-- Shape of Theorem A, display (2) (fixed-remote mean measure): for fixed `d ≥ 2`, `L > 0`, +`0 < ρ < L/4`, compact `B = [bLo, bHi]` and `K = [kLo, kHi]` with `kLo > 0`, there are +`r_* > 0` and `C` (quantified OUTSIDE the universal quantifiers) such that for all +`0 < r ≤ r_*`, `b ∈ B`, `k ∈ K`, frames `R`, indices `j ≤ d` and Borel `E ⊆ D_ρ`, +`| E_(Q_r^W) N_(r,j)(E)/(k r^3) - ∫_E Lambda_j(x;b,k,u) dx | ≤ C r |E|`; and the kernel +`Lambda_j` is continuous on `D_ρ`, bounded above and away from zero uniformly on the +compact parameter sets. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/remote_window_20260924/PROOF.md sha256 a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7 +Anchor: "| E_(Q_r^W) N_(r,j)(E)/(k r^3)" +Interface: `I` (sample paths, laws `Q_r`, `Q_r^W`, weight, kernel `Λ`) is a given object; the statement is about any `Λ` and any laws satisfying `PinnedLawInterface`; the Kac-Rice representation (12) and the contact formula (13) are not transcribed. +Does not claim: the result or its proof (uniform remote nondegeneracy, coupling (5), weight expansion (10), normalizer (11)); identification of `I.Λ` with the kernel (13); the total-variation convergence remark; the multi-witness Section 6; any numerical `C` or `r_*`; uniformity as `ρ → 0`; nonauthor review or acceptance (the landing disposition HOLD_WITH_DOMAIN is transcribed by the register, not here). -/ +def FixedRemoteMeanMeasureA2 (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (ρ bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → 0 < ρ → ρ < L / 4 → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ rStar : ℝ, 0 < rStar ∧ ∃ C : ℝ, + (∃ cΛ : ℝ, 0 < cΛ ∧ ∃ CΛ : ℝ, ∀ j : ℕ, j ≤ d → + ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, ∀ R : OrthonormalBasis (Fin d) ℝ (E d), + ContinuousOn (I.Λ j b k (R 0)) (remoteRegion d L ρ) ∧ + ∀ x ∈ remoteRegion d L ρ, cΛ ≤ I.Λ j b k (R 0) x ∧ I.Λ j b k (R 0) x ≤ CΛ) ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), ∀ j : ℕ, j ≤ d → + ∀ A : Set (E d), MeasurableSet A → A ⊆ remoteRegion d L ρ → + |meanWindowCount d L Ω I r b k R j A / (k * r ^ 3) - ∫ x in A, I.Λ j b k (R 0) x| ≤ + C * r * (volume A).toReal + +/-- Shape of Theorem A, display (3) (fixed-remote count and event bounds): with the same +fixed data and existential `r_* > 0`, `C`, for all `0 < r ≤ r_*`, `b ∈ B`, `k ∈ K`, frames, +indices and Borel `E ⊆ D_ρ`, `E_(Q_r^W) N_(r,j)(E) ≤ C k r^3 |E|` and +`P_(Q_r^W){N_(r,j)(E) ≥ 1} ≤ C k r^3 |E|`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/remote_window_20260924/PROOF.md sha256 a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7 +Anchor: "E_(Q_r^W) N_(r,j)(E) <= C k r^3 |E|," +Interface: `I` is a given object as in `FixedRemoteMeanMeasureA2`. +Does not claim: a matching lower bound on the event probability (the source says this is NOT one); the unnormalized numerator bound (14); the result or its proof; any numerical constant; nonauthor review or acceptance. -/ +def FixedRemoteCountBoundA3 (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (ρ bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → 0 < ρ → ρ < L / 4 → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ rStar : ℝ, 0 < rStar ∧ ∃ C : ℝ, + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), ∀ j : ℕ, j ≤ d → + ∀ A : Set (E d), MeasurableSet A → A ⊆ remoteRegion d L ρ → + meanWindowCount d L Ω I r b k R j A ≤ C * k * r ^ 3 * (volume A).toReal ∧ + ((I.QW r b k R) {ω | 1 ≤ windowCount d (I.F ω) b k r j A}).toReal ≤ + C * k * r ^ 3 * (volume A).toReal + +/-! ### Fixed scaled annulus (d = 2): the height-window candidate (A2) and the all-height +bridge (2) -/ + +/-- Shape of the fixed-annulus height-window candidate statement (A2): `d = 2`, fixed `L > 0`, +compact marks (`b ∈ [bLo, bHi]`, `0 < kLo ≤ k ≤ kHi`), fixed scaled annulus +`K = {A0 ≤ |(u,t)| ≤ B0}` with `1 < A0 < B0`; there are `C, r_* > 0`, uniform in the frame, +the marks, the Borel `E0 ⊆ K` and `j ∈ {0,1,2}`, such that +`E_(Q_r^W) N_j(r E0) ≤ C k r^3 area(E0)` for `0 < r ≤ r_*`, where `N_j(r E0)` counts +index-`j` critical points in `r E0` with height STRICTLY between `b - k r^3` and `b`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md sha256 1fd9fe7141e464fd1c09ebf0318d8a701729c9c61ea24f73f7e20a9cf10c552b +Anchor: "E_Qr^W N_j(r E0) <= C*k*r^3*area(E0)," +Interface: `I` is a given object; `r E0` is embedded in the torus through the frame by `scaledAt`; the six-pin regression (A3)-(A5), the normalizer (A8) and the weighted Kac-Rice formula (A21) are not transcribed. +Does not claim: the result or its proof (contact vector (A10), eigenfloor (A12), cutoff (A13), three-Hessian estimate (A18), inner-strip bound (A26)); the all-index sum or the Markov event bound; an all-height assertion; uniformity in `L`, as `k → 0`, as `B0 → ∞`, or for `d ≥ 3`; any numerical constant; nonauthor review or acceptance (the landing disposition REVIEWED_SCOPED is transcribed by the register, not here). -/ +def FixedAnnulusWindowCandidateA2 (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface 2 L Ω) (bLo bHi kLo kHi A0 B0 : ℝ) : Prop := + 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → 1 < A0 → A0 < B0 → + ∃ C rStar : ℝ, 0 < C ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), ∀ j : ℕ, j ≤ 2 → + ∀ E0 : Set (E 2), MeasurableSet E0 → E0 ⊆ {p : E 2 | A0 ≤ ‖p‖ ∧ ‖p‖ ≤ B0} → + meanWindowCount 2 L Ω I r b k R j (scaledAt 2 R r 0 E0) ≤ + C * k * r ^ 3 * (volume E0).toReal + +/-- Shape of the all-height fixed-annulus bridge candidate statement, display (2): `d = 2`, +fixed `L > 0`, compact marks, fixed scaled annulus `K_AB = {A ≤ |(u,v)| ≤ B}` with +`1 < A < B`; there are `C, r_* > 0`, uniform in the marks, frames, Borel `E ⊆ K_AB` and +`j ∈ {0,1,2}`, such that `E_(Q_r^W) N_j(rE) ≤ C r^3 area(E)` for `0 < r ≤ r_*`, where +`N_j(rE)` counts index-`j` critical points in `rE` at ALL heights. The review interfaces +R1-R7 of the source are the steps of its proof, not parts of this statement. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path frontiers/rn_annulus_bridge_20260925/PROOF.md sha256 d55e2c03bb17e7977ff94130cc1ff21e54840cd1e20dc4e41ea9ad52228beb05 +Anchor: "E_(Q_r^W) N_j(rE) <= C r^3 area(E), 0 0`, all frames and indices, existential constants) -/ + +/-- Shape of row D5-a (P2, punctured pin disks): all heights; for Borel +`E ⊂ {0 < p² + q² ≤ 1/16}` in the endpoint chart, `E_{Q_r^W} N_j(M + rE) ≤ C r³|E|`, and +the same at `S`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path reviews/d5_reconciliation_20260929/RECONCILIATION.md sha256 14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432 +Anchor: "E_{Q_r^W} N_j(M + rE) ≤ C r³|E|" +Interface: `I` is a given object; the chart `M + rE` is `scaledAt` based at the pin. +Does not claim: the P2 source (PUNCTURED_PIN_PROOF.md) or its proof; the P2 continuum review or the proposed register transitions of the reconciliation; any numerical constant; nonauthor review or acceptance. -/ +def D5aPuncturedPinDisks (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface 2 L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ C rStar : ℝ, 0 < C ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), ∀ j : ℕ, j ≤ 2 → + ∀ E0 : Set (E 2), MeasurableSet E0 → + E0 ⊆ {p : E 2 | 0 < ‖p‖ ^ 2 ∧ ‖p‖ ^ 2 ≤ (1 : ℝ) / 16} → + meanAllHeightCount 2 L Ω I r b k R j (scaledAt 2 R r (pinM 2 r (R 0)) E0) ≤ + C * r ^ 3 * (volume E0).toReal ∧ + meanAllHeightCount 2 L Ω I r b k R j (scaledAt 2 R r (pinS 2 r (R 0)) E0) ≤ + C * r ^ 3 * (volume E0).toReal + +/-- Shape of row D5-b (microdisk): all heights; for fixed `κ > 0`, the expected index-`j` +count in the punctured disk of physical radius `κ r²` about each pin is `O(r⁵)`: there are +`C, r_* > 0` with `E_{Q_r^W} N_j(disk) ≤ C r^5` for `0 < r ≤ r_*`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path reviews/d5_reconciliation_20260929/RECONCILIATION.md sha256 14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432 +Anchor: "the nested disks of radius `κr²` about each pin, for fixed `κ`: `O(r⁵)`" +Interface: `I` is a given object; the disk is punctured at the pin (the pin itself is a critical point under the law), following the dimension-lift reading "nested ball" inside the punctured ball `D`. +Does not claim: the P2 source or its nested-disk corollary; any numerical constant; uniformity in `κ`; nonauthor review or acceptance. -/ +def D5bMicrodisk (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface 2 L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∀ κ : ℝ, 0 < κ → ∃ C rStar : ℝ, 0 < C ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), ∀ j : ℕ, j ≤ 2 → + meanAllHeightCount 2 L Ω I r b k R j + {x : E 2 | 0 < ‖x - pinM 2 r (R 0)‖ ∧ ‖x - pinM 2 r (R 0)‖ ≤ κ * r ^ 2} ≤ C * r ^ 5 ∧ + meanAllHeightCount 2 L Ω I r b k R j + {x : E 2 | 0 < ‖x - pinS 2 r (R 0)‖ ∧ ‖x - pinS 2 r (R 0)‖ ≤ κ * r ^ 2} ≤ C * r ^ 5 + +/-- Shape of row D5-c (C1, compact collar): all heights; for fixed `R ≥ 1` and +`0 < η ≤ 1/4`, for Borel `E ⊂ C_{η,R}` in midpoint scaled coordinates, +`E_{Q_r^W} N_j(rE) ≤ C r³|E|`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path reviews/d5_reconciliation_20260929/RECONCILIATION.md sha256 14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432 +Anchor: "for Borel `E ⊂ C_{η,R}`, `η ≤ 1/4`, fixed `R ≥ 1`" +Interface: `I` is a given object; `C_{η,R}` is `collar 2 η R` (definition taken from the dimension-lift source, since RECONCILIATION.md does not define it). +Does not claim: the C1 source (COLLAR_PROOF.md) or its proof; the normalizer (P14) it imports; any numerical constant; nonauthor review or acceptance. -/ +def D5cCollarC1 (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface 2 L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∀ η R0 : ℝ, 0 < η → η ≤ (1 : ℝ) / 4 → 1 ≤ R0 → ∃ C rStar : ℝ, 0 < C ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), ∀ j : ℕ, j ≤ 2 → + ∀ E0 : Set (E 2), MeasurableSet E0 → E0 ⊆ collar 2 η R0 → + meanAllHeightCount 2 L Ω I r b k R j (scaledAt 2 R r 0 E0) ≤ + C * r ^ 3 * (volume E0).toReal + +/-- Shape of row D5-d (C2, the summed pin-neighbourhood bound): all heights; for fixed +`R ≥ 1`, `E_{Q_r^W} N_j(r B_R ∖ {M,S}) ≤ C_R r³`, the scaled ball of radius `R` with the +two pins removed. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path reviews/d5_reconciliation_20260929/RECONCILIATION.md sha256 14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432 +Anchor: "E_{Q_r^W} N_j(r B_R ∖ {M,S}) ≤ C_R r³" +Interface: `I` is a given object. +Does not claim: the composition `D5-a + D5-c at η = 1/4` as a proof; the "collar to the reviewed annulus" remark; any numerical constant; nonauthor review or acceptance. -/ +def D5dSummedPinNeighbourhoodC2 (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface 2 L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∀ R0 : ℝ, 1 ≤ R0 → ∃ C rStar : ℝ, 0 < C ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), ∀ j : ℕ, j ≤ 2 → + meanAllHeightCount 2 L Ω I r b k R j (scaledAt 2 R r 0 + {p : E 2 | ‖p‖ ≤ R0 ∧ p ≠ EuclideanSpace.single (0 : Fin 2) (-(1 : ℝ) / 2) ∧ + p ≠ EuclideanSpace.single (0 : Fin 2) ((1 : ℝ) / 2)}) ≤ C * r ^ 3 + +/-- Shape of row D5-e (I3/I4, intermediate shells): window `I_r`; the dyadic shells and +`{A_0 r ≤ |X| ≤ ρ}` satisfy `E N ≤ C r³(A_0^{−2} + ρ²)`: there are `C, s_0 > 0` such that +for `A_0 ≥ 4` and `A_0 r < ρ ≤ s_0` the expected index-`j` window count in +`{A_0 r ≤ |X| ≤ ρ}` (distance from the midpoint) is at most `C r^3 (A_0^(-2) + ρ^2)`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path reviews/d5_reconciliation_20260929/RECONCILIATION.md sha256 14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432 +Anchor: "E N ≤ C r³(A_0^{−2} + ρ²)" +Interface: `I` is a given object; the side conditions `A_0 ≥ 4`, `A_0 r < ρ ≤ s_0` are taken from the dimension-lift display (1.6), since RECONCILIATION.md states the row without them. +Does not claim: the single-shell display (1.5) (stated in `DimensionLiftIntermediateShellsI`); the I3/I4 source (intermediate_window_20260928) or its proof; any numerical constant; nonauthor review or acceptance. -/ +def D5eIntermediateShellsI3I4 (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface 2 L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ C s0 : ℝ, 0 < C ∧ 0 < s0 ∧ + ∀ A0 ρ r : ℝ, 4 ≤ A0 → 0 < r → A0 * r < ρ → ρ ≤ s0 → + ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), ∀ j : ℕ, j ≤ 2 → + meanWindowCount 2 L Ω I r b k R j {x : E 2 | A0 * r ≤ ‖x‖ ∧ ‖x‖ ≤ ρ} ≤ + C * r ^ 3 * ((A0 ^ 2)⁻¹ + ρ ^ 2) + +/-- Shape of row D5-f (I5, global window first moment and planar event order): window +`I_r`; `E_{Q_r^W} N_{r,j}(T² ∖ {M,S}) ≤ C r³`, and for the event `A_r` that some critical +point other than `M, S` has height in `I_r`, `c r³ ≤ P(A_r) ≤ C r³` with `c > 0`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path reviews/d5_reconciliation_20260929/RECONCILIATION.md sha256 14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432 +Anchor: "E_{Q_r^W} N_{r,j}(T² ∖ {M,S}) ≤ C r³" +Interface: `I` is a given object; `A_r` is the event `{N ≥ 1}` for the all-index torus-wide window count. +Does not claim: the tiling composition `D5-d (R = 4) + I4 + remote Theorem A` as a proof; the lower bound's source (remote_collision Corollary F-) or the I5 source; any numerical constant; nonauthor review or acceptance; any register transition proposed by the reconciliation; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. -/ +def D5fGlobalWindowFirstMomentI5 (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface 2 L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + (∃ C rStar : ℝ, 0 < C ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), ∀ j : ℕ, j ≤ 2 → + meanWindowCount 2 L Ω I r b k R j (torusMinusPins 2 L r (R 0)) ≤ C * r ^ 3) ∧ + (∃ c C rStar : ℝ, 0 < c ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), + c * r ^ 3 ≤ ((I.QW r b k R) {ω | 1 ≤ torusWindowCount 2 L Ω I r b k R ω}).toReal ∧ + ((I.QW r b k R) {ω | 1 ≤ torusWindowCount 2 L Ω I r b k R ω}).toReal ≤ C * r ^ 3) + +/-! ### The dimension lift: Theorems P_d, C_d, I_d, G_d (every fixed d ≥ 2) -/ + +/-- Shape of Theorem P_d (all-height punctured pin ball), display (1.2): fixed `d ≥ 2`, +`L > 0`, compact marks; there are `C, C', r_* > 0` such that for `0 < r ≤ r_*`, all `b, k, R, j` +and every Borel `E ⊆ D = {0 < p² + |q|² ≤ 1/16}` in the endpoint chart, +`E_(Q_r^W) N_j(M + r E) ≤ C r^3 ∫_E omega_d ≤ C' r^3`, and the same about `S`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/d5_dimension_lift_20260929/PROOF.md sha256 6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80 +Anchor: "E_(Q_r^W) N_j(M + r E) <= C r^3 integral_E omega_d(p, q) dp dq <= C' r^3." +Interface: `I` is a given object; the chart is `scaledAt` based at the pin; the weight is `omegaD`. +Does not claim: the result or its proof (devices D1-D5, stable gradient frame, density and conditional moments, weighted Kac-Rice (4.10), crossover); the nested-ball `O(r^5)` corollary (stated for `d = 2` as `D5bMicrodisk`); the planar P2 identification; any numerical constant; uniformity in `d`; nonauthor review or acceptance. -/ +def DimensionLiftPuncturedPinBallP (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ C C' rStar : ℝ, 0 < C ∧ 0 < C' ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), ∀ j : ℕ, j ≤ d → + ∀ E0 : Set (E d), MeasurableSet E0 → + E0 ⊆ {p : E d | 0 < ‖p‖ ^ 2 ∧ ‖p‖ ^ 2 ≤ (1 : ℝ) / 16} → + (meanAllHeightCount d L Ω I r b k R j (scaledAt d R r (pinM d r (R 0)) E0) ≤ + C * r ^ 3 * ∫ p in E0, omegaD d r p ∧ + C * r ^ 3 * ∫ p in E0, omegaD d r p ≤ C' * r ^ 3) ∧ + (meanAllHeightCount d L Ω I r b k R j (scaledAt d R r (pinS d r (R 0)) E0) ≤ + C * r ^ 3 * ∫ p in E0, omegaD d r p ∧ + C * r ^ 3 * ∫ p in E0, omegaD d r p ≤ C' * r ^ 3) + +/-- Shape of Theorem C_d (compact collar), display (1.3): fixed `d ≥ 2`, `L > 0`, compact +marks, fixed `R ≥ 1` and `0 < η ≤ 1/4`; there are `C, r_* > 0` such that for every Borel +`E ⊆ C(η, R)`, `E_(Q_r^W) N_j(r E) ≤ C r^3 |E|` for `0 < r ≤ r_*`, all `b, k, R, j`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/d5_dimension_lift_20260929/PROOF.md sha256 6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80 +Anchor: "E_(Q_r^W) N_j(r E) <= C r^3 |E|." +Interface: `I` is a given object; `C(η, R)` is `collar d η R`. +Does not claim: the consequence (1.4) for the scaled ball with pins removed (stated for `d = 2` as `D5dSummedPinNeighbourhoodC2`); the result or its proof; any numerical constant; nonauthor review or acceptance. -/ +def DimensionLiftCompactCollarC (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∀ η R0 : ℝ, 0 < η → η ≤ (1 : ℝ) / 4 → 1 ≤ R0 → ∃ C rStar : ℝ, 0 < C ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), ∀ j : ℕ, j ≤ d → + ∀ E0 : Set (E d), MeasurableSet E0 → E0 ⊆ collar d η R0 → + meanAllHeightCount d L Ω I r b k R j (scaledAt d R r 0 E0) ≤ + C * r ^ 3 * (volume E0).toReal + +/-- Shape of Theorem I_d (intermediate height-window shells), displays (1.5) and (1.6): +there are `C, s_0 > 0` such that for `0 < r ≤ s/4` and `0 < s ≤ s_0`, +`E_(Q_r^W) N_(r,j)({ s ≤ |X| ≤ 2 s }) ≤ C r^3 [ (r/s)^2 + s^2 ]`, and hence, uniformly for +`A_0 ≥ 4` and `A_0 r < ρ ≤ s_0`, `E_(Q_r^W) N_(r,j)({ A_0 r ≤ |X| ≤ ρ }) ≤ C r^3 (A_0^(-2) + ρ^2)`. +`|X|` is the distance from the midpoint. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/d5_dimension_lift_20260929/PROOF.md sha256 6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80 +Anchor: "E_(Q_r^W) N_(r,j)({ s <= |X| <= 2 s }) <= C r^3 [ (r/s)^2 + s^2 ]," +Interface: `I` is a given object. +Does not claim: the result or its proof (polynomial row rank, Euler suppression, shell ledger, dyadic sum); the planar I3/I4 identification; any numerical constant; nonauthor review or acceptance. -/ +def DimensionLiftIntermediateShellsI (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ C s0 : ℝ, 0 < C ∧ 0 < s0 ∧ + (∀ r s : ℝ, 0 < r → r ≤ s / 4 → 0 < s → s ≤ s0 → + ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), ∀ j : ℕ, j ≤ d → + meanWindowCount d L Ω I r b k R j {x : E d | s ≤ ‖x‖ ∧ ‖x‖ ≤ 2 * s} ≤ + C * r ^ 3 * ((r / s) ^ 2 + s ^ 2)) ∧ + (∀ A0 ρ r : ℝ, 4 ≤ A0 → 0 < r → A0 * r < ρ → ρ ≤ s0 → + ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), ∀ j : ℕ, j ≤ d → + meanWindowCount d L Ω I r b k R j {x : E d | A0 * r ≤ ‖x‖ ∧ ‖x‖ ≤ ρ} ≤ + C * r ^ 3 * ((A0 ^ 2)⁻¹ + ρ ^ 2)) + +/-- Shape of Theorem G_d (global height-window first moment), display (1.7): fixed `d ≥ 2`, +`L > 0`, compact marks; there are `C, r_* > 0` such that +`E_(Q_r^W) N_(r,j)( X minus {M, S} ) ≤ C r^3` for `0 < r ≤ r_*`, all `b, k, R, j`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/d5_dimension_lift_20260929/PROOF.md sha256 6fb94b247e8e45bca9dc02f46670ac6029fb54e0ef906001efe48fe908549b80 +Anchor: "E_(Q_r^W) N_(r,j)( X minus {M, S} ) <= C r^3." +Interface: `I` is a given object. +Does not claim: the tiling proof (P_d, C_d, I_d and the fixed-remote Theorem A); Corollaries F_d and M_d; the planar I5 identification; the `d = 3` SIDE24 application; any regional supersession of historical carriers; any numerical constant; nonauthor review or acceptance. -/ +def DimensionLiftGlobalWindowG (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ C rStar : ℝ, 0 < C ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), ∀ j : ℕ, j ≤ d → + meanWindowCount d L Ω I r b k R j (torusMinusPins d L r (R 0)) ≤ C * r ^ 3 + +/-! ### C6: Fourier cutoff (F4), Palm route (Theorem Q, Corollary Theta) -/ + +/-- Shape of the planar Fourier-cutoff consequence (F4): in `d = 2`, GIVEN the +dimension-matched first-moment hypothesis `G_2` (`E_(Q_r^W) N ≤ C r^3` for small `r`, +uniformly in the marks and frames), there are `C` and `r_* > 0` such that +`E_(Q_r^W) N(N-1) ≤ C r^3 log(1/r)` for `0 < r ≤ r_*`, all `b, k, R`. The source derives +this from Theorem W with `p = 2`; the hypothesis `G_2` is stated as a hypothesis here, not +discharged. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/c6_fourier_cutoff_20260929/PROOF.md sha256 c1692379a3a066589bd2522aaa5b4d480e736b737c8c39a474d1735185793733 +Anchor: "E_Qr^W N(N-1) <= C r^3 log(1/r)," +Interface: `I` is a given object; `N` is `torusWindowCount`; `(N)_2 = N(N-1)` is `Nat.descFactorial N 2`. +Does not claim: Theorem F (the cap `Psi_F` and its tail (F1)-(F2)); Theorem W for general `d` and `p` (F3); that `G_2` holds (the source takes it from the reviewed planar I5 composition); Proposition B; that the logarithm is sharp or removable; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. -/ +def FourierCutoffPlanarSecondFactorialF4 (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface 2 L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + (∃ C0 r0 : ℝ, 0 < r0 ∧ + ∀ r : ℝ, 0 < r → r ≤ r0 → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), + ∫ ω, (torusWindowCount 2 L Ω I r b k R ω : ℝ) ∂(I.QW r b k R) ≤ C0 * r ^ 3) → + ∃ C rStar : ℝ, 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin 2) ℝ (E 2), + ∫ ω, (Nat.descFactorial (torusWindowCount 2 L Ω I r b k R ω) 2 : ℝ) ∂(I.QW r b k R) ≤ + C * r ^ 3 * Real.log (1 / r) + +/-- Shape of Theorem Q (Palm route, factorial moments in every fixed dimension): fixed +`d ≥ 2`, `L > 0`, compact marks; for every fixed integer `q ≥ 2` there are `C_q, r_* > 0` +such that `E_(Q_r^W)[ (N)_q ] ≤ C_q r^3` for `0 < r ≤ r_*`, all `b, k, R`. The radius +threshold is allowed to depend on `q` (the source's quantifier order is "There are +`C_q, r_* > 0` such that for `0 < r ≤ r_*`, all `b, k, R` and every fixed integer `q ≥ 2`"; +the weaker reading with `r_*` after `q` is stated). + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/c6_palm_route_20260929/PROOF.md sha256 aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b +Anchor: "E_(Q_r^W)[ (N)_q ] <= C_q r^3." +Interface: `I` is a given object; `N` is `torusWindowCount`; `(N)_q` is `Nat.descFactorial N q`. +Does not claim: the result or its proof (the cap `Psi` in `d` dimensions, the parametric Rouché tail, the marked Kac-Rice formula with Hölder insertion, the regimes R1-R4); the dependence on [C6L] and [DL] at their bytes; a lower bound for `q ≥ 3`; Corollary P; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. -/ +def PalmRouteFactorialMomentsQ (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∀ q : ℕ, 2 ≤ q → ∃ Cq rStar : ℝ, 0 < Cq ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), + ∫ ω, (Nat.descFactorial (torusWindowCount d L Ω I r b k R ω) q : ℝ) ∂(I.QW r b k R) ≤ + Cq * r ^ 3 + +/-- Shape of Corollary Theta (the C6 order is `r^3`): in every fixed `d ≥ 2`, there are +`c > 0`, `C` and `r_* > 0` such that `c r^3 ≤ E_(Q_r^W)[ N(N-1) ] ≤ C r^3` and +`c r^3 ≤ Q_r^W{ N ≥ 2 } ≤ C r^3` for `0 < r ≤ r_*`, all `b, k, R`. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/c6_palm_route_20260929/PROOF.md sha256 aa37f16040ab9e05bb2e4e3f1d678367497e8dc2cf43eef197a4d6592c2f129b +Anchor: "c r^3 <= E_(Q_r^W)[ N(N-1) ] <= C r^3," +Interface: `I` is a given object; `N` is `torusWindowCount`. +Does not claim: the lower bounds' source ([EDL] (A4)) or the upper bounds' sources (Theorem Q, [DL] Corollary M_d); that the logarithm of the Fourier cutoff is removed as a reviewed fact; any statement about the positions of the pairs; any numerical constant; nonauthor review or acceptance; that the count `torusWindowCount` (every critical point off the pins, degenerate ones included) equals the sum of the index counts: it dominates the source's all-index `N`, and the two agree only off the degenerate locus. -/ +def PalmRouteSecondFactorialTheta (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ c C rStar : ℝ, 0 < c ∧ 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), + c * r ^ 3 ≤ + ∫ ω, (Nat.descFactorial (torusWindowCount d L Ω I r b k R ω) 2 : ℝ) ∂(I.QW r b k R) ∧ + ∫ ω, (Nat.descFactorial (torusWindowCount d L Ω I r b k R ω) 2 : ℝ) ∂(I.QW r b k R) ≤ + C * r ^ 3 ∧ + c * r ^ 3 ≤ ((I.QW r b k R) {ω | 2 ≤ torusWindowCount d L Ω I r b k R ω}).toReal ∧ + ((I.QW r b k R) {ω | 2 ≤ torusWindowCount d L Ω I r b k R ω}).toReal ≤ C * r ^ 3 + +/-! ### C6 rare-cluster consequences: Theorem O (abstract, on laws on ℕ) -/ + +/-- Total-variation distance between two probability mass functions on `ℕ`: half the `l^1` +distance (the source's convention; equal to the supremum over events for probability laws). + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/c6_rare_cluster_laws_20260929/PROOF.md sha256 c52a3cf197b5cf26071a8cc951e15ef3ac564b3d7453d41647f91af540a1b8eb +Anchor: "For probability laws, dTV is the supremum over measurable events, equivalently half the l1 distance on this countable state space." +Interface: none; if the `l^1` series is not summable the `tsum` is `0`. +Does not claim: equality with the event supremum (true for probability laws, not stated). -/ +noncomputable def pmfTV (μ ν : ℕ → ℝ) : ℝ := + (1 / 2) * ∑' n : ℕ, |μ n - ν n| + +/-- The Poisson probability mass function `Pois(lam)(n) = exp(-lam) lam^n / n!` +(for `lam ≥ 0`; `Pois(0) = δ_0`). + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/c6_rare_cluster_laws_20260929/PROOF.md sha256 c52a3cf197b5cf26071a8cc951e15ef3ac564b3d7453d41647f91af540a1b8eb +Anchor: "dTV(Law N, Pois(lambda))" +Interface: none; an explicit formula (the source's `Pois(lambda)`). +Does not claim: that it sums to one (true for `lam ≥ 0`, not stated). -/ +noncomputable def poissonPMF (lam : ℝ) (n : ℕ) : ℝ := + Real.exp (-lam) * lam ^ n / (Nat.factorial n : ℝ) + +/-- Shape of Theorem O (sharp ordinary-Poisson obstruction): for a family of laws +`μ_r` on `ℕ` (the law of the window count `N_r`) which, for all sufficiently small `r` +(that is, for `0 < r ≤ r_0` for some `r_0 > 0`), are probability mass functions with +summable mean satisfying only (H1) `E N_r ≤ A r^3` and (H2) `P(N_r ≥ 2) ≥ a r^3` with +`a > 0`: then, for all sufficiently small `r`, +`(a/2) r^3 ≤ inf_{lambda ≥ 0} dTV(Law N_r, Pois(lambda)) ≤ A r^3`: every nonnegative +intensity is at distance at least `(a/2) r^3`, and some intensity is within `A r^3`. The +hypotheses are imposed only on `0 < r ≤ r_0`, as in the source ("for all sufficiently small +r"); imposing (H2) for every `r > 0` would be unsatisfiable (`a r^3 ≤ 1` fails for large +`r`) and would make the statement hold for every family. The source's remark that the +premises imply `2a ≤ A` is not taken as a hypothesis. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/c6_rare_cluster_laws_20260929/PROOF.md sha256 c52a3cf197b5cf26071a8cc951e15ef3ac564b3d7453d41647f91af540a1b8eb +Anchor: "(a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3." +Interface: the laws `μ_r` are given probability mass functions on `ℕ` (nonnegative, summing to one, with summable mean) for `0 < r ≤ r_0`; the hypotheses (H1), (H2) are hypotheses on that range only; nothing ties `μ_r` to the Gaussian model here (the source says the only analytic inputs are H1, H2, Hq). +Does not claim: the result (Theorem O) or its proof; that `2a ≤ A` (derived in the source, not a hypothesis here); that H1 or H2 hold for the Gaussian window count (the source takes them from [D5] Theorem G_d and [C6] Corollary Theta); the mean-matched statement `lambda = mu`; Theorems C and S, the factorial-moment matching or the independent-replica limit; the conditional-law liminf remark; that the upper half is the source's infimum bound verbatim: `∃ lam ≥ 0, dTV(Law N_r, Pois(lam)) ≤ A r^3` implies `inf_{lam ≥ 0} dTV ≤ A r^3` and is stronger unless the infimum is attained, and the existential form is used because the source's proof exhibits the witness `lam = 0` (the source: The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p); nonauthor review or acceptance. -/ +def RareClusterPoissonObstructionO (μ : ℝ → ℕ → ℝ) (a A : ℝ) : Prop := + 0 < a → + (∃ r0 : ℝ, 0 < r0 ∧ ∀ r : ℝ, 0 < r → r ≤ r0 → + (∀ n : ℕ, 0 ≤ μ r n) ∧ Summable (μ r) ∧ ∑' n : ℕ, μ r n = 1 ∧ + Summable (fun n : ℕ => (n : ℝ) * μ r n) ∧ ∑' n : ℕ, (n : ℝ) * μ r n ≤ A * r ^ 3 ∧ + a * r ^ 3 ≤ 1 - μ r 0 - μ r 1) → + ∃ rStar : ℝ, 0 < rStar ∧ ∀ r : ℝ, 0 < r → r ≤ rStar → + (∀ lam : ℝ, 0 ≤ lam → (a / 2) * r ^ 3 ≤ pmfTV (μ r) (poissonPMF lam)) ∧ + (∃ lam : ℝ, 0 ≤ lam ∧ pmfTV (μ r) (poissonPMF lam) ≤ A * r ^ 3) + +/-! ### Remote collision (Corollary D) and the OPEN witness-collision target -/ + +/-- Shape of Corollary D (second factorial moment in the fixed remote region): fixed `d ≥ 2`, +`L > 0`, `0 < ρ < L/4`, compact marks; there are `C` and `r_* > 0` such that for +`0 < r ≤ r_*`, all marks and frames and every Borel `E ⊆ D_ρ`, +`E_(Q_r^W) N(E)(N(E)-1) ≤ C r^5 |E|` and `Q_r^W{ N(E) ≥ 2 } ≤ C r^5 |E|`, where +`N(E) = sum_j N_j(E)` is the sum of the index-`j` window counts in `E` +(`windowCountIndexSum`, the source's definition). Pairs at every mutual separation inside +`D_ρ`, down to coincidence, are included. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path frontiers/remote_collision_20260928/PROOF.md sha256 b9b8b58fd8266db7ffe6537078003888ef138b445d3289ec5588f116af9050c2 +Anchor: "E_(Q_r^W) N(E)(N(E)-1) <= C r^5 |E|," +Interface: `I` is a given object; the pair Kac-Rice formula (3.1) and the divided-difference coordinates (4.1) are not transcribed. +Does not claim: Theorem C (the ordered near-pair count) or its proof (Lemmas 1-5); Corollaries E and F; a torus-wide second factorial moment or `math.rn-region.witness-collision` in its full sense; pairs with a witness near the pins; equality of `N(E)` with the all-critical-point count `windowCountAll` (they agree only off the degenerate locus); any numerical constant; nonauthor review or acceptance. -/ +def RemoteCollisionSecondFactorialD (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (ρ bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → 0 < ρ → ρ < L / 4 → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ C rStar : ℝ, 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), + ∀ A : Set (E d), MeasurableSet A → A ⊆ remoteRegion d L ρ → + ∫ ω, (Nat.descFactorial (windowCountIndexSum d (I.F ω) b k r A) 2 : ℝ) ∂(I.QW r b k R) ≤ + C * r ^ 5 * (volume A).toReal ∧ + ((I.QW r b k R) {ω | 2 ≤ windowCountIndexSum d (I.F ω) b k r A}).toReal ≤ + C * r ^ 5 * (volume A).toReal + +/-- OPEN OBLIGATION, stated as a TARGET and not as a result. GRAPH node +`math.rn-region.witness-collision` (classification OPEN_ACTIVE, catalog C6): the regional +shrinking pin/witness-collision mechanism. The reconciliation describes what is missing as +a torus-wide second factorial moment of the window count "with pairs where at least one +witness is within fixed `η` of a pin", at every mutual separation (the fixed-remote `P_η` +of Corollary D covers only pairs with both witnesses at distance `≥ η` from the pins). The +Prop below is that target: fixed `d ≥ 2`, `L > 0`, `η > 0`, compact marks; there are `C` and +`r_* > 0` with `E_(Q_r^W) #{ordered window pairs with the first witness within η of a pin} ≤ C r^3` +for `0 < r ≤ r_*`, all `b, k, R`. No source in this packet establishes it; its register +status is OPEN and this declaration does not change that. + +Scope note (statement-shape review 5483861746 on Math- PR #419, 2026-10-11, AMEND_SCOPE_NOTE; +nothing below changes the Prop): (i) the source's §5 states its open obligation for the planar +case and makes no claim for `d > 2`; the formal target quantifies every `d ≥ 2`, so its instances +with `d > 2` are a proposed generalisation carried by this packet, not a transcription of the +source's scope. (ii) `nearPinPairCount` counts ordered pairs whose FIRST witness is within `η` of +a pin; the source's "at least one witness" set is the union of that set with its swap, so (in the +finite case) it has between one and two times as many elements, and a bound `C r^3` here is a +bound `2 C r^3` for the source's count: the orders agree, the functionals do not literally +coincide. (iii) `Set.ncard` is `0` on an infinite set and `PinnedLawInterface` supplies neither +almost-sure finiteness nor measurability of the pair population, so an infinite population would +make the integrand vanish spuriously; reading this Prop as the source's second factorial moment +needs those two facts about the law, which the interface does not provide and this Prop does not +assume. The weighted law `I.QW` is a given field, not the source's six-pin regression. + +Source: d6g8k5htny-coder/Math- commit 6c020d6a72936632f2055122e71a7818a4ff497e path reviews/d5_reconciliation_20260929/RECONCILIATION.md sha256 14342b7eb0ad47c1c99b06be58fb6c901f5c7c72fc70f73e81de3dde50c23432 +Anchor: "A torus-wide second factorial moment `E_{Q_r^W} N(N−1)` for the window count, with pairs where at least" +Interface: `I` is a given object; the pair count is `nearPinPairCount`. +Does not claim: that this target holds, is attainable, or is implied by any reviewed source (the Palm route's Theorem Q is an author-side candidate whose `q = 2` case would imply it; that candidate is stated separately as `PalmRouteFactorialMomentsQ` and is not consumed here); that the lower obstruction `E N(N−1) ≥ 2c r³` is stated here; the optimal order; any status transition of the node. -/ +def WitnessCollisionNearPinPairsTarget (d : ℕ) [NeZero d] (L : ℝ) (Ω : Type) [MeasurableSpace Ω] + (I : PinnedLawInterface d L Ω) (η bLo bHi kLo kHi : ℝ) : Prop := + 2 ≤ d → 0 < L → 0 < η → bLo ≤ bHi → 0 < kLo → kLo ≤ kHi → + ∃ C rStar : ℝ, 0 < rStar ∧ + ∀ r : ℝ, 0 < r → r ≤ rStar → ∀ b ∈ Set.Icc bLo bHi, ∀ k ∈ Set.Icc kLo kHi, + ∀ R : OrthonormalBasis (Fin d) ℝ (E d), + ∫ ω, (nearPinPairCount d L (I.F ω) b k r η (R 0) : ℝ) ∂(I.QW r b k R) ≤ C * r ^ 3 + +end UniversalLaw.Spec.RN diff --git a/specifications/lean_statements_20261010/Specifications/Side24.lean b/specifications/lean_statements_20261010/Specifications/Side24.lean new file mode 100644 index 000000000..d77832273 --- /dev/null +++ b/specifications/lean_statements_20261010/Specifications/Side24.lean @@ -0,0 +1,838 @@ +import Mathlib + +/-! +# SIDE24 coefficient in dimensions two and three (statements only) + +Scientific effect: NONE. Formal progress of every declaration here: `specified`. +Organizational-independence credit of this transcription: 0 (Anthropic / Claude). + +This module transcribes, as definitions, interface structures and `Prop`-valued +specifications, the displayed statements of the Layer 0 note +`coefficients/side24_v1/PROOF.md` (object SIDE24-COEFFICIENT-D23-20260924-v1) and of its +`ENCLOSURE.json`: the exact cone moments `D_1 = 4/3` and `D_2 = 29/6 - sqrt 6`, the +reference coefficient `c_{d,ref}` of display (1), the image constant `E` of display (2), the +relative covariance allowance `epsilon = 10^(-108)` of display (3), the twenty-digit +published endpoints, the headline two-dimensional and three-dimensional enclosures, the +reference comparison (4), the Section 2 derivative and lattice-image bounds, the Section 3 +joint-covariance ordering in the positive-semidefinite order, the Section 4 Gaussian density +comparison and integrand-ratio bounds, and the Section 1 cone-moment identities with the +Gaussian laws of the displayed variances `8/3`, `5/3`, `1`. + +The exact periodic coefficient `c_{d,24}` is "precisely its equation (15.2)" of the parent +(UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1); that formula involves the Gaussian cone expectation +`D_u` of (15.1), which this module cannot write without a conditional-Gaussian construction. +The coefficient is therefore an INTERFACE parameter (`PeriodicCoefficientInterface`): its +defining property (15.2) is a hypothesis field, the cone moment `D_u` is a given field, and the +densities and the conditional variance are pinned to derivatives of `K_24` by hypothesis fields. + +What this module does not establish. It consists only of definitions and structures; it +states nothing as a result and proves nothing. It does not enclose the coefficient, does not +evaluate `Gamma(7/6)`, does not prove any of the displayed inequalities, does not identify an +object satisfying an interface here with the object of the source, and does not touch the +parent's results. The Section 5 Stirling/`Gamma(7/6)` remainder procedure is described in the +source as a method, not displayed as a precise statement, and is NOT specified here. The +arithmetic skeleton of these displays (pairing counts, the image constant, `60E < epsilon`, +odd-block minors, exponent inequalities, the `D_2` algebra, endpoint adjacency) is already +kernel-checked on main as the 31 `UniversalLaw.Side24` targets of `formal/` at origin/main +`f6deeba7`; nothing here re-proves it and nothing here encloses the coefficient. Nothing here +promotes, reclassifies or discharges any claim, premise or obligation of the registers; +alignment of these statements with the source is PENDING_INDEPENDENT_REVIEW. + +Primary sources (Layer 0, byte-pinned): `d6g8k5htny-coder/Math-` commit +`760340e921ac4ceda296b8118da936f1133e956e`, paths `coefficients/side24_v1/PROOF.md` +(sha256 `c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769`, source id +`side24-proof`) and `coefficients/side24_v1/ENCLOSURE.json` (sha256 +`72b6cd92d31394cdaf5da8919a5d548e902228af1f095cc184158a71d8287811`, source id +`side24-enclosure`). Parent clause (15.2) quoted from source id `lifetime-parent`. +-/ + +namespace UniversalLaw.Spec.Side24 + +open MeasureTheory + +/-- The ambient Euclidean space `R^d` of the field; `d ∈ {2, 3}` in the source. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Use its centered variance-one field with the exact normalized covariance" +Interface: none; this is the ambient space `EuclideanSpace ℝ (Fin d)`. +Does not claim: anything about the field or the torus `R^d/(24 Z^d)`. -/ +abbrev E (d : ℕ) : Type := EuclideanSpace ℝ (Fin d) + +/-! ## Exact constants of the note -/ + +/-- The cone moment `D_1 = 4/3` of the reference law for `m = d - 1 = 1`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "D_1=E[A^2 1{A<0}]=4/3." +Interface: none; a rational constant. +Does not claim: the identity with the Gaussian expectation (that is `ConeMomentD1Identity`); the arithmetic `(1/2)(8/3) = 4/3` is kernel-checked on main (`UniversalLaw.Side24.cone_moment_m1_thirds`), not here. -/ +noncomputable def coneMomentD1 : ℝ := 4 / 3 + +/-- The cone moment `D_2 = 29/6 - sqrt 6` of the reference law for `m = d - 1 = 2`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: " = 29/6-sqrt(6)." +Interface: none; an explicit real constant. +Does not claim: the identity with the Gaussian cone expectation (that is `ConeMomentD2Identity` and `ConeMomentD2Algebra`); the rational part `29/6` and `4 sqrt(3/8) = sqrt 6` are kernel-checked on main (`UniversalLaw.Side24.cone_moment_m2_algebra`), not here. -/ +noncomputable def coneMomentD2 : ℝ := 29 / 6 - Real.sqrt 6 + +/-- The reference cone moment `D_m` as a function of `m = d - 1`: `D_1` for `m = 1`, `D_2` for +`m = 2`, and `0` (unspecified by the source) otherwise. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "c_d,ref = Gamma(7/6)*(3/2)^(1/3)*D_(d-1)" +Interface: none. +Does not claim: any value outside `m ∈ {1, 2}`; the value `0` there is a placeholder, not a cone moment. -/ +noncomputable def coneMomentRef (m : ℕ) : ℝ := + if m = 1 then coneMomentD1 else if m = 2 then coneMomentD2 else 0 + +/-- The nonperiodic reference coefficient of display (1): +`c_{d,ref} = Gamma(7/6) (3/2)^(1/3) D_{d-1} / [2 sqrt 3 pi^(d-1) sqrt pi]`, `d = 2, 3`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "/ [2sqrt(3)*pi^(d-1)*sqrt(pi)], d=2,3." +Interface: none; `Real.Gamma`, `Real.sqrt`, `Real.pi` and real powers are Mathlib's. +Does not claim: that this equals the periodic coefficient (the source says it is NOT claimed equal); any value for `d ∉ {2, 3}`; the derivation from the parent formula (that is `ReferenceCoefficientIdentity`); any numerical value of `Gamma(7/6)`. -/ +noncomputable def cRef (d : ℕ) : ℝ := + Real.Gamma (7 / 6) * (3 / 2 : ℝ) ^ ((1 : ℝ) / 3) * coneMomentRef (d - 1) / + (2 * Real.sqrt 3 * Real.pi ^ (d - 1) * Real.sqrt Real.pi) + +/-- The image constant `E = 21175738586478 * 10^(-125)` of display (2), as a rational over a +power of ten. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "= 21175738586478 *10^(-125)." +Interface: none. +Does not claim: the factorisation `1458*(76*24^6+15) = 21175738586478`, kernel-checked on main (`UniversalLaw.Side24.image_constant_value`), not here. -/ +noncomputable def imageConstE : ℝ := 21175738586478 / 10 ^ 125 + +/-- The relative covariance allowance `epsilon = 10^(-108)` of display (3). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "epsilon=10^(-108), since60E Real.exp (-(‖z + latticeVec d n‖ ^ 2) / 2)) + +/-! ## Section 2: the uniform derivative bound and the lattice-image bound -/ + +/-- A tuple of `q` unit directions in `R^d` (the "unit directional differentiations", mixed +directions allowed). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "For q<=6 unit directional differentiations of phi(x)=exp(-|x|^2/2), product-rule" +Interface: none. +Does not claim: anything. -/ +def IsUnitTuple (d q : ℕ) (u : Fin q → E d) : Prop := + ∀ i, ‖u i‖ = 1 + +/-- The product-rule pairing sum `sum_{j=0}^{floor(q/2)} q! r^(q-2j) / [2^j j! (q-2j)!]` +evaluated at `r = |x|`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "q! |x|^(q-2j) / [2^j j! (q-2j)!]." +Interface: none. +Does not claim: that the sum of the coefficients at `q = 6` is `76` (kernel-checked on main, `UniversalLaw.Side24.pairing_sum_six`), nor the bound itself (`DerivativePairingBound`). -/ +noncomputable def pairingSum (q : ℕ) (r : ℝ) : ℝ := + ∑ j ∈ Finset.range (q / 2 + 1), + (Nat.factorial q : ℝ) * r ^ (q - 2 * j) / + ((2 : ℝ) ^ j * (Nat.factorial j : ℝ) * (Nat.factorial (q - 2 * j) : ℝ)) + +/-- Section 2, first display: for `q ≤ 6` and unit directions, the absolute `q`-th directional +derivative of `phi(x) = exp(-|x|^2/2)` is at most `exp(-|x|^2/2)` times the pairing sum. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "pairings bound the absolute derivative by" +Interface: none; the derivative is Mathlib's `iteratedFDeriv ℝ q` applied to the direction tuple. +Does not claim: the bound (statement only); validity for `q > 6` (the source restricts to `q ≤ 6`); nothing here encloses the coefficient. -/ +def DerivativePairingBound : Prop := + ∀ (d q : ℕ), q ≤ 6 → ∀ (x : E d) (u : Fin q → E d), IsUnitTuple d q u → + |iteratedFDeriv ℝ q (Kinf d) x u| ≤ Real.exp (-(‖x‖ ^ 2) / 2) * pairingSum q ‖x‖ + +/-- Section 2: at `|x| ≥ 1` every unit-direction contraction of order `q ≤ 6` of +`phi(x) = exp(-|x|^2/2)` is bounded by `76 |x|^6 exp(-|x|^2/2)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "At |x|>=1 this is at most76 |x|^6 e^(-|x|^2/2)." +Interface: none. +Does not claim: the bound (statement only); the coefficient arithmetic `1+15+45+15=76` and `76` bounding the smaller orders are kernel-checked on main (`UniversalLaw.Side24.pairing_sum_six`, `pairing_sum_le_six`), not here; nothing here encloses the coefficient. -/ +def DerivativeBoundAtLeastOne : Prop := + ∀ (d q : ℕ), q ≤ 6 → ∀ (x : E d), 1 ≤ ‖x‖ → ∀ (u : Fin q → E d), IsUnitTuple d q u → + |iteratedFDeriv ℝ q (Kinf d) x u| ≤ 76 * ‖x‖ ^ 6 * Real.exp (-(‖x‖ ^ 2) / 2) + +/-- Section 2: at zero every unit-direction contraction of order `q ≤ 6` of `phi` has absolute +value at most `15`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "contraction has absolute value at most15." +Interface: none. +Does not claim: the bound (statement only); `5!! = 15` is kernel-checked on main (`UniversalLaw.Side24.sixth_moment_double_factorial`), not here. -/ +def DerivativeBoundAtZero : Prop := + ∀ (d q : ℕ), q ≤ 6 → ∀ (u : Fin q → E d), IsUnitTuple d q u → + |iteratedFDeriv ℝ q (Kinf d) 0 u| ≤ 15 + +/-- The squared Euclidean length `|n|^2 = sum_i n_i^2` of an integer point. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "sum_{n!=0} |n|^6 e^(-288|n|^2)" +Interface: none. +Does not claim: anything. -/ +noncomputable def intNormSq (d : ℕ) (n : Fin d → ℤ) : ℝ := + ∑ i, ((n i : ℤ) : ℝ) ^ 2 + +/-- The lattice-image term `|n|^6 exp(-288 |n|^2)` for `n ≠ 0`, and `0` at `n = 0` +(`288 = 24^2/2`). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "sum_{n!=0} |n|^6 e^(-288|n|^2)" +Interface: none. +Does not claim: `24^2/2 = 288`, kernel-checked on main (`UniversalLaw.Side24.period_exponent`), not here. -/ +noncomputable def imageTerm (d : ℕ) (n : Fin d → ℤ) : ℝ := + if n = 0 then 0 else intNormSq d n ^ 3 * Real.exp (-288 * intNormSq d n) + +/-- Section 2 lattice bound: for `d ≤ 3`, the image family is summable and +`sum_{n ≠ 0} |n|^6 exp(-288|n|^2) ≤ 1458 exp(-288)`. Summability is stated so the `tsum` is +not the vacuous `0`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "<=1458 e^(-288)." +Interface: none. +Does not claim: the bound (statement only); the intermediate `729 sum_j j^9 e^(-288 j^2)` shell count is not specified; `27*27 = 729`, `729*2 = 1458` are kernel-checked on main (`UniversalLaw.Side24.lattice_shell_constant`), not here. -/ +def LatticeImageSumBound : Prop := + ∀ d : ℕ, d ≤ 3 → + Summable (fun n : Fin d → ℤ => imageTerm d n) ∧ + (∑' n : Fin d → ℤ, imageTerm d n) ≤ 1458 * Real.exp (-288) + +/-- Section 2: successive terms `j^9 exp(-288 j^2)`, `j ≥ 1`, have ratio at most +`512 exp(-864)`, and that constant is below `1/2`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "successive terms have ratio at most512e^(-864)<1/2." +Interface: none. +Does not claim: the bounds (statement only); `2^9 = 512`, `288*3 = 864` are kernel-checked on main (`UniversalLaw.Side24.geometric_ratio_constants`), not here. -/ +def SuccessiveTermRatio : Prop := + (∀ j : ℕ, 1 ≤ j → + ((j : ℝ) + 1) ^ 9 * Real.exp (-288 * ((j : ℝ) + 1) ^ 2) ≤ + 512 * Real.exp (-864) * ((j : ℝ) ^ 9 * Real.exp (-288 * (j : ℝ) ^ 2))) ∧ + 512 * Real.exp (-864) < 1 / 2 + +/-- Section 2 exponential facts: `exp(288/125) > 10`, hence `exp(-288) < 10^(-125)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "hence e^(-288)<10^(-125) and the geometric-ratio inequality follows too." +Interface: none; `Real.exp` is Mathlib's. +Does not claim: the inequalities (statement only); the rational Taylor sum through degree 20 exceeding `10` is kernel-checked on main (the `UniversalLaw.Side24` Taylor-sum target in `Ledger.lean`), not here. -/ +def ExponentialBounds : Prop := + 10 < Real.exp (288 / 125) ∧ Real.exp (-288) < 1 / 10 ^ 125 + +/-- Display (2): for `d ∈ {2, 3}`, every unit-direction contraction through total order `6`, +mixed directions included, satisfies `|D^q K_24(0) - D^q K_infty(0)| < E`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "|D^q K_24(0)-D^q K_infty(0)|" +Interface: summability of the theta family is the premise `ThetaSummable d`; derivatives are Mathlib's `iteratedFDeriv ℝ q`. +Does not claim: 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. -/ +def ImageBound : Prop := + ∀ d : ℕ, d = 2 ∨ d = 3 → ThetaSummable d → + ∀ q : ℕ, q ≤ 6 → ∀ u : Fin q → E d, IsUnitTuple d q u → + |iteratedFDeriv ℝ q (K24 d) 0 u - iteratedFDeriv ℝ q (Kinf d) 0 u| < imageConstE + +/-! ## Section 3: the joint covariance of `(G, t, svec H)` in a frame -/ + +/-- Index set of the `svec` coordinates: ordered pairs `(i, j)` with `i ≤ j`, of cardinality +`d(d+1)/2` (`3` for `d = 2`, `6` for `d = 3`). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Use the joint vector (G,t,svec H), where svec lists diagonal Hessian entries and" +Interface: none. +Does not claim: anything. -/ +abbrev SymIndex (d : ℕ) : Type := {p : Fin d × Fin d // p.1 ≤ p.2} + +/-- Index set of the joint vector `(G, t, svec H)`: `d` gradient coordinates, one third +derivative `t`, and `d(d+1)/2` `svec` coordinates; dimension `6` for `d = 2` and `10` for `d = 3`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "sqrt(2) times independent off-diagonal entries. Its dimension is at most10." +Interface: none. +Does not claim: the dimension count `6`, `10`, kernel-checked on main (`UniversalLaw.Side24.pin_determinant_and_joint_dimension`), not here. -/ +abbrev JointIndex (d : ℕ) : Type := Fin d ⊕ Unit ⊕ SymIndex d + +/-- The list of differentiation directions attached to a joint coordinate, in the orthonormal +frame `R` with `u = R 0`: `G_i = ∂_{R i} f`, `t = ∂_u^3 f`, `H_{ij} = ∂_{R i} ∂_{R j} f`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Use the joint vector (G,t,svec H), where svec lists diagonal Hessian entries and" +Interface: none. +Does not claim: anything. -/ +def jetDirs (d : ℕ) (u : E d) (R : Fin d → E d) : JointIndex d → List (E d) + | Sum.inl i => [R i] + | Sum.inr (Sum.inl _) => [u, u, u] + | Sum.inr (Sum.inr p) => [R p.1.1, R p.1.2] + +/-- The `svec` scaling: `1` on gradient, `t` and diagonal Hessian coordinates, `sqrt 2` on +off-diagonal Hessian coordinates. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "sqrt(2) times independent off-diagonal entries. Its dimension is at most10." +Interface: none. +Does not claim: anything. -/ +noncomputable def svecScale (d : ℕ) : JointIndex d → ℝ + | Sum.inr (Sum.inr p) => if p.1.1 = p.1.2 then 1 else Real.sqrt 2 + | _ => 1 + +/-- The derivative of `K` at `x` contracted against a list of directions: +`D^{|l|} K(x)[l_1, …, l_{|l|}]`, through Mathlib's `iteratedFDeriv`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Each covariance entry differs from the reference by at most2E, by (2)." +Interface: none. +Does not claim: differentiability of `K` (if absent the derivative is `0` in Lean). -/ +noncomputable def derivContraction (d : ℕ) (K : E d → ℝ) (x : E d) (l : List (E d)) : ℝ := + iteratedFDeriv ℝ l.length K x (fun i => l.get i) + +/-- The joint covariance matrix of `(G, t, svec H)` at a point of a centred stationary field +with covariance `K`, in the frame `R` with `u = R 0`, through the stationary rule +`Cov(∂^α f, ∂^β f) = (-1)^{|β|} ∂^{α+β} K(0)`, with the `svec` scaling. With `K = K_infty` +this is `C_ref`; with `K = K_24` it is `C_24`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Each covariance entry differs from the reference by at most2E, by (2)." +Interface: none; the covariance is DEFINED by the derivative rule, no field is constructed. +Does not claim: that this matrix is the covariance of any random vector (no field is constructed here); symmetry or positive semidefiniteness of the matrix. -/ +noncomputable def jointCov (d : ℕ) (K : E d → ℝ) (u : E d) (R : Fin d → E d) : + Matrix (JointIndex d) (JointIndex d) ℝ := + Matrix.of fun a b => + svecScale d a * svecScale d b * (-1 : ℝ) ^ (jetDirs d u R b).length * + derivContraction d K 0 (jetDirs d u R a ++ jetDirs d u R b) + +/-- The gradient covariance block `Cov(G_i, G_j) = -∂_{R i} ∂_{R j} K(0)` in the frame `R`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "covariance differentiation gives" +Interface: none. +Does not claim: that it equals the identity for `K_infty` (that is `ReferenceOddAndPinBlocks`). -/ +noncomputable def gradCov (d : ℕ) (K : E d → ℝ) (R : Fin d → E d) : Matrix (Fin d) (Fin d) ℝ := + Matrix.of fun i j => - derivContraction d K 0 [R i, R j] + +/-- The pin covariance block of `V = H u`, coordinates `V_i = ∂_u ∂_{R i} f`: +`Cov(V_i, V_j) = ∂_u ∂_{R i} ∂_u ∂_{R j} K(0)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Thus V=Hu has covariance diag(3,1,...,1), tau^2=Var(t|G=0)=6, and" +Interface: none. +Does not claim: that it equals `diag(3,1,...,1)` for `K_infty` (that is `ReferenceOddAndPinBlocks`). -/ +noncomputable def pinCov (d : ℕ) (K : E d → ℝ) (u : E d) (R : Fin d → E d) : + Matrix (Fin d) (Fin d) ℝ := + Matrix.of fun i j => derivContraction d K 0 [u, R i, u, R j] + +/-- The covariance vector `Cov(t, G_j) = -∂_u^3 ∂_{R j} K(0)` of `t = ∂_u^3 f` with the gradient. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Var(t)=15, Cov(t,G)=(-3,0,...,0)," +Interface: none. +Does not claim: its reference value `(-3, 0, …, 0)` (that is `ReferenceOddAndPinBlocks`). -/ +noncomputable def thirdGradCov (d : ℕ) (K : E d → ℝ) (u : E d) (R : Fin d → E d) : Fin d → ℝ := + fun j => - derivContraction d K 0 [u, u, u, R j] + +/-- The variance `Var(t) = -∂_u^6 K(0)` of `t = ∂_u^3 f`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Var(t)=15, Cov(t,G)=(-3,0,...,0)," +Interface: none. +Does not claim: its reference value `15` (that is `ReferenceOddAndPinBlocks`). -/ +noncomputable def thirdVar (d : ℕ) (K : E d → ℝ) (u : E d) : ℝ := + - derivContraction d K 0 [u, u, u, u, u, u] + +/-- The conditional variance `tau^2 = Var(t | G = 0)` as the Schur complement +`Var(t) - Cov(t,G) Cov(G)^(-1) Cov(G,t)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "tau^2=Var(t|G=0)=6" +Interface: none; the Gaussian conditional variance is written as the Schur complement (a standard identity, used here as the definition). +Does not claim: invertibility of `Cov(G)` (Mathlib's matrix inverse is `0` on a singular matrix); the value `6` (that is `ReferenceOddAndPinBlocks`). -/ +noncomputable def condThirdVariance (d : ℕ) (K : E d → ℝ) (u : E d) (R : Fin d → E d) : ℝ := + thirdVar d K u - + dotProduct (thirdGradCov d K u R) (Matrix.mulVec (gradCov d K R)⁻¹ (thirdGradCov d K u R)) + +/-- The reference joint covariance `C_ref` (kernel `K_infty`) in the frame `R`, `u = R 0`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "(1-epsilon) C_ref <= C_24 <= (1+epsilon) C_ref," +Interface: none. +Does not claim: its displayed block values (`ReferenceOddAndPinBlocks`), or that it is frame independent. -/ +noncomputable def Cref (d : ℕ) (u : E d) (R : Fin d → E d) : + Matrix (JointIndex d) (JointIndex d) ℝ := + jointCov d (Kinf d) u R + +/-- The exact periodic joint covariance `C_24` (kernel `K_24`) in the frame `R`, `u = R 0`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "(1-epsilon) C_ref <= C_24 <= (1+epsilon) C_ref," +Interface: summability of the theta family is a premise wherever this matrix is used. +Does not claim: rotational invariance (the source asserts none for `C_24`). -/ +noncomputable def C24 (d : ℕ) (u : E d) (R : Fin d → E d) : + Matrix (JointIndex d) (JointIndex d) ℝ := + jointCov d (K24 d) u R + +/-- Display (3): for `d ∈ {2, 3}`, in every orthonormal frame `R` (with `u = R 0`), +`(1-epsilon) C_ref ≤ C_24 ≤ (1+epsilon) C_ref` in the positive-semidefinite order. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "The inequalities are in the positive-semidefinite ordering and hold in every" +Interface: summability of the theta family is the premise `ThetaSummable d`; the matrices are defined by the derivative rule of `jointCov`. +Does not claim: 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. -/ +def CovarianceOrdering : Prop := + ∀ d : ℕ, d = 2 ∨ d = 3 → ThetaSummable d → + ∀ (hd : 0 < d) (R : Fin d → E d), Orthonormal ℝ R → + (C24 d (R ⟨0, hd⟩) R - (1 - epsilon) • Cref d (R ⟨0, hd⟩) R).PosSemidef ∧ + ((1 + epsilon) • Cref d (R ⟨0, hd⟩) R - C24 d (R ⟨0, hd⟩) R).PosSemidef + +/-- Section 3: each entry of `C_24` differs from the corresponding entry of `C_ref` by at most +`2E`, in every orthonormal frame. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Each covariance entry differs from the reference by at most2E, by (2)." +Interface: summability of the theta family is the premise `ThetaSummable d`. +Does not claim: the bound (statement only); the spectral-norm bound `20E` (not specified); nonauthor review or acceptance. -/ +def EntrywiseCovarianceBound : Prop := + ∀ d : ℕ, d = 2 ∨ d = 3 → ThetaSummable d → + ∀ (hd : 0 < d) (R : Fin d → E d), Orthonormal ℝ R → + ∀ a b : JointIndex d, + |C24 d (R ⟨0, hd⟩) R a b - Cref d (R ⟨0, hd⟩) R a b| ≤ 2 * imageConstE + +/-- Section 3: `C_ref ≥ I/3` in the positive-semidefinite order, in every orthonormal frame. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Thus C_ref >= I/3 and" +Interface: none. +Does not claim: 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. -/ +def ReferenceCovarianceLowerBound : Prop := + ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ (hd : 0 < d) (R : Fin d → E d), Orthonormal ℝ R → + (Cref d (R ⟨0, hd⟩) R - (1 / 3 : ℝ) • (1 : Matrix (JointIndex d) (JointIndex d) ℝ)).PosSemidef + +/-- Section 1 reference blocks, in every orthonormal frame with `u = R 0`: `Cov(G) = I_d`, +`Var(t) = 15`, `Cov(t, G) = (-3, 0, …, 0)`, `Cov(V) = diag(3, 1, …, 1)` and `tau^2 = 6`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Var(t)=15, Cov(t,G)=(-3,0,...,0)," +Interface: none; the blocks are the derivative-rule matrices of `K_infty`. +Does not claim: the identities (statement only); `15 - 9 = 6` is kernel-checked on main (`UniversalLaw.Side24.conditional_third_derivative_variance`), not here. -/ +def ReferenceOddAndPinBlocks : Prop := + ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ (hd : 0 < d) (R : Fin d → E d), Orthonormal ℝ R → + gradCov d (Kinf d) R = 1 ∧ + thirdVar d (Kinf d) (R ⟨0, hd⟩) = 15 ∧ + thirdGradCov d (Kinf d) (R ⟨0, hd⟩) R = (fun j => if j = ⟨0, hd⟩ then -3 else 0) ∧ + pinCov d (Kinf d) (R ⟨0, hd⟩) R = Matrix.diagonal (fun i => if i = ⟨0, hd⟩ then 3 else 1) ∧ + condThirdVariance d (Kinf d) (R ⟨0, hd⟩) R = 6 + +/-! ## Section 4: Gaussian density comparison -/ + +/-- The centred Gaussian density on `R^n` with covariance `C`: +`(2 pi)^(-n/2) det(C)^(-1/2) exp(-x^T C^(-1) x / 2)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Comparing Gaussian determinants" +Interface: none. +Does not claim: that this is a probability density when `C` is singular (the matrix inverse is then `0` in Lean). -/ +noncomputable def gaussDensity (n : ℕ) (C : Matrix (Fin n) (Fin n) ℝ) (x : Fin n → ℝ) : ℝ := + (2 * Real.pi) ^ (-(n : ℝ) / 2) * C.det ^ (-(1 : ℝ) / 2) * + Real.exp (-(dotProduct x (Matrix.mulVec C⁻¹ x)) / 2) + +/-- The number `n = m(m+1)/2` of independent symmetric-matrix coordinates. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Let C be a centered covariance in n=m(m+1)/2 independent symmetric-matrix" +Interface: none. +Does not claim: anything. -/ +def symCoordDim (m : ℕ) : ℕ := m * (m + 1) / 2 + +/-- Section 4 density comparison: if `(1-epsilon) C0 ≤ C ≤ (1+epsilon) C0` in the +positive-semidefinite order, with `C0` positive definite, then pointwise +`[(1-epsilon)/(1+epsilon)]^(n/2) phi_{(1-epsilon)C0} ≤ phi_C ≤ [(1+epsilon)/(1-epsilon)]^(n/2) phi_{(1+epsilon)C0}`, +`n = m(m+1)/2`, `epsilon = 10^(-108)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "[(1-epsilon)/(1+epsilon)]^(n/2) phi_((1-epsilon)C0)" +Interface: none; `C.PosDef` is stated as a premise although it follows from `C0.PosDef` and the ordering. +Does not claim: 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. -/ +def DensityComparison : Prop := + ∀ (m : ℕ) (C0 C : Matrix (Fin (symCoordDim m)) (Fin (symCoordDim m)) ℝ), + C0.PosDef → C.PosDef → + (C - (1 - epsilon) • C0).PosSemidef → ((1 + epsilon) • C0 - C).PosSemidef → + ∀ x : Fin (symCoordDim m) → ℝ, + ((1 - epsilon) / (1 + epsilon)) ^ ((symCoordDim m : ℝ) / 2) * + gaussDensity (symCoordDim m) ((1 - epsilon) • C0) x ≤ + gaussDensity (symCoordDim m) C x ∧ + gaussDensity (symCoordDim m) C x ≤ + ((1 + epsilon) / (1 - epsilon)) ^ ((symCoordDim m : ℝ) / 2) * + gaussDensity (symCoordDim m) ((1 + epsilon) • C0) x + +/-- The exponent `a = m + 2/3 + n/2` of Section 4, with `m = d - 1`, `n = m(m+1)/2`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "a=m+2/3+n/2, b=d+n/2." +Interface: none. +Does not claim: the values `13/6`, `25/6`, kernel-checked on main (`UniversalLaw.Side24.exponent_values`), not here. -/ +noncomputable def exponentA (d : ℕ) : ℝ := + ((d - 1 : ℕ) : ℝ) + 2 / 3 + (symCoordDim (d - 1) : ℝ) / 2 + +/-- The exponent `b = d + n/2` of Section 4, with `n = (d-1)d/2`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "a=m+2/3+n/2, b=d+n/2." +Interface: none. +Does not claim: the inequalities `a+2b<14`, `2a+b<13`, kernel-checked on main (`UniversalLaw.Side24.exponent_bounds`), not here. -/ +noncomputable def exponentB (d : ℕ) : ℝ := + (d : ℝ) + (symCoordDim (d - 1) : ℝ) / 2 + +/-- Section 4 ratio bounds: for `d ∈ {2, 3}`, +`1 - 13 epsilon ≤ (1-epsilon)^a / (1+epsilon)^b` and `(1+epsilon)^a / (1-epsilon)^b ≤ 1 + 28 epsilon`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "ratio between1-13epsilon and" +Interface: none. +Does not claim: 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. -/ +def IntegrandRatioBounds : Prop := + ∀ d : ℕ, d = 2 ∨ d = 3 → + 1 - 13 * epsilon ≤ (1 - epsilon) ^ exponentA d / (1 + epsilon) ^ exponentB d ∧ + (1 + epsilon) ^ exponentA d / (1 - epsilon) ^ exponentB d ≤ 1 + 28 * epsilon + +/-! ## Section 1: reference densities and the reference coefficient identity -/ + +/-- Section 1: in every orthonormal frame with `u = R 0`, the reference gradient and pin +densities at zero satisfy `p_G(0) p_V(0) = (2 pi)^(-d) / sqrt 3`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3)." +Interface: none; the densities are `gaussDensity` of the derivative-rule blocks of `K_infty`. +Does not claim: the identity (statement only); `det diag(3,1,…,1) = 3` is kernel-checked on main (`UniversalLaw.Side24.pin_determinant_and_joint_dimension`), not here. -/ +def ReferencePinDensityProduct : Prop := + ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ (hd : 0 < d) (R : Fin d → E d), Orthonormal ℝ R → + gaussDensity d (gradCov d (Kinf d) R) 0 * + gaussDensity d (pinCov d (Kinf d) (R ⟨0, hd⟩) R) 0 = + (2 * Real.pi) ^ (-(d : ℝ)) / Real.sqrt 3 + +/-- Ordinary sphere area `|S^(d-1)|`: `2 pi` for `d = 2`, `4 pi` for `d = 3`, `0` otherwise. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Substituting these moments and |S^1|=2pi, |S^2|=4pi into the exact parent coefficient" +Interface: none. +Does not claim: any value outside `d ∈ {2, 3}`. -/ +noncomputable def sphereArea (d : ℕ) : ℝ := + if d = 2 then 2 * Real.pi else if d = 3 then 4 * Real.pi else 0 + +/-- Display (1) as an identity: the parent prefactor `Gamma(7/6) / [24^(1/3) sqrt pi]` times +`p_G(0) p_V(0) = (2 pi)^(-d) / sqrt 3`, times `tau^(4/3)` with `tau = sqrt 6`, times `D_{d-1}`, +times `|S^(d-1)|`, equals `c_{d,ref}` (using `6^(2/3)/24^(1/3) = (3/2)^(1/3)`). + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "This eliminates BOTH the angular and cone integrals for the reference expression." +Interface: none. +Does not claim: the identity (statement only); that the left side is the parent formula (15.2) evaluated at the reference law (the angular integrand is constant by rotational invariance of `K_infty`, a fact not specified here); `6^2*2 = 3*24` is kernel-checked on main (`UniversalLaw.Side24.cube_root_simplification`), not here; nonauthor review or acceptance. -/ +def ReferenceCoefficientIdentity : Prop := + ∀ d : ℕ, d = 2 ∨ d = 3 → + Real.Gamma (7 / 6) / ((24 : ℝ) ^ ((1 : ℝ) / 3) * Real.sqrt Real.pi) * + ((2 * Real.pi) ^ (-(d : ℝ)) / Real.sqrt 3) * (Real.sqrt 6) ^ ((4 : ℝ) / 3) * + coneMomentRef (d - 1) * sphereArea d = + cRef d + +/-! ## Section 1: the cone moments as Gaussian expectations -/ + +/-- Interface for the Gaussian variables of the reference cone moments. GIVEN: a probability +space; the `m = 1` transverse entry `A` with law `N(0, 8/3)`; the `m = 2` coordinates +`s, x, y` (shared trace shift and traceless part) jointly independent with laws `N(0, 5/3)`, +`N(0, 1)`, `N(0, 1)`, stated as a product law. Existence of such variables is not asserted. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Var(s)=5/3 and Var(x)=Var(y)=1. The negative-definite cone is s<-R," +Interface: every field is a given property; the laws are `ProbabilityTheory.gaussianReal` with the displayed variances. +Does not claim: 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`). -/ +structure ConeMomentInterface where + /-- The sample space. -/ + Ω : Type + /-- Its sigma-algebra. -/ + [mΩ : MeasurableSpace Ω] + /-- The probability measure. -/ + P : Measure Ω + /-- `P` is a probability measure. -/ + [isProb : IsProbabilityMeasure P] + /-- Section 1 clause "For m=1 the variance of A is8/3.": the scalar transverse Hessian. -/ + A : Ω → ℝ + /-- Law of `A`: centred Gaussian of variance `8/3`. -/ + A_law : Measure.map A P = ProbabilityTheory.gaussianReal 0 (8 / 3) + /-- Section 1 clause "write A=[[s+x,y],[y,s-x]]": the shared scalar shift `s`. -/ + s : Ω → ℝ + /-- The traceless coordinate `x`. -/ + x : Ω → ℝ + /-- The traceless coordinate `y`. -/ + y : Ω → ℝ + /-- Section 1 clause "s,x,y are independent, Var(s)=5/3 and Var(x)=Var(y)=1": the joint law + of `(s, x, y)` is the product of `N(0,5/3)`, `N(0,1)`, `N(0,1)`. -/ + sxy_law : Measure.map (fun ω => (s ω, x ω, y ω)) P = + (ProbabilityTheory.gaussianReal 0 (5 / 3)).prod + ((ProbabilityTheory.gaussianReal 0 1).prod (ProbabilityTheory.gaussianReal 0 1)) + +/-- The `m = 2` transverse Hessian `A = [[s+x, y], [y, s-x]]`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "For m=2 write A=[[s+x,y],[y,s-x]]. Then s,x,y are independent," +Interface: none. +Does not claim: anything. -/ +noncomputable def transverseMatrix (s x y : ℝ) : Matrix (Fin 2) (Fin 2) ℝ := + Matrix.of ![![s + x, y], ![y, s - x]] + +/-- The negative-definite cone `A < 0` for the `m = 2` matrix, as positive definiteness of `-A`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "R=sqrt(x^2+y^2), and det A=s^2-R^2." +Interface: none. +Does not claim: the source's reduction of the cone to `s < -R` (not specified; the cone is taken by its definition). -/ +def NegDefCone (q : ℝ × ℝ × ℝ) : Prop := + (-(transverseMatrix q.1 q.2.1 q.2.2)).PosDef + +/-- The cone-moment integrand `h(A) = det(A)^2 1{A<0}` for the `m = 2` matrix. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "R=sqrt(x^2+y^2), and det A=s^2-R^2." +Interface: none. +Does not claim: the identity `det A = s^2 - R^2` (not specified separately). -/ +noncomputable def coneIntegrand (q : ℝ × ℝ × ℝ) : ℝ := + Set.indicator {q : ℝ × ℝ × ℝ | NegDefCone q} + (fun q => (transverseMatrix q.1 q.2.1 q.2.2).det ^ 2) q + +/-- Section 1 identity `D_1 = E[A^2 1{A<0}] = 4/3` for `A ~ N(0, 8/3)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "D_1=E[A^2 1{A<0}]=4/3." +Interface: `I : ConeMomentInterface` is given; the law of `A` is its hypothesis field. +Does not claim: 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. -/ +def ConeMomentD1Identity (I : ConeMomentInterface) : Prop := + ∫ ω, Set.indicator (Set.Iio (0 : ℝ)) (fun a => a ^ 2) (I.A ω) ∂I.P = coneMomentD1 + +/-- Section 1 identity `D_2 = E[det(A)^2 1{A<0}] = (1/2)[25/3 - 20/3 + 8 - 8 sqrt(3/8)]` for +`A = [[s+x, y], [y, s-x]]` with the product Gaussian law. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "D_2 = (1/2)[25/3-20/3+8-8sqrt(3/8)]" +Interface: `I : ConeMomentInterface` is given; the joint law is its hypothesis field. +Does not claim: 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. -/ +def ConeMomentD2Identity (I : ConeMomentInterface) : Prop := + ∫ ω, coneIntegrand (I.s ω, I.x ω, I.y ω) ∂I.P = + (1 / 2) * (25 / 3 - 20 / 3 + 8 - 8 * Real.sqrt (3 / 8)) + +/-- Section 1 algebra `(1/2)[25/3 - 20/3 + 8 - 8 sqrt(3/8)] = 29/6 - sqrt 6`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: " = 29/6-sqrt(6)." +Interface: none. +Does not claim: 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. -/ +def ConeMomentD2Algebra : Prop := + (1 / 2 : ℝ) * (25 / 3 - 20 / 3 + 8 - 8 * Real.sqrt (3 / 8)) = coneMomentD2 + +/-- Section 1 moments of the trace shift `s ~ N(0, 5/3)`: `E s^2 = 5/3`, `E s^4 = 25/3`, +`E exp(-s^2/2) = sqrt(3/8)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "E s^2=5/3, E s^4=25/3 and" +Interface: `I : ConeMomentInterface` is given. +Does not claim: 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. -/ +def TraceMoments (I : ConeMomentInterface) : Prop := + ∫ ω, I.s ω ^ 2 ∂I.P = 5 / 3 ∧ + ∫ ω, I.s ω ^ 4 ∂I.P = 25 / 3 ∧ + ∫ ω, Real.exp (-(I.s ω ^ 2) / 2) ∂I.P = Real.sqrt (3 / 8) + +/-- Section 1: `R^2 = x^2 + y^2` has density `exp(-z/2)/2` on `(0, ∞)`, written through its +distribution function `P(R^2 ≤ a) = ∫_0^a exp(-z/2)/2 dz` for `a ≥ 0`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "The variable R^2 has density e^(-z/2)/2." +Interface: `I : ConeMomentInterface` is given. +Does not claim: the law (statement only); measurability of the event (if the event is not measurable, `P` is its outer measure). -/ +def RayleighSquareLaw (I : ConeMomentInterface) : Prop := + ∀ a : ℝ, 0 ≤ a → + (I.P {ω | I.x ω ^ 2 + I.y ω ^ 2 ≤ a}).toReal = ∫ z in (0 : ℝ)..a, Real.exp (-z / 2) / 2 + +/-- Section 1 elementary integral: for `a ≥ 0`, +`∫_0^a (a-z)^2 exp(-z/2) dz/2 = a^2 - 4a + 8 - 8 exp(-a/2)`. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "integral_0^a (a-z)^2 e^(-z/2) dz/2 = a^2-4a+8-8e^(-a/2)." +Interface: none; the integral is Mathlib's interval integral against Lebesgue measure. +Does not claim: the identity (statement only). -/ +def ElementaryConeIntegral : Prop := + ∀ a : ℝ, 0 ≤ a → + ∫ z in (0 : ℝ)..a, (a - z) ^ 2 * Real.exp (-z / 2) / 2 = + a ^ 2 - 4 * a + 8 - 8 * Real.exp (-a / 2) + +/-! ## The exact periodic coefficient as an interface, and the headline statements -/ + +/-- Interface for the exact periodic coefficient `c_{d,24}`, "precisely its equation (15.2)" +of the parent. GIVEN: the coefficient `c24`; the ordinary sphere-area measure on `S^(d-1)`; +the gradient density `p_G(0)`, the pin density `p_{V_u}(0)`, the conditional standard +deviation `tau_u` and the cone moment `D_u` of the parent's (15.1). Hypothesis fields: the +sphere measure is carried by the unit sphere with the areas `2 pi`, `4 pi`; the parent's +positivity clause "Finite-jet rank proves tau_u>0, positive densities p_G(0),p_Vu(0), and +D_u>0"; the densities and `tau_u^2` agree with the Gaussian density formula and the Schur +complement of the derivative-rule blocks of `K_24`; summability of the theta family; and +the parent's display (15.2) +`c_{d,L}= Gamma(7/6) / [24^(1/3) sqrt(pi)] * integral_{S^(d-1)} p_G(0) p_Vu(0) tau_u^(4/3) D_u d sigma(u)` +(source id `lifetime-parent`, Section 15). The cone moment `D_u` is NOT pinned to `K_24` +here: writing it needs the conditional Gaussian law `A_u | V_u = 0`, which this module does +not construct. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "Let c_d,24 be precisely its equation (15.2), using ordinary sphere area, ordered" +Interface: every field is a given object or a hypothesis; nothing is derived; in particular `coneMoment` is characterised only by its docstring. +Does not claim: existence of an object satisfying this interface; that `c24` is the finite-bar asymptotic constant (that reading is conditional on the parent's results, which are not touched); that the parent's "ordered maximum/saddle roles and the full height/gradient pin Jacobian" are encoded beyond the prefactor of (15.2). -/ +structure PeriodicCoefficientInterface where + /-- The exact periodic coefficient `c_{d,24}`, as a function of the dimension. -/ + c24 : ℕ → ℝ + /-- Scope clause "using ordinary sphere area": the surface measure `sigma` on `S^(d-1)`. -/ + sphereMeasure : (d : ℕ) → Measure (E d) + /-- The sphere measure is carried by the unit sphere. -/ + sphereMeasure_support : ∀ d : ℕ, sphereMeasure d ((Metric.sphere (0 : E d) 1)ᶜ) = 0 + /-- Section 1 clause "|S^1|=2pi, |S^2|=4pi": total areas in dimensions `2` and `3`. -/ + sphereMeasure_area : + (sphereMeasure 2 Set.univ).toReal = 2 * Real.pi ∧ (sphereMeasure 3 Set.univ).toReal = 4 * Real.pi + /-- Parent (15.2) ingredient: the gradient density at zero `p_G(0)`. -/ + pG0 : ℕ → ℝ + /-- Parent (15.2) ingredient: the pin density at zero `p_{V_u}(0)` for the direction `u`. -/ + pV0 : (d : ℕ) → E d → ℝ + /-- Parent (15.1) ingredient: `tau_u`, with `tau_u^2 = Var(t_u | G = 0)`. -/ + tau : (d : ℕ) → E d → ℝ + /-- Parent (15.1) ingredient: `D_u = E[(det A_u)^2 1{A_u<0} | V_u=0]`, the Gaussian cone + expectation under the conditional law of the transverse Hessian; GIVEN, not constructed. -/ + coneMoment : (d : ℕ) → E d → ℝ + /-- Parent clause: positive density `p_G(0)`. -/ + pG0_pos : ∀ d : ℕ, d = 2 ∨ d = 3 → 0 < pG0 d + /-- Parent clause: positive density `p_{V_u}(0)` for every unit direction. -/ + pV0_pos : ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ u : E d, ‖u‖ = 1 → 0 < pV0 d u + /-- Parent clause: `tau_u > 0` for every unit direction. -/ + tau_pos : ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ u : E d, ‖u‖ = 1 → 0 < tau d u + /-- Parent clause: `D_u > 0` for every unit direction. -/ + coneMoment_pos : ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ u : E d, ‖u‖ = 1 → 0 < coneMoment d u + /-- The theta family of `K_24` is summable. -/ + theta_summable : ∀ d : ℕ, d = 2 ∨ d = 3 → ThetaSummable d + /-- `p_G(0)` is the centred Gaussian density at zero with the gradient block of `K_24`, in + every orthonormal frame. -/ + pG0_def : ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ R : Fin d → E d, Orthonormal ℝ R → + pG0 d = gaussDensity d (gradCov d (K24 d) R) 0 + /-- `p_{V_u}(0)` is the centred Gaussian density at zero with the pin block of `K_24`, in + every orthonormal frame with `u = R 0`. -/ + pV0_def : ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ (hd : 0 < d) (R : Fin d → E d), Orthonormal ℝ R → + pV0 d (R ⟨0, hd⟩) = gaussDensity d (pinCov d (K24 d) (R ⟨0, hd⟩) R) 0 + /-- `tau_u^2` is the Schur complement `Var(t_u) - Cov(t_u,G) Cov(G)^(-1) Cov(G,t_u)` of the + derivative-rule blocks of `K_24`, in every orthonormal frame with `u = R 0`. -/ + tau_sq : ∀ d : ℕ, d = 2 ∨ d = 3 → ∀ (hd : 0 < d) (R : Fin d → E d), Orthonormal ℝ R → + tau d (R ⟨0, hd⟩) ^ 2 = condThirdVariance d (K24 d) (R ⟨0, hd⟩) R + /-- Parent display (15.2), for `d ∈ {2, 3}`. -/ + eq_15_2 : ∀ d : ℕ, d = 2 ∨ d = 3 → + c24 d = Real.Gamma (7 / 6) / ((24 : ℝ) ^ ((1 : ℝ) / 3) * Real.sqrt Real.pi) * + ∫ u, pG0 d * pV0 d u * tau d u ^ ((4 : ℝ) / 3) * coneMoment d u ∂(sphereMeasure d) + +/-- The headline statement of the source: the two enclosures +`0.07340691930603427103 < c_{2,24} < 0.07340691930603427104` and +`0.04177593184059834334 < c_{3,24} < 0.04177593184059834335`, for any coefficient satisfying +the interface. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "0.07340691930603427103 < c_2,24 < 0.07340691930603427104," +Interface: `I : PeriodicCoefficientInterface` is given; its `c24` is pinned only by the hypothesis fields of the interface (parent (15.2) with a GIVEN cone moment). The interface does not determine `c24`: the Prop is a predicate on the interface object, not a closed statement; its universal closure over all inhabitants is false whenever the interface is inhabited (scaling `coneMoment` 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. +Does not claim: the enclosures (statement only: nothing here encloses the coefficient); that an object satisfying the interface exists or is the coefficient of the source; the parent's results or the persistence interpretation; any finite-radius error estimate, unrestricted remainder or RN/24-jet statement; the arithmetic skeleton (endpoint adjacency, `d3 < d2`, endpoints in `(0, 1/10)`) is kernel-checked on main (31 `UniversalLaw.Side24` targets at origin/main f6deeba7), not here; nonauthor review or acceptance. -/ +def CoefficientEnclosure (I : PeriodicCoefficientInterface) : Prop := + d2Lower < I.c24 2 ∧ I.c24 2 < d2Upper ∧ d3Lower < I.c24 3 ∧ I.c24 3 < d3Upper + +/-- Display (4): `|c_{d,24} / c_{d,ref} - 1| < 10^(-106)` for `d = 2, 3`, for any coefficient +satisfying the interface. + +Source: d6g8k5htny-coder/Math- commit 760340e921ac4ceda296b8118da936f1133e956e path coefficients/side24_v1/PROOF.md sha256 c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769 +Anchor: "|c_d,24/c_d,ref -1| < 10^(-106), d=2,3." +Interface: `I : PeriodicCoefficientInterface` is given; `c_{d,ref}` is the explicit `cRef`. The interface does not determine `c24`: the Prop is a predicate on the interface object, not a closed statement; its universal closure over all inhabitants is false whenever the interface is inhabited (scaling `coneMoment` 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. +Does not claim: the bound (statement only); equality of the periodic and reference constants (the source says it is NOT claimed); `32*10^(-108) < 10^(-106)` is kernel-checked on main (`UniversalLaw.Side24.density_comparison_below_reported`), not here; nothing here encloses the coefficient. -/ +def ReferenceComparison (I : PeriodicCoefficientInterface) : Prop := + ∀ d : ℕ, d = 2 ∨ d = 3 → |I.c24 d / cRef d - 1| < 1 / 10 ^ 106 + +end UniversalLaw.Spec.Side24 diff --git a/specifications/lean_statements_20261010/check.py b/specifications/lean_statements_20261010/check.py new file mode 100644 index 000000000..ace85b64c --- /dev/null +++ b/specifications/lean_statements_20261010/check.py @@ -0,0 +1,1112 @@ +#!/usr/bin/env python3 +"""Source and execution checker for the statement-only Lean package +`specifications/lean_statements_20261010` (standard library only). + + python3 check.py [source] source mode: packet identity against git HEAD, the lexical + refusal list on every module, root imports, the declaration + scan against REGISTER.json (and back), vocabulary, verbatim + anchors, pinned source bytes, dependency lock, toolchain pin + python3 check.py --execute the same, then a fresh `lake build`, `leanchecker`, `#check` + and `#print axioms` of every registered declaration, the + environment audit (every constant Lean added for the five + modules, classified by its ConstantInfo and compared with the + register), real negative controls, a receipt under .lake/evidence/ + python3 check.py --allow-dirty local development only: the clean-worktree requirement is + skipped; a receipt then records the worktree as unverified + and must not be trusted as evidence about any commit + +Scientific effect: NONE. The package holds statements (`def … : Prop` and interface +`structure`s), not proofs. A green check means that the registered statements elaborate against +the pinned Mathlib and that the inventory in REGISTER.json is complete and consistent with the +module text and the byte-pinned Layer 0 sources. Formal progress of every statement here is +`specified`; the checker refuses `proved` and `kernel-checked` for them. It is not a proof of any +claim, not an alignment review (alignment is PENDING_INDEPENDENT_REVIEW), not an acceptance, and +it transcribes dispositions without deciding them. Nothing here can raise any label. + +Two independent stages enforce "statements only". The source scan is a fail-closed pre-Lean +stage over comment-stripped text: whole-token refusals, string literals and guillemet names +refused outright (they can desynchronise any comment lexer), every command keyword at column 0, +explicit result types, no proof-shaped result type, no `Sort`, no `_root_`. It is not a Lean +lexer. The authority is the `--execute` environment audit: Lean's own `ConstantInfo` of every +constant the modules add (`.lake/evidence/Environment.lean`) refuses theorems, axioms, opaque +constants, instances, non-structure inductives, definitions whose type is a proposition and any +transitive axiom outside propext, Classical.choice, Quot.sound (structure- and definition- +generated auxiliaries are tolerated only when range-less or flagged by the environment), and +requires the statement definitions (type syntactically `∀ …, Prop`, no reduction, so a +`Set`-valued definition is auxiliary exactly as the scan's `: Prop` rule has it), structures and +auxiliary definitions to equal the register and the scan exactly. +""" +from __future__ import annotations + +import argparse +import copy +import hashlib +import json +import os +from pathlib import Path, PurePosixPath +import re +import shutil +import subprocess +import sys + +ROOT = Path(__file__).resolve().parent +REPO = ROOT.parents[1] +PACKAGE = 'specifications/lean_statements_20261010' +REPOSITORY = 'd6g8k5htny-coder/Math-' +WORKFLOW = '.github/workflows/lean-specifications.yml' +REFERENCE_LOCK = 'formal/lake-manifest.json' +ROOT_MODULE = 'Specifications' +MODULES = ('Specifications/Field.lean', 'Specifications/Lifetime.lean', 'Specifications/Side24.lean', + 'Specifications/RN.lean', 'Specifications/P15.lean') +NAMESPACE = 'UniversalLaw.Spec.' +TOOLCHAIN = 'leanprover/lean4:v4.34.1' +LEAN_COMMIT = '5045d0056413266e57c625dcd7c365b10e377c52' +MATHLIB_REV = 'd13f23b723b8a846827a245b89c10fc7d3f11612' +LOCK_NAME = 'universal_law_specifications' +REGISTER_OBJECT = 'LEAN-STATEMENT-REGISTER-20261010-v1' +RECEIPT_OBJECT = 'LEAN-STATEMENT-RECEIPT-20261010-v1' +LANDING_SOURCE = 'landing-claims' +FORMAL_PROGRESS = ('none', 'specified', 'proved', 'kernel-checked') +CLAIM_PROGRESS = ('none', 'specified') +CLAIM_KINDS = ('landing_claim', 'graph_node', 'proof_index_result', 'open_obligation', + 'existing_lean_package', 'main_experiment_proof', 'exact_certificate') +STATEMENT_KINDS = ('def_prop', 'structure', 'theorem') +LOCAL_STATEMENT_KINDS = ('def_prop', 'structure') +DISPOSITIONS = ('HOLD_WITH_DOMAIN', 'FAIL_CLOSED', 'EXACT_COUNTEREXAMPLE', 'REVIEWED_SCOPED', 'ENGINEERING_HOLD') +ALLOWED_AXIOMS = frozenset({'propext', 'Classical.choice', 'Quot.sound'}) +PUBLIC_REPOS = frozenset({'d6g8k5htny-coder/main', 'd6g8k5htny-coder/Math-', 'd6g8k5htny-coder/query-', + 'd6g8k5htny-coder/meta-framework', 'd6g8k5htny-coder/Universal-Law-Workspace'}) +TOP_KEYS = frozenset({'schema_version', 'object', 'scientific_effect', 'scientific_status_authority', + 'alignment_status', 'meaning', 'toolchain', 'vocabulary', 'sources', 'claims'}) +SOURCE_KEYS = frozenset({'id', 'repository', 'commit', 'path', 'bytes', 'sha256', 'local_copy'}) +SOURCES_JSON_KEYS = SOURCE_KEYS | {'blob'} +ROW_KEYS = frozenset({'claim_id', 'kind', 'title', 'layer0', 'disposition_transcribed', 'formal_progress', + 'statements', 'kernel_evidence', 'not_established', 'next_step'}) +STATEMENT_KEYS = frozenset({'package', 'module', 'declaration', 'kind', 'formal_progress', 'anchor', 'does_not_claim'}) +EVIDENCE_KEYS = frozenset({'package', 'receipt_or_workflow', 'note'}) +# Lexical refusal list (DESIGN 3b): proof-bearing, trust-affecting or syntax-bending commands and +# terms, plus commands that add declarations under another name (export, alias, irreducible_def), +# `Sort` (Prop spelled as `Sort 0`) and `_root_` (escapes the module namespace). Matched as whole +# identifier tokens of the comment-stripped code, so a docstring may mention them. Not an +# elaboration check: the environment audit in --execute is the Lean-side authority. +REFUSED_TOKENS = ('theorem', 'lemma', 'example', 'instance', 'axiom', 'sorry', 'admit', 'native_decide', + 'unsafe', 'opaque', 'partial', 'macro', 'macro_rules', 'notation', 'infix', 'infixl', + 'infixr', 'prefix', 'postfix', 'syntax', 'elab', 'elab_rules', 'set_option', 'run_cmd', + 'run_tac', 'run_elab', 'implemented_by', 'extern', 'decreasing_by', 'ofReduceBool', + 'trustCompiler', 'initialize', 'builtin_initialize', 'deriving', 'attribute', + 'include', 'omit', 'mutual', 'inductive', 'class', 'local', 'scoped', + 'export', 'alias', 'irreducible_def', 'proof_wanted', 'suppress_compilation', + 'initialize_simps_projections', 'register_simp_attr', 'declare_aesop_rule_sets', + 'add_decl_doc', 'library_note', 'assert_not_exists', 'assert_exists', 'compile_inductive', + 'compile_def', 'recall', 'whatsnew', 'count_heartbeats', 'unseal', 'seal', 'nonrec', + 'declare_syntax_cat', 'binder_predicate', 'register_option', 'register_label_attr', + 'register_hint', 'to_additive', 'simps', 'with_weak_namespace', 'Sort', '_root_') +# Characters that have no place in the code of a statement module: string literals and guillemet +# names (both can hide `/-`, `-/` or `--` from a comment lexer; Lean tokenizes them first), +# `#` commands (`#eval` runs code at build time) and attributes. +REFUSED_LITERALS = ('"', '«', '»', '#', '@[') +# Column-0 commands a statement module may use; everything else is refused. DESIGN 3b also lists +# `section` and `variable`, but no module uses them and a `variable (h : False)` mentioned in a +# body would add a vacuating hypothesis the scan cannot see (only `#check` in --execute shows the +# elaborated signature), so both are refused until a module needs them. +ALLOWED_COMMANDS = frozenset({'import', 'namespace', 'end', 'open', 'def', 'noncomputable', 'abbrev', 'structure'}) +# Lean commands are not indentation-sensitive and `in` chains them, so every command keyword must +# start a line (the only exception: `def`/`abbrev`/`structure` right after `noncomputable ` at +# column 0). `private`, `protected`, `universe`, `section` and `variable` are then refused by the +# allow-list. +COMMAND_KEYWORDS = ALLOWED_COMMANDS | {'private', 'protected', 'universe', 'section', 'variable'} +COMMAND_KEYWORD = re.compile(r"(?≥∧∨↔¬∀∃∈∉⊆⊂∣≈]') +LEAN_OPTIONS_LINE = 'leanOptions = { autoImplicit = false }' +PLACEHOLDER = re.compile(r'<<[^<>\n]*>>') +LABELS = ('Source:', 'Anchor:', 'Interface:', 'Does not claim:') +# A verbatim anchor shorter than this locates nothing (a one-character anchor is verbatim in any +# source); docstring and register anchors are both held to it. +MIN_ANCHOR = 16 +FIXED_FILES = ('.gitignore', 'README.md', 'SCOPE.md', 'REGISTER.json', 'check.py', 'test_check.py', 'replay.sh', + 'lakefile.toml', 'lean-toolchain', 'lake-manifest.json', ROOT_MODULE + '.lean', 'sources/SOURCES.json') +SKIP_DIRS = frozenset({'.lake', '__pycache__'}) +HEX40 = re.compile(r'[0-9a-f]{40}') +HEX64 = re.compile(r'[0-9a-f]{64}') +AXIOM_LINE = re.compile(r"'([^'\s]+)' (?:depends on axioms: \[([^\]]*)\]|does not depend on any axioms)") +AUDIT_LINE = re.compile(r'\bAUDIT (\S+) (\S+)$', re.M) +AUDIT_END = re.compile(r'\bAUDIT END (\d+) (\d+)$', re.M) +AUDIT_CLASSES = frozenset({'statement', 'structure', 'auxiliary', 'generated'}) +LEAN_ERROR = re.compile(r':\d+:\d+: error(?:\([^)]*\))?:') # Lean 4.34 may tag errors: `error(lean.unknownIdentifier):` +BRACKETS = {'(': ')', '[': ']', '{': '}', '⦃': '⦄', '⟨': '⟩'} +CLOSERS = frozenset(BRACKETS.values()) + + +def sha(data: bytes) -> str: + return hashlib.sha256(data).hexdigest() + + +def git_blob(data: bytes) -> str: + return hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest() + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def load_json(text: str): + """Strict JSON: duplicate keys, non-integer numbers and NaN/Infinity are refused.""" + def unique(pairs): + obj = {} + for key, value in pairs: + require(key not in obj, 'duplicate JSON key: ' + key) + obj[key] = value + return obj + + def reject(token): + raise ValueError('non-integer or non-finite JSON number refused: ' + token) + return json.loads(text, object_pairs_hook=unique, parse_float=reject, parse_constant=reject) + + +def safe_path(root: Path, relative) -> Path: + require(isinstance(relative, str) and relative and '\\' not in relative, f'invalid path: {relative!r}') + pure = PurePosixPath(relative) + require(not pure.is_absolute() and '..' not in pure.parts and str(pure) == relative, f'unsafe path: {relative}') + path = root + for part in pure.parts: + path = path / part + require(not path.is_symlink(), f'symlink refused: {relative}') + require(path.is_file() and path.resolve().is_relative_to(root.resolve()), f'missing or escaping file: {relative}') + return path + + +# ----------------------------------------------------------------------------- Lean source scan + +def lex(text: str): + """Comment-stripped code with every newline preserved, plus the docstring blocks `/-- … -/` + as (first_line, last_line, raw_text). Nested block comments are supported; an unterminated + block comment, string literal or guillemet name is refused. A string literal `"…"` (with + backslash escapes) or a guillemet name `«…»` outside a comment is copied verbatim into the + code, so a `/-`, `-/` or `--` inside it never opens or closes a comment and the literal + itself remains visible to the refusal of `"` and `«` in statement modules.""" + out, docs, i, n, depth, line, start = [], [], 0, len(text), 0, 0, None + while i < n: + two = text[i:i + 2] + if depth: + if two == '/-': + depth, i = depth + 1, i + 2 + elif two == '-/': + depth, i = depth - 1, i + 2 + if depth == 0 and start is not None: + docs.append((start[0], line, text[start[1]:i])) + start = None + else: + if text[i] == '\n': + out.append('\n') + line += 1 + else: + out.append(' ') + i += 1 + elif two == '/-': + depth = 1 + if text.startswith('/--', i): + start = (line, i) + i += 2 + elif two == '--': + end = text.find('\n', i) + i = n if end < 0 else end + elif text[i] in '"«': + if text[i] == '"': + j = i + 1 + while j < n and text[j] != '"': + j += 2 if text[j] == '\\' else 1 + else: + j = text.find('»', i) + j = n if j < 0 else j + require(j < n, 'unterminated string literal or guillemet name') + segment = text[i:j + 1] + out.append(segment) + line += segment.count('\n') + i = j + 1 + else: + if text[i] == '\n': + line += 1 + out.append(text[i]) + i += 1 + require(depth == 0, 'unterminated block comment') + return ''.join(out), docs + + +def refused_tokens(code: str) -> list: + words = set(re.findall(r'[A-Za-z_][A-Za-z0-9_]*', code)) + return sorted(set(REFUSED_TOKENS) & words) + + +def header(lines: list, i: int, after: str) -> str: + """Declaration header text: from `after` (the rest of line i past the name) up to the first + bracket-depth-zero `:=`, `where` or leading `| `, or up to the next column-0 line.""" + parts, depth, k, seg = [], 0, i, after + while True: + p = 0 + while p < len(seg): + c = seg[p] + if c in BRACKETS: + depth += 1 + elif c in CLOSERS: + depth -= 1 + elif depth == 0: + if seg.startswith(':=', p): + parts.append(seg[:p]) + return ''.join(parts) + if (p == 0 or seg[p - 1].isspace()) and re.match(r'where(?![A-Za-z0-9_])', seg[p:]): + parts.append(seg[:p]) + return ''.join(parts) + p += 1 + parts.append(seg + '\n') + k += 1 + if k >= len(lines): + return ''.join(parts) + seg = lines[k] + if (seg and not seg[0].isspace()) or (depth == 0 and seg.lstrip().startswith('| ')): + return ''.join(parts) + + +def result_type(hdr: str): + """Text after the last bracket-depth-zero `:` of a header, whitespace-normalized, or None.""" + depth, last = 0, None + for p, c in enumerate(hdr): + if c in BRACKETS: + depth += 1 + elif c in CLOSERS: + depth -= 1 + elif c == ':' and depth == 0 and hdr[p:p + 2] != ':=' and (p == 0 or hdr[p - 1] != ':'): + last = p + return None if last is None else ' '.join(hdr[last + 1:].split()) + + +def docstring_labels(raw: str, name: str, sources: dict) -> dict: + """The four labelled lines DESIGN 3b requires of a specification docstring. The `Source:` line + must name a registered pinned source exactly; the quoted `Anchor:` must be verbatim in it.""" + body = raw[3:-2] + found = {} + for label in LABELS: + hits = re.findall(r'^' + re.escape(label) + r'(.*)$', body, re.M) + require(len(hits) == 1, f'{name}: docstring must carry exactly one `{label}` line (found {len(hits)})') + # `Interface:` and `Does not claim:` may continue on following lines; the two parsed labels may not. + tail = body[re.search(r'^' + re.escape(label), body, re.M).end():] + require(tail.strip(), f'{name}: empty `{label}` entry in docstring') + require(label in ('Interface:', 'Does not claim:') or hits[0].strip(), f'{name}: `{label}` must carry its value on the same line') + found[label] = hits[0].strip() + m = re.fullmatch(r'(\S+) commit ([0-9a-f]{40}) path (\S+) sha256 ([0-9a-f]{64})', found['Source:']) + require(m is not None, f'{name}: `Source:` must read ` commit <40-hex> path sha256 <64-hex>`') + matches = [s for s in sources.values() if (s['repository'], s['commit'], s['path'], s['sha256']) == m.groups()] + require(len(matches) == 1, f'{name}: `Source:` line names no registered pinned source') + a = re.fullmatch(r'"(.+)"', found['Anchor:']) + require(a is not None, f'{name}: `Anchor:` must be one double-quoted verbatim line') + require(len(a.group(1)) >= MIN_ANCHOR, f'{name}: docstring anchor shorter than {MIN_ANCHOR} characters locates nothing') + require(a.group(1) in matches[0]['text'], + f'{name}: docstring anchor is not a verbatim substring of pinned source {matches[0]["id"]}') + return {'source_id': matches[0]['id'], 'doc_anchor': a.group(1)} + + +def depth_zero(text: str) -> str: + """`text` with everything inside brackets blanked, for depth-zero pattern tests.""" + depth, out = 0, [] + for ch in text: + if ch in BRACKETS: + depth += 1 + elif ch in CLOSERS: + depth -= 1 + out.append(ch if depth == 0 and ch not in CLOSERS else ' ') + return ''.join(out) + + +def scan_module(text: str, stem: str, sources: dict) -> list: + """Top-level declarations of one statement module: fully qualified name, Lean keyword, result + type, and kind `def_prop` (result exactly `Prop`), `structure` or None (auxiliary definition). + Refuses the lexical list, string literals and guillemet names, `#` commands and attributes, + any command keyword off column 0 (Lean commands are not indentation-sensitive), `open … in`, + any column-0 command outside the allowed set (`section` and `variable` included, until a + module needs them), imports other than `Mathlib`, a namespace other than + UniversalLaw.Spec., unbalanced scoping, declaration + names with a `_`-prefixed component, definitions without an explicit result type, Prop-valued + result types not written exactly as `Prop`, proof-shaped result types, and registered-kind + declarations without the four labelled docstring lines. A pre-Lean stage: the environment + audit of --execute is the authority on what the modules declare.""" + code, docs = lex(text) + bad = refused_tokens(code) + require(not bad, f'refused token(s) in {stem}: {bad!r}') + literals = [lit for lit in REFUSED_LITERALS if lit in code] + require(not literals, f'refused character(s) in the code of {stem}: {literals!r} (string literals, guillemet names, ' + '`#` commands and attributes have no place in a statement module)') + for m in COMMAND_KEYWORD.finditer(code): + head = code[code.rfind('\n', 0, m.start()) + 1:m.start()] + require(head == '' or (head == 'noncomputable ' and m.group(1) in ('def', 'abbrev', 'structure')), + f'{stem}:{code.count(chr(10), 0, m.start()) + 1}: command keyword `{m.group(1)}` off column 0 ' + '(every command must start its own line; `open … in` and indented commands are refused)') + imports = re.findall(r'^import\s+(\S+)\s*$', code, re.M) + require(imports == ['Mathlib'], f'{stem}: imports must be exactly `import Mathlib`: {imports!r}') + expected_ns = NAMESPACE + stem + doc_end = {end: raw for _, end, raw in docs} + lines = code.split('\n') + stack, namespaces, decls = [], [], [] + for i, line in enumerate(lines): + if not line or line[0].isspace(): + continue + tok = line.split()[0] + require(tok in ALLOWED_COMMANDS, f'{stem}:{i + 1}: column-0 command not allowed in a statement module: {tok!r}') + if tok == 'import': + continue + m = re.match(r'^namespace(?:\s+(\S+))?\s*$', line) + if m: + require(m.group(1), f'{stem}:{i + 1}: anonymous namespace') + namespaces.append(m.group(1)) + stack.append(('namespace', m.group(1))) + continue + m = re.match(r'^end(?:\s+(\S+))?\s*$', line) + if m: + require(stack, f'{stem}:{i + 1}: `end` without an open namespace') + kind, name = stack.pop() + require(name == m.group(1), f'{stem}:{i + 1}: `end {m.group(1)}` does not close `{kind} {name}`') + continue + if tok == 'open': + block, k = [line], i + 1 + while k < len(lines) and lines[k] and lines[k][0].isspace(): + block.append(lines[k]) + k += 1 + require(IN_TOKEN.search(depth_zero('\n'.join(block))) is None, + f'{stem}:{i + 1}: `open … in` is not allowed (it chains a command onto this line)') + continue + m = re.match(r'^(noncomputable )?(def|abbrev|structure)\s+([^\s(\[{⦃:]+)(.*)$', line) + require(m is not None, f'{stem}:{i + 1}: unparsable declaration line: {line[:80]!r}') + require(not any(part.startswith('_') for part in m.group(3).split('.')), + f'{stem}:{i + 1}: declaration name components may not start with `_` (Lean treats them as internal details): {m.group(3)}') + prefix = [n for k, n in stack if k == 'namespace'] + require(prefix and prefix[0] == expected_ns, f'{stem}:{i + 1}: declaration outside namespace {expected_ns}') + fq = '.'.join(prefix + [m.group(3)]) + hdr = header(lines, i, m.group(4)) + res = None if m.group(2) == 'structure' else result_type(hdr) + if m.group(2) == 'structure': + kind = 'structure' + elif res == 'Prop': + kind = 'def_prop' + else: + require(res is not None, f'{fq}: every definition must carry an explicit result type (`: Prop` for a statement)') + require(re.search(r'(?= 0 and j not in doc_end and not lines[j].strip(): + j -= 1 + require(j in doc_end, f'{fq}: registered-kind declaration has no docstring immediately before it') + decl.update(docstring_labels(doc_end[j], fq, sources)) + decls.append(decl) + require(not stack, f'{stem}: unclosed namespace: {stack!r}') + require(namespaces == [expected_ns], f'{stem}: exactly one namespace {expected_ns} is required: {namespaces!r}') + return decls + + +# ----------------------------------------------------------------------------- packet checks + +def git(repo: Path, *args, timeout=120): + return subprocess.run(['git', '-C', str(repo), *args], capture_output=True, text=True, timeout=timeout) + + +def check_identity(repo: Path, allow_dirty: bool): + proc = git(repo, 'rev-parse', 'HEAD') + head = proc.stdout.strip() if proc.returncode == 0 and HEX40.fullmatch(proc.stdout.strip()) else None + if allow_dirty: + return head, 'UNVERIFIED' + require(head is not None, 'not a git repository with a 40-hex HEAD (local development may pass --allow-dirty)') + status = git(repo, 'status', '--porcelain', '--untracked-files=all', '--', PACKAGE, WORKFLOW, REFERENCE_LOCK) + require(status.returncode == 0, 'git status failed: ' + status.stderr.strip()[:300]) + require(status.stdout.strip() == '', + 'packet, workflow or reference lock differ from HEAD (uncommitted or untracked):\n' + status.stdout[:2000]) + require(git(repo, 'ls-files', '--error-unmatch', WORKFLOW).returncode == 0, 'workflow is not tracked: ' + WORKFLOW) + if os.environ.get('GITHUB_ACTIONS') == 'true': + require(os.environ.get('GITHUB_SHA') == head, 'GITHUB_SHA differs from the checked-out HEAD') + require(os.environ.get('GITHUB_REPOSITORY') == REPOSITORY, 'unexpected GITHUB_REPOSITORY') + return head, 'CLEAN' + + +def load_sources(root: Path, repo, reg: dict): + """REGISTER.json sources must equal sources/SOURCES.json (minus the git blob kept there); + every local copy must match bytes, sha256 and blob; when the pinned commit is present in the + repository history the copy must also equal `:`.""" + sj = load_json(safe_path(root, 'sources/SOURCES.json').read_text(encoding='utf-8')) + require(isinstance(sj, dict) and set(sj) == {'schema_version', 'scientific_effect', 'meaning', 'sources'} + and sj['schema_version'] == 1 and sj['scientific_effect'] == 'NONE' and isinstance(sj['sources'], list), + 'sources/SOURCES.json schema') + by_id = {} + for s in sj['sources']: + require(isinstance(s, dict) and set(s) == SOURCES_JSON_KEYS, 'sources/SOURCES.json rows have a fixed schema') + require(s['id'] not in by_id, 'duplicate source id: ' + str(s['id'])) + by_id[s['id']] = s + rows = reg['sources'] + require(isinstance(rows, list) and [r.get('id') for r in rows if isinstance(r, dict)] == list(by_id), + 'REGISTER.json sources must list exactly the sources/SOURCES.json ids in order') + out, history = {}, 0 + for r in rows: + require(set(r) == SOURCE_KEYS, 'register source rows have a fixed schema: ' + str(r.get('id'))) + s = by_id[r['id']] + require(all(r[k] == s[k] for k in SOURCE_KEYS), f'register source {r["id"]} differs from sources/SOURCES.json') + require(r['repository'] in PUBLIC_REPOS, 'nonpublic source repository refused: ' + str(r['repository'])) + require(isinstance(r['commit'], str) and HEX40.fullmatch(r['commit']), 'invalid commit for ' + r['id']) + require(isinstance(r['sha256'], str) and HEX64.fullmatch(r['sha256']), 'invalid sha256 for ' + r['id']) + require(type(r['bytes']) is int and r['bytes'] > 0, 'invalid byte count for ' + r['id']) + require(isinstance(r['local_copy'], str) and r['local_copy'].startswith('sources/') and r['local_copy'] not in + [o['local_copy'] for o in out.values()], 'local copies live under sources/ and are unique: ' + r['id']) + data = safe_path(root, r['local_copy']).read_bytes() + require(len(data) == r['bytes'] and sha(data) == r['sha256'] and git_blob(data) == s['blob'], + f'local copy differs from pinned identity (bytes, sha256, blob): {r["id"]}') + if repo is not None and git(repo, 'cat-file', '-e', f'{r["commit"]}:{r["path"]}').returncode == 0: + blob = subprocess.run(['git', '-C', str(repo), 'cat-file', 'blob', f'{r["commit"]}:{r["path"]}'], + capture_output=True, timeout=120) + require(blob.returncode == 0 and blob.stdout == data, + f'local copy differs from {r["commit"][:12]}:{r["path"]} in git history: {r["id"]}') + history += 1 + out[r['id']] = {**r, 'blob': s['blob'], 'text': data.decode('utf-8')} + return out, history + + +def packet_files(root: Path) -> list: + files = [] + for dirpath, dirnames, filenames in os.walk(root): + for d in dirnames: + require(not (Path(dirpath) / d).is_symlink(), 'symlink directory refused: ' + d) + dirnames[:] = sorted(d for d in dirnames if d not in SKIP_DIRS) + for f in filenames: + p = Path(dirpath) / f + require(not p.is_symlink(), 'symlink refused: ' + p.relative_to(root).as_posix()) + files.append(p.relative_to(root).as_posix()) + return sorted(files) + + +def check_tree(root: Path, sources: dict, modules): + present = packet_files(root) + expected = sorted(set(FIXED_FILES) | set(modules) | {s['local_copy'] for s in sources.values()}) + require(present == expected, 'packet tree differs from the fixed inventory: unexpected ' + + repr(sorted(set(present) - set(expected))) + ', missing ' + repr(sorted(set(expected) - set(present)))) + require('.lake/' in (root / '.gitignore').read_text(encoding='utf-8').splitlines(), '.gitignore must ignore .lake/') + + +def check_lock(root: Path, repo: Path) -> dict: + require((root / 'lean-toolchain').read_bytes() == (TOOLCHAIN + '\n').encode(), 'lean-toolchain must read exactly ' + TOOLCHAIN) + lakefile = (root / 'lakefile.toml').read_text(encoding='utf-8') + for pattern in (r'^name = "' + LOCK_NAME + r'"$', r'^defaultTargets = \["' + ROOT_MODULE + r'"\]$', + r'^\[\[require\]\]$', r'^name = "mathlib"$', + r'^git = "https://github\.com/leanprover-community/mathlib4\.git"$', + r'^rev = "' + MATHLIB_REV + r'"$', r'^\[\[lean_lib\]\]$', r'^name = "' + ROOT_MODULE + r'"$', + '^' + re.escape(LEAN_OPTIONS_LINE) + '$'): + require(re.search(pattern, lakefile, re.M) is not None, 'lakefile.toml lacks the pinned line ' + pattern) + require(lakefile.count('[[require]]') == 1 and lakefile.count('[[lean_lib]]') == 1, + 'lakefile.toml must declare exactly one dependency and one library') + lock = load_json((root / 'lake-manifest.json').read_text(encoding='utf-8')) + require(isinstance(lock, dict) and lock.get('name') == LOCK_NAME, 'lake-manifest.json name must be ' + LOCK_NAME) + ref = load_json(safe_path(repo, REFERENCE_LOCK).read_text(encoding='utf-8')) + require({k: v for k, v in lock.items() if k != 'name'} == {k: v for k, v in ref.items() if k != 'name'}, + 'lake-manifest.json differs from ' + REFERENCE_LOCK + ' beyond the package name') + packages = lock['packages'] + revs = {p['name']: p['rev'] for p in packages} + require(len(revs) == len(packages) and all(isinstance(v, str) and HEX40.fullmatch(v) for v in revs.values()), + 'duplicate or unpinned dependency in lake-manifest.json') + require(revs.get('mathlib') == MATHLIB_REV, 'lake-manifest.json Mathlib rev differs from the pin') + return revs + + +def check_root(root: Path, modules): + code, _ = lex((root / (ROOT_MODULE + '.lean')).read_text(encoding='utf-8')) + lines = [line for line in code.splitlines() if line.strip()] + expected = ['import ' + m[:-5].replace('/', '.') for m in modules] + require(lines == expected, 'root module must import exactly the registered modules in order: ' + repr(lines)) + + +def check_docs(root: Path): + """README.md and SCOPE.md may not carry unfilled `<<…>>` placeholders: a committed packet + whose tables are placeholders would describe itself as more than it is.""" + for name in ('README.md', 'SCOPE.md'): + hits = PLACEHOLDER.findall((root / name).read_text(encoding='utf-8')) + require(not hits, f'{name} still carries unfilled placeholder(s): {hits[:4]!r}') + + +def landing_dispositions(sources: dict) -> dict: + if LANDING_SOURCE not in sources: + return {} + data = json.loads(sources[LANDING_SOURCE]['text']) + out = {} + for c in data['claims']: + require(isinstance(c.get('claim_id'), str) and c['claim_id'] not in out and c.get('disposition') in DISPOSITIONS, + 'LANDING_CLAIMS copy has an unexpected claim row') + out[c['claim_id']] = c['disposition'] + return out + + +def check_register(reg: dict, scans: dict, sources: dict) -> dict: + """Vocabulary and the two-way inventory: every registered statement of this package is a + scanned Prop definition or structure of the named module, and every scanned Prop definition + or structure is registered. Dispositions are transcriptions of LANDING_CLAIMS, never values + decided here. Claim-level and statement-level labels can never say proved or kernel-checked + for this package.""" + require(isinstance(reg, dict) and set(reg) == TOP_KEYS, 'REGISTER.json top-level keys must be exactly ' + repr(sorted(TOP_KEYS))) + require(type(reg['schema_version']) is int and reg['schema_version'] == 1, 'schema_version must be 1') + require(reg['object'] == REGISTER_OBJECT, 'object must be ' + REGISTER_OBJECT) + require(reg['scientific_effect'] == 'NONE', 'scientific_effect must be NONE') + require(reg['scientific_status_authority'] is False, 'scientific_status_authority must be false') + require(reg['alignment_status'] == 'PENDING_INDEPENDENT_REVIEW', + 'alignment_status must be PENDING_INDEPENDENT_REVIEW (a register cannot self-award a review)') + require(isinstance(reg['meaning'], str) and reg['meaning'].strip(), 'meaning must be a nonempty string') + require(reg['toolchain'] == {'lean': TOOLCHAIN, 'lean_commit': LEAN_COMMIT, 'mathlib': MATHLIB_REV}, 'toolchain pins differ') + require(reg['vocabulary'] == {'formal_progress': list(FORMAL_PROGRESS)}, 'vocabulary must be exactly the four formal-progress labels') + claims = reg['claims'] + require(isinstance(claims, list), 'claims must be a list') + landing = landing_dispositions(sources) + by_name = {(mod, d['name']): d for mod, decls in scans.items() for d in decls} + required = {(mod, d['name'], d['kind']) for mod, decls in scans.items() for d in decls if d['kind']} + registered, ids, n_statements = set(), set(), 0 + for row in claims: + require(isinstance(row, dict) and set(row) == ROW_KEYS, 'claim rows have a fixed schema; offending keys: ' + + repr(sorted(set(row) ^ ROW_KEYS)) if isinstance(row, dict) else 'claim row must be an object') + cid = row['claim_id'] + require(isinstance(cid, str) and cid.strip() and cid not in ids, 'claim_id must be a unique nonempty string: ' + repr(cid)) + ids.add(cid) + require(row['kind'] in CLAIM_KINDS, f'{cid}: kind must be one of {CLAIM_KINDS}') + require(isinstance(row['title'], str) and row['title'].strip(), f'{cid}: title must be nonempty') + layer0 = row['layer0'] + require(isinstance(layer0, dict) and set(layer0) == {'source_id', 'heading'}, f'{cid}: layer0 must be {{source_id, heading}}') + sid = layer0['source_id'] + require(sid is None or sid in sources, f'{cid}: layer0.source_id must be null or a registered source id') + require(layer0['heading'] is None or (isinstance(layer0['heading'], str) and layer0['heading'].strip()), + f'{cid}: layer0.heading must be null or a nonempty string') + disp = row['disposition_transcribed'] + require(disp is None or disp in DISPOSITIONS, f'{cid}: disposition_transcribed must be null or a LANDING_CLAIMS disposition') + if row['kind'] == 'landing_claim': + require(cid in landing, f'{cid}: landing_claim rows must carry a LANDING_CLAIMS claim_id (pinned source {LANDING_SOURCE})') + require(disp == landing[cid], f'{cid}: disposition_transcribed must equal the LANDING_CLAIMS disposition verbatim ({landing[cid]!r})') + else: + require(disp is None or (cid in landing and disp == landing[cid]), + f'{cid}: disposition_transcribed must be null or the verbatim LANDING_CLAIMS disposition of this claim_id') + fp = row['formal_progress'] + require(fp in CLAIM_PROGRESS, f'{cid}: claim-level formal_progress must be none or specified; no research claim of this ' + 'programme is proved, and arithmetic skeletons carry their own labels under statements') + require(isinstance(row['statements'], list), f'{cid}: statements must be a list') + has_spec = False + for st in row['statements']: + require(isinstance(st, dict) and set(st) == STATEMENT_KEYS, f'{cid}: statement rows have a fixed schema') + for k in ('package', 'module', 'declaration', 'anchor', 'does_not_claim'): + require(isinstance(st[k], str) and st[k].strip(), f'{cid}: statement {k} must be a nonempty string') + require(st['kind'] in STATEMENT_KINDS, f'{cid}/{st["declaration"]}: statement kind must be one of {STATEMENT_KINDS}') + require(st['formal_progress'] in ('specified', 'proved'), f'{cid}/{st["declaration"]}: a statement is specified or ' + 'proved; never none, and kernel-checked appears only in a run receipt') + if st['package'] == PACKAGE: + require(st['kind'] in LOCAL_STATEMENT_KINDS, f'{cid}/{st["declaration"]}: statements of this package are def_prop or structure, never theorem') + require(st['formal_progress'] == 'specified', f'{cid}/{st["declaration"]}: statements of this package are never proved or kernel-checked') + key = (st['module'], st['declaration']) + require(key in by_name and by_name[key]['kind'] is not None, + f'{cid}: {st["declaration"]} is not declared as a Prop definition or structure in {st["module"]}') + require(by_name[key]['kind'] == st['kind'], f'{cid}: {st["declaration"]} is a {by_name[key]["kind"]}, registered as {st["kind"]}') + src = sources[by_name[key]['source_id']] + require(len(st['anchor']) >= MIN_ANCHOR, f'{cid}: anchor of {st["declaration"]} shorter than {MIN_ANCHOR} characters locates nothing') + require(st['anchor'] in src['text'], f'{cid}: anchor of {st["declaration"]} is not a verbatim substring of pinned source ' + f'{src["id"]} (the source its docstring names)') + registered.add((st['module'], st['declaration'], st['kind'])) + else: + require((st['kind'] == 'theorem') == (st['formal_progress'] == 'proved'), + f'{cid}/{st["declaration"]}: outside this package a theorem is proved and a def_prop or structure is specified') + require(sid is not None, f'{cid}: statements outside this package need a layer0 source for their anchor') + require(len(st['anchor']) >= MIN_ANCHOR, f'{cid}: anchor of {st["declaration"]} shorter than {MIN_ANCHOR} characters locates nothing') + require(st['anchor'] in sources[sid]['text'], f'{cid}: anchor of {st["declaration"]} is not a verbatim substring of pinned source {sid}') + has_spec = has_spec or st['kind'] == 'def_prop' + n_statements += 1 + require(fp != 'specified' or has_spec, f'{cid}: specified requires a def_prop statement of the claim itself') + require(isinstance(row['kernel_evidence'], list), f'{cid}: kernel_evidence must be a list') + for ev in row['kernel_evidence']: + require(isinstance(ev, dict) and set(ev) == EVIDENCE_KEYS and all(isinstance(ev[k], str) and ev[k].strip() for k in EVIDENCE_KEYS), + f'{cid}: kernel_evidence rows are {{package, receipt_or_workflow, note}} with nonempty strings') + for k in ('not_established', 'next_step'): + require(isinstance(row[k], str) and row[k].strip(), f'{cid}: {k} must be a nonempty string') + missing = sorted(required - registered) + require(not missing, f'unregistered declaration(s): every Prop definition and structure of the modules must appear in ' + f'REGISTER.json; {len(missing)} missing, first {missing[:8]!r}') + return {'claims': len(claims), 'statements': n_statements, 'registered_local_statements': len(registered)} + + +def source_check(root: Path = ROOT, repo: Path = REPO, allow_dirty: bool = False, modules=MODULES) -> dict: + root, repo = Path(root).resolve(), Path(repo).resolve() + head, worktree = check_identity(repo, allow_dirty) + reg_raw = safe_path(root, 'REGISTER.json').read_bytes() + reg = load_json(reg_raw.decode('utf-8')) + require(isinstance(reg, dict) and set(reg) == TOP_KEYS, 'REGISTER.json top-level keys must be exactly ' + repr(sorted(TOP_KEYS))) + sources, history = load_sources(root, repo if head else None, reg) + if os.environ.get('GITHUB_ACTIONS') == 'true': + require(history == len(sources), f'in CI every pinned source must be verified against its commit in git history ' + f'(fetch-depth 0): {history} of {len(sources)} verified') + check_tree(root, sources, modules) + revs = check_lock(root, repo) + check_root(root, modules) + scans = {m: scan_module(safe_path(root, m).read_text(encoding='utf-8'), PurePosixPath(m).stem, sources) for m in modules} + stats = check_register(reg, scans, sources) + check_docs(root) + decls = [d for m in modules for d in scans[m]] + return {'scientific_effect': 'NONE', 'formalization_status': 'specified', 'alignment_status': 'PENDING_INDEPENDENT_REVIEW', + 'checked_commit': head, 'worktree': worktree, 'register_sha256': sha(reg_raw), + 'module_sha256': {m: sha((root / m).read_bytes()) for m in modules}, 'dependency_revisions': revs, + 'declarations': len(decls), 'prop_definitions': sum(d['kind'] == 'def_prop' for d in decls), + 'structures': sum(d['kind'] == 'structure' for d in decls), 'auxiliary_definitions': sum(d['kind'] is None for d in decls), + 'sources': len(sources), 'sources_verified_in_git_history': history, **stats, 'scans': scans} + + +# ----------------------------------------------------------------------------- Lean execution + +def lake_binary(): + found = shutil.which('lake') + if found: + return found + candidate = Path.home() / '.elan/bin/lake' + return str(candidate) if candidate.is_file() else None + + +def lean_env(lake: str) -> dict: + return dict(os.environ, PATH=f'{Path(lake).parent}{os.pathsep}{os.environ.get("PATH", "")}') + + +def run(command, label, out: Path, cwd: Path, env, expect_success=True, timeout=1800): + proc = subprocess.run(command, cwd=cwd, text=True, stdout=subprocess.PIPE, stderr=subprocess.STDOUT, + timeout=timeout, env=env, encoding='utf-8', errors='replace') + (out / (label + '.log')).write_text(proc.stdout, encoding='utf-8') + print(label + ': exit ' + str(proc.returncode), flush=True) + require((proc.returncode == 0) == expect_success, 'unexpected process outcome: ' + label + '\n' + proc.stdout[-6000:]) + return proc.stdout + + +def check_lean_version(text: str) -> str: + """The last line of `lake env lean --version` (lake may print `info:` lines first) must be the + pinned release exactly: version, release commit and Release build.""" + lines = [line.strip() for line in text.splitlines() if line.strip()] + require(lines and all(line.startswith('info:') for line in lines[:-1]), 'unexpected output around the Lean version: ' + repr(text[:200])) + version = lines[-1] + require(re.fullmatch(r'Lean \(version 4\.34\.1, [^()\r\n]+, commit ' + LEAN_COMMIT + r', Release\)', version) is not None, + 'running Lean is not the pinned release: ' + repr(version[:160])) + return version + + +def dependency_state(root: Path, revs: dict) -> dict: + actual = {} + for name, rev in revs.items(): + pkg = root / '.lake' / 'packages' / name + require(pkg.is_dir() and not pkg.is_symlink(), 'dependency checkout missing: ' + name) + actual[name] = subprocess.check_output(['git', '-C', str(pkg), 'rev-parse', 'HEAD'], text=True).strip() + status = subprocess.check_output(['git', '-C', str(pkg), 'status', '--porcelain', '--untracked-files=normal'], text=True) + require(status.strip() == '', f'dependency worktree is not clean: {name}: {status.strip()[:200]}') + require(actual == revs, 'checked-out dependency revisions differ from lake-manifest.json: ' + repr(actual)) + return actual + + +def audit_axioms(text: str, targets: list) -> dict: + require(isinstance(targets, list) and targets and len(set(targets)) == len(targets), 'empty or duplicate target list') + records = {} + for match in AXIOM_LINE.finditer(text): + name, raw = match.groups() + require(name in targets and name not in records, 'unexpected or duplicate axiom report: ' + name) + axioms = [] if raw is None or not raw.strip() else [a.strip() for a in raw.split(',')] + require(all(re.fullmatch(r"[A-Za-z_][A-Za-z0-9_.']*", a) for a in axioms), 'malformed axiom name') + require(set(axioms) <= ALLOWED_AXIOMS, 'forbidden transitive axiom: ' + repr(sorted(set(axioms) - ALLOWED_AXIOMS))) + records[name] = sorted(set(axioms)) + require(not AXIOM_LINE.sub('', text).strip(), 'unrecognized audit output') + require(set(records) == set(targets), 'missing axiom report: ' + repr(sorted(set(targets) - set(records)))) + return records + + +def signatures(log: str, names: list) -> dict: + """One elaborated signature per name from the `#check` log: the block starting at the line + that begins with the name, continued by indented lines (the pretty printer wraps).""" + lines, found = log.splitlines(), {} + for n in names: + starts = [i for i, line in enumerate(lines) if re.match(re.escape(n) + r'(?![A-Za-z0-9_.])', line)] + require(len(starts) == 1, 'no unique elaborated signature reported for ' + n) + i = starts[0] + 1 + while i < len(lines) and lines[i] and lines[i][0].isspace(): + i += 1 + found[n] = '\n'.join(lines[starts[0]:i]) + return found + + +def kernel_audit(lake: str, root: Path, env: dict, out: Path, names: list, root_module: str = ROOT_MODULE): + """`#print axioms` (parsed, allowed set only) and `#check` (logged, one signature per name). + Returns the axiom records and the elaborated signatures.""" + audit = out / 'Audit.lean' + audit.write_text('import ' + root_module + '\n' + ''.join(f'#print axioms {n}\n' for n in names), encoding='utf-8') + axioms = audit_axioms(run([lake, 'env', 'lean', str(audit)], 'axioms', out, root, env), names) + types = out / 'Types.lean' + types.write_text('import ' + root_module + '\n' + ''.join(f'#check {n}\n' for n in names), encoding='utf-8') + log = run([lake, 'env', 'lean', str(types)], 'types', out, root, env) + require(LEAN_ERROR.search(log) is None, 'the elaborated-types log reports an error') + return axioms, signatures(log, names) + + +AUDIT_PROGRAM = ''' +/-! Every constant of the audited modules (or, when `auditModules` is empty, every constant of +this file under `auditPrefix`) is classified by Lean's own `ConstantInfo`, never by source text: +`structure`; `statement` (a definition whose type is syntactically `Prop` or `∀ …, Prop`, read +through `forallTelescope` without reduction, so that a `Set`-valued definition, whose type +unfolds to `… → Prop`, is `auxiliary` exactly as the scan's `: Prop` rule has it; the reducing +`isProp` still decides `REFUSED:proof`); `auxiliary` (any +other definition); `generated` (what `structure` and `def` elaboration add: projections, +recursors, `casesOn`, `noConfusion`, `injEq`/`inj`/`sizeOf_spec`, `_sizeOf_*`, `_flat_ctor`, +`match_*`, `_proof_*`; tolerated only when range-less, flagged by the environment, or an +internal name under its structure); or `REFUSED:` for a theorem, axiom, opaque or quotient +constant, an inductive type that is not a structure, a constructor or recursor of a foreign +type, an instance, a definition whose type is a proposition (a proof), or any transitive axiom +outside propext, Classical.choice and Quot.sound. One `AUDIT ` line per constant, +then `AUDIT END `; any refusal makes the file fail. -/ +run_cmd do + let env ← getEnv + let allowed : Array Name := #[``propext, ``Classical.choice, ``Quot.sound] + let mut constants : Array ConstantInfo := #[] + if auditModules.isEmpty then + constants := env.constants.foldStage2 (fun acc _ ci => if auditPrefix.isPrefixOf ci.name then acc.push ci else acc) #[] + else + let mut found : Array Name := #[] + for (modName, data) in env.header.moduleNames.zip env.header.moduleData do + if auditModules.contains modName then + found := found.push modName + constants := constants ++ data.constants + unless found.size == auditModules.size do + throwError "AUDIT REFUSED: audited module not imported: {auditModules.filter (!found.contains ·)}" + let structs : Array Name := constants.filterMap fun ci => match ci with + | .inductInfo _ => if isStructure env ci.name then some ci.name else none + | _ => none + let underStructure (n : Name) : Bool := structs.any fun s => s != n && s.isPrefixOf n + let generatedTheorem (n : Name) : Bool := structs.any fun s => + let c := (getStructureCtor env s).name + n == c ++ `injEq || n == c ++ `inj || n == c ++ `sizeOf_spec + let mut refused := 0 + for ci in constants do + let n := ci.name + let axioms ← collectAxioms n + let bad := axioms.filter (!allowed.contains ·) + let range ← findDeclarationRanges? n + let inst ← liftCoreM (isInstance n) + let (statement, proof) ← liftTermElabM do + pure ((← forallTelescope ci.type fun _ body => pure body.isProp), (← isProp ci.type)) + let kind : String := + match ci with + | .inductInfo _ => if structs.contains n then "structure" else "REFUSED:inductive" + | .ctorInfo v => if structs.contains v.induct then "generated" else "REFUSED:constructor" + | .recInfo _ => if underStructure n then "generated" else "REFUSED:recursor" + | .axiomInfo _ => "REFUSED:axiom" + | .opaqueInfo _ => "REFUSED:opaque" + | .quotInfo _ => "REFUSED:quot" + | .thmInfo _ => + if (underStructure n && env.isProjectionFn n) || (range.isNone && (generatedTheorem n || n.isInternalDetail)) + then "generated" else "REFUSED:theorem" + | .defnInfo _ => + if (underStructure n && (env.isProjectionFn n || isAuxRecursor env n || isNoConfusion env n + || n.isInternalDetail || range.isNone)) || (range.isNone && n.isInternalDetail) then "generated" + else if inst then "REFUSED:instance" + else if proof then "REFUSED:proof" + else if statement then "statement" + else "auxiliary" + let verdict : String := + if kind.startsWith "REFUSED" || bad.isEmpty then kind + else "REFUSED:axioms:" ++ ",".intercalate (bad.map toString).toList + if verdict.startsWith "REFUSED" then refused := refused + 1 + logInfo m!"AUDIT {verdict} {n}" + logInfo m!"AUDIT END {constants.size} {refused}" + if refused > 0 then throwError "AUDIT REFUSED {refused} constant(s); see the AUDIT lines" +''' + + +def audit_source(root_module: str, module_names: list, declarations: str = '') -> str: + """The environment-audit file: imports, the audited module list (empty: this file), optional + control declarations, and the fixed audit program.""" + modules = '#[' + ', '.join('`' + m for m in module_names) + ']' + return (f'import Lean\nimport {root_module}\nopen Lean Meta Elab Command\n\n' + f'/-- Modules whose constants are audited; empty means the constants of this file under `auditPrefix`. -/\n' + f'def auditModules : Array Name := {modules}\n' + f'/-- Namespace of the control declarations audited when `auditModules` is empty. -/\n' + f'def auditPrefix : Name := `{CONTROL_NS[:-1]}\n{declarations}{AUDIT_PROGRAM}') + + +def parse_audit(log: str) -> dict: + """`AUDIT ` lines as {class: set(names)}; the `AUDIT END` line must account for + every line and every refusal.""" + classes, seen = {}, set() + for m in AUDIT_LINE.finditer(log): + verdict, name = m.groups() + require(name not in seen, 'duplicate audit line for ' + name) + seen.add(name) + classes.setdefault(verdict, set()).add(name) + ends = AUDIT_END.findall(log) + require(len(ends) == 1, 'the environment audit did not report exactly one END line') + total, refused = int(ends[0][0]), int(ends[0][1]) + require(total == len(seen), f'environment audit reported {total} constants but {len(seen)} AUDIT lines') + require(refused == sum(len(v) for k, v in classes.items() if k.startswith('REFUSED')), 'environment audit refusal count differs from its lines') + return classes + + +def environment_audit(lake: str, root: Path, env: dict, out: Path, module_names: list, expected: dict, + root_module: str = ROOT_MODULE) -> dict: + """Lean's own inventory of the modules (see AUDIT_PROGRAM) must pass with no refusal, and its + statements, structures and auxiliary definitions must equal the register and the scan.""" + path = out / 'Environment.lean' + path.write_text(audit_source(root_module, module_names), encoding='utf-8') + classes = parse_audit(run([lake, 'env', 'lean', str(path)], 'environment', out, root, env)) + require(set(classes) <= AUDIT_CLASSES, 'environment audit refused constant(s): ' + repr(sorted(set(classes) - AUDIT_CLASSES))) + for kind in ('statement', 'structure', 'auxiliary'): + got, want = classes.get(kind, set()), set(expected[kind]) + require(got == want, f'environment audit: Lean\'s {kind} constants differ from the register and scan: ' + f'unexpected {sorted(got - want)[:8]!r}, missing {sorted(want - got)[:8]!r}') + return {k: len(v) for k, v in sorted(classes.items())} + + +HONEST_CONTROL = ''' +namespace UniversalLaw.Spec.Control +/-- Control: a parametrized statement. -/ +def IsTwo (n : Nat) : Prop := 0 < n ∧ n = 2 +/-- Control: an interface with a Prop field (its projection is a generated theorem). -/ +structure Iface (d : Nat) where + x : Nat + pos : 0 < x +/-- Control: an extending interface. -/ +structure Ext (d : Nat) extends Iface d where + y : Nat + hy : y = x +/-- Control: an auxiliary definition. -/ +def helper (x : Nat) : Nat := x + 1 +/-- Control: an auxiliary abbreviation. -/ +abbrev E (d : Nat) : Type := Fin d → Nat +end UniversalLaw.Spec.Control +''' +HIDDEN_CONTROL = ''' +namespace UniversalLaw.Spec.Control +def Opener : Prop := "/-" = "x" +theorem smuggledBetweenStrings : True := trivial +def Closer : Prop := "-/" = "x" + def HiddenIndented : Prop := False +def proofAsDef : (2 : Nat) + 2 = 4 := rfl +def usesSorry : Prop := sorry +axiom hidden : False +instance controlInstance : Inhabited Nat := ⟨0⟩ +end UniversalLaw.Spec.Control +''' +CONTROL_NS = 'UniversalLaw.Spec.Control.' + + +def lean_control_environment(lake: str, root: Path, env: dict, out: Path, root_module: str = ROOT_MODULE) -> str: + """Two Lean-side controls of the environment audit itself, run over the constants of a control + file (never a packet module): an honest file must be classified exactly, and a file hiding a + theorem between string literals, an indented Prop definition, a proof written as `def`, a + `sorry`, an `axiom` and an `instance` must be refused for exactly those reasons, with the + hidden Prop definition reported as a statement the register does not carry.""" + honest = out / 'controls' / 'ControlEnvironmentHonest.lean' + honest.write_text(audit_source(root_module, [], HONEST_CONTROL), encoding='utf-8') + classes = parse_audit(run([lake, 'env', 'lean', str(honest)], 'control_environment_honest', out, root, env)) + require(classes.get('statement') == {CONTROL_NS + 'IsTwo'} and classes.get('structure') == {CONTROL_NS + 'Iface', CONTROL_NS + 'Ext'} + and classes.get('auxiliary') == {CONTROL_NS + 'helper', CONTROL_NS + 'E'} and set(classes) == AUDIT_CLASSES, + 'honest environment control was not classified exactly: ' + repr({k: sorted(v) for k, v in classes.items()})) + hidden = out / 'controls' / 'ControlEnvironmentHidden.lean' + hidden.write_text(audit_source(root_module, [], HIDDEN_CONTROL), encoding='utf-8') + log = run([lake, 'env', 'lean', str(hidden)], 'control_environment_hidden', out, root, env, expect_success=False) + require('AUDIT REFUSED' in log and LEAN_ERROR.search(log) is not None, 'hidden environment control failed for the wrong reason') + classes = parse_audit(log) + expected = {'REFUSED:theorem': {CONTROL_NS + 'smuggledBetweenStrings'}, 'REFUSED:proof': {CONTROL_NS + 'proofAsDef'}, + 'REFUSED:axioms:sorryAx': {CONTROL_NS + 'usesSorry'}, 'REFUSED:axiom': {CONTROL_NS + 'hidden'}, + 'REFUSED:instance': {CONTROL_NS + 'controlInstance'}, + 'statement': {CONTROL_NS + 'Opener', CONTROL_NS + 'Closer', CONTROL_NS + 'HiddenIndented'}} + require(classes == expected, 'hidden environment control was not refused for exactly the expected reasons: ' + + repr({k: sorted(v) for k, v in classes.items()})) + return ('REFUSED_BY_ENVIRONMENT_AUDIT (theorem between string literals, proof as def, sorry, axiom, instance; the indented ' + 'Prop definition surfaces as a statement absent from any register) and the honest control classified exactly') + + +def lean_control_type_error(lake: str, root: Path, env: dict, out: Path, root_module: str = ROOT_MODULE) -> str: + path = out / 'ControlTypeError.lean' + path.write_text('import ' + root_module + '\n\nnamespace UniversalLaw.Spec.Control\n\n' + '/-- Negative control: a Prop whose body has type ℕ. Lean must refuse it. -/\n' + 'def TypeErrorControl : Prop := (1 : Nat)\n\nend UniversalLaw.Spec.Control\n', encoding='utf-8') + log = run([lake, 'env', 'lean', str(path)], 'control_type_error', out, root, env, expect_success=False) + require('type mismatch' in log.lower() and LEAN_ERROR.search(log) is not None, + 'type-error control failed for the wrong reason: ' + log[-600:]) + return 'REJECTED_BY_LEAN' + + +def lean_control_missing(lake: str, root: Path, env: dict, out: Path, name: str, root_module: str = ROOT_MODULE) -> str: + path = out / 'ControlMissing.lean' + path.write_text('import ' + root_module + '\n#check ' + name + '\n', encoding='utf-8') + log = run([lake, 'env', 'lean', str(path)], 'control_missing_declaration', out, root, env, expect_success=False) + require(('unknown identifier' in log.lower() or 'unknown constant' in log.lower()) and LEAN_ERROR.search(log) is not None, + 'missing-declaration control failed for the wrong reason: ' + log[-600:]) + return 'REJECTED_BY_LEAN' + + +def control_row(module: str, declaration: str, anchor: str) -> dict: + """A register row naming `declaration` with a real anchor, so that the only defect of the row + is the declaration it names (the anchor itself passes the length and verbatim checks).""" + require(len(anchor) >= MIN_ANCHOR, 'control row anchor must itself be a valid anchor') + return {'claim_id': 'negative-control', 'kind': 'graph_node', 'title': 'negative control row (never committed)', + 'layer0': {'source_id': None, 'heading': None}, 'disposition_transcribed': None, 'formal_progress': 'specified', + 'statements': [{'package': PACKAGE, 'module': module, 'declaration': declaration, 'kind': 'def_prop', + 'formal_progress': 'specified', 'anchor': anchor, 'does_not_claim': 'anything'}], + 'kernel_evidence': [], 'not_established': 'everything', 'next_step': 'none'} + + +def expect_refusal(fn, fragment: str, label: str): + try: + fn() + except ValueError as exc: + require(fragment in str(exc), f'negative control {label} was refused for the wrong reason: {exc}') + return 'REFUSED_BY_SOURCE_CHECK' + raise ValueError('negative control escaped the source check: ' + label) + + +def source_controls(root: Path, reg: dict, scans: dict, sources: dict, modules, ctrl: Path): + """The source-side negative controls (no Lean): an injected theorem, a register row naming a + missing declaration, an unregistered Prop definition spliced into a module copy, and a + register copy labelling a statement `proved`. Each mutation is written under `ctrl` and run + through the same functions the packet is held to; the packet itself is never modified. + Returns the outcomes, the missing declaration name and the injected module text.""" + module = modules[0] + stem = PurePosixPath(module).stem + ns = NAMESPACE + stem + outcomes = {} + injected = f'import Mathlib\n\nnamespace {ns}\n\ntheorem injected : True := trivial\n\nend {ns}\n' + outcomes['injected_theorem'] = expect_refusal(lambda: scan_module(injected, stem, sources), 'refused token', 'injected_theorem') + local = [d for d in scans[module] if d['kind'] == 'def_prop'] + require(local, 'no registered Prop definition available to derive the controls') + src = sources[local[0]['source_id']] + missing = ns + '.DoesNotExistControl' + bad = copy.deepcopy(reg) + bad['claims'].append(control_row(module, missing, local[0]['doc_anchor'])) + outcomes['missing_declaration'] = expect_refusal(lambda: check_register(bad, scans, sources), 'is not declared', 'missing_declaration') + text = safe_path(root, module).read_text(encoding='utf-8') + tail = text.rfind('\nend ' + ns) + require(tail > 0, 'module copy has no closing namespace line') + extra = (f'\n/-- Negative control.\n\nSource: {src["repository"]} commit {src["commit"]} path {src["path"]} sha256 {src["sha256"]}\n' + f'Anchor: "{local[0]["doc_anchor"]}"\nInterface: none.\nDoes not claim: anything. -/\n' + 'def UnregisteredControl : Prop := True\n') + mutated = text[:tail] + extra + text[tail:] + (ctrl / 'Unregistered.lean.txt').write_text(mutated, encoding='utf-8') + mutated_scans = dict(scans, **{module: scan_module(mutated, stem, sources)}) + outcomes['unregistered_declaration'] = expect_refusal(lambda: check_register(reg, mutated_scans, sources), + 'unregistered declaration', 'unregistered_declaration') + proved = copy.deepcopy(reg) + rows = [st for row in proved['claims'] for st in row['statements'] if st['package'] == PACKAGE] + require(rows, 'no registered statement available to derive the proved-label control') + rows[0]['formal_progress'] = 'proved' + outcomes['proved_label'] = expect_refusal(lambda: check_register(proved, scans, sources), 'never proved', 'proved_label') + (ctrl / 'register_mutations.json').write_text(json.dumps({'missing_declaration': bad['claims'][-1], 'proved_label': rows[0]}, + indent=2, sort_keys=True) + '\n', encoding='utf-8') + return outcomes, missing, injected + + +def negative_controls(lake: str, root: Path, env: dict, out: Path, reg: dict, scans: dict, sources: dict, modules) -> dict: + """Real controls: the four source-side ones, Lean runs showing that the injected theorem is + accepted by Lean (so the scan is the refusal point) and that a type error and a missing + declaration are refused by Lean, and the two controls of the environment audit itself.""" + ctrl = out / 'controls' + ctrl.mkdir() + outcomes, missing, injected = source_controls(root, reg, scans, sources, modules, ctrl) + path = ctrl / 'InjectedTheorem.lean' + path.write_text(injected.replace('import Mathlib', 'import ' + ROOT_MODULE), encoding='utf-8') + run([lake, 'env', 'lean', str(path)], 'control_injected_theorem', out, root, env) + outcomes['injected_theorem'] += ' (Lean accepts the file: exit 0, so the scan is the refusal point)' + outcomes['type_error'] = lean_control_type_error(lake, root, env, out) + outcomes['missing_declaration'] += ' and ' + lean_control_missing(lake, root, env, out, missing) + outcomes['environment_audit'] = lean_control_environment(lake, root, env, out) + return outcomes + + +def execute(root: Path, repo: Path, info: dict, allow_dirty: bool, modules=MODULES) -> dict: + root, repo = Path(root).resolve(), Path(repo).resolve() + lake = lake_binary() + require(lake is not None, 'lake not found on PATH or in ~/.elan/bin') + env = lean_env(lake) + out = root / '.lake' / 'evidence' + require(not (root / '.lake').is_symlink() and not out.is_symlink() and not (root / '.lake' / 'build').is_symlink(), 'symlink build directory') + if out.exists(): + shutil.rmtree(out) + out.mkdir(parents=True) + version = check_lean_version(run([lake, 'env', 'lean', '--version'], 'version', out, root, env)) + revs = dependency_state(root, info['dependency_revisions']) + if (root / '.lake' / 'build').exists(): + shutil.rmtree(root / '.lake' / 'build') # fresh build of this package only; dependency caches are untouched + log = run([lake, 'build'], 'build', out, root, env) + require(LEAN_ERROR.search(log) is None and re.search(r'\berror\b', log, re.I) is None, 'build log reports an error') + require('declaration uses' not in log, 'build log reports sorry or a similar escape') + require('is a proposition; use' not in log, 'build log reports a definition whose type is a proposition (a proof written as def)') + run([lake, 'env', 'leanchecker', ROOT_MODULE], 'leanchecker', out, root, env) + names = [d['name'] for m in modules for d in info['scans'][m] if d['kind']] + require(names, 'no registered declarations to audit') + axioms, sigs = kernel_audit(lake, root, env, out, names) + expected = {'statement': [d['name'] for m in modules for d in info['scans'][m] if d['kind'] == 'def_prop'], + 'structure': [d['name'] for m in modules for d in info['scans'][m] if d['kind'] == 'structure'], + 'auxiliary': [d['name'] for m in modules for d in info['scans'][m] if d['kind'] is None]} + environment = environment_audit(lake, root, env, out, [m[:-5].replace('/', '.') for m in modules], expected) + reg = load_json(safe_path(root, 'REGISTER.json').read_text(encoding='utf-8')) + sources, _ = load_sources(root, repo if info['checked_commit'] else None, reg) + outcomes = negative_controls(lake, root, env, out, reg, info['scans'], sources, modules) + after = source_check(root, repo, allow_dirty, modules) + require(after['register_sha256'] == info['register_sha256'] and after['module_sha256'] == info['module_sha256'] + and after['checked_commit'] == info['checked_commit'], 'packet changed during execution') + declarations = [{'name': d['name'], 'module': m, 'kind': d['kind'], 'lean': d['lean'], 'source_id': d['source_id'], + 'doc_anchor': d['doc_anchor'], 'line': d['line'], 'signature': sigs[d['name']], 'axioms': axioms[d['name']], + 'formal_progress': 'specified'} for m in modules for d in info['scans'][m] if d['kind']] + (out / 'declarations.json').write_text(json.dumps(declarations, indent=2, ensure_ascii=False) + '\n', encoding='utf-8') + receipt = {'schema_version': 1, 'object': RECEIPT_OBJECT, 'scientific_effect': 'NONE', 'scientific_status_authority': False, + 'package': PACKAGE, 'formalization_status': 'specified', 'alignment_status': 'PENDING_INDEPENDENT_REVIEW', + 'meaning': 'The registered statements elaborated against the pinned Mathlib at the checked commit and depend on ' + 'no axiom outside propext, Classical.choice and Quot.sound; the environment audit found no theorem, ' + 'axiom, opaque constant, instance or proof-typed definition in the modules and Lean\'s own inventory ' + 'of Prop-former definitions, structures and auxiliary definitions equals the register and the scan. ' + 'This is kernel evidence that the statements are well formed, not that they are true, aligned with ' + 'their sources, reviewed or accepted. No claim, premise or obligation changes status because of this receipt.', + 'checked_commit': info['checked_commit'], 'worktree': info['worktree'], + 'receipt_trust': 'CLEAN worktree at checked_commit' if info['worktree'] == 'CLEAN' else + 'LOCAL_ONLY: the worktree was not verified against checked_commit; not evidence about any commit', + 'repository': os.environ.get('GITHUB_REPOSITORY'), 'workflow_run_id': os.environ.get('GITHUB_RUN_ID'), + 'workflow_run_attempt': os.environ.get('GITHUB_RUN_ATTEMPT'), 'lean_version': version, 'toolchain': TOOLCHAIN, + 'dependency_revisions': revs, 'register_sha256': info['register_sha256'], 'module_sha256': info['module_sha256'], + 'declarations': len(names), 'prop_definitions': info['prop_definitions'], 'structures': info['structures'], + 'auxiliary_definitions': info['auxiliary_definitions'], 'claims': info['claims'], 'statements': info['statements'], + 'axioms': axioms, 'environment_audit': environment, 'lean_options': {'autoImplicit': False}, + 'negative_controls': outcomes, 'build': 'PASS', 'leanchecker': 'PASS', 'dependency_worktrees': 'CLEAN'} + receipt['logs'] = {p.name: sha(p.read_bytes()) for p in sorted(out.glob('*.log'))} + (out / 'receipt.json').write_text(json.dumps(receipt, indent=2, sort_keys=True) + '\n', encoding='utf-8') + return receipt + + +def main(argv=None) -> int: + p = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter) + p.add_argument('mode', nargs='?', choices=['source'], + help='optional; `source` names the default mode (DESIGN 3b), so `check.py source` and `check.py` are the same check') + p.add_argument('--root', type=Path, default=ROOT, help='packet directory') + p.add_argument('--repo', type=Path, default=REPO, help='repository root holding formal/lake-manifest.json') + p.add_argument('--execute', action='store_true', help='fresh build, leanchecker, #check/#print axioms, negative controls, receipt') + p.add_argument('--allow-dirty', action='store_true', help='local development only: skip the clean-worktree requirement') + args = p.parse_args(argv) + try: + info = source_check(args.root, args.repo, args.allow_dirty) + if info['worktree'] != 'CLEAN': + print('WORKTREE_UNVERIFIED: packet identity against HEAD was not checked (--allow-dirty); nothing printed here ' + 'is evidence about a commit', file=sys.stderr) + summary = {k: v for k, v in info.items() if k not in ('scans', 'module_sha256')} + if args.execute: + receipt = execute(args.root, args.repo, info, args.allow_dirty) + print(json.dumps({'passed': True, 'formalization_status': receipt['formalization_status'], + 'alignment_status': receipt['alignment_status'], 'checked_commit': receipt['checked_commit'], + 'worktree': receipt['worktree'], 'declarations': receipt['declarations'], + 'negative_controls': receipt['negative_controls']}, indent=2, sort_keys=True)) + else: + print('SOURCE_CHECK_PASS (statements only: not a Lean build, not a proof, not a review, not an acceptance)') + print(json.dumps(summary, indent=2, sort_keys=True)) + except (ValueError, KeyError, TypeError, OSError, UnicodeDecodeError, subprocess.SubprocessError) as exc: + print('LEAN_SPEC_CHECK_FAIL: ' + str(exc), file=sys.stderr) + return 1 + return 0 + + +if __name__ == '__main__': + sys.exit(main()) diff --git a/specifications/lean_statements_20261010/lake-manifest.json b/specifications/lean_statements_20261010/lake-manifest.json new file mode 100644 index 000000000..95a952ce3 --- /dev/null +++ b/specifications/lean_statements_20261010/lake-manifest.json @@ -0,0 +1,117 @@ +{ + "version": "1.2.0", + "packagesDir": ".lake/packages", + "packages": [ + { + "name": "plausible", + "url": "https://github.com/leanprover-community/plausible", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "118aa17ee84656b8bd727fef7c458ee8c833385c", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "name": "LeanSearchClient", + "url": "https://github.com/leanprover-community/LeanSearchClient", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "ddf04cf3949fa556442341e87d47f9f6e6074707", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "name": "importGraph", + "url": "https://github.com/leanprover-community/import-graph", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "e928b72544873815af278d38681b31c0293588e3", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "name": "proofwidgets", + "url": "https://github.com/leanprover-community/ProofWidgets4", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "106ff4fafc74ef4ac99d81dbf3ab399118f497a5", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.lean" + }, + { + "name": "aesop", + "url": "https://github.com/leanprover-community/aesop", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "355695d523e41d0554926416cba2a2b3544fbbc9", + "manifestFile": "lake-manifest.json", + "inputRev": "master", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "name": "Qq", + "url": "https://github.com/leanprover-community/quote4", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "6a489d9af5d0c47e5b259e2e8bcdfc1811b5a259", + "manifestFile": "lake-manifest.json", + "inputRev": "master", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "name": "batteries", + "url": "https://github.com/leanprover-community/batteries", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "f2effa3d803fda822b1f97b806c47cf2adfbcbc2", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "name": "Cli", + "url": "https://github.com/leanprover/lean4-cli", + "type": "git", + "subDir": null, + "scope": "leanprover", + "rev": "e92c9f15fdfacc8536f31cfb3b7ad26c3c8cd204", + "manifestFile": "lake-manifest.json", + "inputRev": "v4.34.0", + "inherited": true, + "configFile": "lakefile.toml" + }, + { + "name": "mathlib", + "url": "https://github.com/leanprover-community/mathlib4.git", + "type": "git", + "subDir": null, + "scope": "", + "rev": "d13f23b723b8a846827a245b89c10fc7d3f11612", + "manifestFile": "lake-manifest.json", + "inputRev": "d13f23b723b8a846827a245b89c10fc7d3f11612", + "inherited": false, + "configFile": "lakefile.lean" + } + ], + "name": "universal_law_specifications", + "lakeDir": ".lake", + "fixedToolchain": true +} diff --git a/specifications/lean_statements_20261010/lakefile.toml b/specifications/lean_statements_20261010/lakefile.toml new file mode 100644 index 000000000..15113ca7c --- /dev/null +++ b/specifications/lean_statements_20261010/lakefile.toml @@ -0,0 +1,12 @@ +name = "universal_law_specifications" +version = "0.1.0" +defaultTargets = ["Specifications"] + +[[require]] +name = "mathlib" +git = "https://github.com/leanprover-community/mathlib4.git" +rev = "d13f23b723b8a846827a245b89c10fc7d3f11612" + +[[lean_lib]] +name = "Specifications" +leanOptions = { autoImplicit = false } diff --git a/specifications/lean_statements_20261010/lean-toolchain b/specifications/lean_statements_20261010/lean-toolchain new file mode 100644 index 000000000..ba8ebf2db --- /dev/null +++ b/specifications/lean_statements_20261010/lean-toolchain @@ -0,0 +1 @@ +leanprover/lean4:v4.34.1 diff --git a/specifications/lean_statements_20261010/replay.sh b/specifications/lean_statements_20261010/replay.sh new file mode 100755 index 000000000..df96ad30c --- /dev/null +++ b/specifications/lean_statements_20261010/replay.sh @@ -0,0 +1,20 @@ +#!/usr/bin/env bash +# Local replay of the statement-only package check. Scientific effect: NONE. +# Order: source check (fails closed before any Lean), unit tests in both Python modes, then the +# Lean execution (fresh lake build, leanchecker, #check / #print axioms of every registered +# declaration, the environment audit of every constant the modules add, the negative controls, +# receipt under .lake/evidence/). +# Extra arguments are passed to check.py (local development: --allow-dirty; a receipt produced +# that way records the worktree as unverified and is not evidence about any commit). +set -euo pipefail +ROOT="$(cd "$(dirname "${BASH_SOURCE[0]}")/../.." && pwd)" +P="$ROOT/specifications/lean_statements_20261010" +cd "$ROOT" +export PATH="$HOME/.elan/bin:$PATH" +export PYTHONDONTWRITEBYTECODE=1 +mkdir -p "$P/.lake" +python3 -B -S "$P/check.py" "$@" +python3 -B -S "$P/test_check.py" 2>&1 | tee "$P/.lake/tests.log" +python3 -B -O -S "$P/test_check.py" 2>&1 | tee "$P/.lake/tests-optimized.log" +python3 -B -S "$P/check.py" --execute "$@" +echo 'PASS: statements elaborated and inventoried; formal progress specified; alignment PENDING_INDEPENDENT_REVIEW; scientific effect NONE.' diff --git a/specifications/lean_statements_20261010/sources/SOURCES.json b/specifications/lean_statements_20261010/sources/SOURCES.json new file mode 100644 index 000000000..3bbc39a27 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/SOURCES.json @@ -0,0 +1,237 @@ +{ + "schema_version": 1, + "scientific_effect": "NONE", + "meaning": "byte-pinned Layer 0 copies anchored by the statement register; publication of availability only, never acceptance", + "sources": [ + { + "id": "lifetime-parent", + "repository": "d6g8k5htny-coder/Math-", + "commit": "c39164254ad1529ed49543743598d5dcb24bab98", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "local_copy": 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No existing proof, review disposition, register, graph, +lemma flag, premise, prize or numerical certificate is changed. + +## 1. Model, results and exact logical scope + +Use the ORIGINAL normalized periodized Gaussian field on X=R^d/(T Z^d), for a +FIXED d>=2 and T>0. Its covariance is the normalized sum of +exp(-|x-y+Tn|^2/2). Put omega=2pi/T. The Fourier variances are + + a_n = c_T exp(-omega^2 |n|^2/2), n in Z^d, a_n>0. + +Real-valuedness imposes conjugacy of the complex Fourier coefficients. No +independence of their conditional laws will be used. All orthonormal original +pin frames are allowed; the lattice used for Fourier truncation is FIXED in the +torus coordinates and is NOT rotated with that pin frame. + +Retain the exact original value/gradient observations at M=-(r/2)u and S=(r/2)u, +with f(M)=b, f(S)=b-kr^3 and both gradients zero. Let Q_r be their continuous +Gaussian regression, equivalently the bounded-target nonsingular frame U_r=v_r +of LP. Birth and positive-gap marks range over the original compact sets, with +k>=k_->0. Use the full typed endpoint weight W_r and full Z_r=E_Qr W_r: + + dQ_r^W=(W_r/Z_r)dQ_r. + +N counts critical points other than M,S with values in I_r=(b-kr^3,b), all +indices. C(f) counts ALL real critical points on the torus, including the pins +and all heights. The two are not interchangeable. Constants depend on fixed +d,T and compact parameter ranges; no growing-dimension, k->0 or numerical +uniformity is claimed. Reduce r_* as needed within the parent validity regime. + +**Theorem F (count cap, no G_d input).** There is a measurable field functional +Psi_F>=C(f) such that uniformly in this original pin family + + Q_r(Psi_F>x) <= C exp(-c x^(2/d)), + Q_r^W(Psi_F>x) <= C exp(-c x^(2/d)), (F1) + +for x>=1, after changing constants. Thus all its fixed moments are uniformly +finite under either law. In particular, for some a>0, + + sup E_Qr^W exp(a Psi_F^(2/d)) < infinity. (F2) + +The same statements hold for the unconditioned field, without the pin/tilt +steps. In d=2 this is a uniform EXPONENTIAL tail for the total critical count, +not just a finite collection of moments. + +**Theorem W (window consequence).** Let G_d denote the dimension-matched +hypothesis E_Qr^W N<=C r^3. Given G_d, for every fixed integer p>=2, + + E_Qr^W (N)_p <= C_p r^3 [log(1/r)]^(d(p-1)/2). (F3) + +For d=2, G_2 is the reviewed planar I5 composition under the complete D5 +reading rule, including its actual full-depth P2 continuum review. Hence + + E_Qr^W N(N-1) <= C r^3 log(1/r), (F4) + E_Qr^W (N)_p <= C_p r^3 [log(1/r)]^(p-1). + +This is a NEW candidate improvement over the integrated Taylor-radius bound +r^3[log(1/r)/log log(1/r)]^2. For d>=3 only (F1)-(F2) are asserted without G_d; +the factorial composition remains explicitly conditional on that still-separate +first-moment input. No assertion that #141 or #145 is accepted is made here. + +**Proposition B (an inference boundary, not a Gaussian counterexample).** The +factor [log(1/r)]^(d(p-1)/2) cannot in general be removed using only G_d, a +nonnegative count cap, and the tail/moment statements (F1)-(F2). Section 8 gives +exact rare-count variables attaining that factor. This does not prove that the +factor is optimal for this Gaussian field and does not refute the Palm route. + +## 2. Inputs, including their conditioning boundaries + +LP Sections 3-5 provide the uniform covariance sandwich for U_r, bounded v_r, + + sup v_r^T Cov(U_r)^(-1)v_r < infinity, + ||W_r||_(L2(Q_r)) <= C r^2, Z_r>=c r^2. (F5) + +The matrix/moment facts are consumed in the original law; no witness is added. +C6L supplies the compact-analytic-set/holomorphic-degree argument and the +protected covering of the torus. C6S Section 4 explicitly extends its +polynomial-symbol independence and complex small-ball density argument to +fixed d. Section 4 below restates why the needed density estimate holds; no +uniform lower covariance bound is asserted at a real pinned critical point. +I5 and D5R supply only the planar small-window mean for (F4). Their original +source and reviewer identities are in SOURCE_MAP.json. Historical narrower +algebra-only HOLDs are preserved, not counted as full analytic acceptance. + +The new ingredients are a cutoff of the FULL conditioned field in its original +Fourier coordinates, uniform Gaussian-frequency error bounds, and conversion +of each truncated gradient to a Laurent system. The Taylor truncation tail of +C6S and the new witness-conditioning claims of #145 are not inputs. + +## 3. Uniform Fourier error AFTER conditioning + +Let c_n(f) be the complex Fourier coefficient of a realization. Define + + f_[m](z)=sum_(|n|_infinity<=m) c_n(f) exp(i omega n.z), + P_m(z)=grad f_[m](z), F(z)=grad f(z). (F6) + +This is the Fourier truncation of f drawn under Q_r, not the prior field +conditioned on a truncated set of observations. f_[m] generally does NOT satisfy +the original pin equations; no argument below requires it to do so. F, N, W +and Z always refer to the original, exactly pinned field. + +For a real standardized prior Fourier coordinate xi, Gaussian regression gives + + |E[xi | U_r=v_r]|^2 <= v_r^T Cov(U_r)^(-1)v_r, + Var(xi | U_r=v_r) <= 1. (F7) + +The first is covariance Cauchy-Schwarz after whitening U_r, and the second is +variance reduction by orthogonal projection. Thus, for every fixed s>=1, + + ||c_n||_(Ls(Q_r)) <= C_s sqrt(a_n), (F8) + +uniformly in m,r and marks/frames, with an immaterial factor for real and +imaginary components. The coefficients need NOT be conditionally independent. +This bound includes the conditional mean, not just the centered residual. + +Fix a complex tube |Im z|<=Y, with real parts taken modulo the torus. For every +fixed derivative order j, Minkowski and absolute convergence give + + ||sup |D^j(f-f_[m])|||_(Ls(Q_r)) + <= C_s sum_(|n|_infinity>m) (1+|n|)^j + exp(-omega^2 |n|^2/4 + omega Y |n|). + +To bound this tail put A=omega^2/4 and B=omega Y. Since +Bv <= A v^2/2+B^2/(2A), and |n|^2>m^2 in the omitted set, + + exp(-A|n|^2+B|n|) + <= exp(B^2/(2A)) exp(-A m^2/4) exp(-A|n|^2/4). + +The remaining polynomial-weighted lattice sum is finite. In particular, there +is a fixed a>0, chosen smaller if necessary, such that for the fixed tube used +by the cover, + + R_m := sup |F-P_m|, + E_Qr R_m^2 <= C exp(-4a m^2), + E_Qr [sup ||DF||]^2 <= C. (F9) + +The constants absorb the finitely many small cutoffs. This also proves uniform +absolute convergence of the relevant complex derivative series. Differentiating +a finite Fourier sum and taking its complex analytic extension commute. No +Cauchy radius tending to infinity is needed. + +## 4. Protected annuli and small-gradient volume + +Choose 0=7eta/8 with +radii eta_j=eta/8; set eta_0=eta. Their real balls cover the torus. For every +centre c_j take fixed complex test radii rho in [eta_j,2eta_j], and write + + A'_j={eta_j/2<=|z-c_j|<=3eta_j}. + +Its real slice stays at least eta/4 from the original pins. In each chart choose +real lifts of the centre. Distances near the torus seam are understood using +lifts; the local balls have diameter less than T and no aliasing occurs there. + +For z=x+itu, t=|Im z|>0, define the REAL 2d-dimensional observation + + G(z)=(Re F(z), Im F(z)/t). + +It extends continuously at t=0, with direction u retained, to (grad f(x),H_f(x)u). +For t>0 the evaluation symbols at z, conjugate z, and the real pin sites are +independent exponential-polynomial characters of Z^d. At t=0 the extra symbols +at remote x are i b.n and -(u.n)(c.n), independent because their linear and +quadratic parts cannot cancel and u is nonzero. At r=0 the pin frame consists +of its nonduplicated contact jets at the midpoint. Independence of characters +with polynomial coefficients follows by separation of the joint generalized +eigenspaces of the commuting lattice shifts. This is the C6L Lemma E/C6S +Section 4 argument, not an assumed isotropic GOE law. + +The joint covariance of the pin frame and G is therefore positive at each +point of the compact annulus/mark/direction/frame parameter set, including the +contact limit. Continuity and the Schur complement give a UNIFORM covariance +floor for G under Q_r. Mean shifts cannot increase the Gaussian density supremum. +Consequently, uniformly on these annuli for 0=1, introduce w_l=exp(i omega z_l), l=1,...,d. Each component of +P_m is a Laurent polynomial with exponent support in [-m,m]^d. Multiplication +of each component by the nonvanishing monomial (w_1...w_d)^m clears the negative +exponents and gives an ordinary polynomial Q_(m,l) of total degree at most 2dm. +Extra zeros on coordinate hyperplanes w_l=0 are irrelevant to the original +complex torus but may be counted in an upper bound. + +The exponential map is injective on every closed test ball of radius at most +2eta: equality of exponentials implies an original coordinate difference in +T Z^d; the distance between two points in the ball is at most 4eta=1 for which (F12) holds on at least one test +sphere, or infinity if none works. Let J be the finite number of cover balls, + + Psi_F=(2d)^d sum_j k_j^d. (F14) + +On the event all k_j are finite, C(f)<=Psi_F. The successful radius can vary +with j and f; all of them still cover the fixed real balls. Boundaries of one +cover ball need not carry any zero: a strict Rouché success excludes them. + +Measurability is explicit. For fixed m the sphere minimum of +|P_m|-|F-P_m| is continuous in rho and in the compact analytic field norm. +Strict success at some rho in the closed interval can be detected at a rational +radius in its interior, by continuity (also at endpoint success). The event +{k_j<=m} is the UNION of success events for cutoffs 1,...,m, not just success at +cutoff m; monotonicity of individual Fourier tests is not assumed. + +## 6. Gaussian cutoff tail and stretched-exponential count tail + +Fix j and define h_m=exp(-a m^2), lambda_m=exp(a m^2/(4d)) and + + delta_m=h_m/lambda_m. + +Let L_j be the supremum of ||DF|| on the fixed ball of radius 3eta_j. For large +m, delta_m<=eta_j/2. If k_j>m, the cutoff-m Rouché test fails at EVERY radius, +so each sphere contains z_rho with |P_m(z_rho)|<=|F-P_m|(z_rho). On +{R_m<=h_m,L_j<=lambda_m}, this gives |F(z_rho)|<=2h_m and |F|<=3h_m on the ball +of radius delta_m around it. + +Take radii eta_j+2i delta_m within [eta_j,2eta_j]. There are at least +eta_j/(2delta_m) such radii. Their centres are separated by at least 2delta_m, +so the OPEN balls have disjoint interiors; touching boundaries have zero volume. +All lie in A'_j. Thus failure forces sublevel volume at least + + c_d eta_j delta_m^(2d-1). + +Combining this lower volume with (F11) and Markov gives + + Q_r(k_j>m,R_m<=h_m,L_j<=lambda_m) + <= C h_m lambda_m^(2d-1)(1+a m^2) + = C(1+a m^2) exp[-a(2d+1)m^2/(4d)]. (F15) + +The other two events are bounded using (F9): + + Q_r(R_m>h_m) <= C exp(-2a m^2), + Q_r(L_j>lambda_m) <= C exp[-a m^2/(2d)]. (F16) + +Absorb the fixed polynomial and finite small-cutoff range into constants: + + Q_r(k_j>m) <= C exp(-c m^2). (F17) + +This proves k_jx, some k_j exceeds (x/[(2d)^d J])^(1/d); integer floors change only +constants for x large. A finite union gives the Q_r part of (F1). + +Under the ORIGINAL tilt, covariance Cauchy-Schwarz and (F5) give + + Q_r^W(A) <= ||W_r/Z_r||_2 Q_r(A)^(1/2) <= C Q_r(A)^(1/2). + +The square root changes c, not the power x^(2/d). This proves the other part +of (F1). For Y=Psi_F^(2/d), integrate its exponential tail to obtain (F2) for +sufficiently small a. All constants are uniform only in the specified original +pin family. No further witness-conditioned cap bound has been proved. + +## 7. Factorial moments: exactly one mean and a tail + +For an integer p>=2, N<=Psi_F gives the pathwise inequality + + (N)_p <= N Psi_F^(p-1) + <= Lambda^(p-1) N + Psi_F^p 1_{Psi_F>Lambda}. (F18) + +By Cauchy-Schwarz under Q_r^W, its uniformly bounded 2p-th cap moment and (F1), + + E_Qr^W[Psi_F^p 1_{Psi_F>Lambda}] + <= C_p exp(-c Lambda^(2/d)). (F19) + +Assume G_d. With L_r=log(1/r)>=1 choose Lambda=(A L_r)^(d/2), where A is a fixed +large constant making cA>=3. Then (F18)-(F19) give (F3). Each p has its own +finite constant; no uniform assertion for growing p is made. In d=2 insert +the dimension-matched planar I5, yielding (F4). + +The planar conclusion is asymptotically stronger than the Taylor bound because +L_r/[L_r/log L_r]^2=(log L_r)^2/L_r ->0. In fixed general d the analogous ratio +is (log L_r)^d/L_r^(d/2)->0. Neither comparison makes a numerical radius promise. +G_d for d>=3 remains a separate analytic dependency and is NOT proved by (F1). + +## 8. Why a marked estimate is still needed for O(r^3) + +The following is an EXACT abstract construction, not a Gaussian field. Fix d>=2 +and an integer m>=2. Set r_m=2^(-m^2), n_m=m^d, and let + + N_m=Psi_m=n_m with probability 2^(-3m^2)/n_m, + N_m=Psi_m=0 otherwise. + +Then E N_m=r_m^3 and for 0<=xx)=2^(-3m^2)/n_m <= exp[-2(log 2) x^(2/d)]. + +Above n_m the tail is zero. Also E exp[(log 2)Psi_m^(2/d)]<=2 uniformly. For +fixed integer p>=2, eventually n_m>=p and + + E (N_m)_p = r_m^3 product_(j=1)^(p-1)(n_m-j) + ~ r_m^3 m^(d(p-1)) + = constant * r_m^3[log(1/r_m)]^(d(p-1)/2). (F20) + +Thus even an exponential planar cap tail does not turn an unconditional small +mean into an O(r^3) factorial bound. The exact size-biased identity is + + E(N)_p = (E N) E_sizebiased[(N-1)_(p-1)], + dP_sizebiased = N dQ_r^W / E N. + +A uniform bound on that SIZE-BIASED moment, or an appropriate marked Kac-Rice +estimate, would be an additional input. This identifies the role of #145's +proposed marked argument, not a proof of it. The previous #146 obstruction for +the Taylor-cap tail is credited as motivation; (F20) is a new simpler exact +construction matched to the present Fourier-tail class. The true Gaussian C6 +order may still be Theta(r^3). + +## 9. Review and execution boundaries + +New analytic targets: (i) the FULL conditioned Fourier coefficient bound, +including its mean; (ii) keeping truncation distinct from exact pin constraints; +(iii) exponential-coordinate injectivity and isolated-zero Bezout, even for +nongeneric coefficient systems; (iv) the two cutoffs and all three probabilities +in (F15)-(F16); (v) the correct dimension power and separate G_d dependency. + +The finite checker verifies Laurent clearing and degree counts, rational Gaussian +regression inequalities, completing-square algebra, the probability exponents, +measurability's finite-union convention, and the exact inference-boundary +example. It does not establish Gaussian uniformity, complex analytic zero-set +facts, Kac-Rice or nonauthor acceptance. No hidden import or old missing runner +is used. The exact dependency identities are in SOURCE_MAP.json; local packet +replay does not claim a full upstream checkout, and the hosted workflow separately +checks the pinned upstream bytes. + +The statement is a source-consumer extension, not a novelty/priority claim. +There is no new persistent-pair law, Poisson approximation, same-index refinement, +quantitative SARD rate, numerical C103 certificate, status promotion, or resumed +stopped process. References and the actual limited reconnaissance scope are in +RECONNAISSANCE.md. diff --git a/specifications/lean_statements_20261010/sources/c6-palm-route/PROOF.md b/specifications/lean_statements_20261010/sources/c6-palm-route/PROOF.md new file mode 100644 index 000000000..89eb8adf0 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/c6-palm-route/PROOF.md @@ -0,0 +1,721 @@ +# C6 sharp: the torus-wide factorial moments of the window count are `O(r^3)` in every fixed dimension + +**Object:** CL-C6-PALM-20260929-v1. +**Version:** v1.4, 29 September 2026 (provenance refresh after Codex review 5358203608: the [DL] identity in §2 and the +status sentences point at the merged bytes on `main`; no mathematical text changed). v1.4: R3a (§6.2) states the axis-safe determinant-weight bound +`W_r |det H_X| / Z_r <= C K^(3d)` valid throughout the strip `|v| <= s^(1/8)`, in place of a bound cited on the axis +alone (OpenAI review 5357899713); the [DL] status sentences updated. v1.3: Lemma 5.1 rewritten with a canonical regression kernel for every `(X, h)` +and the two-measure argument, and the Morse genericity cited from [LP] §8 (OpenAI comment 5896041000). v1.1: clarifications after OpenAI review 5356233690 of §4 (the ratio +`zeta_0 <= eta'/8` in Lemma M' (b), the witness lift convention, `Phi = oo` at a vanishing floor, the finite-union +measurability convention of [C6L] v1.3, Proposition 4.5' and the remark after Lemma 4.3). v1.2: Corollary P restated +as a size-biased statement about the second factorial moment only, and the order of constants in §6, after OpenAI +comment 5895460246. No theorem or ledger change in either. +**Author:** Anthropic Claude (Claude Code session `session_015wNj8LPTKXsaT68G3DgPPh`), 29 September 2026. +**Disposition:** author-side proof candidate. **Nonauthor analytic review is required.** This note stacks on [C6L], +now merged on `main` with the OpenAI lane's complete nonauthor acceptance at its planar scope, and on [DL], now +merged on `main` via Math-#141 with a source-exposed OpenAI nonauthor acceptance of its continuum rows (§2); this note +itself carries OpenAI review 5358116559 (ACCEPT of §§5–7 and Theorem Q at `06bc0d6`), see `README.md`. +**Scientific effect:** NONE. No `STATUS`, `PROOF_INDEX` verdict, `GRAPH` node, claim, `lemma_closed` flag, catalog +entry, prize or source body changes. + +## 0. Why this note exists + +Catalog entry C6 (`reviews/candidates_pending_20260928/CANDIDATES.md`) asks for the optimal order of the torus-wide +second factorial moment of the window count, "with the pin-neighborhood collisions included". Two things are on the +table today. + +- [C6L] (`frontiers/c6_factorial_moment_20260929/PROOF.md`, merged via Math-#140) proves + `E_(Q_r^W)[N(N-1)] <= C r^3 log^2(1/r)` in `d = 2` through a pathwise Rouché–Bézout cap `Psi` with a uniform exponential tail (its Lemma M), and its §9 names + the two statements that would remove the logarithm: `(M')`, the tail of `Psi` under the additional conditioning on + one window witness, and `(H)`, a Hölder insertion into the reviewed first-moment proofs. +- [DL] (`frontiers/d5_dimension_lift_20260929/PROOF.md`, Math-#141) supplies the first moment `E_(Q_r^W) N <= C r^3` + and each of its regional intensity bounds in every fixed `d >= 2`. + +This note proves `(M')` and carries out `(H)`. The result is `E_(Q_r^W)[N(N-1)] <= C r^3` in every fixed `d >= 2`, +hence, with the lower bound (A4) of [EDL], `E_(Q_r^W)[N(N-1)] = Theta(r^3)`: the C6 order is `r^3`, and the +logarithm of [C6L] is removed. The same argument gives `E_(Q_r^W)[(N)_q] <= C_q r^3` for every fixed integer `q >= 2`. + +The mechanism is the Palm route of [C6L] §9. Since `N(N-1) <= N Psi` pathwise, the marked Kac–Rice formula turns the +pair count into a first-moment integral whose mark is `Psi`. The conditional law at a witness is a Gaussian +regression on the pin frame plus the witness frame. Every regional first-moment proof already controls that regression +through one frame with an explicit inverse-covariance bound and an explicit Gaussian penalty. The two new facts are: + +1. the tail of `Psi` under any such regression law is governed by two numbers, the complex sup of the mean's gradient + and Hessian and a floor for the residual covariance of the complexified gradient on the test annuli, and both are + polynomial in the same degeneracy variables that the penalty of that region already absorbs (§4, §5, §6); +2. a witness inside a test annulus forces a zero of the complexified gradient there, and the packing argument of + [C6L] Lemma M survives the exclusion of a fixed slab of radii around it (§4). + +Nothing about the pins, the weight, the normalizer or the Kac–Rice framework changes. + +## 1. Setting and results + +### 1.1 Model + +Fix `d >= 2`, `m = d - 1`, and `L > 0`. `f` is the centered variance-one Gaussian field on `X = R^d/(L Z^d)` with the +exact normalized periodized Gaussian covariance of [LP] §1; [C6L] writes the period as `T`. Fix compact +`B = [b_-, b_+]` and `K = [k_-, k_+]` with `0 < k_-`. Frames `R` range over `O(d)`, `u = R e_1` is the axial direction, +`y in R^m` are the transverse coordinates. For small `r > 0` the pins are + + M = -(r/2) u, S = (r/2) u, f(M) = b, f(S) = b - k r^3, grad f(M) = grad f(S) = 0. + +`Q_r` is Gaussian regression on these `2(d+1)` observations. `F_j(H) = |det H|` if `H` is nonsingular with `j` +negative eigenvalues and `0` otherwise. `W_r = F_d(H_M) F_(d-1)(H_S)`, `Z_r = E_(Q_r) W_r`, `dQ_r^W = (W_r/Z_r) dQ_r`, +`I_r = (b - k r^3, b)`. `N` is the number of critical points of `f` in `X minus {M, S}` with height in `I_r`, all +indices; `(N)_q = N(N-1)...(N-q+1)`. Constants `C, c, r_*` depend on `d, L, B, K` and on the fixed cover parameter +`eta` of §3; never on `r, b, k, R` or the witness position. No constant is evaluated numerically. Nothing is uniform as +`k -> 0`, as marks grow, or as `d` or `L` vary. + +`K` also denotes a fixed multiple of `1 + ||f||_(C^6(X))`, as in the sources. The gradient `F = grad f` is extended to +`C^d` (§3); `|.|` is the Hermitian norm on `C^d`. + +### 1.2 Statements + +**Theorem Q (factorial moments, every fixed dimension).** There are `C_q, r_* > 0` such that for `0 < r <= r_*`, all +`b, k, R` and every fixed integer `q >= 2`, + + E_(Q_r^W)[ (N)_q ] <= C_q r^3. (1.1) + +In particular `E_(Q_r^W)[N(N-1)] <= C r^3`. + +**Corollary Theta.** In every fixed `d >= 2`, + + c r^3 <= E_(Q_r^W)[ N(N-1) ] <= C r^3, c r^3 <= Q_r^W{ N >= 2 } <= C r^3. (1.2) + +The lower bounds are [EDL] (A4). The upper bound on the event is [DL] Corollary M_d, and it also follows from (1.1) +by `1{N >= 2} <= N(N-1)/2`. This settles the order asked for in catalog C6: it is `r^3`, with pin-neighbourhood pairs +included, and the logarithm of [C6L] Theorem C6-L is removed. + +**Corollary P (pair intensity has order one relative to the singles).** `E_(Q_r^W)[N(N-1)] / E_(Q_r^W) N` is bounded +above and below by positive constants. Equivalently, under the size-biased (Palm) law `dP^sb = N dQ_r^W / E_(Q_r^W) N`, +the expected number of further window critical points, `E_sb[N - 1] = E_(Q_r^W)[N(N-1)] / E_(Q_r^W) N`, is of order +one. This is a statement about the size-biased law, not about `Q_r^W` conditioned on `N >= 1`. For `q >= 3` Theorem +Q gives only the upper bound `E_(Q_r^W)[(N)_q] <= C_q r^3`; no lower bound of order `r^3` is claimed for `q >= 3` +(it would need higher-cluster lower events, which are not supplied, and it can fail for a count that never exceeds +two). + +### 1.3 What the results are not + +They are moment bounds under the tilted pinned law with existential constants. They say nothing about the +distribution of the pairs' positions, nothing about elder selection, nothing outside the fixed torus and compact +marks, and nothing about numerical constants. §9 lists the non-claims. The note is conditional on [C6L] and [DL] at +the exact bytes of §2. [C6L] carries the OpenAI lane's complete nonauthor acceptance at its planar scope (its +`d`-dimensional restatement in §3 is this note's own obligation); [DL] has xAI acceptance of its algebraic identities +and, at Math-#141 head `c709854` and the wording-only successors, OpenAI source-exposed nonauthor acceptance of every +continuum row (reviews 5357858391, 5357882570: Theorems P_d, C_d, I_d, G_d), merged on `main` at `e1ca400` with +identical bytes. A defect in either blocks the corresponding step here. + +### 1.4 The planar case depends on this note alone + +For `d = 2`, every regional bound that §6 takes from [DL] is the planar statement it lifts, and each of those has a +nonauthor verdict on file: the pin ball (P10)–(P12), (P18)–(P20) (xAI 5894512272 and #111), the collar (C4)–(C19) +(`reviews/d5_collar_count_20260928/`), the shells (I9)–(I33) (`reviews/d5_i5_planar_f_20260928/`), the remote +region [RM] Theorem A (main#76), the normalizer [LP] (5.5) (Math-#106), and [C6L] (merged, review 5355682335). The +correspondence row by row is `frontiers/d5_dimension_lift_20260929/CONTINUUM_CROSSWALK.md`. Hence Theorem Q and +Corollary Theta at `d = 2`, that is the planar C6 order `E_(Q_r^W)[N(N-1)] = Theta(r^3)`, are conditional on the +steps of this note only (§§3–7), not on the `d >= 3` read of [DL]. The every-`d` statement additionally needs [DL]. + +## 2. Sources consumed, with the exact interface + +Identities (bytes, sha256, git blob, branch head) are in `SOURCE_MAP.json`. + +| Tag | Source | What is consumed | +|---|---|---| +| [C6L] | `frontiers/c6_factorial_moment_20260929/PROOF.md` v1.3 on `main` `bf7c45f` (merged via Math-#140; previously pinned at v1.2, see `SOURCE_MAP.json` for the byte comparison) | The entire extension and moments §4; the cover, `m_j*`, `Psi` and measurability §5; Lemma R (steps 1–4), Lemma M (steps (a)–(d) as a template), Lemma D (the reduction to `G = (Re F, Im F/t)` and its `t = 0` extension), Lemma E; the pathwise cap §2; the Palm route statement §9. All in `d = 2`; §3 below restates them in `d` dimensions with the two exponent changes. | +| [DL] | `frontiers/d5_dimension_lift_20260929/PROOF.md` on `main` `e1ca400` (merged via Math-#141; bytes identical to the reviewed branch head `ea35953`, blob `9d82c707`; every earlier pin and byte comparison in `SOURCE_MAP.json`) | Theorems P_d, C_d, I_d, G_d and the frames, densities, determinant bounds and ledgers of their proofs: §4 (pin ball: `Y`, (4.2)–(4.12)), §5 (collar: `G` of (5.2)–(5.7), the axial strip §5.4, the remainder §5.6), §6 (shells: `Z` of (6.3)–(6.15), the axial strip §6.5), §7 (tiling). The letter `G` of [DL] §5 is written `Gc` here. | +| [LP] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | The model in every `d`; §2 distinct-site jet rank and positive spectrum; (4.1) uniform `C^q` moments; (5.5) `Z_r >= z_* r^2`. | +| [PP], [CP], [IW] | `reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md`, `.../COLLAR_PROOF.md`, `frontiers/intermediate_window_20260928/PROOF.md` | The planar templates behind [DL], cited where [DL] cites them; in particular (P18) and (I33) for the weighted Kac–Rice formula with a field mark, and the degree-five frames of [CP] §4 and [IW] §3. | +| [RM] | `frontiers/remote_window_20260924/PROOF.md` | Theorem A in every `d` (reviewed, main#76); §2 uniform remote nondegeneracy of `(U_r, Y_x)`; §3 coupling and uniform moments under the extra witness conditioning (5); §4–§5 the weight and the intensity (12)–(14). | +| [EDL] | `frontiers/elder_dimension_lift_20260928/PROOF.md` | (A4): `Q_r^W(N_r >= 2) >= c r^3` and `E[N_r(N_r-1)] >= c r^3` in every fixed `d`. Used only in Corollary Theta. | + +External framework: Armentano, Azaïs and León, arXiv:2304.07424v3, Theorems 2.2 and 7.1 with Remark 8, exactly as +cited in (P18), (I33), [RM] §5 and [DL] §4.7. Standard Gaussian-process facts used with a citation in the text: +the Sudakov–Fernique comparison inequality and the Borell–TIS inequality (Adler–Taylor, *Random Fields and Geometry*, +Theorems 2.2.3 and 2.1.1), and the refined Bézout inequality (Fulton, *Intersection Theory*, Example 12.3.1). + +Steps of [C6L] whose proofs are dimension-free are cited and not reproved; the two places where the dimension enters +(Bézout's exponent, and the volume exponents in Lemma D and in the packing) are written out. + +## 3. The cap `Psi` in `d` dimensions + +### 3.1 Entire extension + +By Poisson summation, `f = sum_n sigma_n (xi_n cos(n'.x) + zeta_n sin(n'.x))` with `n' = 2 pi n / L`, i.i.d. standard +`xi, zeta`, and `sigma_n^2 > 0` proportional to `exp(-|n'|^2/2)` ([LP] §2, [C6L] §4). Since +`sum_n sigma_n |n'|^k e^(|n'| Y) < oo` for every `k, Y`, `f` extends almost surely to an entire function on `C^d`, +real on `R^d`, with `f(conj z) = conj f(z)`, and for every `k, p, Y` + + E sup_( |Im z| <= Y ) |partial^k f(z)|^p < oo. (3.1) + +Write `F = grad f : C^d -> C^d` for the complexified gradient and `DF` for its complex Jacobian. Every Gaussian +regression of `f` on finitely many bounded linear functionals has an entire mean and an entire residual, because the +regression coefficients `z -> Cov(f(z), Lambda_i) = Lambda_i[K(z - .)]` are entire. + +### 3.2 Lemma R_d (Rouché–Bézout cap) + +For `c in C^d` and `rho > 0` let `Omega_rho(c) = { z in C^d : |z - c| < rho }`. + +**Lemma R_d.** Let `F` be holomorphic near the closure of `Omega_rho(c)`, `c` real, and let `P: C^d -> C^d` be a +polynomial map of degree at most `m`. If `|F - P| < |P|` on `partial Omega_rho(c)`, then `F` has at most `m^d` zeros +in the real ball `B_R(c, rho)`. + +*Proof.* Steps 1–4 of [C6L] Lemma R with two changes of exponent. Step 1: `Z(P)` meets the sphere nowhere, so +`Z(P) cap Omega_rho` is a compact analytic subset of an open set in `C^d` and is finite (a compact analytic set of +positive dimension in `C^d` is excluded by the maximum principle for the coordinate functions on its irreducible +components). Step 2: the refined Bézout inequality bounds the isolated zeros of `P`, counted with multiplicity, by the +product of the `d` degrees, at most `m^d`. Step 3: the homotopy `P + t(F - P)` has no zero on the sphere, so the +Brouwer degrees of `P` and `F` on `Omega_rho` agree; `|F| >= |P| - |F - P| > 0` on the sphere makes `Z(F) cap +Omega_rho` finite as in step 1; the local degree of a holomorphic map at an isolated zero is its multiplicity, at +least `1`. Step 4: real zeros in `B_R(c, rho)` are among the complex ones. ∎ + +### 3.3 The cover, the degrees, and `Psi` + +Translate so that the midpoint `o = (M + S)/2` is the origin. Fix `eta in (0, L/20)`. + +- `c_0 = o`, `eta_0 = eta`. +- `c_1, ..., c_J` form an `eta/16`-net of `{ |x| >= 7 eta / 8 }` in `X`, with `eta_j = eta/8`. + +For any choice of `rho_j in [eta_j, 2 eta_j]` the real balls `B_R(c_j, rho_j)` cover `X`. The complex annuli + + A'_j = { z in C^d : eta_j / 2 <= |z - c_j| <= 3 eta_j } + +have real slices at distance at least `eta/2` from `o` (for `j >= 1`, `|c_j| >= 7 eta/8` and `|x - c_j| <= 3 eta / 8`), +hence at distance at least `eta/4` from both pins when `r <= eta/2`. All complex balls used below have radius at most +`4 eta < L/5`, inside the injectivity range of the torus. + +Let `T_m^(j)` be the degree-`m` Taylor polynomial of `F` at `c_j`, and + + m_j* = min{ m >= 1 : exists rho in [eta_j, 2 eta_j] with |F - T_m^(j)| < |T_m^(j)| on partial Omega_rho(c_j) }, + Psi = sum_(j=0)^J (m_j*)^d, (3.2) + +with `m_j* = oo` if no degree works. Measurability of `m_j*` is [C6L] §5 in its v1.3 form: success at degree `k` is +not asserted monotone in `k`, `m_j*` is the first successful degree, and + + { m_j* <= m } = union_(k=1)^m union_( rho in Q cap [eta_j, 2 eta_j] ) { min_(partial Omega_rho(c_j)) ( |T_k^(j)| - |F - T_k^(j)| ) > 0 }, + +a finite union over degrees of countable unions over rational radii of measurable events (the set of successful +radii at a fixed degree is relatively open by continuity of the sphere minimum in `rho`, so it is nonempty iff it +contains a rational radius). The tail argument of §4 uses only that `{m_j* > m}` implies failure at degree `m`. +By Lemma R_d applied to each ball `B_R(c_j, rho_j)` with an admissible `rho_j`, + + Psi >= #{ critical points of f on X } >= N + 2, (3.3) + +pathwise, the pins being critical points. Hence, for every integer `q >= 2`, + + N(N-1) <= N Psi, (N)_q <= N Psi^(q-1). (3.4) + +### 3.4 Lemma E_d + +**Lemma E_d.** Let `w_1, ..., w_K in C^d` be pairwise distinct modulo `L Z^d` and `P_1, ..., P_K` polynomials. If +`sum_k P_k(n) e^(2 pi i n . w_k / L) = 0` for all `n in Z^d`, then every `P_k = 0`. + +*Proof.* [C6L] Lemma E with `Z^2` replaced by `Z^d` and the two shifts by the `d` coordinate shifts; the joint +generalized eigenspaces of the commuting shifts for distinct eigenvalue `d`-tuples are independent on the finite +shift-invariant span of `{ n^alpha chi_k(n) }`. ∎ + +Consequently ([RM] §2, [LP] §2, [C6L] Lemma D): any finite family of derivative functionals of `f`, at finitely many +sites of `C^d` that are pairwise distinct modulo `L Z^d`, and with independent symbols at each site, has positive +definite covariance; the covariance of a real-linear combination is `sum_n sigma_n^2 |ell(e^(i n'.))|^2`. + +## 4. The parametric Rouché tail + +### 4.1 Regression laws and their two invariants + +Let `Lambda = (Lambda_1, ..., Lambda_l)` be a finite frame of bounded linear functionals of the field (values or +derivatives at finitely many points, or finite linear combinations of them) whose covariance under `Q_r` is +invertible, and let `tau in R^l`. Write + + Q' = Q_r( . | Lambda = tau ) + +for the Gaussian regression law, `m = E_(Q') f` for its mean (an entire function), and `g = f - m` for its residual +under `Q'` (a centered Gaussian field whose covariance is the `Q_r`-covariance minus the regression term). Every +conditional law used in a first-moment proof is of this form, with `Lambda` the frame of that proof. + +Two numbers describe `Q'` for the purposes of this section. Fix a real point `X in X` (the witness) and +`zeta_0 = eta/64`, so that `zeta_0 <= eta_j / 8` for every `j` (the smallest radius is `eta_j = eta/8`; for the ball +around the pins `zeta_0 = eta_0/64`). + +**Lifts of the witness.** Each ball `Omega_(4 eta_j)(c_j)` is read in the chart of `C^d` centred at a lift of +`c_j`; since `8 eta_j < 2L/5 < L`, at most one lift `X~_j in R^d` of `X` satisfies `|X~_j - c_j| <= 4 eta_j`. When +such a lift exists, `rho_X = |X~_j - c_j|` and `|z - X|` means `|z - X~_j|` for `z` in that chart; when none exists, +no radius is excluded in Lemma M' (b) and the floor (4.2) is taken over all of `A'_j`, because every lift of `X` is +then at distance more than `eta_j` from the annulus. Distances to the pins are read in the same charts; the cover +geometry of §3.3 keeps the real slices of the annuli at distance at least `eta/4` from every lift of `M` and `S`. + + mu(Q') = max_j sup_( z in Omega_(4 eta_j)(c_j) ) ( |grad m(z)| + ||D^2 m(z)|| ), (4.1) + + phi(Q'; X) = inf { det Cov_(Q')( G(z) ) : z in A'_j for some j, |z - X| >= zeta_0 / 2 }, (4.2) + +where, for `z = x + i t u` with `u` a real unit vector and `t = |Im z| > 0`, + + G(z) = ( Re F(z), Im F(z) / t ) in R^(2d), + +extended continuously to `t = 0` by `G = (grad f(x), D^2 f(x) u)` ([C6L] Lemma D). `Cov_(Q')(G(z))` is the residual +covariance of `G(z)`, which does not depend on `tau`. Put `Phi(Q'; X) = max(1, phi^(-1/2))`, with `Phi = oo` when +`phi = 0`; in that case every bound of this section that carries `Phi` is vacuous, and it is the applications in §6 +that prove `phi > 0` with an explicit floor in each regime. Nothing here assumes `phi > 0` from the words "regression +law". + +### 4.2 Three lemmas about regression laws + +**Lemma 4.1 (mean).** For every `Y > 0` and multi-index `alpha` there is `C_(Y, alpha)`, depending only on the prior +field, such that for `|Im z| <= Y` + + |partial^alpha m(z)| <= C_(Y, alpha) ( 1 + beta ), + beta := ( sum_i Var_(Q_r) Lambda_i )^(1/2) . ||Cov_(Q_r)(Lambda)^(-1)|| . |tau - E_(Q_r) Lambda|. (4.3) + +*Proof.* Regression is transitive: `m(z) = E_(Q_r) f(z) + Cov_(Q_r)(f(z), Lambda) Cov_(Q_r)(Lambda)^(-1) +(tau - E_(Q_r) Lambda)`, and the same identity holds for `partial^alpha` in place of `f`, real and imaginary parts +separately. The first term: `E_(Q_r) partial^alpha f(z) = Cov(partial^alpha f(z), E_r) Cov(E_r)^(-1) e` with +`Cov(E_r)^(-1)` and `e` bounded ([PP] §3, [DL] §4.1) and, by Cauchy–Schwarz, +`|Cov(Re partial^alpha f(z), E_(r,i))| <= Var(Re partial^alpha f(z))^(1/2) Var(E_(r,i))^(1/2) <= C_(Y,alpha)`, since +`Var(Re partial^alpha f(z)) <= E|partial^alpha f(z)|^2 <= sum_n sigma_n^2 |n'|^(2|alpha|) e^(2|n'|Y)` and the frame +variances are bounded. The second term: each entry of `Cov_(Q_r)(partial^alpha f(z), Lambda)` is bounded by +`C_(Y,alpha) Var_(Q_r)(Lambda_i)^(1/2)` by Cauchy–Schwarz again, residual variances being at most prior ones. ∎ + +**Lemma 4.2 (residual floor).** For every `z`, + + Cov_(Q')( G(z) ) >= lambda_min( Cov_(Q_r)( (G(z), Lambda) ) ) I_(2d). (4.4) + +*Proof.* The residual covariance of `G(z)` given `Lambda` is the Schur complement of the block `Cov_(Q_r)(Lambda)` in +the joint covariance; for a symmetric matrix `N >= c I` and any partition, `min_y (x, y)^T N (x, y) = x^T (Schur) x +>= c |x|^2`. ∎ + +**Lemma 4.3 (residual sup moments).** For every compact complex domain `D` inside `{ |Im z| <= Y }`, every `p` and +every `|alpha| <= 6`, + + E_(Q') sup_D |partial^alpha g(z)|^p <= C_(p, D, alpha), (4.5) + +with `C` depending only on the prior field. + +*Proof.* Fix a real component `X'(z) = Re partial^alpha g(z)` (or `Im`) and let `X(z)` be the same component of the +prior field. Since `g` is the regression residual of the prior on the total frame `(E_r, Lambda)` with standardized +coefficient vector `C(z)`, `E(X'(z) - X'(z'))^2 = E(X(z) - X(z'))^2 - |C(z) - C(z')|^2 <= E(X(z) - X(z'))^2` and +`Var X'(z) <= Var X(z)`. Sudakov–Fernique gives `E sup_D X' <= E sup_D X` and `E sup_D (-X') <= E sup_D (-X)`; +Borell–TIS turns the mean of the supremum and the maximal variance into all moments. `E sup_D |X| < oo` is (3.1). ∎ + +*Remark (a proof without comparison).* Each Fourier coordinate of the residual `g` after any finite Gaussian +regression is centered with variance at most its prior variance `sigma_n^2`, so its `L^p` norm is at most `C_p +sigma_n`; summing `sigma_n |n'|^k e^(|n'| Y)` by Minkowski, as in (3.1), bounds every derivative supremum on the tube +uniformly in the regressing frame, without independence between the residual coordinates and without any inverse +Gram bound (Math-#145 review 5356233690, item 1). Either proof gives (4.5). + +**Lemma 4.4 (conditional moments of `K`).** `E_(Q')[ K^n ] <= C_n ( 1 + beta )^n` for every `n`, with `beta` from +(4.3). *Proof.* `K <= C(1 + ||m||_(C^6(X)) + ||g||_(C^6(X)))`; Lemma 4.1 with `Y = 0` bounds the first norm by +`C(1 + beta)`, Lemma 4.3 the moments of the second. ∎ + +### 4.3 Lemma D' (density on the annuli, forced zeros excluded) + +**Lemma D'.** For `z in A'_j` with `|z - X| >= zeta_0 / 2` and `t = |Im z|`, + + Q'( |F(z)| <= 2 epsilon ) <= min( 1, C_d Phi(Q'; X) epsilon^(2d) t^(-d) ), (4.6) + +and for every `j`, + + integral_( {z in A'_j : |z - X| >= zeta_0/2} ) Q'( |F(z)| <= 2 epsilon ) d^(2d) z + <= C_d Phi(Q'; X) epsilon^(2d) ( 1 + log(1/epsilon) ). (4.7) + +*Proof.* Under `Q'`, `G(z)` is Gaussian with covariance `Cov_(Q')(G(z))` of determinant at least `phi`, so its density +is at most `(2 pi)^(-d) phi^(-1/2)`. The event forces `|Re F| <= 2 epsilon` and `|Im F|/t <= 2 epsilon / t`, a set of +`R^(2d)`-volume at most `(4 epsilon)^d (4 epsilon / t)^d`. This is (4.6). For (4.7), with `A = C_d Phi epsilon^(2d)`, +integrate `min(1, A |y|^(-d))` over `y = Im z in R^d`, `|y| <= 3 eta_j`, and over `Re z` in a set of bounded volume: +the region `|y| <= A^(1/d)` contributes at most `omega_d A`, and the region `A^(1/d) <= |y| <= 3 eta_j` at most +`A |S^(d-1)| log( (3 eta_j)^d / A ) / d <= C A (1 + log(1/epsilon))`, because `A >= c epsilon^(2d)`; if +`A^(1/d) > 3 eta_j` the whole integral is at most the volume, which is at most `C A`. ∎ + +The radial exponent `(d - 1) - d = -1` is the same in every `d`: the single logarithm of [C6L] Lemma M (c) is not a +planar accident. Nothing else in Lemma D' depends on `d` beyond the volume exponents. + +### 4.4 Lemma M' (parametric tail) + +**Lemma M'.** There are `m_0`, depending only on `eta`, and `C`, depending only on `d`, `eta`, `L` and the prior +field, such that for every regression law `Q'` as in §4.1, every witness point `X`, every `j` and every `m >= m_0`, + + Q'( m_j* > m ) <= C ( 1 + mu(Q')^2 + Phi(Q'; X) ) . m . 2^(-m/(2d)). (4.8) + +*Proof.* Fix `j`, write `c = c_j`, `eta' = eta_j`, `S = sup_(Omega_(4 eta')(c)) |F|`, `Lip = sup_(Omega_(3 eta')(c)) +||DF||`, all under `Q'` (so `F = grad m + grad g`). + +**(a) Remainder.** [C6L] (6.1) is pathwise and dimension-free: on `Omega_(2 eta')(c)`, `|F - T_m| <= S 2^(-m)`. + +**(b) Failure means near-zeros on every admissible sphere, with a slab excluded.** Put + + lambda = 2^(m/(4d)), epsilon = 2 lambda 2^(-m), delta = epsilon / lambda = 2^(1-m), + +and take `m_0` so large that `delta <= min( zeta_0 / 2, eta' / 8 )` for `m >= m_0`. If `m_j* > m` then for every +`rho in [eta', 2 eta']` some `z_rho in partial Omega_rho(c)` has `|T_m(z_rho)| <= |F - T_m|(z_rho) <= S 2^(-m)`, hence +`|F(z_rho)| <= 2 S 2^(-m)`. On the event `{ S <= lambda, Lip <= lambda }`, `|F| <= 2 epsilon` on each ball +`B(z_rho, delta)`. + +Let `rho_X = |X~_j - c|` for the lift of §4.1 (if there is none, discard nothing). Among the radii +`rho_i = eta' + 2 i delta`, `0 <= i <= floor(eta'/(2 delta))`, discard those with `|rho_i - rho_X| < zeta_0`. At +most `zeta_0 / delta + 1` are discarded, so at least `eta'/(2 delta) - zeta_0/delta - 1` remain, and this is at +least `eta'/(4 delta)` exactly when `eta'/4 - zeta_0 >= delta`, which holds because `zeta_0 <= eta'/8` (§4.1) and +`delta <= eta'/8` give `eta'/4 - zeta_0 >= eta'/8 >= delta`. (The weaker `zeta_0 <= eta'/4` would not suffice.) For a +retained radius, every `z in B(z_(rho_i), delta)` satisfies `| |z - c| - rho_X | >= zeta_0 - delta >= zeta_0/2`, hence +`|z - X| >= zeta_0/2`; and `eta'/2 <= rho_i - delta <= |z - c| <= rho_i + delta <= 3 eta'`, so the ball lies in `A'_j`. +The retained balls are pairwise disjoint (centres on spheres `2 delta` apart). Therefore the volume + + V := Leb_(2d) { z in A'_j : |z - X| >= zeta_0/2, |F(z)| <= 2 epsilon } + >= ( eta' / (4 delta) ) omega_(2d) delta^(2d) = ( omega_(2d) / 4 ) eta' delta^(2d-1). (4.9) + +**(c) Markov.** By (4.7), `E_(Q') V <= C_d Phi epsilon^(2d) (1 + log(1/epsilon))`. Hence + + Q'( m_j* > m, S <= lambda, Lip <= lambda ) + <= 4 E V / ( omega_(2d) eta' delta^(2d-1) ) + <= C Phi lambda^(2d-1) epsilon ( 1 + log(1/epsilon) ) / eta' + = C Phi lambda^(2d) 2^(1-m) ( 1 + log(1/epsilon) ) / eta' + <= C Phi m 2^(-m/2), + +since `lambda^(2d) = 2^(m/2)` and `log(1/epsilon) <= m log 2`. + +**(d) Truncation.** `S <= mu + sup |grad g|` and `Lip <= mu + sup ||D^2 g||` on the relevant complex balls, so by +Lemma 4.3 with `p = 2`, `E_(Q') S^2 + E_(Q') Lip^2 <= C (1 + mu^2)`, and Markov gives +`Q'(S > lambda) + Q'(Lip > lambda) <= C (1 + mu^2) 2^(-m/(2d))`. + +Adding (c) and (d), and using `2^(-m/2) <= 2^(-m/(2d))` and `m >= 1`, gives (4.8). ∎ + +For `d = 2` and the unconditioned `Q_r` (empty witness frame, `mu` and `Phi` bounded, `lambda = 2^(m/8)`), this is +[C6L] Lemma M with the slab exclusion, which costs a factor `2` in (4.9). + +### 4.5 Moments of `Psi` under a regression law + +**Proposition 4.5.** For every `p >= 1` there is `C_p`, depending only on `p, d, eta, L` and the prior field, such that +for every regression law `Q'` as in §4.1 and every witness point `X`, + + E_(Q')[ Psi^p ] <= C_p ( 1 + mu(Q')^2 + Phi(Q'; X) ). (4.10) + +*Proof.* `E[(m_j*)^(pd)] = sum_(m >= 0) ((m+1)^(pd) - m^(pd)) Q'(m_j* > m) <= m_0^(pd) + sum_(m >= m_0) pd (m+1)^(pd-1) +Q'(m_j* > m)`, and by (4.8) the series is at most `C (1 + mu^2 + Phi) sum_m pd (m+1)^(pd-1) m 2^(-m/(2d)) < oo`. Then +`Psi^p <= (J+1)^(p-1) sum_j (m_j*)^(pd)`. The finite control `TL` verifies that the summand ratio falls below +`2^(-1/(4d))` from an explicit index on, for `d = 2, ..., 6` and `p <= 4`. ∎ + +**Proposition 4.5' (logarithmic form).** With `A = 1 + mu(Q')^2 + Phi(Q'; X)` finite, for every `p >= 1`, + + E_(Q')[ Psi^p ] <= C'_p ( 1 + log A )^(pd). (4.11) + +*Proof.* (4.8) and `Q' <= 1` give `Q'(m_j* > m) <= min(1, C A m 2^(-m/(2d)))`. Choose `M_0 = C_0 (1 + log A)` with +`C_0 = C_0(d, C)` so large that `C A m 2^(-m/(4d)) <= 1` for all `m >= M_0`; then `Q'(m_j* > m) <= 2^(-m/(4d))` for +`m >= M_0`. Split the sum of the proof of Proposition 4.5 at `M_0`: the part below `M_0` is at most `M_0^(pd)`, the +part above is at most `sum_(m >= M_0) pd (m+1)^(pd-1) 2^(-m/(4d)) <= C_p`. ∎ + +This form was pointed out by the OpenAI lane (Math-#145 comment 5895432912). It follows from the same premises as +Proposition 4.5 and turns the second factor of (5.3) into `C_q (1 + log(1 + mu^2 + Phi))^(d(q-1))`. The proof of +Theorem Q below uses only the polynomial form (4.10); the logarithmic form reduces the losses recorded in §6 but is +not needed for `O(r^3)`, since every regime's penalty already absorbs the polynomial factor. + +## 5. Kac–Rice with a global mark, and the Hölder insertion + +### 5.1 Marked formula + +For a compact `B` in `X minus {M, S}`, a Borel set of heights `I`, an index `j`, and a nonnegative `sigma(f)`-measurable +random variable `Xi`, define the marked window count `N_j^Xi(B, I) = sum_X Xi 1{f(X) in I} 1{H_X in O_j}`, the sum over +critical points `X in B`, with `O_j` the open set of nonsingular symmetric matrices of index `j`. + +**Lemma 5.1.** Under `Q_r`, for every such `B, I, j` and `Xi`, + + E_(Q_r) N_j^Xi(B, I) = integral_B integral_I p_( grad f(X), f(X) | pins )(0, h) + E_(Q_r)[ Xi F_j(H_X) | grad f(X) = 0, f(X) = h ] dh dX, (5.1) + +both sides possibly `+oo`. The same holds without the height variable (all heights) with +`p_( grad f(X) | pins )(0) E[ Xi F_j(H_X) | grad f(X) = 0 ]`. + +**Kernel convention.** The conditional expectation in (5.1) is taken with respect to the canonical Gaussian regression +kernel: for every `X in B` and every `h in R`, `Q_(X,h)` is the law of `f` given `(grad f(X), f(X)) = (0, h)` defined by +the explicit regression formulas (mean `E_(Q_r) f + Cov_(Q_r)(f, (grad f(X), f(X))) Sigma_X^(-1) ((0,h) - E_(Q_r)(grad +f(X), f(X)))`, residual covariance the Schur complement), which exist for every `(X, h)` because `Sigma_X` is +nondegenerate on the compact `B` off the pins ([LP] §2) and depend continuously on `(X, h)`. `Q_X` denotes the same +with the gradient alone. These kernels are fixed once; no almost-everywhere version of a conditional expectation is +evaluated at the single value `grad f(X) = 0`. + +*Proof.* Fix `r`, a compact `B` off the pins, `I` and `j`. Define two measures on the field space (the Borel +`sigma`-field of `C(X)`): + + Lc(A) = E_(Q_r)[ 1_A(f) N_j(B, I) ], + Rc(A) = integral_B integral_I p_( grad f(X), f(X) | pins )(0, h) E_(Q_(X,h))[ 1_A(f) F_j(H_X) ] dh dX. + +Both are finite: `Lc(C(X)) = E_(Q_r) N_j(B, I) < oo` by the unweighted Kac–Rice formula on `B` ([LP] §9, [RM] §5, +(P18), (I33)), whose right side is `Rc(C(X))`; so the two total masses agree. For a bounded continuous function `g` +of finitely many jets of `f` at finitely many fixed sites, the weighted formula of the sources, with the auxiliary +Gaussian field of arXiv:2304.07424v3 Theorem 7.1 and Remark 8 taken constant in `X` and its conditional law read +through the kernel `Q_(X,h)`, gives `integral g dLc = integral g dRc`. These `g` form a vector space closed under +products which generates the Borel `sigma`-field of `C(X)` (a continuous field is determined by its values on a +countable dense set). Two finite measures whose integrals agree on such a class agree on the generated `sigma`-field +(monotone class theorem for functions: the bounded measurable `g` with `integral g dLc = integral g dRc` form a +vector space closed under bounded monotone limits containing the multiplicative class). Hence `Lc = Rc` on all Borel +`A`, which is (5.1) for indicators; simple functions by linearity, and nonnegative `Xi` by monotone convergence on +both sides (`Xi ∧ n` increasing to `Xi`). The all-height statement is the same construction with the kernels `Q_X` +and the unweighted gradient formula; the extension from compact `B` to `X minus {M, S}` is monotone convergence +along an exhaustion by compact punctures. ∎ + +Almost surely under `Q_r`, `f` is Morse with distinct critical values, hence every critical point off the pins is +nondegenerate: this is [LP] §8 ("Pinned genericity and global event measurability"), whose countable-exhaustion +argument for the pinned law transfers to `Q_r^W` by absolute continuity and `0 < Z_r < oo`. No density statement for +the nonlinear functional `det D^2 f` is used or needed. Hence `N = sum_j N_j(X minus {M,S}, I_r)` almost surely. + +### 5.2 The Hölder insertion + +Apply Lemma 5.1 with `Xi = W_r Psi^(q-1)` and, by (3.4), + + E_(Q_r^W)[ (N)_q ] <= Z_r^(-1) E_(Q_r)[ W_r N Psi^(q-1) ] + = Z_r^(-1) sum_j E_(Q_r) N_j^( W_r Psi^(q-1) )( X minus {M,S}, I_r ). (5.2) + +Inside (5.1), for the conditional law `Q' = Q_r( . | grad f(X) = 0, f(X) = h )` (or without the height in the +all-height regions), conditional Hölder gives + + E_(Q')[ W_r F_j(H_X) Psi^(q-1) ] <= E_(Q')[ (W_r F_j(H_X))^2 ]^(1/2) E_(Q')[ Psi^(2(q-1)) ]^(1/2). (5.3) + +The first factor is controlled exactly as the unmarked expectation in the source proofs: each of them bounds +`W_r F_j(H_X) <= |det H_M det H_S det H_X|` pathwise by a deterministic geometric factor times `K^(3d)`, so +`E_(Q')[(W_r F_j)^2]^(1/2)` is at most the same geometric factor times `E_(Q')[K^(6d)]^(1/2) <= C (1 + beta)^(3d)` +by Lemma 4.4 (the source's own bound is `C(1 + beta)^(3d)` for the unmarked `E_(Q')[K^(3d)]`, so the form is +unchanged). The second factor is at most `C_q (1 + mu + Phi^(1/2))` by Proposition 4.5. So the marked intensity in +each region is the unmarked intensity bound of the source with two extra polynomial factors in `beta`, `mu` and `Phi`, +which §6 computes region by region and the region's Gaussian penalty absorbs. + +## 6. The regimes + +The tiling is [DL] §7: the two punctured pin balls of scaled radius `1/4` (Theorem P_d, all heights), the compact +collar `C(1/4, 4)` in midpoint scaled coordinates (Theorem C_d, all heights), the dyadic shells `4 r <= |X| <= s_0` +(Theorem I_d, window), and the remote region `|X| >= s_0` ([RM] Theorem A, window). + +**Order of constants.** `eta` is fixed first (§3.3), which fixes the cover, the annuli `A'_j`, `zeta_0 = eta/64` and +the joint-floor constant `c_J` of Lemma 6.1. The collar radius `R = 4` and the shell cutoff `s_0` are fixed next, with +`s_0 <= eta/4`, so that every witness of the local regimes lies inside the core of the ball around the pins and every +Taylor scale `sigma in {r, s}` of Lemma 6.3 is at most `s_0`. Lemma 6.3 then reduces `s_0`, and Lemma 6.2 and the +source proofs reduce `r_*`, until the remainder conditions (`C sigma^6 <= (1/2)(c sigma^10)^(1/2)` in Lemma 6.3, `r <= +c|v|^2` and `s <= c|v|^3` in Lemma 6.2) hold; these reductions depend on `d, L, B, K, eta` only, never on `r`, `s`, the +witness or the height. This is the same order as in [DL] §7 and [IW] §10, with `eta` added at the front. In the +remote regime the exclusion of the witness is read in the local periodic lifts of §4.1. In each region the source proof +fixes one frame `Lambda` and one target `tau`, bounds `||Cov_(Q_r)(Lambda)^(-1)||`, `Var_(Q_r) Lambda_i` and +`|tau - E_(Q_r) Lambda|`, and derives the density and the conditional moments from them. The table records, for each +regime, the parameter `beta` of (4.3) and a floor `lambda_J` for `lambda_min Cov_(Q_r)((G(z), Lambda))` over the +`z` of (4.2); then `mu <= C(1 + beta)` by Lemma 4.1 and `Phi <= max(1, lambda_J^(-d))` by Lemma 4.2. + +| Regime | Frame `Lambda` (source) | `beta <=` | `lambda_J >=` | Penalty in the source | +|---|---|---|---|---| +| R1 pin ball, `X = M + r(p,q)`, all heights | `Y` of [DL] (4.2)–(4.5) | `C (1 + chi)`, `chi = |p|/Delta` | `c` | `e^(-c chi^2)` (region I); `e^(-c/(2r^2))` (region II) | +| R2a collar, `|v| <= r^(1/3)` | raw `grad f(X)`, [DL] §5.4 | `C r^(-10)` | `c r^10` | `exp[-c r^(-2/3)]` | +| R2b collar, `r^(1/3) <= |v| <= v_0` | `Gc` of [DL] (5.2)–(5.3) | `C |v|^(-4)` | `c |v|^4` | `exp[-c/|v|^2]` | +| R2c collar, `|v| >= v_0` | `Gc`, fixed floor, [DL] §5.6 | `C` | `c` | none needed | +| R3a shell, `|v| <= s^(1/8)` | raw `(grad f(X), f(X))`, [DL] §6.5 | `C s^(-10)` | `c s^10` | `exp[-c s^(-1/4)]` | +| R3b shell, `s^(1/8) <= |v| <= 2` | `Z` of [DL] (6.4), (6.11) | `C |v|^(-6)` | `c |v|^6` | `exp(-c/|v|^2)` (or a constant for `|v| >= v_0`) | +| R4 remote, `|X| >= s_0` | `Y_x = (grad f(x), f(x))`, [RM] §2–§3 | `C` | `c` | none needed | + +### 6.1 The joint floors: a common lemma + +Let `Jo` denote the vector of independent derivative monomials of `f` at `o` of orders `0` to `5` that are not +contact functionals, i.e. all monomials except `1, x, x^2, x^3, y_i, x y_i` ([IW] §3.1, [DL] §6.1); `Jo` contains the +jets `A_4, T_3, S_0, C_3, D_3` of [DL]. Let `J5` be the full vector of monomials of order `0` to `5` at `o`. + +**Lemma 6.1.** There is `c_J > 0` such that, for all `z in A'_j` (every `j`), all `t in [0, 3 eta_j]`, all frames and +`r <= r_*`, + + Cov_(Q_r)( (G(z), Jo) ) >= c_J I, Cov( (G(z), J5) ) >= c_J I (prior law). (6.1) + +The same holds with `Jo` taken at `M` instead of `o`. + +*Proof.* At `t > 0` the functionals of `G(z)` are evaluations of `grad f` at the two non-real points `z, conj z`, +distinct from each other and from every real point modulo `L Z^d`; at `t = 0` they are `(grad f(x), D^2 f(x) u)` at +the real point `x` with `|x - o| >= eta/2`, whose symbols `i a.n' - (u.n')(b.n')` vanish identically only for +`a = b = 0`. The monomials at `o` (or `M`) have independent symbols. By Lemma E_d the prior covariance of +`(G(z), J5)` is positive definite at every parameter, and by continuity and compactness of `(z, u, t, frame)` it has +a uniform floor. Under `Q_r` the covariance of `(G(z), Jo)` is the Schur complement of `Cov(E_r)` in +`Cov((G(z), Jo, E_r))`; at `r = 0` the frame `E_r` is replaced by `E_0`, whose monomials are exactly the ones excluded +from `Jo`, so `(G(z), Jo, E_0)` is a family of the type just treated and its covariance is positive definite; `E_r -> +E_0` in `L^2` uniformly ([PP] §3, [DL] §4.1), so the Schur complement is continuous on `r in [0, r_*]` and compactness +gives the floor. ∎ + +**Lemma 6.2 (frames written on jets).** Suppose `Lambda = a + B Jo + Rm` with `a` deterministic under `Q_r`, `B` a +matrix with `sigma_min(B)^2 >= lambda_B > 0`, and `||Rm||_(L^2(Q_r)) <= rho_R`. If `rho_R <= (1/2) ( c_J min(1, +lambda_B) )^(1/2)`, then + + lambda_min Cov_(Q_r)( (G(z), Lambda) ) >= (1/4) c_J min(1, lambda_B). (6.2) + +*Proof.* For unit `(alpha, gamma)`, `Var(alpha.G + gamma.Lambda)^(1/2) >= Var(alpha.G + gamma.B Jo)^(1/2) - +|gamma| rho_R`, and `Var(alpha.G + gamma.B Jo) = (alpha, B^T gamma)^T Cov((G, Jo)) (alpha, B^T gamma) >= c_J (|alpha|^2 ++ |B^T gamma|^2) >= c_J min(1, lambda_B)`. ∎ + +**Lemma 6.3 (raw frames at a coalescing witness).** Let `V` be the degree-five observation vector of [CP] §4 (three +sites at scale `r`: `V_r = (f(ra), r grad f(ra), f(rb), r grad f(rb), f(rc), r grad f(rc))`) or of [IW] §3 / [DL] §6.1 +(pins at scale `r`, witness at scale `s`: `V_(r,s,w) = (S_s U_r, f(sw), s grad f(sw))`), with scale `sigma = r` resp. +`sigma = s`. Then, under the prior law, `Cov((G(z), V)) >= c sigma^10 I`, and consequently, under `Q_r`, +`Cov_(Q_r)((G(z), grad f(X)))` resp. `Cov_(Q_r)((G(z), grad f(X), f(X)))` is at least `c sigma^10 I`. + +*Proof.* The sources write `V = Bc D_sigma J5 + Rm` with `Bc` of uniformly positive smallest row singular value, +`D_sigma = diag(sigma^|alpha|)`, and `||Rm||_(L^p) <= C sigma^6`. For unit `(alpha, gamma)`, by (6.1), +`Var(alpha.G + gamma.Bc D_sigma J5) >= c_J (|alpha|^2 + |D_sigma Bc^T gamma|^2) >= c_J (|alpha|^2 + sigma^10 +c' |gamma|^2) >= c sigma^10`, and the remainder `C sigma^6 |gamma|` is at most half of `(c sigma^10)^(1/2)` for +small `sigma`. Under `Q_r`, take the Schur complement in the pin rows of `V` (a principal submatrix keeps the floor, a +Schur complement of a matrix `>= c I` is `>= c I`), then unscale the gradient rows by `sigma^(-1) >= 1`. ∎ + +### 6.2 The regimes one by one + +**R1 (pin ball).** `Lambda = Y`, `tau = 0`. [DL] (4.5): `c_0 I <= Cov_(Q_r)(Y) <= C_0 I`; mean `dvec + O(1)` with +`|dvec| <= C chi`; so `beta <= C(1 + chi)`. Joint floor: `Y = dvec + B J_M + R` with `det(B B^T) >= c_d` ([DL] (4.4)), +bounded entries, hence `sigma_min(B)^2 >= c'_d`, and `||R||_(L^2) <= C r`; Lemma 6.2 gives `lambda_J >= c` for small +`r`. The witness is inside the core of `A'_0` (`|X| <= r`), so the slab exclusion is vacuous. + +Marked intensity. From [DL] (4.8): `(W_r F_j)^2 <= K^(6d) r^12 |q|^4 (1 + p^2/|q|^2)^6`; by Lemma 4.4, +`E_(Q')[K^(6d)]^(1/2) <= C(1 + chi^(3d))`; by Proposition 4.5, `E_(Q')[Psi^(2(q-1))]^(1/2) <= C_q (1 + chi)`. Region I +(`|q| >= r|p|`, `t = |p|/|q|`, `chi^2 in [t^2/2, t^2]`): + + rho_j^(W, Psi)(X) <= C r^(3-d) |q|^(2-d) sup_t (1 + t^2)^3 (1 + t^(3d)) (1 + t) e^(-c t^2/2) <= C' r^(3-d) |q|^(2-d). + +Region II (`|q| < r|p|`, `chi <= 1/r`), from [DL] (4.9) and (4.12) with the extra factor `(1 + chi) <= 2/r`: + + rho_j^(W, Psi)(X) <= C r^(-3-4d) e^(-c/(2r^2)) (r|p|)^(2-d) <= C r^(3-d) (r|p|)^(2-d), + +by the `n`-th term of the exponential series with `2n >= 6 + 3d`. Integration as in [DL] §4.8 with Lemma D5 there: + + E_(Q_r) N_j^(W_r Psi^(q-1))( M + rD, all heights ) / Z_r <= C r^3. (6.3) + +The `S`-centred ball is identical ([DL] §4.9). + +**R2a (collar, `|v| <= r^(1/3)`).** `Lambda = grad f(X)` raw, `tau = 0`; [DL] (5.1): `Sigma_X >= c r^10 I`, variances +bounded, mean bounded; `beta <= C r^(-10)`; Lemma 6.3 with `sigma = r`: `lambda_J >= c r^10`, so `Phi <= C r^(-10d)`. +[DL] §5.4 bounds the density by `C r^(-5d) exp[-c r^(-2/3)]` and `|det H_M det H_S det H_X| <= K^(3d)`. Marked: + + rho_j^(W, Psi)(X) <= C r^(-2) r^(-5d) exp[-c r^(-2/3)] . (1 + r^(-10))^(3d) . (r^(-10) + r^(-5d)) + <= C r^(-12-40d) exp[-c r^(-2/3)] <= C r^(3-d), + +by the `n`-th term of the exponential series with `2n/3 >= 15 + 39d`. + +**R2b (collar, `r^(1/3) <= |v| <= v_0`).** `Lambda = Gc`, `tau = 0`; [DL] (5.3): `c|v|^4 I <= Cov_(Q_r)(Gc) <= C|v|^2 +I`, mean bounded; `beta <= C |v|^(-4)`. Joint floor: `Gc = (6k D(u), 0) + B Jo + R` with `B B^T >= (|v|^4/4) I` +([DL] §5.5) and `||R||_(L^2) <= C r`; Lemma 6.2 applies because `C r <= (1/2)(c_J |v|^4/4)^(1/2)` for `|v| >= r^(1/3)` +and small `r`; `lambda_J >= c |v|^4`, `Phi <= C |v|^(-4d)`. Marked, from [DL] (5.4)–(5.6): + + rho_j^(W, Psi)(X) <= C r^(-2) . r^(-(d+1)) |v|^(-2d) e^(-c/|v|^2) . r^6 |v|^(-6) . |v|^(-12d) . (|v|^(-4) + |v|^(-2d)) + <= C r^(3-d) |v|^(-(16d+10)) e^(-c/|v|^2) <= C' r^(3-d). + +**R2c (collar, `|v| >= v_0`).** All of `beta, Phi` are bounded ([DL] §5.6: fixed singular-value floor of `B`); the +marked intensity is `C r^(3-d)` times a constant. + +Integrating R2a–R2c over the collar (`dX = r^d du dv`) gives `Z_r^(-1) E_(Q_r) N_j^(W_r Psi^(q-1))(collar) <= C r^3`, and +with (6.3) the scaled ball of radius `4` minus the pins has marked count at most `C r^3` ([DL] (1.4)). + +**R3a (shell, `|v| <= s^(1/8)`).** `Lambda = (grad f(X), f(X))` raw, `tau = (0, h)`; [DL] (6.1): `Sigma_X >= c s^10 +I_(d+1)`, variances and means bounded, `|h - E f(X)| <= C`; `beta <= C s^(-10)`; Lemma 6.3 with `sigma = s`: +`lambda_J >= c s^10`, `Phi <= C s^(-10d)`. [DL] §6.5: joint density at most `C s^(-5(d+1)) exp[-c s^(-1/4)]`. For the determinant +weight use the axis-safe bound of [DL] §6.4, valid throughout the strip `|v| <= s^(1/8)` including `v = 0`: the endpoint +short columns `H_M e_x`, `H_S e_x`, of size at most `Lr/2` by [LP] (5.3), do not depend on the witness position, so +Hadamard ([DL] Lemma D2 with `e_x`) gives `|det H_M| |det H_S| <= C r^2 K^(2d)` for every `X` in the strip; with the +trivial `|det H_X| <= K^d` and `Z_r >= z_* r^2` ([LP] (5.5)), `W_r |det H_X| / Z_r <= C K^(3d)` on the whole strip, +with no division by `|v|`. (A bound stated only at `v = 0` would not control nonzero `v`; this is the reading of +[DL] §6.4 recorded in OpenAI review 5357882570.) The marked intensity per unit volume and height is at most +`C s^(-5(d+1)) exp[-c s^(-1/4)] (1 + s^(-10))^(3d) (s^(-10) + s^(-5d)) <= C`, and the strip contributes at most +`C r^3 s^2` to the marked shell count exactly as in [DL] §6.5. + +**R3b (shell, `s^(1/8) <= |v| <= 2`).** `Lambda = Z`, `tau = (0, 0, (h - bbar)/s^3)` bounded; [DL] (6.11): +`Cov_(Q_r)(Z) >= c B B^T` with `sigma_min(B) >= c |v|^3`, `||Cov(Z)^(-1)|| <= C |v|^(-6)`, `Cov(Z) <= C |v|^2 I`, mean +bounded; `beta <= C |v|^(-6)`. Joint floor: `Z = (det.) + B Jo + O_(L^p)(s)` with `lambda_B >= c |v|^6` and remainder +`C s <= (1/2)(c_J c |v|^6)^(1/2)` for `|v| >= s^(1/8)` and small `s`; `lambda_J >= c |v|^6`, `Phi <= C |v|^(-6d)`. +Marked, from [DL] (6.10), (6.12), (6.13): + + Lambda_(r,j)^(W, Psi)(X, h) <= C s^(-(d+4)) |v|^(-(d+4)) e^(-c/|v|^2) . s^2 (r + s^2)^2 |v|^(-5) . |v|^(-18d) . (|v|^(-6) + |v|^(-3d)) + <= C s^(-(d+2)) (r + s^2)^2 |v|^(-(22d+15)) e^(-c/|v|^2) <= C' s^(-(d+2)) (r + s^2)^2, + +uniformly down to `|v| = s^(1/8)`. Integration over the shell and the window gives `C r^3 [(r/s)^2 + s^2]` as in [DL] +(6.15), and the dyadic sum `C r^3 (A_0^(-2) + s_0^2)` as in [DL] (1.6). + +**R4 (remote, `|X| >= s_0`).** `Lambda = Y_x = (grad f(x), f(x))`, `tau = (0, h)`. [RM] §2: `Cov((U_r, Y_x))` and its +Schur complements are uniformly bounded and positive on `O(d) x D_rho x [0, r_*]`; so `beta <= C`. Joint floor: the +family `(G(z), Y_x, U_r)`, at sites `o` (contact limit `E_0` at `r = 0`), `x`, and `z, conj z` (or `x' = Re z` at +`t = 0`) with `|z - x| >= zeta_0/2` and `|x' - o| >= eta/2`, is positive definite at every parameter by Lemma E_d and +continuous on the compact parameter set `{ x in D_(s_0), z in union A'_j, |z - x| >= zeta_0/2, t, u, frame, r in [0, +r_*] }`, so `lambda_J >= c` and `Phi <= C`. Here the slab exclusion of Lemma M' (b) is what keeps `z` away from the +forced zero of `F` at the witness. [RM] §3 (5) gives uniform moments of the conditioned field, so +`E_(Q')[(W_r F_j(H_x))^2]^(1/2) <= C r^2` as in [RM] (10), (14); with `Z_r >= z_* r^2` and the window of length +`k r^3`, + + Z_r^(-1) E_(Q_r) N_j^(W_r Psi^(q-1))( D_(s_0), I_r ) <= C r^3. + +### 6.3 Summary of §6 + +In every regime the marked intensity is at most a constant times the unmarked bound of the source, after absorbing +the two extra factors `(1 + beta)^(3d)` (in place of the source's `(1 + beta)^(3d)`, unchanged in form) and +`(1 + beta + lambda_J^(-d/2))` into the source's own Gaussian penalty. The radial ledgers are untouched: the pin ball +keeps `r^(3-d) omega_d`, the collar `r^(3-d)`, the shells `s^(-(d+2))(r + s^2)^2`, the remote region `r^3` per unit +volume. The finite control `LG` records the exponents of both extra factors and the resulting absorption powers for +`d = 2, ..., 6`. + +## 7. Proof of Theorem Q and the corollaries + +*Theorem Q.* By (3.4), (5.2) and Lemma 5.1 on each region of the tiling, with monotone convergence over compact +punctures of the pins and boundaries assigned once, + + E_(Q_r^W)[ (N)_q ] <= sum_j sum_(regions) Z_r^(-1) E_(Q_r) N_j^(W_r Psi^(q-1))( region, heights ) + <= (d+1) [ 2 C r^3 + C r^3 + C r^3 (4^(-2) + s_0^2) + C r^3 ] = C_q r^3, + +the four terms being the two pin balls (6.3), the collar (R2), the shells (R3, all-height counts dominate window +counts in the first two regions), and the remote region (R4). Every intensity bound of §6 is uniform in `b, k, R, h in +I_r`, and the constants depend on `q` only through Proposition 4.5. ∎ + +*Corollary Theta.* The lower bounds are [EDL] (A4), valid in every fixed `d`. The upper bound on `E N(N-1)` is (1.1) +with `q = 2`, and `Q_r^W{N >= 2} <= E N(N-1)/2`. ∎ + +*Corollary P.* `E_(Q_r^W) N >= Q_r^W(N >= 1) >= Q_r^W(N >= 2) >= c r^3` by [EDL] (A4), and `E N <= C r^3` by [DL] (1.7); +divide (1.2) by these. The size-biased identity `E[(N)_q] = E N . E_sb[(N-1)_(q-1)]` is the definition of `P^sb` +(Math-#146). The two-sided statement is for the second factorial moment only. ∎ + +## 8. Remarks on the mechanism + +- The logarithm of [C6L] came from the pathwise inequality `N(N-1) <= lambda N + Psi^2 1{Psi > lambda}` and the + stretched-exponential tail of `Psi`. Here `Psi` is placed inside the Kac–Rice integrand instead, and only its + conditional second moment is needed. That moment is polynomial in the same degeneracy variables that each region's + Gaussian penalty already absorbs; this is why no logarithm survives. +- The forced zero of `F` at the witness is handled geometrically, by discarding a fixed slab of test radii around + `|X - c_j|` in the packing. Because the retained balls stay at distance `zeta_0/2` from `X`, the residual covariance + of `G(z)` on them has a uniform floor by compactness, and no quantitative expansion of `F` near `X` is needed. +- The improvement route of Math-#140 comment 5893989971 (growing complex radius, `exp(-c m log m)` degree tail) is not + used and not needed here; it would sharpen Lemma M' but the polynomial form (4.8) already suffices for (1.1). +- In `d = 2` the argument uses only the planar sources [PP], [CP], [IW], [RM] through [DL]'s restatement, and [C6L]'s + planar lemmas directly; the dimension enters only through the exponents `m^d`, `epsilon^(2d) t^(-d)`, + `delta^(2d-1)`, `lambda = 2^(m/(4d))`. + +## 9. What this note does not establish + +- Anything about the fixed-`eta` remote pair kernel of `frontiers/remote_collision_20260928/PROOF.md` beyond + consistency: Theorem Q does not reprove its Corollary D (`O(r^5)` pairs with both witnesses remote) and does not + sharpen it; that source is not consumed here. +- The distribution of `N` beyond its factorial moments, or a Poisson or compound-Poisson approximation with rate. +- Elder selection; uniformity in `L`, in `d`, as `k -> 0` or as marks grow; numerical constants; any register, + catalog or status change. The catalog entry C6 and the GRAPH node `math.rn-region.witness-collision` are not moved + by this file; a source-bound reconciliation after nonauthor review of [C6L], [DL] and this note is a separate act. +- Organizational independence: all lanes share one GitHub account. [C6L], [DL] and this note are all Claude-authored, + in two sessions. [C6L] has its nonauthor verdict; [DL] and this note still need theirs before consumption. + +## 10. Finite controls + +`palm_exact_check.py` (standard library, exact rational arithmetic; `-O` mode identical) checks six groups and rejects +seven mutants: + +- `PK` the slab-excluded packing of Lemma M' (b): `zeta_0 <= eta'/8`, the count inequality `eta'/4 - zeta_0 >= delta`, + the retained count at least `eta'/(4 delta)`, every retained ball inside the annulus, and every point of a retained + ball at distance at least `zeta_0/2` from every point at distance `rho_X` from the centre, on grids of `rho_X`, for + `delta = 2^(1-m)` and both radii `eta_0, eta_j`; +- `EX` the exponent arithmetic of Lemma M' for `d = 2, ..., 6`: `lambda^(2d) 2^(-m) = 2^(-m/2)`, `lambda^(-2) = + 2^(-m/(2d))`, and the final exponent `1/(2d)`; +- `TL` the moment series of Proposition 4.5: an explicit index from which the summand ratio is below `2^(-1/(4d))`, + for `d = 2, ..., 6` and `p <= 4`, as an exact rational inequality; +- `PW` the pathwise inequalities (3.4) on integer grids with `Psi >= N + 2`; +- `SC` the Schur-complement floor of Lemma 4.2 and the residual-metric domination of Lemma 4.3 on random rational + covariance matrices (positivity certified by exact `LDL^T`); +- `LG` the regime ledger of §6: for each regime the `r`-exponent is that of [DL], and the absorption powers of R1 II + and R2a are `2n >= 6 + 3d` and `2n/3 >= 15 + 39d`. + +The mutants are `no-slab`, `weak-zeta` (the insufficient ratio `zeta_0 = eta'/4`), `lambda-too-large`, +`series-ratio`, `factorial-power`, `schur-upper`, `forget-mark-power`. +`RESULTS.json` is the exact output. These checks verify counting, exponents, identities and inequalities of finite +matrices only. They do not verify the Gaussian, compactness, regression, Rouché or Kac–Rice steps, which are the +written arguments above and in the sources. + +## 11. Attribution and reconnaissance + +The cap `Psi`, Lemma M and Lemma D are those of [C6L]; the regional frames, densities and penalties are those of the +planar D5 sources as restated in [DL]; the Kac–Rice framework is arXiv:2304.07424v3 as cited by every source. The +contribution of this note is the parametric form of the tail (Lemma M', Proposition 4.5), the slab exclusion, the +identification of the two invariants `mu` and `Phi` with the frame parameters that the first-moment proofs already +control (Lemmas 4.1–4.4, 6.1–6.3), and the assembly through the marked Kac–Rice formula (Lemma 5.1). Nothing is +claimed as new beyond this. `RECONNAISSANCE.md` records the external check and its failure mode. diff --git a/specifications/lean_statements_20261010/sources/c6-rare-cluster/PROOF.md b/specifications/lean_statements_20261010/sources/c6-rare-cluster/PROOF.md new file mode 100644 index 000000000..2ab625ced --- /dev/null +++ b/specifications/lean_statements_20261010/sources/c6-rare-cluster/PROOF.md @@ -0,0 +1,145 @@ +# Rare-cluster consequences of the sharp C6 moment bounds + +**Object:** OA-C6-RARE-CLUSTER-20260929-v1. **Author:** OpenAI / ChatGPT, current owner-authorized session, 29 September 2026. +**Disposition:** author-side conditional proof candidate; nonauthor review OPEN. **Scientific effect: NONE.** No source proof, status register, Boolean, prize or acceptance disposition is changed. Same GitHub account as the other model lanes: no organizational-independence claim. + +## 1. Exact interface and dependency boundary + +Use the count N=N_r under the original determinant-tilted pinned law Q_r^W of [C6, Section 1.1]: all-index critical points, excluding the two conditioned endpoints, with heights strictly between b-k r^3 and b. Fix d>=2, the torus side L, compact birth marks and compact positive gap marks k>=k_->0. The source covariance, conditioning, full normalizer and frames are unchanged. All constants below inherit this scope; none is numerical or uniform as d,L or the marks escape it. + +Put e=r^3. The ONLY analytic inputs to this note are + + (H1) E N <= A e, + (H2) P(N>=2) >= a e, where a>0, + (Hq) E (N)_q <= A_q e for every fixed integer q>=2. + +Here (N)_q=N(N-1)...(N-q+1), and A_1=A. [D5, Theorem G_d], summed over the fixed d+1 indices, gives H1. [C6, Corollary Theta] gives H2, and [C6, Theorem Q] gives Hq. Thresholds may be decreased for each finite collection of moment orders; no uniform-in-q assertion is needed. These premises imply 2a<=A. + +Both sources are pinned to commit 78ac1100217279e7b692f0959f86eee34cd1910e, with full byte identities in SOURCE_PINS.json. Their recorded nonauthor reviews do not constitute a review of this new note. The parent Gaussian proofs are not revalidated here. A defect in an input blocks the corresponding application, not the elementary abstract implication. + +[C6] frontiers/c6_palm_route_20260929/PROOF.md, Theorem Q and Corollary Theta. +[D5] frontiers/d5_dimension_lift_20260929/PROOF.md, Theorem G_d. +[LOCAL] frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md, Section 7, already proves the exact-mean Bernoulli-configuration obstruction. That result is prior in-project work, not a new claim of this note. + +For probability laws, dTV is the supremum over measurable events, equivalently half the l1 distance on this countable state space. Norms of finite or signed measures written ||.||_1 mean full l1, without the factor 1/2. + +## 2. Conditional counts and size-biased counts are different + +Write p=P(N>0), mu=E N, let J have law F=Law(N | N>0), and let Y have size-biased law P(Y=n)=n P(N=n)/mu. Then + + a e <= p <= A e, 2a e <= mu <= A e. (2.1) + +For each integer k>=1, define B_k=sum_{j=1}^k S(k,j) A_j, where S(k,j) are the nonnegative Stirling numbers of the second kind. The pointwise identity n^k=sum_j S(k,j)(n)_j gives + + E N^k <= B_k e, + E J^k <= B_k/a, E Y^k <= B_{k+1}/(2a). (2.2) + +Consequently both families are tight and every fixed power is uniformly integrable (use the next power in Markov's tail bound). Moreover + + P(J>=2) >= a/A, + E J-1 >= a/A, + 2a/A <= E(Y-1)=E(N)_2/mu <= A_2/(2a). (2.3) + +The exact difference between the two excess means is + + E(Y-1) - E(J-1) = Var(J)/E J >= 0. (2.4) + +Indeed P(Y=n)=n F(n)/E J, so E Y=E J^2/E J. This proves (2.4). It does not assert that the two sampling laws are equal: J taking 1 and 3 with probabilities 1/2 each has mean excess 1, while Y has mean excess 3/2. The new use of all-order C6 bounds is (2.2) and its compactness consequences, not merely the distinction already stated in C6's Corollary P. + +## 3. Sharp ordinary-Poisson obstruction, including optimized intensity + +**Theorem O.** Under H1 and H2 alone, for all sufficiently small r, + + (a/2) r^3 <= inf_{lambda>=0} dTV(Law N, Pois(lambda)) <= A r^3. (3.1) + +The same order holds for lambda=mu. Thus an ordinary Poisson law cannot achieve o(r^3) error, even by fitting an arbitrary intensity. Absolute distance still tends to zero; this is an obstruction at the rare-event scale, not a claim that the distance is bounded away from zero. + +**Proof.** Take e<=min{a/(4A^2),1/(4A)}. If lambda<=2Ae, then + + P(Pois(lambda)>=2) <= lambda^2/2 <= 2A^2 e^2, + +by the second factorial moment of a Poisson variable. Testing {n>=2} gives distance at least ae/2. If lambda>2Ae, test {n>0}; monotonicity and 1-exp(-x)>=x-x^2/2 give + + 1-exp(-lambda)-p >= 1-exp(-2Ae)-Ae + >= Ae-2A^2 e^2 >= Ae/2 >= ae/2. + +The upper bound in (3.1) uses Pois(0)=delta_0 and dTV(Law N,delta_0)=p. For mean matching, the triangle inequality through delta_0 gives at most p+1-exp(-mu)<=2Ae, and the same lower bound applies. This proves the theorem. Notice that Hq is not needed. + +Conditioning a mean-matched Poisson variable on positivity does not cure the discrepancy: its probability of being >=2 tends to zero as mu->0, whereas F([2,infinity))>=a/A. Therefore the conditional TV discrepancy has liminf at least a/A. + +## 4. Canonical r-dependent compound Poissonization: sharp r^6 error + +Let C_r have the compound Poisson law CP(p,F): K~Pois(p), independent positive integer marks J_1,J_2,... with law F, and C_r=sum_{i=1}^K J_i. The empty sum is zero. + +**Theorem C.** Under H1 and H2, for p<=1, + + p^2/3 <= dTV(Law N, Law C_r) <= p(1-exp(-p)) <= p^2. (4.1) + +In particular the error for THIS approximant is Theta(r^6), with bounds a^2 r^6/3 and A^2 r^6. Its mean is exactly mu. + +**Proof.** Law N=(1-p)delta_0+pF. Apply the same mark-summing probability kernel to a Bernoulli(p) count and a Pois(p) count. Total variation contracts under that kernel. The Bernoulli law has its sole positive difference at count 1, of mass p-p exp(-p), so their TV distance is exactly p(1-exp(-p))<=p^2. Positivity of all marks means P(C_r=0)=exp(-p). Testing the zero event gives + + dTV >= exp(-p)-1+p >= p^2/2-p^3/6 >= p^2/3. + +The Taylor inequalities follow from their signed integral remainders on p>=0. Finally E C_r=p E J=E N. + +**Essential limitation.** F is the actual, generally unknown, r-dependent conditional count law. This is a universal re-expression/Poissonization of a rare occurrence, not an explicitly identified cluster law, an asymptotic independence theorem, or a spatial Poisson-cluster process. No novelty is asserted for this elementary generic probability construction. The cluster rate is p, NOT mu: replacing it by mu changes the mean to mu E J in general. The r^6 lower bound is for CP(p,F), not a minimax lower bound over every conceivable compound Poisson parameterization. + +## 5. Factorial-moment matching to second rare-event order + +Let a_j=E(N)_j. Under Hq, for each fixed integer q>=1, + + E(C_r)_q = sum_{pi in Partitions({1,...,q})} product_{B in pi} a_{|B|}. (5.1) + +For q=1 this equals mu. For q>=2, + + 0 <= E(C_r)_q-E(N)_q = O_q(e^2)=O_q(r^6). (5.2) + +**Proof.** Conditional on K, expand the ordered selections of q distinct items in the sum of its independent integer-valued marks. A set partition of the q labels specifies which labels belong to the same mark; if it has j blocks its expectation is E(K)_j times the product of the mark factorial moments. Since E(K)_j=p^j and p E(J)_b=a_b, summing gives (5.1). All terms are nonnegative and moments finite, so Tonelli applies. No moment-generating-function analyticity is assumed. The single-block partition contributes a_q. Every other partition has at least two blocks; each a_b=O(e), and there are finitely many partitions for fixed q, proving (5.2). In particular the second-moment correction is exactly mu^2. + +## 6. Subsequential finite cluster-intensity measures, not a unique limit + +Define a finite measure on the positive integers by + + nu_r(n)=r^(-3) P(N_r=n), lambda_r=nu_r(N_+)=p/r^3. + +Then a<=lambda_r<=A, nu_r({2,3,...})>=a, and sum n^k nu_r(n)<=B_k for k>=1 (use B_0=A for mass). + +**Theorem S.** Every sequence r_i->0 has a subsequence and a finite measure nu on N_+ such that for every fixed nonnegative integer k, + + sum_{n>=1} n^k |nu_{r_i}(n)-nu(n)| -> 0. (6.1) + +Its mass lambda is in [a,A], and nu({2,3,...})>=a. Along that subsequence the positive conditional laws converge to nu/lambda in TV and all fixed moments. The size-biased laws converge to n nu(n)/sum_m m nu(m), also in TV and all fixed moments. + +**Proof.** Each coordinate is bounded by A. Diagonal selection gives coordinatewise convergence. Fatou preserves the moment bounds. For every k>=0 and M>=1, + + sum_{n>M} n^k nu_r(n) <= B_{k+1}/M, + +and the same holds for nu. On the finite set n<=M coordinatewise convergence gives weighted l1 convergence. First take the subsequence limit and then M->infinity. The SAME selected subsequence works for every k because it was selected only by coordinates and has all these tail bounds. Mass and the multiple-count lower bound pass to the limit by k=0 convergence. Denominators for the two normalized laws stay bounded below by a and 2a, respectively, so their claims follow from (6.1), using one extra power for size bias. + +There is also the exact signed-measure identity + + Law N_r = delta_0 + r^3 (nu_r-lambda_r delta_0). (6.2) + +Thus (6.1) gives a first-order signed-measure expansion along each selected subsequence. It gives neither uniqueness nor a rate for nu_r->nu. + +**Sharp non-implications.** Set P(N_r>0)=r^3 and let the positive count alternate between 2 and 3 on successive dyadic intervals of r. H1,H2 and every Hq hold, but nu_r alternates between delta_2 and delta_3. Taking N_r=2 Bernoulli(r^3) also shows that no positive lower bound for the third or higher factorial moments follows. Finally, all polynomial moments need not imply any exponential moment: let P(J=2^j) be proportional to 2^(-j^2), j>=1. Every polynomial moment is finite, whereas E exp(tJ)=infinity for each t>0. The rare mixture with probability r^3 satisfies H1,H2,Hq. These examples concern what the premises alone imply, not counterexamples to the specific Gaussian model having additional structure. + +## 7. Independent-replica limit, with an explicit extra assumption + +Take m_r=floor(t/r^3) INDEPENDENT copies of the entire count N_r, for fixed t>=0, and let T_r be their sum. Along a subsequence from Theorem S, + + dTV(Law T_r, CP(t nu)) + <= t A^2 r^3 + A r^3 + t ||nu_r-nu||_1 -> 0. (7.1) + +Here CP(t nu) denotes the compound Poisson law with finite Levy measure t nu, with no jump of size zero. + +**Proof.** Couple each copy to its Section 4 approximant: the total error is at most m_r p^2<=t A^2 r^3. The sum of the independent approximants has Levy measure m_r r^3 nu_r. Two finite Levy measures eta,zeta can be coupled by sharing their pointwise minimum and independent residual Poisson jumps; their compound count laws have TV distance at most ||eta-zeta||_1. Apply + + ||m_r r^3 nu_r-t nu||_1 <= r^3 A + t ||nu_r-nu||_1. + +No independence of windows or spatial cells in ONE field has been proved. Such a use needs a new dependence estimate. Equation (7.1) is a theorem about deliberately independent experiments, not a new spatial claim about the original torus. + +## 8. Review and verification boundary + +Requested nonauthor slices: (A) optimized-intensity Poisson bound and exact binary-kernel constants; (B) factorial partitions, weighted-l1 compactness and the independent-replica coupling; (C) source interface and non-claims. The checker covers exact finite identities, rational probes and explicit false variants only. Neither it nor a green workflow proves H1/H2/Hq, validates all infinite-dimensional arguments, or promotes any register. There is no Lean formalization of this packet. External primary-source reconnaissance is recorded separately, without claiming a comprehensive novelty audit. diff --git a/specifications/lean_statements_20261010/sources/cumulative-transfer-correction/CUMULATIVE_TRANSFER_CORRECTION.md b/specifications/lean_statements_20261010/sources/cumulative-transfer-correction/CUMULATIVE_TRANSFER_CORRECTION.md new file mode 100644 index 000000000..044ac5fda --- /dev/null +++ b/specifications/lean_statements_20261010/sources/cumulative-transfer-correction/CUMULATIVE_TRANSFER_CORRECTION.md @@ -0,0 +1,56 @@ +# Correction to the cumulative transfer statement (D2) + +**Author:** OpenAI / ChatGPT. **Scientific effect:** NONE. Separate reviewer confirmation requested. + +This additive correction replaces only the cumulative paragraph and its proof in Section D of NOTE.md at source ad35e46d15c2815c36746442808a1626a9724e8a (SHA256530dd3efaa965c850ea9e6575f42d3952c3efe6a89b2b5355b6afb4285c9d37e). The original source stays unchanged for provenance. Sections B/C and the density theorem D1 are not changed. + +Nonauthor review PR32, commit189628119f65d845e270759e1b2b1e84ddb4cfe3, correctly found that saying only 'monotonicity and the same comparison/envelope assumptions' could drop the necessary normalization h/(kappa*r^m)->1. Two-sided comparisons do not determine the sharp constant. The correction explicitly retains that ratio hypothesis while dropping only derivative assumptions. + +## Corrected cumulative theorem + +Let (Z,lambda) be sigma-finite, alpha>-1, m>0, beta=(alpha+1)/m, and r0>0. Let all integrands be jointly measurable. Assume, for lambda-almost every z: + +1. kappa(z)>0; 0<=B(r,z)<=G(z); B(r,z)->B0(z) as r->0. +2. The weighted negative moment integral G(z)*kappa(z)^(-beta) dlambda is finite. +3. h(r,z) is continuous and strictly increasing for 0 1. + +4. Uniformly in r,z, h(r,z)>=c0*kappa(z)*r^m for a fixed c0>0. + +Neither a derivative limit nor a lower derivative bound is required for this cumulative theorem. Define + + N(ell)=integral_Z integral_0^r0 r^alpha B(r,z) 1{h(r,z)<=ell} dr dlambda. + +Then + + lim_(ell->0) N(ell)/ell^beta + = 1/(alpha+1) * integral_Z B0(z)*kappa(z)^(-beta) dlambda. + +If the displayed coefficient is positive, this limit is a positive asymptotic equivalence. If it is zero, only the corresponding o(ell^beta) conclusion is asserted. + +## Proof + +At each fixed z and sufficiently small ell, let R=R(ell,z) be the inverse of h. The explicitly retained ratio hypothesis gives + + R^(alpha+1) / (ell/kappa(z))^beta -> 1. + +Also, + + R^(-alpha-1) integral_0^R r^alpha B(r,z) dr + -> B0(z)/(alpha+1), + +by convergence of B at zero and alpha>-1. Therefore the normalized inner integral converges to B0(z)*kappa(z)^(-beta)/(alpha+1). + +For domination, the lower comparison implies h(r,z)<=ell only if r<=(ell/(c0*kappa(z)))^(1/m). Thus, also when the inverse would exceed r0, + + ell^(-beta) integral_0^r0 r^alpha B(r,z)1{h<=ell}dr + <= c0^(-beta) G(z)*kappa(z)^(-beta)/(alpha+1). + +This is integrable by assumption2. Dominated convergence proves the result. + +## Why the added precision matters + +For h(r)=2*kappa*r^m with constant amplitude1, the actual coefficient is multiplied by2^(-beta). This example satisfies suitable fixed two-sided comparison bounds but violates h/(kappa*r^m)->1. For alpha=1,m=3,kappa=1 and R=1/2, ell=1/4 and N=1/8: the actual cube is1/512, whereas the uncorrected candidate constant (ell^(2/3)/2)^3 is1/128. The ratio hypothesis excludes precisely this mismatch. + +This corrects an underspecified theorem statement; it is not an excuse to infer a density asymptotic from a cumulative asymptotic. D1 still requires its separate derivative controls. The Gaussian contact, thin-belt, whole-annulus, and global persistence statuses are untouched. diff --git a/specifications/lean_statements_20261010/sources/d5-dimension-lift/PROOF.md b/specifications/lean_statements_20261010/sources/d5-dimension-lift/PROOF.md new file mode 100644 index 000000000..9d82c707f --- /dev/null +++ b/specifications/lean_statements_20261010/sources/d5-dimension-lift/PROOF.md @@ -0,0 +1,601 @@ +# D5 in every fixed dimension: punctured pin balls, the compact collar, intermediate height-window shells, and the global window first moment + +**Object:** CL-D5-DIMENSION-LIFT-20260929-v1. +**Author:** Anthropic Claude (Claude Code session `session_015wNj8LPTKXsaT68G3DgPPh`), 29 September 2026. +**Disposition:** author-side proof candidate. **Nonauthor analytic review is required.** +**Scientific effect:** NONE. No `STATUS`, `PROOF_INDEX` verdict, `GRAPH` node, claim, `lemma_closed` flag, prize or source body changes. + +## 0. Why this note exists + +Every D5 estimate of the last three days is planar. The punctured endpoint disk [PP], the compact collar [CP], the intermediate shells and the global first moment (I5) [IW], and the planar halves of Corollary F [RC] and Theorem L [LOCAL] all fix `d = 2`. The fixed-remote Theorem A [RM] and the lower bounds of [RC] (F-) and [EDL] (A4) hold in every fixed `d`. [RC] states the consequence exactly: for `d > 2` the matching upper bound on the window event "stays conditional on a dimension-matched global first moment", and no source supplies one. The program's own application is `d = 3` (SIDE24, `L = 24` in [LP] §1). + +This note supplies that first moment in every fixed `d >= 2`. It lifts the three planar D5 components and their composition. The planar proofs are followed step by step; every step that is not literally dimension-free is replaced here by a written `d`-dimensional argument. Five devices do all the new work (§3). Nothing else changes: the model, the pins, the weight, the normalizer, the Kac–Rice framework and the regression estimates are those of the sources. + +## 1. Setting and results + +### 1.1 Model + +Fix `d >= 2`, `m = d - 1`, and `L > 0`. `f` is the centered variance-one Gaussian field on `X = R^d/(L Z^d)` with the exact normalized periodized Gaussian covariance of [LP] §1. Fix compact `B = [b_-, b_+]` and `K = [k_-, k_+]` with `0 < k_-`. Frames `R` range over `O(d)`; `u = R e_1` is the axial direction and `y in R^m` are the transverse coordinates. For small `r > 0` the pins are + + M = -(r/2) u, S = (r/2) u, f(M) = b, f(S) = b - k r^3, grad f(M) = grad f(S) = 0. + +`Q_r` is Gaussian regression on these `2(d+1)` observations (not an adjacency or elder law). For a symmetric matrix `H`, `F_j(H) = |det H|` if `H` is nonsingular with `j` negative eigenvalues, and `0` otherwise. Retain the original weight, the full normalizer and the tilted law + + W_r = F_d(H_M) F_(d-1)(H_S), Z_r = E_(Q_r) W_r, dQ_r^W = (W_r/Z_r) dQ_r, + +and the between-pin height window `I_r = (b - k r^3, b)`. `N_j(B)` counts index-`j` critical points of `f` in a Borel set `B` at all heights; `N_(r,j)(B)` counts those with height in `I_r`. Constants `C, c, r_*, s_0` depend on `d, L, B, K` and on the fixed chart radii named in each statement; never on `r, s, b, k, R, j` or the Borel set. No constant is evaluated numerically. Nothing is uniform as `k -> 0`, as marks grow, or as `d` or `L` vary. + +Throughout, `K` also denotes a fixed multiple of `1 + ||f||_(C^6(X))` (the sources use the same letter for the mark interval; the meaning is always clear from context), and `L <= K` a Hessian Lipschitz constant on the relevant chart. + +### 1.2 Statements + +Write a point of the scaled chart around `M` as `X = M + r (p, q)` with `p in R`, `q in R^m`, and put + + D = { (p, q) : 0 < p^2 + |q|^2 <= 1/16 }, + omega_d(p, q) = max(|q|, r |p|)^(2-d) (omega_2 = 1). (1.1) + +**Theorem P_d (all-height punctured pin ball).** There are `C, r_* > 0` such that for `0 < r <= r_*`, all `b, k, R, j` and every Borel `E` contained in `D`, + + E_(Q_r^W) N_j(M + r E) <= C r^3 integral_E omega_d(p, q) dp dq <= C' r^3. (1.2) + +The same holds for the punctured ball about `S` (§4.9). For a nested ball `p^2 + |q|^2 <= kappa^2 r^2` (physical radius `kappa r^2`), the right side is `O(r^5)`. For `d = 2` this is [PP] (P2). + +**Theorem C_d (compact collar).** In midpoint scaled coordinates `X = r (u, v)`, `u in R`, `v in R^m`, fix `R >= 1` and `0 < eta <= 1/4`, and let `C(eta, R)` be the closed ball of radius `R` with the two open balls of radius `eta` about `(-1/2, 0)` and `(1/2, 0)` removed. There are `C, r_* > 0` such that for every Borel `E` contained in `C(eta, R)`, + + E_(Q_r^W) N_j(r E) <= C r^3 |E|. (1.3) + +Consequently, for every Borel subset `E` of the scaled ball of radius `R` with the two pins removed, + + E_(Q_r^W) N_j(r E) <= C r^3 (1 + integral_(E cap pin balls) omega_d) <= C' r^3. (1.4) + +For `d = 2` these are [CP] (C1), (C2). + +**Theorem I_d (intermediate height-window shells).** There are `C, s_0 > 0` such that for `0 < r <= s/4` and `0 < s <= s_0`, + + E_(Q_r^W) N_(r,j)({ s <= |X| <= 2 s }) <= C r^3 [ (r/s)^2 + s^2 ], (1.5) + +and hence, uniformly for `A_0 >= 4` and `A_0 r < rho <= s_0`, + + E_(Q_r^W) N_(r,j)({ A_0 r <= |X| <= rho }) <= C r^3 ( A_0^(-2) + rho^2 ). (1.6) + +For `d = 2` these are [IW] (I3), (I4). + +**Theorem G_d (global height-window first moment).** There are `C, r_* > 0` such that + + E_(Q_r^W) N_(r,j)( X minus {M, S} ) <= C r^3. (1.7) + +For `d = 2` this is [IW] (I5). + +**Corollary F_d.** Let `A_r` be the event that some critical point other than `M, S` has height in `I_r`. In every fixed `d >= 2`, + + c r^3 <= Q_r^W(A_r) <= C r^3. (1.8) + +The lower half is [RC] Corollary (F-), which holds in every fixed `d`. The upper half is Markov's inequality applied to (1.7), summed over the `d+1` indices. This discharges the sentence in [RC] §1 that leaves the `d > 2` upper half conditional. + +**Corollary M_d (window multiplicity in every dimension).** Let `N_r` count all window critical points other than `M, S`. In every fixed `d >= 2`, + + c r^3 <= Q_r^W{ N_r >= 2 } <= C r^3, E_(Q_r^W)[ N_r (N_r - 1) ] >= c r^3. (1.9) + +The lower bounds are [EDL] (A4). The upper bound is `1{N >= 2} <= N/2` and (1.7). In particular a torus-wide `O(r^5)` second factorial moment is excluded in every `d`, the global window point process is at total-variation distance `Theta(r^3)` from its exact-mean Bernoulli approximation (the elementary identity of [LOCAL] §7 is dimension-free), and no upper bound on the second factorial moment is claimed. + +### 1.3 What the results are not + +They are first moments and event probabilities. They are not an all-height intermediate or global count, a torus-wide second factorial moment, a shrinking-separation collision estimate outside the fixed remote region, an elder-selection statement, a numerical constant, a 24-jet certificate, or a status transition. §8 lists the non-claims. + +## 2. Sources consumed, with the exact interface + +| Tag | Source (identities in `SOURCE_MAP.json`) | What is consumed | +|---|---|---| +| [LP] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | §2 positive Fourier spectrum and distinct-site jet rank; §3 contact frame `U_r`, uniform regression; (4.1) uniform `C^q` moments of every order under `Q_r`; (5.1) `|alpha_i|, ||beta_i|| <= M_3/2`; (5.3) `det H_i / r = alpha_i det A_i - r beta_i^T adj(A_i) beta_i`; (5.5) `Z_r >= z_* r^2`. All in every fixed `d`; accepted as A1–A4 in the D1 reconciliation. | +| [PP] | `reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md` | The planar template for Theorem P_d: (P3)–(P20). | +| [CP] | `reviews/d5_local_collar_20260928/COLLAR_PROOF.md` | The planar template for Theorem C_d: (C3)–(C20). | +| [IW] | `frontiers/intermediate_window_20260928/PROOF.md` | The planar template for Theorems I_d, G_d: (I6)–(I34). | +| [RM] | `frontiers/remote_window_20260924/PROOF.md` | Theorem A in every `d`, reviewed (main#76): the outer tile in §7. | +| [RC] | `frontiers/remote_collision_20260928/PROOF.md` | (F-) lower bound; the conditional sentence for `d > 2`. | +| [EDL] | `frontiers/elder_dimension_lift_20260928/PROOF.md` | (A4) `Q^W(N_r >= 2) >= c r^3` in every `d`. | +| [LOCAL] | `frontiers/window_multiplicity_laws_20260928/LOCAL_MULTIPLICITY.md` | The total-variation identity of §7 only. | + +External framework: Armentano, Azaïs and León, arXiv:2304.07424v3, Theorems 2.1, 2.2 and 7.1 with Remarks 7–8, exactly as in [LP] §9, [RM] §5, [PP] §9, [CP] §9 and [IW] §9. The zero-counted field is `grad f : X -> R^d`; parameter and value dimensions agree. + +Planar steps that are dimension-free are cited by their source label and not reproved. A reviewer should read each cited step in its source; this note is not self-contained without them. + +## 3. Five devices + +The planar proofs use, at several places, the two-dimensional facts that a `2 x 2` matrix with two small columns has a small determinant, that a `2 x 2` symmetric matrix is a scalar plus a small correction, and that a bounded intensity integrates to `O(r^3)` over a disk of area `r^2`. The following replace them. + +**Lemma D1 (two soft directions).** Let `H` be a real `d x d` matrix with `||H||_op <= K`, and let `e, e'` be unit vectors with `sigma = ||e wedge e'|| > 0` (the sine of their angle). Then + + |det H| <= ||H e|| ||H e'|| K^(d-2) / sigma, (3.1) + +and every unit vector `w` in `span(e, e')` satisfies `||H w|| <= ( ||H e|| + ||H e'|| ) / sigma`. + +*Proof.* Let `0 <= sigma_1 <= ... <= sigma_d` be the singular values of `H`. Then `|det H| = prod sigma_i <= sigma_1 sigma_2 K^(d-2)`. The singular values of the second exterior power `Lambda^2 H` are the products `sigma_i sigma_j`, `i < j`, so its smallest singular value is `sigma_1 sigma_2`. Apply it to the unit decomposable 2-vector `omega = (e wedge e')/sigma`: `sigma_1 sigma_2 <= ||Lambda^2 H omega|| = ||H e wedge H e'|| / sigma <= ||He|| ||He'|| / sigma`. For the second claim write `w = a e + b e'`; in the plane, Cramer's rule gives `a = ||w wedge e'|| / ||e wedge e'||`, so `|a| <= 1/sigma`, and likewise `|b| <= 1/sigma`. ∎ + +For `d = 2` and `e = e_x`, (3.1) is the planar two-column bound of [PP] §8.1 and [CP] (C18). The finite control `X` checks `det(H)^2 ||u_1 wedge u_2||^2 <= ||Hu_1 wedge Hu_2||^2 ||H||_F^(2(d-2))` on random rational matrices and the equality case on a diagonal family. + +**Lemma D2 (one soft direction).** For a unit vector `e`, `|det H| <= ||H e|| K^(d-1)`. *Proof.* Hadamard's inequality in an orthonormal basis containing `e`; or `sigma_1 <= ||He||`. ∎ + +**Lemma D3 (jet functionals on `R^m`).** Let `beta in R^m`. Coordinates on symmetric matrices, symmetric 3-tensors, and vectors are their independent entries. + +- The linear map `S -> S beta`, `Sym_m -> R^m`, has Gram matrix `G_beta` (over independent entries) with `G_beta >= (|beta|^2 / 2) I_m`. +- The functional `C -> (1/2) beta^T C beta` on `Sym_m` has squared norm `>= |beta|^4 / 4`. +- The functional `D -> D[beta, beta, beta] / 12` on symmetric 3-tensors has squared norm `>= |beta|^6 / 144`. + +*Proof.* For `xi in R^m`, `xi . S beta = sum_i xi_i beta_i S_ii + sum_(i= (1/4) sum_(i,j) (xi_i beta_j + xi_j beta_i)^2 + = (1/4) || xi beta^T + beta xi^T ||_F^2 = (1/2)( |xi|^2 |beta|^2 + (xi . beta)^2 ) >= (1/2) |beta|^2 |xi|^2. + +For the other two, an independent entry with multiplicity `mu in {1, 2}` (resp. `{1, 3, 6}`) carries the coefficient `mu beta_i beta_j / 2` (resp. `mu beta_i beta_j beta_k / 12`); since `mu^2 >= mu`, the squared norm is at least `(1/4) sum_(all ordered) (beta_i beta_j)^2 = |beta|^4/4` (resp. `|beta|^6 / 144`). ∎ + +The finite control `N` checks all three on random rational vectors. + +**Lemma D4 (transverse block determinant).** Let `S in Sym_m`, `hat v` a unit vector, `epsilon = hat v^T S hat v`, `w = S hat v - epsilon hat v` (so `w` is orthogonal to `hat v`), and `S'` the compression of `S` to `hat v^perp`. Then + + det S = epsilon det S' - w^T adj(S') w, |det S| <= |epsilon| ||S||^(m-1) + |S hat v|^2 ||S||^(m-2), (3.2) + +with `det S' = 1`, `adj(S') = 0` when `m = 1`, and `adj(S') = 1` when `m = 2`. + +*Proof.* In an orthonormal basis beginning with `hat v`, `S = [[epsilon, w^T], [w, S']]`, and the first identity is the cofactor expansion (Schur's formula when `S'` is invertible, and by continuity otherwise). Then `||adj(S')||_op <= ||S'||^(m-2)` and `|w| <= |S hat v|`. ∎ + +The finite control `TB` checks the identity and the bound on random rational matrices for `m = 2, 3, 4`. + +**Lemma D5 (pin weight).** For `d >= 2` and `0 < a <= 1/4`, + + integral_(|q| <= a) |q|^(2-d) dq = |S^(m-1)| a (with |S^0| = 2), hence + integral_D omega_d <= (1/2) |S^(m-1)| (1/4), integral_(p^2+|q|^2 <= kappa^2 r^2) omega_d <= 2 kappa r |S^(m-1)| kappa r. + +*Proof.* Polar coordinates in `R^m`: the radial exponent is `(2-d) + (m-1) = 0`. Since `omega_d <= |q|^(2-d)`, both bounds follow. ∎ + +The finite control `W` checks the radial exponent and the nested-ball count exponent `3 + 2 = 5` for `d = 2, ..., 6`. + +## 4. Proof of Theorem P_d + +We follow [PP] §§3–10. Coordinates: translate so that `M = 0` and `S = (r, 0)`; `X = (rp, rq)`, `|p|, |q| <= 1/4`, `(p, q) != 0`. + +### 4.1 Uniform endpoint regression ([PP] §3) + +Let `g(x) = f(x, 0)` and `h(x) = grad_y f(x, 0) in R^m`. The unconditional observation vector + + E_r = ( g(0), g'(0), h(0), [g'(r) - g'(0)]/r, (12/r^3)[ g(r) - g(0) - (r/2)(g'(r) + g'(0)) ], [h(r) - h(0)]/r ) + +has `2 + m + 1 + 1 + m = 2(d+1)` coordinates, is invertibly equivalent to the raw pins for each `r > 0`, has target `e = (b, 0, 0, 0, -12k, 0)` independent of `r`, and contact limit `E_0 = (f, f_x, grad_y f, f_xx, -f_xxx, grad_y f_x)(0)`. Taylor's integral formula gives `E_r - E_0 = O_(L^s)(r)` for every finite `s`, uniformly over frames and locations. The contact functionals are distinct one-site derivative monomials, so [LP] §2 gives positive definite covariance; compactness of `O(d)` makes the floor uniform. The regression coupling `f_(Q_r) = f + Cov(f, E_r) Cov(E_r)^(-1) (e - E_r)` and [LP] (4.1) give, for every fixed `n, s`, uniformly bounded `s`-th moments of the `C^n` norm on the whole torus. + +Define the midpoint jets (all at `0 = M`) + + J = ( A_4, T_3, S_0, C_3 ) = ( f_xxxx, grad_y f_xx in R^m, D_y^2 f in Sym_m, D_y^2 f_x in Sym_m ). + +Their independent entries are distinct derivative monomials of orders `4, 3, 2, 3`, none in `E_0`. So the covariance of `J` after conditioning on `E_r` is uniformly sandwiched and its mean is bounded: a Schur-complement statement for the exact field, as in [PP]. + +### 4.2 Relative remainders ([PP] §4) + +[PP] (P4) is a statement about the axial function `g` alone and is unchanged: + + g'(rp) = 6 k r^2 p(p-1) + (r^3 A_4 / 12) p(p-1)(2p-1) + O(r^4 |p| K). + +Since `h(0) = h(r) = 0` and `h''(0) = T_3`, + + h(rp) = (r^2/2) p(p-1) T_3 + O(r^3 |p| K), h'(rp) = r (p - 1/2) T_3 + O(r^2 K), + +componentwise, by the same two-point Hermite argument. Expanding in `y = rq`, + + f_x(X)/r^2 = 6k p(p-1) + (r A_4/12) p(p-1)(2p-1) + (p - 1/2) q . T_3 + (1/2) q^T C_3 q + epsilon_1, + grad_y f(X)/r = (r/2) p(p-1) T_3 + S_0 q + epsilon_2, (4.1) + +with the pathwise bounds + + |epsilon_1| <= C K ( r^2 |p| + r |q| + r |p| |q|^2 + r |q|^3 ), + ||epsilon_2|| <= C K ( r^2 |p| + r |p| |q| + r |q|^2 ), (4.1') + +which are [PP] (P7) with `|q|` the Euclidean norm of `q in R^m`. Since (P7) is written for two scalars, the +`d`-dimensional remainders are derived here rather than cited. Let `h(x) = grad_y f(x, 0) in R^m`, so `h(0) = h(r) = 0` +and `h''(0) = T_3`. Every bound below is pathwise, `K` dominating the derivatives of `f` through order five on the +chart. + +- *Axial rows.* [PP] (P4) concerns the scalar `g(x) = f(x, 0)` alone and is unchanged: + `g'(rp) = 6 k r^2 p(p-1) + (r^3 A_4/12) p(p-1)(2p-1) + O(r^4 |p| K)`. +- *Transverse rows along the axis.* Componentwise, `h_i(0) = h_i(r) = 0` gives `h_i(x) = x(x - r) h_i''(xi_i)/2` with + `xi_i` in the convex hull of `{0, r, x}`, hence `|xi_i| <= r` for `x = rp`, `|p| <= 1/4`, and + `h(rp) = (r^2/2) p(p-1) T_3 + O(r^3 |p| K)`. For the derivative, `h(r) = 0` and Taylor's formula at `0` give + `h'(0) = -(r/2) T_3 + O(r^2 K)`, so `h'(rp) = r (p - 1/2) T_3 + O(r^2 K)`. Both are vector statements: each + component obeys the one-dimensional estimate and the norm of a vector is at most `sqrt(m)` times its largest + component. +- *Transverse Taylor expansion.* With `y = r q`, expand in `y` about `y = 0` at fixed `x = rp`: + + f_x(X) = g'(rp) + y . h'(rp) + (1/2) y^T D_y^2 f_x(rp, 0) y + R_3, |R_3| <= (1/6) |y|^3 sup ||D_y^3 f_x|| <= C K r^3 |q|^3, + grad_y f(X) = h(rp) + D_y^2 f(rp, 0) y + R_2, ||R_2|| <= (1/2) |y|^2 sup ||D_y^3 f|| <= C K r^2 |q|^2, + + the suprema over the segment from `(rp, 0)` to `X`, and `D_y^2 f_x(rp, 0) = C_3 + O(r |p| K)`, + `D_y^2 f(rp, 0) = S_0 + O(r |p| K)` by the Lipschitz bound over the axial distance `r|p|`. +- *Collecting.* Insert the four expansions and divide by `r^2` and `r` respectively: + + epsilon_1 = O(r^2 |p| K) + O(r |q| K) + O(r |p| |q|^2 K) + O(r |q|^3 K), + epsilon_2 = O(r^2 |p| K) + O(r |p| |q| K) + O(r |q|^2 K), + + where the four terms of `epsilon_1` come from the axial remainder, `r q . O(r^2 K)/r^2`, `(1/2) q^T O(r|p|K) q` and + `R_3/r^2`, and the three terms of `epsilon_2` from the axial remainder `O(r^3|p|K)/r`, `O(r|p|K) q` and `R_2/r`. + This is (4.1'). The only place the dimension enters is the constant, through `sqrt(m)` and the operator norms of + the tensors `C_3`, `S_0`, `D_y^3 f`; the powers of `r`, `|p|`, `|q|` are those of (P7). + +The ratios used in §4.3, `r^2|p|/Delta <= r`, `r|q|/Delta <= r`, `r|p||q|^2/Delta <= r|p||q|`, `r|q|^3/Delta <= r|q|^2`, +`r|p||q|/Delta <= r|p|`, `r|q|^2/Delta <= r|q|`, all follow from `Delta >= |q|` and `Delta >= r|p|`, exactly as in the +planar continuum record `reviews/d5_punctured_pin_continuum_claude_20260929/REVIEW.md` (its note n2). The finite +control `RO` checks the expansions (4.1) exactly on a pinned cubic in `d = 3` (there the remainders vanish). + +### 4.3 A stable gradient frame on the whole punctured ball ([PP] §5) + +Set `Delta = (|q|^2 + r^2 p^2)^(1/2) > 0`, `alpha = r p / Delta in [-1, 1]`, `beta = q / Delta in R^m`, so `alpha^2 + |beta|^2 = 1`, and + + Y = ( f_x(X) / (r^2 Delta), grad_y f(X) / (r Delta) ) in R^d. + +From (4.1), + + Y = ( 6k p(p-1) / Delta, 0 ) + B J + R, ||R||_(L^s(Q_r)) <= C_s r, (4.2) + +where the `d x (independent entries of J)` matrix `B = B(p, alpha, beta, Delta)` acts by + + row 1: a(p) alpha A_4 + c(p) beta . T_3 + (Delta/2) beta^T C_3 beta, + rows 2..d: b(p) alpha T_3 + S_0 beta, + a = (p-1)(2p-1)/12, b = (p-1)/2, c = p - 1/2. (4.3) + +The remainder is uniform even when `p` or `q` is arbitrarily small, exactly as in [PP]: `Delta >= |q|` and `Delta >= r|p|` give `r^2|p|/Delta <= r` and `r|q|/Delta <= r`. + +**Rank and floor.** On `|p| <= 1/4`, `|a| >= 1/32`, `|b| >= 3/8`, `|c| >= 1/4` and `b^2 <= 25/64`. Let `B_core` be `B` without the `C_3` block. The `C_3` block only adds a positive semidefinite term to `B B^T`, so `B B^T >= B_core B_core^T`. The `A_4` column contributes `a^2 alpha^2` to the `(1,1)` entry; the `T_3` block contributes the rank-one-plus-diagonal matrix `[[c^2 |beta|^2, c b alpha beta^T], [c b alpha beta, b^2 alpha^2 I_m]]`; the `S_0` block contributes `diag(0, G_beta)` with `G_beta >= (|beta|^2/2) I_m` by Lemma D3. Hence + + B_core B_core^T >= N := [[ a^2 alpha^2 + c^2 |beta|^2, c b alpha beta^T ], [ c b alpha beta, (b^2 alpha^2 + |beta|^2/2) I_m ]]. + +`N` is positive definite on `alpha^2 + |beta|^2 = 1`, and its Schur complement on the first coordinate is + + a^2 alpha^2 + c^2 |beta|^2 - c^2 b^2 alpha^2 |beta|^2 / (b^2 alpha^2 + |beta|^2/2) + = a^2 alpha^2 + c^2 |beta|^4 / (2 b^2 alpha^2 + |beta|^2) + >= min{ a^2/2, c^2 / (4(b^2 + 1)) } >= 1/2048, + +by the case split `|beta|^2 >= 1/2` (second term at least `c^2 (1/4)/(b^2 + 1)`) or `alpha^2 >= 1/2` (first term). Since `det(N) = (Schur complement) (b^2 alpha^2 + |beta|^2/2)^m` and `b^2 alpha^2 + |beta|^2/2 >= min(b^2, 1/2) >= 9/64`, + + det(B B^T) >= det(B_core B_core^T) >= det(N) >= (9/64)^(d-1) / 2048. (4.4) + +For `d = 2` the right side is `9/131072`, the constant of [PP] (P9). The entries of `B` are bounded on the chart, so (4.4) gives a uniform floor `lambda_min(B B^T) >= c_d > 0` on the whole compact parameter set `{|p| <= 1/4} x S^(d-1)`. The finite control `G` checks rank `d`, the block-Schur lower bound and (4.4) on random rational points of the sphere, including `alpha = 0` and `beta = 0`. + +By §4.1, `Cov_(Q_r)(B J) = B Cov(J) B^T >= c B B^T >= c c_d I_d`. The centered `L^2` triangle inequality applied to (4.2), with `r_*` reduced to absorb the `O(r)` remainder, gives + + c_0 I_d <= Sigma := Cov_(Q_r)(Y) <= C_0 I_d. (4.5) + +The axis `q = 0`, `p != 0` (`beta = 0`) is covered by the `A_4, T_3` columns, and the transverse direction `p = 0` (`alpha = 0`) by the `T_3, S_0` blocks. This is a `2(d+1) + d` observation all-height problem; no witness value is pinned. + +### 4.4 Density and the full conditional moment ([PP] §6) + +Put `chi = |p| / Delta`. The mean of `Y` is `dvec + O(1)` with `dvec = (6k p(p-1)/Delta, 0)`, and `(9 k_-/2) chi <= |dvec| <= C chi`. The raw gradient map `grad f(X) -> Y` has Jacobian `(r^2 Delta) (r Delta)^m = r^(d+1) Delta^d`. With `|dvec + z|^2 >= |dvec|^2/2 - |z|^2` and (4.5) in the `d`-dimensional Gaussian density, + + p_( grad f(X) | pins )(0) <= C r^(-(d+1)) Delta^(-d) exp(-c chi^2). (4.6) + +Let `K` dominate the Hessian supremum and its Lipschitz constant on the whole chart. For every fixed `n`, + + E_(Q_r)[ K^n | grad f(X) = 0 ] <= C_n (1 + chi^n). (4.7) + +The argument is [PP] (P12) verbatim: the cross-covariance of `f` and its derivatives through order three with `Y` is bounded by Cauchy–Schwarz and (4.5); the Gaussian residual `f - m_f - C(x) Sigma^(-1)(Y - m_Y)` is independent of `Y` with uniformly bounded `C^3` moments; conditioning `Y = 0` replaces the mean by a function of `C^3` norm at most `C(1 + chi)`. + +### 4.5 The original normalizer, once only + +[LP] (5.5): `Z_r >= z_* r^2` in every fixed `d`. (For `d = 2` [PP] (P14) is an alternative proof.) It is never replaced by a witness-conditioned quantity. + +### 4.6 Two pathwise determinant estimates ([PP] §8) + +Let `A = M`, `B = S`, `C = X` be three exact critical points, `|B - A| = r`, `|C - A| =: d_X = r (p^2 + |q|^2)^(1/2) <= r/4`, `ell = |B - C| <= 5r/4`. The critical-segment identity [PP] (P15), `||H_U e|| <= L |V - U| / 2` for `e = (V-U)/|V-U|`, is one-dimensional along the segment and holds in every `d`. + +**Angle-sensitive bound.** Let `sigma = |sin angle(B - A, C - A)| > 0`, i.e. `q != 0`. Apply Lemma D1 at `A` with `e = e_AB`, `e' = e_AC`: `|det H_A| <= (Lr/2)(L d_X/2) K^(d-2) / sigma`, and `||H_A w|| <= L (r + d_X)/(2 sigma)` for unit `w` in the triangle plane `P`. At `B`, `||H_B e_BA|| <= Lr/2` by the segment `BA`; for the unit vector `w` in `P` orthogonal to `e_BA`, transport gives `||H_B w|| <= ||H_A w|| + L r <= L(r + d_X)/(2 sigma) + L r <= 13 L r / (8 sigma)`; so by Lemma D1 with `sigma = 1`, `|det H_B| <= (13/16) L^2 r^2 K^(d-2) / sigma`. At `C`, the sine of the angle is `r sigma / ell`, and the segments `CA`, `CB` give `|det H_C| <= (L d_X/2)(L ell/2) K^(d-2) ell / (r sigma) <= (25/64) L^2 r d_X K^(d-2) / sigma`. Multiplying, and using `sigma <= 1`, + + |det H_M det H_S det H_X| <= L^6 r^4 d_X^2 K^(3(d-2)) / sigma^4 + = L^6 K^(3(d-2)) r^6 |q|^2 ( 1 + p^2/|q|^2 )^3. (4.8) + +For `d = 2` this is [PP] (P16). Only the `K^(3(d-2))` factor is new; it is the price of the `d - 2` directions in which no Hessian is forced small. + +**Angle-free bound.** Lemma D2 with the short segment `AC` at `A` and at `C`, and the segment `BA` at `B`: + + |det H_M det H_S det H_X| <= (1/8) L^3 K^(3(d-1)) r d_X^2 = (1/8) L^3 K^(3d-3) r^3 (p^2 + |q|^2). (4.9) + +This holds on collinear triangles. The two factors of `d_X` are load-bearing, as in [PP]. + +### 4.7 Weighted Kac–Rice ([PP] §9) + +Unchanged: the counted field is `grad f : X -> R^d`; the nonnegative mark is `W_r 1{H_X in O_j}`, where `O_j` is the open set of nonsingular symmetric matrices of index `j`; the witness determinant enters once through the Jacobian factor; the conditional gradient covariance is positive definite on compact punctures by distinct-site nondegeneracy ([LP] §2), and continuous Gaussian regression supplies the remaining hypotheses of arXiv:2304.07424v3 Theorem 7.1 (Remarks 7–8). Truncation and monotone convergence remove the growth restriction. Thus the intensity per unit physical volume is + + rho_j^W(X) = Z_r^(-1) p_( grad f(X) | pins )(0) E_(Q_r)[ W_r F_j(H_X) | grad f(X) = 0 ]. (4.10) + +### 4.8 The crossover, and the integrable weight ([PP] §10) + +**Region I: `|q| >= r |p|`.** Here `q != 0`. Put `t = |p|/|q|`, so `r^2 t^2 <= 1`, `chi^2 = t^2/(1 + r^2 t^2) in [t^2/2, t^2]`, and `|q|^2 Delta^(-d) <= |q|^(2-d)` because `Delta >= |q|`. Insert (4.8), dropping types, and (4.6), (4.7) with `n = 3d`, and [LP] (5.5) into (4.10): + + rho_j^W(X) <= C r^(-2) r^(-(d+1)) Delta^(-d) e^(-c chi^2) r^6 |q|^2 (1 + t^2)^3 (1 + chi^(3d)) + <= C r^(3-d) |q|^(2-d) sup_t (1 + t^2)^3 (1 + t^(3d)) e^(-c t^2/2) + <= C' r^(3-d) |q|^(2-d). (4.11) + +**Region II: `|q| < r |p|`.** Here `p != 0`, `Delta >= r|p|`, `Delta^2 <= 2 r^2 p^2`, `chi^2 in [1/(2r^2), 1/r^2]`, and `p^2 + |q|^2 <= 2 p^2 = 2 (r|p|)^2 / r^2`. Insert (4.9) instead: + + rho_j^W(X) <= C r^(-2) r^(-(d+1)) (r|p|)^(-d) e^(-c chi^2) r^3 (r|p|)^2 r^(-2) (1 + chi^(3d)) + <= C r^(-2-4d) e^(-c/(2r^2)) (r|p|)^(2-d) + <= C r^(3-d) (r|p|)^(2-d), (4.12) + +the middle step using `1 + chi^(3d) <= 2 r^(-3d)`, and the last step `e^(-c/(2r^2)) <= n! (2/c)^n r^(2n)` with `2n >= 5 + 3d`. The longitudinal axis `q = 0` is included; no division by `q` is made. + +**Integration.** In both regions `rho_j^W(X) <= C r^(3-d) omega_d(p, q)` with `omega_d` from (1.1). Apply (4.10) on increasing compact punctures and use monotone convergence. Since `dX = r^d dp dq`, + + E_(Q_r^W) N_j(M + rE) = integral_(rE) rho_j^W <= C r^3 integral_E omega_d(p, q) dp dq, + +which is (1.2). Lemma D5 gives the finite right side and the `O(r^5)` nested-ball bound. For `d = 2`, `omega_2 = 1` and this is [PP] (P2) with a bounded intensity; for `d >= 3` the intensity is unbounded on the axis but its weight is integrable, which is exactly why the count still has order `r^3`. ∎ + +### 4.9 Reflection to `S` + +[PP] §10.1 applies verbatim: use the reflected local field `tilde f(x, y) = f(S - x e_1 + R_y y)` for any orthogonal map `R_y` of the transverse space (a reflection of one transverse axis keeps the orientation if desired; only orthogonality matters), with `tilde b = b - k r^3`, `tilde k = -k`. Sections 4.1–4.4 use only boundedness of the targets and `|k| >= k_-`; the deterministic estimates are index-free; the weighted Kac–Rice argument uses the relabeled original mark `F_(d-1)(H_0) F_d(H_r)`, the same random variable as `W_r`; and the original `Z_r` is retained. The two physical balls of radius `r/4` about `M` and `S` are disjoint. + +## 5. Proof of Theorem C_d + +We follow [CP] §§3–10 in midpoint coordinates `X = r(u, v)`, `u in R`, `v in R^m`, with the exact endpoint cubic `H_r(x) = b - k r^3/2 - (3k r^2/2) x + 2k x^3` and the midpoint jets `T_3 = grad_y f_xx(0) in R^m`, `C_3 = D_y^2 f_x(0) in Sym_m`, `S_0 = D_y^2 f(0) in Sym_m`. + +### 5.1 Degree-five value-and-gradient interpolation at three sites ([CP] §3) + +Let `a = (-1/2, 0)`, `b = (1/2, 0)`, `c = (u, v)` be the three sites of the collar; pairwise distances are at least `min(1, eta)`. Let `P_5` be the polynomials on `R^d` of total degree at most five; `dim P_5 = C(d+5, 5) >= 3(d+1)`. Consider + + B_c : P_5 -> R^(3(d+1)), P -> ( P(a), grad P(a), P(b), grad P(b), P(c), grad P(c) ). + +**Lemma.** `B_c` is onto for every `c` in `C(eta, R)`, including collinear configurations and `c = 0`, and its smallest row singular value has a uniform positive lower bound on `C(eta, R)` in the fixed monomial basis. + +*Proof.* Choose a unit direction `w` such that the three projections `t_i = w . x_i` are distinct: for each of the three pairs the directions orthogonal to `x_i - x_j` form a great subsphere of `S^(d-1)`, a closed set with empty interior, so such `w` exists. Complete `w` to an orthonormal basis `w, w_1, ..., w_m` and write `z_j = w_j . x`, `z_(i,j) = w_j . x_i`. With the degree-two Lagrange polynomials `L_i(t)` at `t_1, t_2, t_3`, the polynomials + + h_i(t) = [ 1 - 2 L_i'(t_i)(t - t_i) ] L_i(t)^2, + d_i(t) = (t - t_i) L_i(t)^2, + e_(i,j)(t, z) = (z_j - z_(i,j)) L_i(t)^2, j = 1, ..., m, + +have total degree at most five. At the three sites, `h_i` has value `delta_(ij)` and zero gradient; `d_i` has zero value, `w`-derivative `delta_(ij)` and zero transverse derivatives; `e_(i,j)` has zero value and `w`-derivative and transverse derivative `delta_(ij)` in the `w_j` direction. These `3(1 + 1 + m) = 3(d+1)` polynomials prescribe arbitrary values and gradients, so `B_c` is onto in the original coordinates. Its entries are polynomials in `(u, v)`; surjectivity at every point of the compact collar gives a positive minimum of the least eigenvalue of `B_c B_c^T`. A single continuous choice of `w` is not needed. ∎ + +### 5.2 The coarse uniform floor ([CP] §4) + +Collect the `3(d+1)` observations `V_r = ( f(ra), r grad f(ra), f(rb), r grad f(rb), f(rc), r grad f(rc) )`. With `J_5` the `C(d+5, 5)` midpoint Taylor coefficients `D^alpha f(0)/alpha!` and `D_r = diag(r^|alpha|)`, Taylor's formula through total degree five gives `V_r = B_c D_r J_5 + R_r`, `||R_r||_(L^s) <= C_s r^6` (the gradient entries carry `r . O(r^5)`), using [LP] (4.1) for the sixth derivatives. `Cov(J_5)` is uniformly positive definite (distinct one-site monomials, [LP] §2, compactness in the frame). For a unit row `t`, `std(t B_c D_r J_5) >= c r^5 ||t B_c|| >= c' r^5`, and the `L^2` triangle inequality gives `Cov(V_r) >= c r^10 I_(3(d+1))` for small `r`. Delete the auxiliary witness value (a principal submatrix keeps the floor), then pass from the scaled entries `r grad f` to the physical entries `grad f` by the congruence `V |-> diag(1, r^(-1) I_d, ...) V`, which multiplies the covariance by `r^(-2) >= 1` on the gradient block and, by the Schur variational identity `Cov(grad f(X) | pins) = min over affine predictors`, keeps the conditional covariance of the physical witness gradient at least `c r^10 I_d` (this is a statement about the congruence and the variational formula, not eigenvalue monotonicity under an arbitrary entrywise scaling), and take the Schur complement on the `2(d+1)` pins. Hence + + Sigma_X := Cov_(Q_r)( grad f(X) ) >= c r^10 I_d, Sigma_X <= C I_d. (5.1) + +Only `2(d+1) + d` observations are conditioned; no witness value and no height window appear. + +### 5.3 The pinned drift ([CP] §5) + +With `D(u) = u^2 - 1/4`, the endpoint pins and Taylor's formula give, uniformly on `|u|, |v| <= R`, + + G_1 := f_x(ru, rv)/r^2 = 6k D(u) + u v . T_3 + (1/2) v^T C_3 v + O_(L^s)(r), + G_2 := grad_y f(ru, rv)/r = S_0 v + O_(L^s)(r), (5.2) + +exactly as [CP] (C7), with `v . T_3`, `v^T C_3 v` and `S_0 v` replacing the planar products. Since (C7) is stated for +two scalars, the vector form is derived in §6.3: (5.2) is (6.3) with `s = r` and `epsilon = 1` (so `d_e = D(u)`), on +the compact chart `|u|, |v| <= R`, whose `O(sK)` and `O(s^2 K)` remainders are `O(rK)`; the terms `(r/2) D(u) T_3`, +`r u C_3 v` and `(r/2) D_3[v, v]` of the transverse line are absorbed into the `O_(L^s)(r)` of the second line of +(5.2), and the inputs are the vector-valued endpoint identities (6.2). Hence `|E_(Q_r) G_1 - 6k D(u)| <= C(|v| + r)` and `Var_(Q_r)(G_1) <= C(|v|^2 + r^2)`. On `|v| <= v_0 <= eta/2` the collar gives `|D(u)| >= eta^2/4`, so `|E_(Q_r) G_1| >= d_0 > 0` after reducing `v_0` and `r_*`. The one-coordinate Mahalanobis bound `m^T Sigma^(-1) m >= m_1^2 / Sigma_11` [CP] (C10) supplies the factor `exp[-c/(r^2 + |v|^2)]` in the joint `d`-dimensional density. + +### 5.4 Very near the axis: `|v| <= r^(1/3)` ([CP] §6) + +By (5.1) the density prefactor is at most `C r^(-5d)`, so + + p_( grad f(X) | pins )(0) <= C r^(-5d) exp[-c r^(-2/3)]. + +Regress the whole field on `grad f(X)`: `||Sigma_X^(-1)|| <= C r^(-10)`, bounded cross-covariances, bounded targets, and the independent residual give `E_(Q_r)[ K^(3d) | grad f(X) = 0 ] <= C r^(-30d)`. The product of three absolute determinants is at most `K^(3d)`. Divide once by [LP] (5.5): + + rho_j^W(X) <= C r^(-2-35d) exp[-c r^(-2/3)] <= C r^(3-d), + +by the `n`-th term of the exponential series with `2n/3 >= 5 + 34d`. The exact midpoint and the whole axis `v = 0` are covered. + +### 5.5 The rest of the near-axis strip: `r^(1/3) <= |v| <= v_0` ([CP] §7) + +Write `G = (G_1, G_2) = (6k D(u), 0) + B J + R`, `J = (T_3, C_3, S_0)`, `||R||_(L^2) <= C r`, with + + row 1: u v . T_3 + (1/2) v^T C_3 v, rows 2..d: S_0 v. + +The `C_3` block acts only on row 1 and the `S_0` block only on rows `2..d`, so by Lemma D3 + + B B^T >= diag( |v|^4/4, (|v|^2/2) I_m ) >= (|v|^4/4) I_d (|v| <= sqrt 2), + +and `trace(B B^T) <= C_R |v|^2`. The conditional covariance of `J` is uniformly sandwiched (§4.1, now at the midpoint), so `Cov(B J) >= c |v|^4 I_d`. Since `r/|v|^2 <= r^(1/3)`, the centered remainder is at most half the leading standard deviation for small `r`, and + + c |v|^4 I_d <= Cov_(Q_r)(G) <= C |v|^2 I_d. (5.3) + +The physical gradient map is `diag(r^2, r I_m)` with determinant `r^(d+1)`. Using §5.3 and (5.3), + + p_( grad f(X) | pins )(0) <= C r^(-(d+1)) |v|^(-2d) exp[-c/|v|^2], (5.4) + E_(Q_r)[ K^(3d) | grad f(X) = 0 ] <= C |v|^(-12d), (5.5) + +the latter by regression on `G` with `||Cov(G)^(-1)|| <= C |v|^(-4)`. + +**Three soft Hessians in the collar.** The segment `MS` gives `||H_M e_x|| <= Lr/2`. From `grad f(X) = grad f(M) = 0`, `||H_M (X - M)|| <= L |X - M|^2 / 2 <= C_R L r^2`; since `X - M = r((u + 1/2) e_x + v)`, subtracting the controlled `e_x` component gives `||H_M v|| <= C_R L r`, hence `||H_M hat v|| <= C_R L r / |v|`. The plane `P = span(e_x, hat v)` is orthonormal because `v` is transverse. Lemma D1 with `sigma = 1` gives `|det H_M| <= C_R L^2 r^2 K^(d-2) / |v|`, and for every unit `w in P`, `||H_M w|| <= C_R L r / |v|` (using `|v| <= R`). Transport to `S` and to `X` (distances at most `(R+1) r`) keeps `||H_S w||, ||H_X w|| <= C_R L r / |v|` on `P`, so `|det H_S|, |det H_X| <= C_R L^2 r^2 K^(d-2) / |v|^2`. Therefore + + |det H_M det H_S det H_X| <= C_R K^(3d) r^6 |v|^(-6) (5.6) + +(we used `|v|^(-5) <= 2 |v|^(-6)` for `|v| <= 2` and `L <= K`). For `d = 2` this is [CP] (C18). Combining [LP] (5.5), (5.4), (5.5), (5.6): + + rho_j^W(X) <= C r^(3-d) |v|^(-(14d+6)) exp[-c/|v|^2] <= C' r^(3-d). (5.7) + +The radius powers are `-2 - (d+1) + 6 = 3 - d`. No height-window factor appears. + +### 5.6 Transverse compact remainder: `|v| >= v_0` ([CP] §8) + +`B` has a fixed positive singular-value floor, the `O(r)` error is absorbed uniformly, the reduced mean, inverse covariance and conditional `C^3` moments are bounded, the density is `O(r^(-(d+1)))`, (5.6) gives `O(r^6)`, and division by `Z_r` leaves `rho_j^W <= C r^(3-d)`. No assertion that `D(u)` is separated from zero is needed here. + +### 5.7 Integration and synthesis ([CP] §§9–10) + +Weighted Kac–Rice on an open neighbourhood of the compact collar (distinct-site nondegeneracy, or the local polynomial floor), with the mark `W_r 1{H_X in O_j}`, gives (4.10) there. Since `dX = r^d du dv` and `rho_j^W <= C r^(3-d)` on all three strips, `E_(Q_r^W) N_j(rE) <= C r^3 |E|`, which is (1.3). + +For (1.4), take `eta = 1/4`, partition a Borel subset of the scaled ball into its intersections with the two punctured pin balls of scaled radius `1/4` (Theorem P_d in endpoint coordinates; translation preserves volume) and the remaining collar (this theorem); assign boundaries once. Take the minimum of the radius cutoffs and the maximum of the constants. Choosing `R` larger than the inner radius of a scaled annulus supplies an overlapping cover of it. Every constant and cutoff of §5 depends on the fixed `d`, `R`, `eta`, `L`, `B` and the mark compacts, as stated in §1.1; nothing in this section is uniform as `d` grows. ∎ + +## 6. Proof of Theorem I_d + +We follow [IW] §§3–10 with `epsilon = r/s in (0, 1/4]`, the endpoint frame `U_r` of [IW] (I7) extended by the `2m` transverse rows `L_r(0) in R^m`, `L_r'(0) in R^m` (affine interpolants of `grad_y f` at `+-r/2`), the scaled vector + + V_(r,s,w) = ( S_s U_r, f(sw), s grad f(sw) ), S_s = diag(1, s, s^2, s^3, s I_m, s^2 I_m), w = (u, v) in A = {1 <= |w| <= 2}, + +which has `2(d+1) + (d+1)` coordinates, and the endpoint-law inputs [IW] (I6), which are [LP] (4.1) and (5.5) in every `d`. + +### 6.1 Polynomial row rank through `epsilon = 0` ([IW] §3.1) + +For `epsilon > 0` the three sites `(-epsilon/2, 0)`, `(epsilon/2, 0)`, `w` are distinct and §5.1 gives rank `3(d+1)` on `P_5`, hence rank `3(d+1)` for the transformed rows (the Hermite transform is invertible). At `epsilon = 0` the `2(d+1)` contact rows `(f, f_x, f_xx, f_xxx, grad_y f, grad_y f_x)(0)` are independent. Polynomials of degree at most five annihilated by all of them are spanned by the monomials other than `1, x, x^2, x^3, y_j, x y_j`. To control the remaining `d + 1` rows: + +- if `v != 0`, rotate the transverse frame so `v = |v| e_1` (the contact functionals are covariant under transverse rotations), and use `y_1^2, y_1^3, x y_1^2, y_j y_1^2 (j = 2..m)`. Their value/gradient matrix at `w` is block triangular: the `y_j`-derivative block is `|v|^2 I_(m-1)`, and the `3 x 3` block on `(value, d/dx, d/dy_1)` has determinant `-|v|^6`, so the total determinant is `-|v|^(2d+2) != 0`; +- if `v = 0`, then `|u| >= 1`, and `x^4, x^5, x^2 y_j` have value/gradient determinant `u^8 . u^(2m) = u^(2d+6) != 0`. + +So the `(3(d+1)) x dim P_5` matrix `B_(epsilon, w)` has full row rank also at confluence. Its entries are polynomial in `epsilon`, hence continuous at `epsilon = 0`; compactness of `[0, 1/4] x A` supplies a uniform positive smallest row singular value. Degree four fails: on the axis, the annihilated monomials of degree at most four have proportional value and `x`-derivative at `(u, 0)` (only `x^4` contributes), so the witness rows have rank at most `d < d + 1`. The finite control `R` checks the two explicit families, a general rational transverse direction with a rational basis of its orthogonal complement, the degree-five rank `d + 1` and the degree-four rank `d` on the axis, for `d = 2, 3, 4`. + +### 6.2 Taylor remainder, floor and conditional floor ([IW] §3.2) + +[IW]'s divided-difference argument is one-dimensional along the axis and applies to each transverse component: every scaled endpoint coordinate is a bounded functional on `C^3` uniformly for `0 <= epsilon <= 1/4`, with the mass-one kernel identity for `H'''(0)` unchanged. Hence `V = B_(epsilon, w) D_s J_5 + R`, `||R||_(L^p) <= C_p s^6`, the standard-deviation floor `c s^5 ||a B||`, and + + Cov(V_(r,s,w)) >= c s^10 I, Sigma_X := Cov_(Q_r)( grad f(X), f(X) ) >= c s^10 I_(d+1), (6.1) + +by unscaling and the Schur variational identity, exactly as [IW] (I10), (I11). + +### 6.3 Midpoint expansions and the height coordinate ([IW] §4) + +Let `T_3 = grad_y f_xx(0) in R^m`, `C_3 = D_y^2 f_x(0) in Sym_m`, `D_3 = D_y^3 f(0)` (symmetric 3-tensor), `S_0 = D_y^2 f(0) in Sym_m`. The endpoint constraints give, pathwise with `K` a fixed multiple of `1 + ||f||_(C^6)`, + + |f(0) - b_bar| <= C r^4 K, f_x(0) = -3k r^2/2 + O(r^4 K), grad_y f(0) = -r^2 T_3/8 + O(r^4 K), + |f_xx(0)| + ||grad_y f_x(0)|| <= C r^2 K, f_xxx(0) = 12k + O(r^2 K), (6.2) + +by the same averaged endpoint identities as [IW] (I12), applied to each transverse component. Since (I12) is stated for +two transverse scalars, the vector-valued identities are derived here. Write `g(x) = f(x e_1)` and +`h(x) = grad_y f(x e_1) in R^m`, so the pins read `g(-r/2) = b`, `g(r/2) = b - k r^3`, `g'(±r/2) = 0`, `h(±r/2) = 0`. +Taylor's formula at `0` through order four, with `K` bounding the fifth derivatives, gives for each of `g` and each +component `h_i`: + + phi(r/2) - phi(-r/2) = r phi'(0) + (r^3/24) phi'''(0) + O(r^5 K), + phi(r/2) + phi(-r/2) = 2 phi(0) + (r^2/4) phi''(0) + O(r^4 K), + phi'(r/2) - phi'(-r/2) = r phi''(0) + O(r^3 K), + phi'(r/2) + phi'(-r/2) = 2 phi'(0) + (r^2/4) phi'''(0) + O(r^4 K). + +For `phi = g`: the third line gives `f_xx(0) = O(r^2 K)`; the fourth gives `g'(0) = -(r^2/8) g'''(0) + O(r^4 K)`; +inserting this into the first, whose left side is `-k r^3`, gives `-(r^3/12) g'''(0) = -k r^3 + O(r^5 K)`, hence +`f_xxx(0) = 12k + O(r^2 K)` and `f_x(0) = -3k r^2/2 + O(r^4 K)`; the second gives `f(0) = b - k r^3/2 - (r^2/8) f_xx(0) ++ O(r^4 K) = b_bar + O(r^4 K)`. For `phi = h_i`, both left sides vanish: the first line gives `h_i'(0) = O(r^2 K)`, +that is `grad_y f_x(0) = O(r^2 K)`, and the second gives `h_i(0) = -(r^2/8) h_i''(0) + O(r^4 K)`, that is +`grad_y f(0) = -(r^2/8) T_3 + O(r^4 K)`. Each is a one-dimensional identity applied to one component; the vector norm +costs a factor `sqrt(m)` in the constant. This is (6.2). + +The expansions (6.3) below hold for any bounded `(u, v)` and any `epsilon <= 1`, with constants depending on the +bound; §5.3 uses them with `s = r`, `epsilon = 1` on `|u|, |v| <= R`. With `d_e = u^2 - epsilon^2/4`, at `X = s(u, v)`, + + f_x(X)/s^2 = 6k d_e + u v . T_3 + (1/2) v^T C_3 v + O(sK), + grad_y f(X)/s = S_0 v + (s/2) d_e T_3 + s u C_3 v + (s/2) D_3[v, v] + O(s^2 K), + [f(X) - b_bar]/s^3 = k(2u^3 - 3 epsilon^2 u/2) + v^T S_0 v/(2s) + (d_e/2) v . T_3 + (u/2) v^T C_3 v + (1/6) D_3[v,v,v] + O(sK). (6.3) + +The row operation `Z = ( f_x/s^2, grad_y f/s, [f - b_bar - (s/2) v . grad_y f]/s^3 )` removes the `S_0/s` term and the `C_3` term from the height row: + + Z = ( 6k d_e, 0, k(2u^3 - 3 epsilon^2 u/2) ) + B J + O_(L^p)(s), J = (T_3, C_3, D_3, S_0), + row 1: u v . T_3 + (1/2) v^T C_3 v, + rows 2..d: S_0 v, + row d+1: (d_e/4) v . T_3 - (1/12) D_3[v, v, v]. (6.4) + +The map from `Z` to `(f_x, grad_y f, f - b_bar)` is triangular with determinant `s^(2 + m + 3) = s^(d+4)`. The target at height `y in I_r` and zero gradient is `tau = (0, 0, (y - b_bar)/s^3)`, `|tau_(d+1)| <= (k_+/2) epsilon^3`, bounded through `epsilon -> 0`. The finite control `RO` checks (6.3)–(6.4) exactly on a pinned cubic in `d = 3`: no negative power of `s`, `S_0` and `C_3` absent from the height row, and the displayed coefficients. + +### 6.4 Euler suppression and the transverse determinants ([IW] §5) + +Suppose the pins hold, `grad f(X) = 0` and `f(X) = y in I_r`. Taylor expansion about `0` through degree three and Euler's identity for homogeneous polynomials give, in every `d`, + + 3[f(X) - f(0)] - X . grad f(X) = 2 grad f(0) . X + (1/2) X^T H_0 X + O(s^4 K), (6.5) + +the cubic terms cancelling because their Euler degree is three (finite control `EU` in `d = 3`). Here `(1/2) X^T H_0 X = (s^2/2)[ u^2 f_xx(0) + 2u grad_y f_x(0) . v + v^T S_0 v ]`. By (6.2), `|y - b_bar| <= k_+ r^3/2` and `r <= s/4 <= 1`, every term other than `(s^2/2) v^T S_0 v` is `O(K(r^2 s + s^4))`, so for `v != 0`, with `hat v = v/|v|`, + + | hat v^T S_0 hat v | <= C K (r^2/s + s^2) / |v|^2 <= C K (r + s^2) / |v|^2. (6.6) + +This is [IW] (I18) for the quadratic form in the direction `v`. It controls one entry of `S_0`, not the matrix. The second piece of information is the transverse gradient equation: from `grad_y f(X) = 0` and the second line of (6.3), `S_0 v = O(sK)`, that is + + | S_0 hat v | <= C s K / |v|. (6.7) + +**Endpoint determinants.** Let `A_i = D_y^2 f(i)` for `i in {M, S}`; then `A_i = S_0 + O(rK)` by the Lipschitz bound over distance `r/2`, so `|hat v^T A_i hat v| <= C K (r + s^2)/|v|^2` (absorbing `rK <= 4 rK/|v|^2`) and `|A_i hat v| <= C s K/|v|`. Lemma D4 gives + + |det A_i| <= C K^(d-2) (r + s^2)/|v|^2 + C K^(d-3) s^2 K^2/|v|^2 <= C K^(d-1) (r + s^2)/|v|^2 + +(the second term is absent when `d = 2`, where `A_i` is the scalar `epsilon_i`). With [LP] (5.3) and (5.1), `|det H_i| = r |alpha_i det A_i - r beta_i^T adj(A_i) beta_i| <= r K |det A_i| + r^2 K^d`, hence + + |det H_M|, |det H_S| <= C r K^d (r + s^2) / |v|^2. (6.8) + +For `d = 2` this is [IW] (I19). The second term of Lemma D4, of size `s^2/|v|^2`, is of the same order as the `s^2` part of the Euler bound; that coincidence is what makes the lift work without loss. + +**Witness determinant.** As in [IW] (I20), `||H_M e_x|| <= Lr/2` and `||H_M (X - M)|| <= C K s^2` give `||H_M hat v|| <= C K s/|v|`; transport over distance at most `3s` gives `||H_X e_x|| <= C K s` and `||H_X hat v|| <= C K s/|v|`. Lemma D1 on the orthonormal pair `(e_x, hat v)`: + + |det H_X| <= C s^2 K^d / |v|. (6.9) + +Combining (6.8), (6.9), dividing once by [LP] (5.5) and using `r^2/s <= r`, + + W_r |det H_X| / Z_r <= C K^(3d) s^2 (r + s^2)^2 |v|^(-5). (6.10) + +**Axis-safe bound, valid throughout the strip `|v| <= s^(1/8)`, including `v = 0`.** The two endpoint short columns `H_M e_x`, `H_S e_x`, of size at most `Lr/2` by [LP] (5.3), do not involve the witness position, so Lemma D2 (Hadamard with `e_x`) gives `|det H_M|, |det H_S| <= C r K^(d-1)` for every `X`; with the trivial `|det H_X| <= K^d`, `W_r |det H_X| <= C r^2 K^(3d)` and, dividing once by [LP] (5.5), `W_r |det H_X| / Z_r <= C K^(3d)` on the whole strip, with no division by `|v|`. A bound stated only at `v = 0` would not control nonzero `v`; it is this uniform bound that §6.5 uses. + +### 6.5 Small axial strip ([IW] §6) + +On `A` and `|v| <= v_0`, `|u|` is bounded below and `d_e >= c > 0`; the first line of (6.3) gives `|E_(Q_r)[f_x(X)/s^2]| >= c_0` and `Var <= C(|v|^2 + s^2)`; the one-coordinate Mahalanobis bound applies to the full `(d+1)`-dimensional exponent. On `|v| <= s^(1/8)`, (6.1) gives the joint density bound `C s^(-5(d+1)) exp[-c s^(-1/4)]`, regression on `Y_X = (grad f(X), f(X))` with `||Sigma_X^(-1)|| <= C s^(-10)` gives `E[K^(3d) | grad f(X) = 0, f(X) = y] <= C s^(-30d)`, and with the axis-safe bound of §6.4 (`W_r |det H_X| / Z_r <= C K^(3d)` throughout the strip) the intensity per unit volume and height is bounded by `C s^(-N) exp[-c s^(-1/4)] <= C`. The strip has volume `O(s^d) <= O(s^2)` and the window length is `k r^3`, so it contributes at most `C r^3 s^2` to the shell count. + +### 6.6 Transverse region: `s^(1/8) <= |v| <= 2` ([IW] §7) + +In (6.4) the three jet blocks `C_3`, `S_0`, `D_3` act on disjoint row sets, so by Lemma D3 + + B B^T >= diag( |v|^4/4, (|v|^2/2) I_m, |v|^6/144 ), det(B B^T) >= c |v|^(2d+8), sigma_min(B) >= c |v|^3, + +and `||B|| <= C|v|`. The conditional covariance of `J` is uniformly sandwiched, so the leading standard deviation of `a . Z` is at least `c ||aB|| >= c |v|^3 ||a||`, while the centered `O(s)` error is at most `C s ||a||`; since `s/|v|^3 <= s^(5/8)`, the error is at most half the leading term for small `s`, giving the relative bound + + Cov_(Q_r)(Z) >= c B B^T, det Cov_(Q_r)(Z) >= c |v|^(2d+8), ||Cov_(Q_r)(Z)^(-1)|| <= C |v|^(-6), Cov_(Q_r)(Z) <= C |v|^2 I. (6.11) + +With the Jacobian `s^(d+4)` and the Gaussian penalty `exp(-c/|v|^2)` from §6.5 (or a constant for `|v| >= v_0`), + + p_( grad f(X), f(X) | pins )(0, y) <= C s^(-(d+4)) |v|^(-(d+4)) exp(-c/|v|^2), (6.12) + E_(Q_r)[ K^(3d) | grad f(X) = 0, f(X) = y ] <= C |v|^(-18d), (6.13) + +the latter by regression on `Z` with (6.11) and the bounded target `tau`. + +### 6.7 The shell ledger ([IW] §8) + +Insert (6.10) into the conditional expectation of the weighted density and apply (6.12), (6.13): + + Lambda_(r,j)(X, y) <= C s^(-(d+2)) (r + s^2)^2 |v|^(-(19d+9)) exp(-c/|v|^2) <= C' s^(-(d+2)) (r + s^2)^2. (6.14) + +Every inverse power of `|v|` is absorbed by its Gaussian factor uniformly down to `s^(1/8)`. Integrate over the physical shell (volume `O(s^d)`) and the window (length `k r^3`): + + E N_(r,j)(shell, transverse) <= C r^3 (r + s^2)^2 / s^2 <= 2C r^3 [ (r/s)^2 + s^2 ], (6.15) + +by `(a + b)^2 <= 2a^2 + 2b^2`; equivalently `2[(r/s)^2 + s^2] - (r + s^2)^2/s^2 = (r/s - s)^2 >= 0` (finite control `L`). Adding the axial contribution of §6.5 proves (1.5). The `r` powers are `-2 + 2 + 3 = 3` and the `s` powers `-(d+4) + 2 + d = -2` in every `d`. + +### 6.8 Kac–Rice with height disintegration, and the dyadic sum ([IW] §§9–10) + +For each `r, s > 0` the closed shell is separated from the pins; (6.1) gives nondegeneracy of `(grad f(X), f(X))`; the marked critical-point formula with the mark `W_r 1{H_X in O_j} 1{f(X) in I_r}` and height disintegration gives + + E_(Q_r^W) N_(r,j)(B) = integral_B integral_(I_r) Lambda_(r,j)(X, y) dy dX, + Lambda_(r,j)(X, y) = Z_r^(-1) p_( grad f(X), f(X) | pins )(0, y) E_(Q_r)[ W_r F_j(H_X) | grad f(X) = 0, f(X) = y ], + +exactly as [IW] (I33), with the same framework citation. The dyadic sum [IW] (I34) is dimension-free: `sum_j (r/s_j)^2 <= 4/(3 A_0^2)` and `sum_j s_j^2 <= (4/3) rho^2` for `s_j = 2^j A_0 r`. This proves (1.6). ∎ + +## 7. Proof of Theorem G_d and the corollaries + +Use one `(Q_r, W_r, Z_r)` throughout, distances from the midpoint, and the tiling of [IW] §10 and its review (N1): + +| Region | Source | Parameters | Heights | +|---|---|---|---| +| `|X| <= 4r`, pins removed | Theorem C_d, (1.4) with `R = 4` | fixed | all heights, which contain `I_r` | +| `4r <= |X| <= s_0` | Theorem I_d, (1.6) with `A_0 = 4`, `rho = s_0` | fixed `s_0` | `I_r` | +| `|X| >= s_0` | [RM] Theorem A, "fix `0 < rho < L/4`", reviewed ACCEPT in every fixed `d` (main#76) | `rho = s_0` | `I_r` | + +Boundaries are assigned once and `r` is taken below all three cutoffs. No annulus is left uncovered, no regional normalizer is multiplied, and no pin-density Jacobian is inserted. Summing the three bounds gives (1.7). + +*Corollary F_d.* `Q_r^W(A_r) <= E_(Q_r^W) sum_j N_(r,j)(X minus {M, S}) <= (d+1) C r^3` by Markov; the lower bound is [RC] (F-), whose proof uses only [RM] (A) and the fixed-remote second factorial moment (D), both valid in every fixed `d`. ∎ + +*Corollary M_d.* [EDL] (A4) gives `Q_r^W(N_r >= 2) >= c r^3` and `E[N_r(N_r - 1)] >= c r^3` in every fixed `d`. For the upper bound, `1{N >= 2} <= N/2` and (1.7). The total-variation statement is [LOCAL] §7 (an identity for any finite point process with mean mass at most one) applied with (1.7) and (A4). ∎ + +## 8. What this note does not establish + +- An all-height intermediate or global `O(r^3)` count. The height window in (6.6) is load-bearing, as [IW] and its reviews record; the all-height shell method gives `O(r)`, not `O(r^3)`. +- A torus-wide second factorial moment upper bound, or any collision estimate outside the fixed remote region of [RC]. +- Elder selection; D1 Theorem A covers it separately. +- Uniformity as `k -> 0`, as marks grow, as `L` or `d` vary, or any numerical `C`, `c`, `r_*`, `s_0`. +- Any recovery of historical RN/JETMOD/24-jet carriers, or any change to their `BLOCKED_ABSENT` / `OPEN_HISTORICAL` classifications. Theorem G_d in `d = 3` is an analytic, existential-constant statement about the same kind of object those carriers were meant to certify numerically; whether the registers should record a regional supersession is a separate source-bound reconciliation, not something this note asserts. +- Organizational independence: every lane shares one GitHub account. + +The candidate's own review obligations are listed in `README.md`. A defect in any of the five devices, or in any planar step cited here as dimension-free, blocks consumption of the corresponding theorem. + +## 9. Finite controls + +`lift_exact_check.py` (standard library, exact rational and polynomial arithmetic) checks the nine groups listed in its docstring and rejects ten deliberate mutations: the exterior-power order, the two frame blocks, the interpolation degree, the row-operation constant, the Euler coefficient, the adjugate sign, one soft direction instead of two, a non-summable shell factor, and a non-integrable pin weight. `RESULTS.json` is its exact output, identical in normal and optimized modes. These checks verify identities, ranks, floors and exponents. They do not verify the Gaussian, compactness, regression, or Kac–Rice steps, which are the written arguments above and in the sources. + +## 10. Attribution and reconnaissance + +The mechanisms are those of the planar sources; the contribution here is the written `d`-dimensional replacement of each planar-specific step and the five devices of §3, which are classical linear algebra (exterior powers, Hadamard, Schur complements) applied at the right places. Nothing is claimed as new beyond this explicit application. `RECONNAISSANCE.md` records the limited external check and its failure mode. diff --git a/specifications/lean_statements_20261010/sources/d5-reconciliation/RECONCILIATION.md b/specifications/lean_statements_20261010/sources/d5-reconciliation/RECONCILIATION.md new file mode 100644 index 000000000..4062d76bd --- /dev/null +++ b/specifications/lean_statements_20261010/sources/d5-reconciliation/RECONCILIATION.md @@ -0,0 +1,163 @@ +# D5 pin-neighborhood first moment — source-bound reconciliation + +**Object:** D5-RECONCILIATION-20260929-v2.1. +**Version v2 (guard repair).** This version answers OpenAI review +[5893615810](https://github.com/d6g8k5htny-coder/Math-/pull/138#issuecomment-5893615810): +- the P2 continuum record is now mandatory and byte-bound; +- real delete, change and symlink negatives are added; +- the transition dependency chains are machine-readable; +- the identity inventory now covers every record §3 names. + +**v2.1** answers OpenAI review +[5355021293](https://github.com/d6g8k5htny-coder/Math-/pull/138#pullrequestreview-5355021293): +- symlinked parent directories are rejected too; +- every mandatory evidence file is proposed as a graph node, with its SHA256 as `fingerprint`, reached by a + `required` edge from each consuming node. A `coverage_source` string alone is not byte invalidation. + +The proposal itself is unchanged. +**Reconciler:** Anthropic Claude, Claude Code session `session_017Mi3hxjaxV45x6zo6o1ee3`. + +**Effect:** a register-reconciliation *proposal*. +- It introduces **no new mathematical claim**. +- It binds the existing D5 sources to exact bytes and to their nonauthor verdicts, and says which D5 text is now stale. +- It splits off the one genuinely open item. +- The register edits in §6 are for a **non-Claude lane** to check and land. `lemma_closed`, prizes and premises are + untouched. + +## 1. What D5 claims, and what the register says + +**D5**, as the register uses it, is the **all-height first moment near the pins**. The fixed law is: +- the original six pins `M = (−r/2, 0)`, `S = (r/2, 0)`, `f(M) = b`, `f(S) = b − kr³`, `∇f(M) = ∇f(S) = 0`; +- the Gaussian regression `Q_r` and the original weight `W_r = F_2(H_M)F_1(H_S)`, with `dQ_r^W = (W_r/Z_r) dQ_r`; +- the planar periodized field with fixed `T`, compact `b`, `0 < k_− ≤ k ≤ k_+`, and all frames in `O(2)`. + +The main `STATUS.md` row (main `f43f243`) reads: *"the inner microdisk bound and collar to the reviewed annulus are not +complete, and the claimed summed pin-neighborhood bound remains AMEND."* + +`PROOF_INDEX.md` lists three D5 bullets: pin neighborhoods, the intermediate scale `r ≪ |x| ≪ ρ`, and the shrinking +multiple-witness collision. + +## 2. Reconciled statements + +Scaled coordinates are `X = r(u, v)` from the midpoint, so the pins are `(±1/2, 0)`. `N_j` counts index-`j` critical +points. Constants are existential: no numerical `C` or `r_*`. + +| ID | Statement | Heights | +|---|---|---| +| **D5-a** (P2) | `E_{Q_r^W} N_j(M + rE) ≤ C r³|E|` for Borel `E ⊂ {0 < p² + q² ≤ 1/16}`, and the same at `S` | all | +| **D5-b** (microdisk) | the nested disks of radius `κr²` about each pin, for fixed `κ`: `O(r⁵)` | all | +| **D5-c** (C1) | `E_{Q_r^W} N_j(rE) ≤ C r³|E|` for Borel `E ⊂ C_{η,R}`, `η ≤ 1/4`, fixed `R ≥ 1` | all | +| **D5-d** (C2), the **summed pin-neighborhood bound** | the same on `{|(u,v)| ≤ R} ∖ {(±1/2, 0)}`, so `E_{Q_r^W} N_j(r B_R ∖ {M,S}) ≤ C_R r³` | all | +| **D5-e** (I3/I4) | the dyadic shells and `{A_0 r ≤ |X| ≤ ρ}`: `E N ≤ C r³(A_0^{−2} + ρ²)` | window `I_r` | +| **D5-f** (I5) | `E_{Q_r^W} N_{r,j}(T² ∖ {M,S}) ≤ C r³`; planar `c r³ ≤ P(A_r) ≤ C r³` | window `I_r` | + +D5-d with `R ≥ B` covers the reviewed fixed annulus `A ≤ |x| ≤ B` of PR28. That is the "collar to the reviewed annulus". + +## 3. Components, exact bytes and nonauthor verdicts + +All paths are on Math- `main` `e947ade51defcec1fb75444efbe91ac7fcf43f60`. The P2 continuum record landed via #139 +at blob `4904e3d9…`. Items marked *external* are cited but not byte-checked: `main#76` and the #105 comments. + +| Component | Source (author) | Blob / SHA256 | Nonauthor verdicts (record) | +|---|---|---|---| +| P2 punctured pin disks | `reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md` (OpenAI) | `d8acf0bc…` / `f972e50b…` | xAI/Grok #111 (`reviews/d5_punctured_pin_nonauthor_20260928/`): algebra (P4), (P9), (P17), (P20) ACCEPT, continuum HOLD. Claude (`reviews/d5_punctured_pin_continuum_claude_20260929/`, landed via #139, `REVIEW.md` blob `4904e3d9…`, **mandatory** in the checker): continuum and **(P2) ACCEPT existential**, committing the #105 review 5872932723 (*external*). | +| C1 collar, C2 synthesis | `reviews/d5_local_collar_20260928/COLLAR_PROOF.md` (OpenAI) | `b5647907…` / `794babe0…` | `reviews/d5_collar_count_20260928/` (#119; lane not named in the file): C1 **ACCEPT existential**, C2 **ACCEPT corollary**. Claude #105 review 5872932723 (*external*): priorities (i)–(vii) confirmed at the analytic level (§4). | +| Reviewed fixed annulus | `frontiers/rn_annulus_bridge_20260925/PROOF.md` (PR28) | `6f317515…` / `d55e2c03…` | `reviews/pr28_annulus_bridge_nonauthor_20260925/`: R1–R7 ACCEPT | +| I3, I4, I5 | `frontiers/intermediate_window_20260928/PROOF.md` (OpenAI) | `f53a527c…` / `b3eb9456…` | xAI/Grok `reviews/d5_i5_planar_f_20260928/`: I3/I4 **ACCEPT existential**, I5 ACCEPT. Claude `reviews/d5_intermediate_window_claude_20260928/`: I3/I4 ACCEPT, I5 ACCEPT with the explicit tiling (N1). | +| Remote Theorem A (D4) | `frontiers/remote_window_20260924/PROOF.md` | `b383bfcc…` / `a332bae9…` | STATUS D4 ACCEPT (main#76, *external*) | +| `P_η` fixed-separation factorial moment | `frontiers/remote_collision_20260928/PROOF.md` (Claude) | `7b48a88e…` / `b9b8b58f…` | xAI/Grok `reviews/d5_remote_collision_grok_20260928/`; `reviews/d5_offpin_second_moment_20260928/`: `P_η` ACCEPT | + +**Consumption graph.** + + D5-a ← P2 + D5-b ← P2 (nested-disk corollary) + D5-c ← C1, which imports the normalizer (P14) and the §3 moments from P2 + D5-d ← D5-a + D5-c at η = 1/4 + D5-e ← I3, I4 + D5-f ← D5-d (R = 4) + I4 (A_0 = 4, ρ = s_0) + remote Theorem A (ρ = s_0) + +`reconciliation_check.py` checks the tiling exactly on a rational grid. + +## 4. The #105 collar verdict, committed + +My analytic review of `COLLAR_PROOF.md` on #105 (5872932723, runner replay 5872950920, 2026-09-28, at the same bytes) +confirmed all seven requested priorities: + +1. **Interpolation.** Degree five, with cardinal properties at three sites, including collinear triples. +2. **Floor versus error.** The `r⁵` floor dominates the `r⁶` error: `V_r = B_c D_r J_5 + O(r⁶)`. +3. **Schur floor.** Deleting the auxiliary value and applying the Schur floor gives `Σ_X ≥ c r^{10} I`. +4. **Whole-field moments.** They hold at the coarse powers `r^{−60}` and `|v|^{−24}`. +5. **Drift.** `|D| ≥ η²/4` on the collar, via the joint-density inequality (C10). +6. **Split and ledgers.** The `r^{1/3}` split and the ledgers (C11), (C13), (C14), (C19) hold. +7. **Cover.** `η = 1/4` and `R > A` give compatibility with the pin disk and with PR28's annulus. + +The `RESULTS.json` flag `continuum_verified: false` in `reviews/d5_collar_count_20260928/` is read here as a statement +about what code verifies. Its `REVIEW.md` table records the analytic ACCEPT. + +## 5. What remains open + +- **D5-open (catalog C6): shrinking multiple-witness collision.** + - **Target.** A torus-wide second factorial moment `E_{Q_r^W} N(N−1)` for the window count, with pairs where at least + one witness is within fixed `η` of a pin. `P_η` covers every pair with both witnesses at distance `≥ η`, at any + mutual separation. + - **What is known.** A validated lower obstruction `E N(N−1) ≥ 2 P(N ≥ 2) ≥ 2c r³` (C2 of #116, recorded in + `docs/integration/2026-09-29-reviewed-window-and-inverse.md`). **No upper bound.** The optimal order is open. +- Numerical constants; uniformity in `T`, in `R ↑` or `η ↓`, and as `k ↓ 0`; `d > 2`; elder pairing. None of these is + claimed by D5. + +## 6. Proposed register transitions (for a non-Claude lane) + +| Surface | Current | Proposed | +|---|---|---| +| main `STATUS.md` | D5 in the not-accepted table (text in §1) | Move to the accepted table: **"D5 — pin neighborhoods / microdisk (first moment)"** with the scope of D5-a…D5-f, existential constants, fixed `T`, compact marks with `k ≥ k_− > 0`, all frames and indices, `d = 2`. Add a not-accepted row **"D5-collision (C6)"** with the §5 text. | +| `PROOF_INDEX.md` | first two D5 bullets under "Open obligations" | Move them to the reviewed section, citing §3's records. Keep the third bullet (shrinking collision) open and point it to C6. | +| `GRAPH.json` | `math.rn-region.pin-collision`, `.mesoscopic-scaled-annulus` and `.intermediate-r-to-rho` are OPEN_ACTIVE | `PROVED_REVIEWED` at the D5-d, D5-d and D5-e scopes respectively. This is realized only by actual source nodes and `required` edges to every evidence component, as declared in `PROPOSED_TRANSITIONS.json`, and never by `coverage_source` metadata. **`math.rn-region.witness-collision` stays OPEN_ACTIVE.** | + +The machine-readable form is `PROPOSED_TRANSITIONS.json`. +- **Dependency chains.** Every transition that uses P2 lists the continuum record, its checker and #111 in its + chain: the D5 row, `pin-collision` and `mesoscopic-scaled-annulus`. +- **Graph nodes.** It proposes real GRAPH nodes, in the D1 `reading_rule_component` form: one per mandatory evidence + file, with `fingerprint` equal to its SHA256 and `source` equal to its path. It also proposes a D5 aggregate node + `math.d5-pin-neighborhood-first-moment`, which stands for the STATUS row. +- **Required edges.** 30 `required: true` edges run from each consuming node to each of its components. +- **Enforcement.** The checker requires every chain to be realized by nodes with matching fingerprints and by the + required edges. A `coverage_source` string alone is not byte invalidation. +- **Declarative only.** `PROPOSED_TRANSITIONS.json` is declarative and is not executed (`"executed": false`, + checked). This PR makes no GRAPH or registry edit. + - A later graph change must create actual source nodes and `required` edges to every mandatory evidence component, + or else a separately tested extension of the existing reverse-impact source reader. + - `coverage_source` metadata is not read by that reader, so it cannot serve as an invalidation repair. + +These transitions rest only on the verdicts in §3. The only Claude-authored mathematical input is `P_η` +(`remote_collision`), and it has xAI/Grok verdicts. The two Claude *review* records (P2 continuum, I3–I5) are joined by +xAI/Grok records on the same objects: algebra and (P17) for P2; I3/I4/I5 for the intermediate window. + +**Independence caveat.** +- For the P2 continuum steps specifically, the only full-depth nonauthor verdict is Claude's. #111 held them. The + integrating lane may want its own check of (P10)–(P12), (P14) and (P18) before landing D5-a. Those are the exact + steps listed in the continuum record. +- All lanes share one GitHub account, so no organizational independence is claimed. + +## 7. Checks + +`reconciliation_check.py` (stdlib only, run from the repository root) has six checks. +- **IDENTITIES.** All 17 sources and records named in §3 exist as regular files, with no symlink anywhere on the path + (file or parent directory), and with the stated SHA256 and git blob. The P2 continuum record and its three companion files are mandatory. +- **VERDICTS.** The exact verdict rows are present, including the continuum record's (P10)–(P20) and (P2) rows. They + are checked, not a generic ACCEPT substring. +- **NEGATIVES.** These are real filesystem faults on a temporary copy of the complete packet. Each is rejected: + - delete the continuum record; + - change one byte, keeping every verdict row; + - replace it by a symlink to identical bytes; + - replace its parent directory by a symlink to an identical directory. +- **TRANSITIONS.** `PROPOSED_TRANSITIONS.json` names only inventoried files. Every P2-dependent chain contains the + continuum record, its checker and #111. Every chain is realized by component nodes whose fingerprint equals the + inventory SHA256, and by required edges. `witness-collision` and C6 stay open. +- **TILING.** The D5-f tiling covers an exact rational grid of `T² ∖ {M,S}`. +- **OPEN_C6.** The collision item is still recorded as open. + +There are eight mutants and each must fail. `optional-continuum` re-creates the v1 defect. The others are +`allow-symlink`, `no-hash`, `drop-dependency`, `drop-edge`, `stale-fingerprint`, `drop-collar` and `close-c6`. + + python -B -S reviews/d5_reconciliation_20260929/reconciliation_check.py # from the repo root; prints RESULTS.json + python -B -S reviews/d5_reconciliation_20260929/reconciliation_check.py --mutant M # exit 1 diff --git a/specifications/lean_statements_20261010/sources/downstream-graph/GRAPH.json b/specifications/lean_statements_20261010/sources/downstream-graph/GRAPH.json new file mode 100644 index 000000000..220b56867 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/downstream-graph/GRAPH.json @@ -0,0 +1,1041 @@ +{ + "schema_version": 1, + "object": "DOWNSTREAM-HARD-GATE-20260925-v1", + "campaign": { + "main_issues": [ + 86, + 90 + ], + "disposition": "research-integrity control; scientific effect NONE" + }, + "terminal_classifications": [ + "PROVED_REVIEWED", + "SUPERSEDED_NONBLOCKING", + "REFUTED", + "BLOCKED_ABSENT" + ], + "non_discharge_tokens": [ + "GREEN_CI", + "HASH_MATCH", + "ARCHITECTURAL_ADMISSION", + "NUMERICAL_EXPERIMENT", + "SAME_AUTHOR_REVIEW", + "NAVIGATION_SUCCESS", + "AUTHOR_SELF_CHECK" + ], + "nodes": { + "hist.rnu_env.py": { + "layer": "D0", + "kind": "carrier", + "classification": "BLOCKED_ABSENT", + "controlling": false, + "fingerprint": "ABSENT", + "notes": "Historical D3 carrier; no successor claims byte identity" + }, + "hist.CL_ANTHROPIC_BUNDLE_2026-09-17_v5.zip": { + "layer": "D0", + "kind": "carrier", + "classification": "BLOCKED_ABSENT", + "controlling": false, + "fingerprint": "ABSENT", + "notes": "Historical archive absent" + }, + "hist.allcell_fdz_enclosures.json": { + "layer": "D0", + "kind": "carrier", + "classification": "BLOCKED_ABSENT", + "controlling": false, + "fingerprint": "ABSENT", + "notes": "Historical cell carrier absent" + }, + "hist.ENV-RESCOV": { + "layer": "D0", + "kind": "predicate", + "classification": "OPEN_HISTORICAL", + "controlling": false, + "fingerprint": "status-rn-unif-ordered-wall-1", + "notes": "OPEN for historical D3; qualitatively rederived on newer routes where proved" + }, + "hist.ALLCELL-FDZ-Q4": { + "layer": "D0", + "kind": "predicate", + "classification": "OPEN_HISTORICAL", + "controlling": false, + "fingerprint": "status-rn-unif-ordered-wall-2", + "notes": "Not a dependency of the fixed-remote theorem" + }, + "hist.LOGQ-TAIL": { + "layer": "D0", + "kind": "predicate", + "classification": "OPEN_HISTORICAL", + "controlling": false, + "fingerprint": "status-rn-unif-ordered-wall-3", + "notes": "Replaced only inside named successors" + }, + "hist.SYM-Fw-jet": { + "layer": "D0", + "kind": "predicate", + "classification": "OPEN_HISTORICAL", + "controlling": false, + "fingerprint": "status-rn-unif-ordered-wall-4", + "notes": "Analogous interface rederived; no historical certificate identity" + }, + "hist.CH-LIFT": { + "layer": "D0", + "kind": "predicate", + "classification": "OPEN_ACTIVE", + "controlling": false, + "fingerprint": "status-rn-unif-ch-lift", + "notes": "Still mathematically relevant for shrinking charts" + }, + "hist.Piece-2-annulus": { + "layer": "D0", + "kind": "predicate", + "classification": "OPEN_ACTIVE", + "controlling": false, + "fingerprint": "status-rn-unif-piece-2", + "notes": "Active mathematical blocker for shrinking/pin-collision regions" + }, + "hist.OBL-H5-JETMOD": { + "layer": "D0", + "kind": "predicate", + "classification": "OPEN_HISTORICAL", + "controlling": false, + "fingerprint": "status-rn-unif-jetmod", + "notes": "No 2026-09-24 proof claims certified 24-jet enclosure" + }, + "hist.lemma_closed": { + "layer": "D0", + "kind": "register", + "classification": "FALSE", + "controlling": false, + "fingerprint": "lemma_closed=false", + "notes": "Scientific effect NONE; this gate never flips the register" + }, + "math.uniform-matrix-cap-lifetime": { + "layer": "D1", + "kind": "candidate", + "author_provider": "OpenAI", + "classification": "PROVED_REVIEWED", + "controlling": false, + "review_issue": 63, + "fingerprint": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "source": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "notes": "Parent Gaussian selection / Kac-Rice / elder chain. Theorems A, B, C accepted at existential scope, read with the four required reading-rule component nodes (congruence erratum, section 9 replacement v1.1, cap sections 1-5, reconciliation record with W1 and the embedding radius).", + "review_source": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "review_disposition": "ACCEPT_AT_EXISTENTIAL_SCOPE", + "review_providers": [ + "xAI/Grok via Cursor (D1-A-E, section 9 v1.1, partial slices)", + "Anthropic/Claude (A1-A7, cap, section 9 v1.1, sections 11-12)" + ], + "scope": "fixed d>=2 and L; compact births and positive gap marks for Theorems A and B, with B and K of positive length for Theorem B (singleton intervals give zero mark-window intensity); embedding radius r < L/(4 sqrt 2); all frames; Theorem C leading term only; no numerical C, r_*, z_* or coefficient; no d>=3 lower bound; no RN/24-jet", + "reading_rule": [ + "math.d1-component.congruence-erratum", + "math.d1-component.section9-replacement-v1_1", + "math.d1-component.marked-cylinder-cap", + "math.d1-component.reconciliation-record" + ] + }, + "math.d1-component.congruence-erratum": { + "layer": "D1", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.uniform-matrix-cap-lifetime", + "component_role": "amendment", + "author_provider": "OpenAI", + "fingerprint": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028", + "source": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md", + "ledger_items": [ + "A3-E1" + ], + "review_basis": [ + { + "review": "C1", + "provider": "Anthropic", + "verdict": "ACCEPT", + "depth": "full" + }, + { + "review": "G3", + "provider": "xAI", + "verdict": "ACCEPT", + "depth": "full" + } + ], + "review_source": "reviews/d1_chain_reconciliation_20260928/LEDGER.json", + "scope": "congruence factor D_r = diag(r^-1/2, I) in parent section 5; replaces the displayed factor" + }, + "math.d1-component.section9-replacement-v1_1": { + "layer": "D1", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.uniform-matrix-cap-lifetime", + "component_role": "amendment", + "author_provider": "OpenAI", + "fingerprint": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f", + "source": "reviews/d1_section9_borel_repair_20260925/REPAIR.md", + "ledger_items": [ + "D1-B-E2" + ], + "review_basis": [ + { + "review": "C1", + "provider": "Anthropic", + "verdict": "ACCEPT", + "depth": "full" + }, + { + "review": "G2", + "provider": "xAI", + "verdict": "ACCEPT", + "depth": "full" + } + ], + "review_source": "reviews/d1_chain_reconciliation_20260928/LEDGER.json", + "scope": "replaces parent section 9 (Borel elder mark in weighted pair Kac-Rice), C1-C6" + }, + "math.d1-component.marked-cylinder-cap": { + "layer": "D1", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.uniform-matrix-cap-lifetime", + "component_role": "deterministic_support", + "author_provider": "OpenAI", + "fingerprint": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc", + "source": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "ledger_items": [ + "CAP" + ], + "review_basis": [ + { + "review": "C1", + "provider": "Anthropic", + "verdict": "ACCEPT", + "depth": "full" + }, + { + "review": "G3", + "provider": "xAI", + "verdict": "ACCEPT", + "depth": "full" + } + ], + "review_source": "reviews/d1_chain_reconciliation_20260928/LEDGER.json", + "scope": "deterministic separating cap, sections 1-5 (lines 1-168) only; sections 6-8 are not consumed" + }, + "math.d1-component.reconciliation-record": { + "layer": "D1", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.uniform-matrix-cap-lifetime", + "component_role": "reading_rule_record", + "author_provider": "Anthropic", + "source": "reviews/d1_chain_reconciliation_20260928/", + "ledger_items": [ + "S11", + "S12", + "R-W1", + "R-embedding-radius" + ], + "review_basis": [ + { + "review": "C2", + "provider": "Anthropic", + "verdict": "ACCEPT", + "depth": "full", + "item": "S11, S12 (record section 6)" + }, + { + "item": "R-W1", + "raised_in": "C1", + "raised_by_provider": "Anthropic", + "discharged_in": "RECONCILIATION.md section 5" + }, + { + "item": "R-embedding-radius", + "raised_in": "G3", + "raised_by_provider": "xAI", + "discharged_in": "RECONCILIATION.md section 5" + } + ], + "technical_checks_non_discharge": [ + { + "review": "https://github.com/d6g8k5htny-coder/Math-/pull/126#pullrequestreview-5345687857", + "provider": "OpenAI", + "role": "author provider of the parent; A4, section 12 and register binding; never discharge" + } + ], + "review_source": "reviews/d1_chain_reconciliation_20260928/LEDGER.json", + "scope": "nonauthor record binding the reading rule (E1, E2, W1, r < L/(4 sqrt 2)) and the interface-to-review ledger; the whole directory is bound" + }, + "math.lifetime-remainder": { + "layer": "D2", + "kind": "candidate", + "classification": "AUTHOR_SIDE_CANDIDATE", + "controlling": false, + "review_issue": 67, + "fingerprint": "380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a", + "source": "frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md", + "notes": "Imports named #63 interfaces (D1-A-E); not parent collapse; does not consume Theorem A (1.1)", + "review_disposition": "ACCEPT", + "review_provider": "xAI/Grok via Cursor", + "review_source": "https://github.com/d6g8k5htny-coder/main/issues/67#issuecomment-5841782206", + "scope": "Theorem R at its stated existential O(1)-remainder scope; no numerical constants or radii" + }, + "math.side24-coefficient": { + "layer": "D3", + "kind": "candidate", + "classification": "AUTHOR_SIDE_CANDIDATE", + "controlling": false, + "review_issue": 65, + "fingerprint": "c06daccc4ba4b9168522b9888b76a7d599934fc3b91bd753ee5d492262917769", + "source": "coefficients/side24_v1/PROOF.md", + "notes": "Evaluates parent Eq15.2; parent reconciled in reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md", + "review_disposition": "ACCEPT", + "review_provider": "xAI/Grok via Cursor", + "review_source": "https://github.com/d6g8k5htny-coder/main/issues/65#issuecomment-5841269490", + "scope": "coefficient calculation for SIDE24 d=2,3; persistence reading via reconciled parent Theorem C; no finite-radius band" + }, + "math.rn-count-interface": { + "layer": "D4", + "kind": "candidate", + "classification": "AUTHOR_SIDE_CANDIDATE", + "controlling": false, + "review_issue": 67, + "fingerprint": "aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab", + "source": "frontiers/three_fronts_20260924/RN_COUNT_INTERFACE.md", + "notes": "Probability-to-count interface; not an expected-count rate" + }, + "math.rn-fixed-remote-window": { + "layer": "D4", + "kind": "candidate", + "classification": "AUTHOR_SIDE_CANDIDATE", + "controlling": false, + "review_issue": 76, + "fingerprint": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", + "source": "frontiers/remote_window_20260924/PROOF.md", + "notes": "Cubic mean on fixed spatial exclusion and between-pin height window" + }, + "math.rn-region.fixed-remote": { + "layer": "D4", + "kind": "region", + "classification": "COVERED_BY_CANDIDATE", + "controlling": false, + "fingerprint": "dist>=rho and between-pin height", + "notes": "Covered by rn-fixed-remote-window candidate only", + "coverage_source": "math.rn-fixed-remote-window" + }, + "math.rn-region.mesoscopic-scaled-annulus": { + "layer": "D5", + "kind": "region", + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "x=r y with A<=|y|<=B away from pins", + "notes": "Reviewed only at D5-d with R >= B (covers the reviewed fixed annulus); see the complete D5 reconciliation reading rule. No d>2, numerical, elder-selection, JETMOD, or witness-collision closure.", + "scope": "D5-d with R >= B (covers the reviewed fixed annulus)", + "review_source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "review_disposition": "ACCEPT_AT_STATED_SCOPE", + "reading_rule": [ + "math.d5-component.punctured-pin-proof", + "math.d5-component.punctured-pin-review-111", + "math.d5-component.punctured-pin-continuum-review", + "math.d5-component.punctured-pin-continuum-check", + "math.d5-component.collar-proof", + "math.d5-component.collar-review", + "math.d5-component.annulus-proof", + "math.d5-component.annulus-review" + ] + }, + "math.rn-region.pin-collision": { + "layer": "D5", + "kind": "region", + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "near +-e1/2 scaled pin sites", + "notes": "Reviewed only at D5-a, D5-b, D5-d; see the complete D5 reconciliation reading rule. No d>2, numerical, elder-selection, JETMOD, or witness-collision closure.", + "scope": "D5-a, D5-b, D5-d", + "review_source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "review_disposition": "ACCEPT_AT_STATED_SCOPE", + "reading_rule": [ + "math.d5-component.punctured-pin-proof", + "math.d5-component.punctured-pin-review-111", + "math.d5-component.punctured-pin-continuum-review", + "math.d5-component.punctured-pin-continuum-check", + "math.d5-component.collar-proof", + "math.d5-component.collar-review" + ] + }, + "math.rn-region.intermediate-r-to-rho": { + "layer": "D5", + "kind": "region", + "classification": "PROVED_REVIEWED", + "controlling": false, + "fingerprint": "r<<|x|<2, numerical, elder-selection, JETMOD, or witness-collision closure.", + "scope": "D5-e (I3/I4, window I_r)", + "review_source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "review_disposition": "ACCEPT_AT_STATED_SCOPE", + "reading_rule": [ + "math.d5-component.intermediate-window-proof", + "math.d5-component.i5-review-xai", + "math.d5-component.i5-review-claude" + ] + }, + "math.rn-region.witness-collision": { + "layer": "D5", + "kind": "region", + "classification": "OPEN_ACTIVE", + "controlling": false, + "fingerprint": "eta->0 mutual witness separation", + "notes": "Off-pin fixed-separation second-moment evidence is reviewed and sharp global order is reviewed and merged through Math-#145, but the regional shrinking pin/witness-collision mechanism and its source-bound graph fold remain open.", + "scope": "catalog C6", + "review_source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md", + "review_disposition": "OPEN", + "reading_rule": [ + "math.d5-component.offpin-second-moment-review" + ] + }, + "math.rn-mesoscopic-reduction": { + "layer": "D5", + "kind": "candidate", + "classification": "AUTHOR_SIDE_REDUCTION", + "controlling": false, + "fingerprint": "Math-PR7-draft", + "source": "external:Math-PR7", + "notes": "Legacy author-side PR7 route. Regionally superseded/nonblocking for the reviewed fixed-annulus between-pin height-window subregion; still unresolved for scopes outside that theorem." + }, + "math.p15-full-price": { + "layer": "D6", + "kind": "candidate", + "classification": "AUTHOR_SIDE_CANDIDATE", + "controlling": false, + "review_issue": 74, + "fingerprint": "full-price-20260924", + "source": "frontiers/full_price_20260924/PROOF.md", + "notes": "Full probability cube under demand>=2; review open. Price-boundary refutation is a scope boundary, not a required premise." + }, + "math.p15-price-boundary": { + "layer": "D6", + "kind": "candidate", + "classification": "REFUTED", + "controlling": false, + "fingerprint": "498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b", + "source": "frontiers/three_fronts_20260924/P15_PRICE_BOUNDARY.md", + "notes": "Unrestricted same-palette transformed-price extension is false" + }, + "eng.hard-gate": { + "layer": "D7", + "kind": "integrity_control", + "classification": "ENGINEERING_CONTROL", + "controlling": false, + "fingerprint": "downstream-hard-gate-20260925-v1", + "source": "frontiers/downstream_gate_20260925/", + "notes": "This package; never mathematical discharge" + }, + "eng.main-pr87-crosswalk": { + "layer": "D0", + "kind": "engineering", + "classification": "HOLD", + "controlling": false, + "fingerprint": "main-PR87-3d4969ab-ci-unexpected-file", + "source": "https://github.com/d6g8k5htny-coder/main/pull/87", + "notes": "Human D0 crosswalk exists; math_status_check rejects unlisted file until allowlist patch lands" + }, + "eng.d0-packet-allowlist-fix": { + "layer": "D7", + "kind": "engineering", + "classification": "AUTHOR_SIDE_CANDIDATE", + "controlling": false, + "fingerprint": "84b7ad724e4da1c5b4b396c90ae03d4ddedc37771a6e45e1082a45fd80d7ff4a", + "source": "frontiers/downstream_gate_20260925/patches/d0_packet_allowlist.patch", + "notes": "Portable fix verified locally; Cursor cannot push to main (403); awaiting agent/owner with write" + }, + "math.rn-selector-region-crosswalk": { + "layer": "D4", + "kind": "inventory", + "classification": "ENGINEERING_CONTROL", + "controlling": false, + "fingerprint": "rn-selector-region-crosswalk-20260925-v1", + "source": "frontiers/downstream_gate_20260925/SELECTOR_REGION.json", + "notes": "Maps historical selectors onto fixed-remote vs open complements" + }, + "math.rn-fixed-annulus-window": { + "layer": "D5", + "kind": "candidate", + "classification": "AUTHOR_SIDE_CANDIDATE", + "controlling": false, + "fingerprint": "081abc13c5e66342c13df2af2bd6b114f320486f", + "source": "frontiers/rn_thin_tube_20260925/FIXED_ANNULUS_CANDIDATE.md", + "review_source": "reviews/pr22_fixed_annulus_nonauthor_20260925/REVIEW.md", + "scope": "d=2; fixed L; fixed scaled annulus 10; all frames; between-pin height window", + "notes": "Source-bound xAI/Grok technical ACCEPT at exact scoped theorem only. No global D5/RN/JETMOD, k->0, d>=3, pin-neighborhood, intermediate-scale, collision, or elder-selection closure.", + "review_disposition": "ACCEPT", + "review_provider": "xAI/Grok via Cursor", + "scientific_status_unchanged": true + }, + "math.rn-region.fixed-annulus-window": { + "layer": "D5", + "kind": "region", + "classification": "COVERED_BY_CANDIDATE", + "controlling": false, + "fingerprint": "fixed-d2-annulus-between-pin-window-reviewed", + "notes": "Covered by math.rn-fixed-annulus-window only at its exact declared scope.", + "review_disposition": "ACCEPT_AT_SCOPED_SOURCE", + "coverage_source": "math.rn-fixed-annulus-window" + }, + "regional.fixed-annulus.high-jet-route": { + "layer": "D0", + "kind": "regional_supersession", + "classification": "SUPERSEDED_NONBLOCKING", + "controlling": false, + "fingerprint": "regional-bypass-fixed-annulus-window-v1", + "scope": "fixed d=2 scaled annulus; compact positive gaps; between-pin height window", + "notes": "For this scope only, the reviewed analytic route bypasses CH-LIFT / Piece-2 / OBL-H5-JETMOD. Historical predicates remain OPEN/ABSENT in their original replay scopes." + }, + "math.d5-pin-neighborhood-first-moment": { + "layer": "D5", + "kind": "candidate", + "classification": "PROVED_REVIEWED", + "controlling": false, + "scope": "D5-a..D5-f of RECONCILIATION.md section 2; existential constants; fixed T; compact marks with k >= k_- > 0; d = 2", + "source": "reviews/d5_reconciliation_20260929/RECONCILIATION.md" + }, + "math.d5-component.punctured-pin-proof": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "proof", + "author_provider": "OpenAI", + "fingerprint": "f972e50bc07674ebce9d971d1a1bd4dba2f8036dde37ecb57d8a44a35650f76a", + "source": "reviews/d5_local_collar_20260928/PUNCTURED_PIN_PROOF.md", + "review_basis": [ + { + "review": "#111", + "provider": "xAI", + "verdict": "ACCEPT algebra; continuum HOLD", + "depth": "algebraic" + }, + { + "review": "d5_punctured_pin_continuum_claude_20260929", + "provider": "Anthropic", + "verdict": "ACCEPT existential", + "depth": "full" + } + ] + }, + "math.d5-component.punctured-pin-review-111": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "xAI", + "fingerprint": "668d74825f156c929177bc1a3481cf421de4dd7874e7b25b2aece6c10aa7dc0e", + "source": "reviews/d5_punctured_pin_nonauthor_20260928/REVIEW.md" + }, + "math.d5-component.punctured-pin-continuum-review": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "Anthropic", + "fingerprint": "124c472cc064ba24021cb4bb8da1002836a4ce26c8df506228f5c20609f6575a", + "source": "reviews/d5_punctured_pin_continuum_claude_20260929/REVIEW.md" + }, + "math.d5-component.punctured-pin-continuum-check": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_checker", + "author_provider": "Anthropic", + "fingerprint": "57bc87d11271b47ebccec6ea73abf823fa90e3a8c3c7d7cd2165bdc6f7e4175c", + "source": "reviews/d5_punctured_pin_continuum_claude_20260929/pp_continuum_check.py" + }, + "math.d5-component.collar-proof": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "proof", + "author_provider": "OpenAI", + "fingerprint": "794babe0fd039c3a93c401c0b8978316331af009fad45f953e63fbe7e1e31aab", + "source": "reviews/d5_local_collar_20260928/COLLAR_PROOF.md", + "review_basis": [ + { + "review": "d5_collar_count_20260928", + "provider": "not named in record", + "verdict": "ACCEPT existential", + "depth": "full" + }, + { + "review": "Math-#105 comment 5872932723 (external)", + "provider": "Anthropic", + "verdict": "CONFIRMED", + "depth": "full" + } + ] + }, + "math.d5-component.collar-review": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "not named in record", + "fingerprint": "8c8c30cc3b19601a7421341d5da8cb6c982fbc1fcc5ae4791acafa11cc81ee4e", + "source": "reviews/d5_collar_count_20260928/REVIEW.md" + }, + "math.d5-component.annulus-proof": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "proof", + "author_provider": "OpenAI", + "fingerprint": "d55e2c03bb17e7977ff94130cc1ff21e54840cd1e20dc4e41ea9ad52228beb05", + "source": "frontiers/rn_annulus_bridge_20260925/PROOF.md", + "review_basis": [ + { + "review": "pr28_annulus_bridge_nonauthor_20260925", + "provider": "xAI", + "verdict": "ACCEPT R1-R7", + "depth": "full" + } + ] + }, + "math.d5-component.annulus-review": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "xAI", + "fingerprint": "69ea7a479a9e8a5150cbe07b373772ab22413a1d81194d5a6a78adf6ada5f03f", + "source": "reviews/pr28_annulus_bridge_nonauthor_20260925/REVIEW.md" + }, + "math.d5-component.intermediate-window-proof": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "proof", + "author_provider": "OpenAI", + "fingerprint": "b3eb9456d7058b5b78149cfca4e072679beda85cd1a0293d209664a542db66cf", + "source": "frontiers/intermediate_window_20260928/PROOF.md", + "review_basis": [ + { + "review": "d5_i5_planar_f_20260928", + "provider": "xAI", + "verdict": "ACCEPT existential", + "depth": "full" + }, + { + "review": "d5_intermediate_window_claude_20260928", + "provider": "Anthropic", + "verdict": "ACCEPT", + "depth": "full" + } + ] + }, + "math.d5-component.i5-review-xai": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.d5-pin-neighborhood-first-moment", + "component_role": "review_record", + "author_provider": "xAI", + "fingerprint": "527d66e72b079ef7c854987f7e9d44f2252b81aa0e8d9e7ef3b45deb02388659", + "source": "reviews/d5_i5_planar_f_20260928/REVIEW.md" + }, + "math.d5-component.i5-review-claude": { + "layer": "D5", + "kind": "reading_rule_component", + "classification": 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"reading_rule_component", + "classification": "PROVED_REVIEWED", + "controlling": false, + "component_of": "math.rn-region.witness-collision", + "component_role": "review_record", + "author_provider": "nonauthor", + "fingerprint": "43beb4ef7da5bf71e15164d5c4bc1862db387d83f906aebdf9831625b1ca6369", + "source": "reviews/d5_offpin_second_moment_20260928/REVIEW.md" + } + }, + "edges": [ + { + "from": "math.lifetime-remainder", + "to": "math.uniform-matrix-cap-lifetime", + "required": true, + "relation": "imports_named_interfaces" + }, + { + "from": "math.side24-coefficient", + "to": "math.uniform-matrix-cap-lifetime", + "required": true, + "relation": "evaluates_eq_15_2" + }, + { + "from": "math.uniform-matrix-cap-lifetime", + "to": "math.d1-component.congruence-erratum", + "required": true, + "relation": "reads_with_congruence_erratum" + }, + { + "from": "math.uniform-matrix-cap-lifetime", + "to": "math.d1-component.section9-replacement-v1_1", + "required": true, + "relation": "reads_section9_as_replaced_v1_1" + }, + { + "from": "math.uniform-matrix-cap-lifetime", + "to": "math.d1-component.marked-cylinder-cap", + "required": true, + "relation": "consumes_cap_sections_1_5" + }, + { + "from": "math.uniform-matrix-cap-lifetime", + "to": "math.d1-component.reconciliation-record", + "required": true, + "relation": "reads_with_reconciled_rule_W1_radius" + }, + { + "from": "math.rn-fixed-remote-window", + "to": "math.rn-count-interface", + "required": true, + "relation": "supplies_numerator_on_restricted_region" + }, + { + "from": "math.rn-fixed-remote-window", + "to": "math.rn-region.fixed-remote", + "required": false, + "relation": "covers_region" + }, + { + "from": "math.rn-mesoscopic-reduction", + "to": "math.rn-fixed-remote-window", + "required": true, + "relation": "successor_scale_after_fixed_remote" + }, + { + "from": "math.rn-mesoscopic-reduction", + "to": "math.rn-region.mesoscopic-scaled-annulus", + "required": false, + "relation": "targets_region" + }, + { + "from": "math.rn-mesoscopic-reduction", + "to": "hist.CH-LIFT", + "required": false, + "relation": "reopens_scaled_chart_problem" + }, + { + "from": "math.rn-mesoscopic-reduction", + "to": "hist.Piece-2-annulus", + "required": false, + "relation": "addresses_partial_piece" + }, + { + "from": "math.p15-full-price", + "to": "math.p15-price-boundary", + "required": false, + "relation": "boundary_counterexample_not_required_premise" + }, + { + "from": "hist.ENV-RESCOV", + "to": "hist.rnu_env.py", + "required": true, + "relation": "historical_certificate_carrier" + }, + { + "from": "hist.ALLCELL-FDZ-Q4", + "to": "hist.allcell_fdz_enclosures.json", + "required": true, + "relation": "historical_certificate_carrier" + }, + { + "from": "hist.SYM-Fw-jet", + "to": "hist.CL_ANTHROPIC_BUNDLE_2026-09-17_v5.zip", + "required": false, + "relation": "historical_bundle_identity" + }, + { + "from": "eng.main-pr87-crosswalk", + "to": "eng.d0-packet-allowlist-fix", + "required": true, + "relation": "ci_unblock_depends_on_allowlist_patch" + }, + { + "from": "math.rn-selector-region-crosswalk", + "to": "math.rn-fixed-remote-window", + "required": false, + "relation": "inventories_covered_region" + }, + { + "from": "math.rn-selector-region-crosswalk", + "to": "math.rn-count-interface", + "required": false, + "relation": "inherits_no_markov_reversal" + }, + { + "from": "math.rn-fixed-annulus-window", + "to": "math.rn-count-interface", + "required": true, + "relation": "supplies_expected_count_on_reviewed_scoped_region" + }, + { + "from": "math.rn-fixed-annulus-window", + "to": "math.rn-region.fixed-annulus-window", + "required": false, + "relation": "covers_reviewed_subregion" + }, + { + "from": "math.rn-fixed-annulus-window", + "to": "regional.fixed-annulus.high-jet-route", + "required": false, + "relation": "regionally_supersedes_historical_high_jet_route" + }, + { + "from": "regional.fixed-annulus.high-jet-route", + "to": "hist.CH-LIFT", + "required": false, + "relation": "regional_bypass_only" + }, + { + "from": "regional.fixed-annulus.high-jet-route", + "to": "hist.Piece-2-annulus", + "required": false, + "relation": "regional_bypass_only" + }, + { + "from": "regional.fixed-annulus.high-jet-route", + "to": "hist.OBL-H5-JETMOD", + "required": false, + "relation": "regional_bypass_only" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.punctured-pin-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.punctured-pin-review-111", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.punctured-pin-continuum-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.punctured-pin-continuum-check", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.collar-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.collar-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.annulus-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.annulus-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.intermediate-window-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.i5-review-xai", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.i5-review-claude", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.d5-pin-neighborhood-first-moment", + "to": "math.d5-component.remote-window-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.punctured-pin-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.punctured-pin-review-111", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.punctured-pin-continuum-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.punctured-pin-continuum-check", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.collar-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.pin-collision", + "to": "math.d5-component.collar-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.punctured-pin-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.punctured-pin-review-111", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.punctured-pin-continuum-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.punctured-pin-continuum-check", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.collar-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.collar-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.annulus-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.mesoscopic-scaled-annulus", + "to": "math.d5-component.annulus-review", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.intermediate-r-to-rho", + "to": "math.d5-component.intermediate-window-proof", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.intermediate-r-to-rho", + "to": "math.d5-component.i5-review-xai", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.intermediate-r-to-rho", + "to": "math.d5-component.i5-review-claude", + "required": true, + "relation": "requires_evidence" + }, + { + "from": "math.rn-region.witness-collision", + "to": "math.d5-component.offpin-second-moment-review", + "required": true, + "relation": "requires_evidence" + } + ] +} diff --git a/specifications/lean_statements_20261010/sources/elder-lower-density-gap/ELDER_LOWER_AND_DENSITY_GAP.md b/specifications/lean_statements_20261010/sources/elder-lower-density-gap/ELDER_LOWER_AND_DENSITY_GAP.md new file mode 100644 index 000000000..aaefc8da9 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/elder-lower-density-gap/ELDER_LOWER_AND_DENSITY_GAP.md @@ -0,0 +1,205 @@ +# A direct planar elder-failure lower bound and the sharp compact-window density loss + +Object: OA-ELDER-LOWER-DENSITY-GAP-20260928-v1. +Author: OpenAI / ChatGPT, foreground research session, 28 September 2026. +Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR ANALYTIC REVIEW REQUIRED. +Scientific effect: NONE. This does not change any historical certificate or governing status. + +## 1. Exact elder probability and conclusions + +Use the exact planar periodized Gaussian field and the original typed pair law +Q_r^W defined in LOCAL_MULTIPLICITY.md. The torus L is fixed, birth b is in a fixed +compact interval, k is in [k_-,k_+] with k_->0, and frames range over O(2). The six +pins are M=-(r/2)u, S=(r/2)u, f(M)=b, f(S)=b-k r^3, and both gradients zero. +W_r=F_2(H_M)F_1(H_S) and the FULL normalizer is Z_r=E_Qr W_r. This is not a law +conditioned on adjacency. + +Let p_r(b,k,R) be the probability that the global ordinary superlevel elder death +partner of M is S. This is **exactly** the p_r definition in [LP, Section 1], not +a new selector. On the generic locus its death height is the maximin + + d_f(M)=sup_{gamma(0)=M, f(gamma(1))>f(M)} min_t f(gamma(t)), (E1) + +with the empty supremum convention of [LP, Section 8]. A path to a point strictly +above the birth height, entirely above f(S), therefore prevents S from being M's +elder death partner. No gradient-trajectory or saddle-saddle transversality theorem +is required for this sufficient obstruction. + +**Theorem E1 (direct lower bound).** Uniformly on the stated compact planar model, +there are c,r_*>0 such that + + 1-p_r(b,k,R) >= c r^3, 0= b-k r^3/4 > f(S), + d_f(M)-f(S) >= 3k r^3/4. (E3) + +It does not use a subtraction of competing witness-intensity bounds, numerical +rungs, an asymptotically fitted exponent, or an assumed independence of witnesses. +In particular a window-count lower bound alone would not prove (E2); the explicit +path below is essential. + +**Corollary E2 (two-sided selection rate, with a separate upper input).** If the +precise Theorem A upper bound in [LP] is consumed, then in d=2 + + c r^3 <= 1-p_r(b,k,R) <= C r^3. (E4) + +The lower proof below is independent of that upper theorem. A defect in the upper +input would affect (E4)'s upper half, not the direct lower mechanism (E2). + +## 2. A rational polygonal path with strict quantitative margins + +Use the local scaled cubic + + G_k(X,Z)=k(2X^3-3X/2-1/2-3Z^2/4-XZ^2). + +Consider the three-segment path + + (-1/2,0) -> (-3/4,0) -> (-3/4,9/4) -> (-1,9/4). (E5) + +Every segment lies in B(0,3): the largest endpoint squared norm is 97/16<9, +and the ball is convex. On the three segments, parameterized consecutively by +v in [0,1], the exact restricted polynomials divided by k are + + g_1(v)=-3v^2/16-v^3/32, + g_2(v)=-7/32, + g_3(v)=-7/32+51v/64-9v^2/32-v^3/32. (E6) + +The first is nonincreasing, the second constant, and + + g_3'(v)=(51-36v-6v^2)/64 >= 9/64>0. + +It follows that the minimum on the entire path is -7k/32, while its final value +is +17k/64. The path passes through a level region, but its proof does not rely on +that level remaining exactly constant under perturbation. + +If a field F on B(0,3) obeys + + ||F-G_k||_infinity < k/64, F(-1/2,0)=0, + +then the entire path has F-values above -15k/64>-k/4 and the endpoint has value +above 16k/64=k/4>0. Consequently, for the physical field with +F(X,Z)=(f(rX,rZ)-b)/r^3, the same path scaled by r connects M to a point with +value above b, entirely above b-k r^3/4. Definition (E1) proves (E3). + +This is stable C^0 connectivity, not a statement that a prescribed path is a +separatrix. It remains a sufficient elder obstruction even without identifying +the exact saddle where the merger occurs. The existence of the two extra saddles +proved in the local multiplicity module is useful additional information, but is +not needed for the path argument. + +## 3. Positive probability of the path under the actual tilted Gaussian law + +For completeness, the ingredients from the local module can be isolated without +using its global upper bound or its extra-saddle implicit-function argument. +With s=f_zz(0)/r, a=f_xxz(0), beta=f_xzz(0), c=f_zzz(0), the exact six pins give + + (f(rX,rZ)-b)/r^3 + =2kX^3-3kX/2-k/2+(s/2)Z^2 + +(a/2)(X^2-1/4)Z+(beta/2)XZ^2+(c/6)Z^3 + +O_{C^2(B3)}(r||f||C4). (E7) + +The term -aZ/8 retains both transverse gradient pins. The target values +s=-3k/2, a=0, beta=-2k, c=0 recover G_k. Choose a fixed small epsilon and the +physical jet box + + |f_zz+(3k/2)r|= c_2 r^4 + +on this event. The full normalizer obeys 0 A_0>0 (E9) + +uniformly on the compact parameters, where sigma is the ordinary surface measure +and pi_r is the density of the endpoint observation frame. Selected intensity has +the same factors times p_r. There is no factor 1/2 for the differently typed, +ordered maximum/saddle pair. + +Changing variables ell=k r^3 at fixed k gives the exact density-gap version + + nu_cand(ell)-nu_eld(ell) + =(ell^(-1/3)/3) integral_B integral_K integral_S + k^(-2/3) A_{(ell/k)^(1/3)}(b,k,u) + [1-p_{(ell/k)^(1/3)}(b,k,u)] d sigma dk db. (E10) + +For sufficiently small ell all these radii are below the common cutoff. The +uniform positive lower and finite upper bounds for A_r, and (E2)/(E4), imply + + c_{B,K} ell^(2/3) + <= nu_cand(ell)-nu_eld(ell) + <= C_{B,K} ell^(2/3). (E11) + +The lower follows directly from (E2); the upper consumes the already identified +parent upper theorem. The k factor in its bounding integral is k^(-5/3), since +r^3=ell/k. It is finite and has positive integral on the fixed gap window. + +Thus the parent's compact-window O(ell^(2/3)) **selection-density loss** is optimal +in order in the planar model; it cannot be replaced by o(ell^(2/3)) under these +assumptions. This does not prove sharpness of the different O(1) remainder in a +leading density expansion, and does not remove the compact-mark restriction. +Combining with nu_cand(ell)~c ell^(-1/3) gives a rejected-pair fraction of order ell. +Integrating (E11) from zero to epsilon gives a cumulative rejected-pair intensity +of order epsilon^(5/3), versus total compact-window intensity of order + epsilon^(2/3). No leading numerical constant is determined by these inequalities. + +## 5. Relation to the lower-rate campaign and review boundaries + +The new direct event proves a uniform, all-small-r positive lower constant for +the exact typed pair law and ordinary elder selector defined in [LP]. Fixing any +birth and any positive gap mark in these ranges gives the corresponding one-mark +specialization. This is an existential analytic lower theorem, not a collection +of finite-r numerical checks. It does not rely on any disputed window-saddle +Cauchy-Schwarz estimate or on its numerical budget subtraction. + +A historical artifact using the symbol q(r,b) is not automatically identified +with this p_r: its covariance, scaling convention, pin meanings, tilt and selector +must match. No historical numerical c,r_0, finite-band certificate, lower-campaign +registry bit, or prize is advanced here. In particular an adjacency-conditioned +law is a different law and is excluded from the statement. + +The ordinary elder obstruction itself uses only the maximin definition and the +strict polygonal path, not the SARD-G no-saddle-connection theorem. The local +Gaussian jet-density and extra-conditioned C^4 control are the core new analytic +obligations. The parent upper theorem and weighted density identity are separately +named dependencies for (E4) and (E11). This separation permits a reviewer to accept +or reject the direct lower mechanism without inheriting a broader parent verdict. diff --git a/specifications/lean_statements_20261010/sources/landing-claims/LANDING_CLAIMS.json b/specifications/lean_statements_20261010/sources/landing-claims/LANDING_CLAIMS.json new file mode 100644 index 000000000..a8719fcd7 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/landing-claims/LANDING_CLAIMS.json @@ -0,0 +1,383 @@ +{ + "schema_version": 1, + "object": "MATH-LANDING-CLAIMS-20260925-v1", + "repository": "d6g8k5htny-coder/Math-", + "audited_default_commit": "760340e921ac4ceda296b8118da936f1133e956e", + "landing_path": "README.md", + "live_queue": { + "repo": "d6g8k5htny-coder/main", + "issue": 86, + "url": "https://github.com/d6g8k5htny-coder/main/issues/86", + "required_state": "closed" + }, + "allowed_dispositions": [ + "HOLD_WITH_DOMAIN", + "FAIL_CLOSED", + "EXACT_COUNTEREXAMPLE", + "REVIEWED_SCOPED", + "ENGINEERING_HOLD" + ], + "claims": [ + { + "claim_id": "side24-coefficient", + "advertised_label": "Gaussian lifetime coefficient", + "claim_type": "derived numerical enclosure of imported expression", + "statement_path": "coefficients/side24_v1/PROOF.md", + "statement_heading": "Theorem", + "source": { + "commit": "760340e921ac4ceda296b8118da936f1133e956e", + "blob": "44b66f04f89fcd87383b3603fa69f1feb64cdddd" + }, + "domain": "exact normalized SIDE24 periodized Gaussian model; dimensions d=2 and d=3; coefficient defined by parent main#63 Eq.15.2", + "measure": "scalar coefficient in the parent per-unit-volume lifetime-density normalization", + "required_dependencies": [ + { + "kind": "support", + "id": "main#63:eq15.2-parent", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "source": { + "commit": "c39164254ad1529ed49543743598d5dcb24bab98", + "blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2" + }, + "review": { + "kind": "github_issue", + "repo": "d6g8k5htny-coder/main", + "number": 63, + "url": "https://github.com/d6g8k5htny-coder/main/issues/63", + "comments": [ + "https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5841570965", + "https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5873873903" + ], + "reconciliation": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md" + }, + "required_for": "persistence interpretation" + } + ], + "review": { + "kind": "github_issue", + "repo": "d6g8k5htny-coder/main", + "number": 65, + "url": "https://github.com/d6g8k5htny-coder/main/issues/65" + }, + "disposition": "REVIEWED_SCOPED", + "disposition_reason": "Coefficient calculation verified (xAI/Grok C1-C6, main#65); its persistence interpretation as the leading coefficient of the parent Theorem C (eq. 15.2) is reconciled in reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md. Arithmetic enclosure of the stated expression only; no finite-radius band." + }, + { + "claim_id": "lifetime-remainder", + "advertised_label": "Quantitative lifetime density", + "claim_type": "derived theorem", + "statement_path": "frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md", + "statement_heading": "Theorem R", + "source": { + "commit": "760340e921ac4ceda296b8118da936f1133e956e", + "blob": "247b3ecf80bfbe896948d5d489b2d5842a81c481" + }, + "domain": "each fixed torus d>=2,L>0; all birth/gap marks, orientations and separations; finite ordinary H0 bars with essential global maximum excluded", + "measure": "per-unit-volume lifetime density with respect to d ell", + "required_dependencies": [ + { + "kind": "support", + "id": "main#63:matrix-cap-lifetime", + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "source": { + "commit": "c39164254ad1529ed49543743598d5dcb24bab98", + "blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2" + }, + "review": { + "kind": "github_issue", + "repo": "d6g8k5htny-coder/main", + "number": 63, + "url": "https://github.com/d6g8k5htny-coder/main/issues/63", + "comments": [ + "https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5841570965", + "https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5873873903" + ], + "reconciliation": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md" + }, + "required_for": "marked Kac-Rice, global elder, regression and parent coefficient interfaces" + }, + { + "kind": "support", + "id": "marked-cylinder-cap", + "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md", + "source": { + "commit": "c39164254ad1529ed49543743598d5dcb24bab98", + "blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1" + }, + "review": { + "kind": "github_issue", + "repo": "d6g8k5htny-coder/main", + "number": 63, + "url": "https://github.com/d6g8k5htny-coder/main/issues/63", + "comments": [ + "https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5841570965", + "https://github.com/d6g8k5htny-coder/main/issues/63#issuecomment-5873873903" + ], + "reconciliation": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md" + }, + "required_for": "deterministic sufficient pairing event" + } + ], + "review": { + "kind": "github_issue", + "repo": "d6g8k5htny-coder/main", + "number": 67, + "url": "https://github.com/d6g8k5htny-coder/main/issues/67" + }, + "disposition": "REVIEWED_SCOPED", + "disposition_reason": "Theorem R accepted at its stated existential O(1)-remainder scope (xAI/Grok R1-R6, main#67); its parent imports (D1-A-E: section 8 comparison, section 9 as replaced by v1.1, section 10, (14.1), (15.2)) are reconciled in reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md. Theorem R does not consume Theorem A's cubic bound (1.1). 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excludes the c_v<=p_v price theorem and broader parent acceptance" + }, + "required_for": "actual capacity restrictions and setwise same-palette full-block coverage only; Theorem F derives its transformed-price budget separately" + } + ], + "review": { + "kind": "local_file", + "path": "reviews/p15_full_price_nonauthor_20260926/REVIEW.md", + "source": { + "commit": "1e1114f5eb8ef8cdbcde591bf74126274c250f88", + "blob": "07db19f826763b8faa7414ba2e403d9c45bc0b6e" + }, + "disposition": "ACCEPT", + "scope": "Theorem F on the realized disjoint capacity/clutter family with a_i>=1, d_i>=2, all independent p in [0,1]^X, c_v<=phi(p_v), same palette K>=K_H(d), sharp rho*=1/[3-log(3e-2)]" + }, + "disposition": "REVIEWED_SCOPED", + "disposition_reason": "source-bound xAI/Grok nonauthor technical review ACCEPTS Theorem F at its exact realized-family scope; no arbitrary-downset, demand-one, or unrestricted P15-prize transfer" + }, + { + "claim_id": "downstream-hard-gate", + "advertised_label": "Downstream hard gate", + "claim_type": "engineering integrity control", + "statement_path": "frontiers/downstream_gate_20260925/README.md", + "statement_heading": "What it enforces", + "source": { + "commit": "2e0a7b37d18277c5965e9a65adabcb26c7e44234", + "blob": "6ff49b966ab41dc4db771d48e547cc60da11764f" + }, + "domain": "machine dependency graph and repository-source transitions", + "measure": "not a mathematical measure; loss-only revalidation/control state", + "required_dependencies": [], + "review": { + "kind": "github_issue", + "repo": "d6g8k5htny-coder/main", + "number": 90, + "url": "https://github.com/d6g8k5htny-coder/main/issues/90" + }, + "disposition": "ENGINEERING_HOLD", + "disposition_reason": "local integrity semantics exist; repository-wide external controlling-source binding remains unresolved" + } + ], + "forbidden_global_inferences": [ + "global RN closure", + "unrestricted P15 prize closure", + "unconditional acceptance of the lifetime parent", + "lemma_closed=true" + ], + "meaning": "machine-readable landing inventory; source/review identity and scope only, never theorem acceptance by itself" +} diff --git a/specifications/lean_statements_20261010/sources/lifetime-parent/UNIFORM_MATRIX_CAP_AND_LIFETIME.md b/specifications/lean_statements_20261010/sources/lifetime-parent/UNIFORM_MATRIX_CAP_AND_LIFETIME.md new file mode 100644 index 000000000..dfed3b8d3 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/lifetime-parent/UNIFORM_MATRIX_CAP_AND_LIFETIME.md @@ -0,0 +1,494 @@ +# Uniform matrix-cap selection and unrestricted short-lifetime asymptotics + +**Object:** UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1. +**Author:** OpenAI / ChatGPT, 24 September 2026. +**Disposition:** author-side mathematical candidate; no nonauthor acceptance, formal verification, numerical finite-band certificate, or program-level closure is asserted. + +This is an additive continuation of MARKED-CYLINDER-CAP-20260924-v1, SHA256 `0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc`, reproduced unchanged in `sources/`. It addresses the matrix-probability, angular-uniformity, compact-mark, and lifetime-composition interfaces of campaign #61. It does **not** extend the predecessor's numerical constants to new parameters. Instead it derives new, non-numerically-evaluated constants by a uniform Gaussian argument. + +## 1. Model, convention, and the three theorems + +Fix a dimension d >= 2 and a side length L > 0. On the flat torus X = R^d/(L Z^d), let f be the centered stationary Gaussian field of variance one with covariance + + K_L(z) = sum_{n in Z^d} exp(-|z+Ln|^2/2) + / sum_{n in Z^d} exp(-|Ln|^2/2). + +In particular d=3, L=24 is an actual three-dimensional SIDE24 application. No replacement by a nonperiodic covariance is made. Fix compact intervals B=[b_-,b_+] and K=[k_-,k_+] with 00, use the embedded local cylinder centered at the origin with pins + + M=-(r/2)u, S=(r/2)u, + f(M)=b, f(S)=s=b-k r^3, + grad f(M)=grad f(S)=0. + +Q_{r,b,k,R} is the continuous Gaussian regression law at these 2(d+1) observations, not a conditioning on a positive-probability event. A maximum has d negative Hessian eigenvalues; the relevant saddle has d-1 negative and one positive eigenvalue. Set + + W_r = |det H_M det H_S| 1{H_M<0, index(H_S)=d-1}, + Z_r = E_Q W_r, dQ^W = (W_r/Z_r)dQ. + +The index counts negative eigenvalues. The FULL normalizer Z_r has no extra pin-density, coordinate Jacobian, or adjacency condition. Let p_r(b,k,R) be the Q^W probability that the **global ordinary superlevel elder death partner** of M is S. + +**Theorem A (uniform selection candidate).** There exist r_*>0 and C=z_* r^2 and the full transverse-matrix boundary layer, including simultaneous small transverse eigenvalues. It does not presume isotropy under arbitrary rotations of the torus. + +**Theorem B (compact-mark density candidate).** Suppose B and K have positive lengths. Count ordered maximum/saddle pairs with 00, + 0 <= nu_cand(ell)-nu_eld(ell) <= C_{B,K} ell^(2/3), (1.2) + +as ell decreases to zero. An exact finite-dimensional Gaussian integral for c_{B,K} appears in Section 11. + +Theorem B is a COMPACT MARK WINDOW result, especially its quantitative difference estimate. Theorem C below separately removes the mark and distance restrictions for the LEADING asymptotic. There is no stated numerical value of C, r_*, z_*, or c_{B,K}, and no claim that r_*=1/20. The new asymptotic normalizer proof does not replace or re-certify the historical H3 finite-band floor. + +**Theorem C (unrestricted finite-lifetime leading density candidate).** For the same fixed torus and dimension, let nu_cand^all(ell) count ALL ordered maximum/index-(d-1)-saddle pairs with height difference ell>0, and let nu_eld^all(ell) count all finite ordinary superlevel H0 persistence bars, both in expectation per unit volume. There is a positive finite constant c_{d,L} such that + + nu_cand^all(ell) ~ c_{d,L} ell^(-1/3), + nu_eld^all(ell) ~ c_{d,L} ell^(-1/3). (1.3) + +All birth heights, all positive scaled-gap marks and all spatial separations on the torus are included in (1.3). The essential global-maximum class is excluded from the finite-lifetime count. The constant is an explicitly identified finite Gaussian/angular integral, with the birth and gap integrals analytically eliminated in Section 15. No elementary closed form or numerical enclosure is claimed. Crucially, the compact-window O(ell^(2/3)) DIFFERENCE estimate in (1.2) is not asserted for the unrestricted densities. Theorem C has a separate domination and off-diagonal proof, not a change of label on Theorem B. + +## 2. Positive Fourier spectrum and finite-jet rank + +Poisson summation gives + + K_L(z)=sum_{n in Z^d} a_n exp(2 pi i n.z/L), + a_n=exp(-2 pi^2 |n|^2/L^2) / sum_j exp(-2 pi^2 |j|^2/L^2)>0. + +Thus every Fourier mode has positive variance, and sum_n sqrt(a_n)(1+|n|)^q is finite for every q. A real sine/cosine expansion therefore converges in every C^q norm in every finite L^p norm. In particular the field has a smooth version and all finite moments of its global derivative suprema exist. + +Any finite list of distinct derivative evaluation functionals at distinct sites is linearly independent. Indeed a vanishing-variance linear combination must annihilate every Fourier mode because every a_n is positive. As a distribution it is a finite sum of derivatives of point masses with every Fourier coefficient zero, so it is zero. Smooth test functions supported at one site and with arbitrarily prescribed finite jets show that every coefficient is zero. At one site in a rotated frame the same assertion also follows from a polynomial P(R^T n) vanishing on all n in Z^d: the invertible rotation reduces this to a polynomial vanishing on the full integer lattice, hence the polynomial is zero. + +Only lists without repeated derivative functionals are called distinct. A Hessian is parametrized by its independent symmetric entries, not by d^2 allegedly independent entries. + +These arguments justify positive covariance for the original pins at each pair of distinct points and for every residual jet list modulo the pins used below. They also explain why finitely many observed modes or a numerically positive sampled covariance would not suffice for this proof. + +## 3. An exact nonsingular contact frame for all orientations + +Write x for the axial coordinate and y_1,...,y_m for the transverse coordinates, m=d-1. On the line y=0 set a=-r/2, c=r/2. Order the original observations as + + O_r=(f(a), f_x(a), f(c), f_x(c), + f_y1(a),f_y1(c), ..., f_ym(a),f_ym(c)). + +This is a permutation and orthogonal re-expression of the specified physical pins. Define U_r=T_r O_r by four axial rows + + U0 = (f(a)+f(c))/2, + U1 = (f(c)-f(a))/r, + U2 = (f_x(c)-f_x(a))/r, + U3 = (6/r^2)[f_x(a)+f_x(c)-2(f(c)-f(a))/r], (3.1) + +and, for every transverse direction j, two rows + + Vj0=(f_yj(a)+f_yj(c))/2, + Vj1=(f_yj(c)-f_yj(a))/r. + +The transformation is invertible for r>0 and has exact absolute determinant + + |det T_r| = (12/r^4)(1/r)^m = 12 r^(-(d+3)). (3.2) + +The prescribed target becomes exactly + + v_r=(b-k r^3/2, -k r^2, 0, 12k, 0,...,0). (3.3) + +This is not an approximate transformation, and the factor 12 in (3.2) is retained in the lifetime ledger. + +Taylor formulas with integral remainders give mean-square convergence, uniformly over frames, of U_r to the contact jet + + U0*=(f, f_x, f_xx, f_xxx, + f_y1,f_xy1,...,f_ym,f_xym) at 0. (3.4) + +The potentially dangerous U3 cancellation is an averaged third derivative, not a subtraction of uncontrolled random values. Uniform convergence of its covariances, and of cross-covariances with any fixed finite derivative order, follows either from those integral formulas or from the rapidly summable Fourier series. The target v_r converges uniformly on B x K to v_0=(b,0,0,12k,0,...,0). + +The functionals in (3.4) are distinct. Hence their covariance Sigma_0(R) is positive definite by Section 2. Its entries are continuous in R. The orthogonal group is compact, so its least eigenvalue has a positive uniform minimum. Consequently Sigma_r(R) remains uniformly positive definite for all sufficiently small r; Sigma_r and Sigma_r^{-1} converge uniformly. + +This compactness argument establishes existence, not an evaluated interval enclosure. It never equates Sigma_r at different orientations. Full torus rotational invariance is neither assumed nor true in general. + +Let A_r=D_y^2 f(M), represented by its m(m+1)/2 independent entries. Append these entries to U_r. At r=0 the extra functionals are f_yi_yj, i<=j; none duplicates a functional in (3.4). The enlarged covariance is therefore uniformly positive definite at contact and for small r. Its Schur complement shows that the Q-law of A_r has covariance uniformly bounded above and uniformly bounded away from zero. Its mean is uniformly bounded. In symmetric-matrix Lebesgue coordinates it has a density satisfying + + density_Q(A) <= C_0 exp(-c_0 ||A||_F^2), (3.5) + +for fixed positive C_0,c_0. This remains valid after any change of eigenvector orientation. Orthogonal invariance of the density is not required. + +## 4. Uniform conditional derivative moments and the independent residual + +Because Sigma_r^{-1} and v_r are uniformly bounded, the regression energy v_r^T Sigma_r^{-1}v_r is uniformly bounded. For each unconditioned standardized Fourier coefficient xi, covariance Cauchy-Schwarz bounds its conditional mean by the square root of this energy, and its conditional centered variance by one. Minkowski's inequality and the summable Fourier coefficients then give, for every finite p and derivative order q, + + sup_{r,b,k,R} || ||f||_{C^q(X)} ||_{L^p(Q)} < infinity. (4.1) + +The same proof applies to a centered residual after further conditioning on A_r. Write, in one full-field Gaussian coupling, + + f(z)=mu_r(z)+B_r(z).(A_r-E_Q A_r)+g_r(z), (4.2) + +where g_r is jointly independent of the entire symmetric matrix A_r. The regression coefficient B_r and its derivatives through order four are uniformly bounded, by the uniform inverse covariance from Section 3 and bounded derivative cross-covariances. The residual has uniformly bounded C^4 moments of every finite order. + +Independence in (4.2) is not asserted separately for Hessian entries, endpoint variables, or sampled derivative maxima. All residual evaluations are orthogonal to A_r; Gaussian independence on a countable dense collection and continuity give independence of the random smooth field. The residual's internal correlations are retained. + +Let J_r=1+||g_r||_{C^4(X)}, with a fixed coordinate norm. The partial-block operator norms of the cap theorem differ from this coordinate norm by finite constants depending only on d. Hence, for M3,M4 on the local cylinder, + + M3 <= K_0(J_r+||A_r||_F), + M4 <= K_0(J_r+||A_r||_F), + J_r >= 1, J_r independent of A_r, + sup ||J_r||_p < infinity for every finite p. (4.3) + +No inverse Hessian moment is used. Gaussian pin covariance and random field Hessians are different objects. + +## 5. Endpoint identities and the actual full normalizer + +For either pin i in {M,S}, put + + alpha_i=f_xx(i)/r, beta_i=grad_y f_x(i)/r, + A_i=D_y^2 f(i), + H_i=[[r alpha_i, r beta_i^T], [r beta_i,A_i]]. + +The two gradient pins imply the vector and scalar average bounds + + |alpha_i| <= M3/2, ||beta_i|| <= M3/2, + ||A_S-A_M||_op <= r M3. (5.1) + +For example integral_a^c f_xx=0; subtract this average at an endpoint and integrate the M3-Lipschitz bound. For beta use the average of grad_y f_x and its vector derivative. This does not invoke a simultaneous vector Rolle zero. + +The exact height identity gives a weighted mean of f_xxx equal to 12k on [a,c]. Variation of f_xxx over that interval is at most rM4. The same average calculation yields + + |alpha_M+6k| <= r M4/2, + |alpha_S-6k| <= r M4/2. (5.2) + +The block determinant identity, valid even when alpha_i or A_i is zero, is + + det H_i/r = alpha_i det A_i + -r beta_i^T adj(A_i) beta_i. (5.3) + +By (4.1),(5.1), det H_i/r and W_r/r^2 have uniformly bounded moments of every finite order. + +Let A_0 be D_y^2 f(0) under regression on U0*=v_0. Its law is a full-support nondegenerate Gaussian on symmetric matrices. Finite-dimensional Gaussian regression gives uniform convergence of the means and covariances of A_M to those of A_0. Equations (5.1),(5.2) imply that the congruence-scaled endpoint Hessians converge, in probability uniformly over parameters, to + + diag(-6k,A_0) at M, diag(6k,A_0) at S. + +Here the congruence uses diag(sqrt(r),I), so its off-diagonal entries are sqrt(r) beta_i. The limiting matrices are nonsingular almost surely. This follows from the full matrix density and k>=k_->0. Thus the type indicator converges to 1{A_0<0}. Uniform integrability from (5.3) gives + + Z_r/r^2 -> z_0(b,k,R) + := (6k)^2 E[det(A_0)^2 1{A_0<0}], (5.4) + +uniformly on the compact parameter set. + +For completeness, uniformity can be checked by a subsequence argument: any allegedly bad sequence of parameters has a convergent subsequence; continuous Gaussian means/covariances, (5.2), and uniform integrability give the stated limit on that subsequence. The discontinuity set of the type indicator has Gaussian probability zero. Equation (3.5) supplies a common integrable majorant. + +The right side of (5.4) is continuous and strictly positive: the Gaussian law has positive density on an open negative-definite matrix ball. Compactness therefore yields + + 00, with eigenvalues + + 00. + +Determinant monotonicity on positive-definite matrices therefore gives + + |det H_M| <= r (h/2) det B. (6.1) + +This is a canceled-pivot bound: no estimate of 1/alpha_M is taken. + +From ||A_S+B||<=rh, the ordered singular values of A_S are at most lambda_j+rh. The adjugate norm is at most the product of the largest m-1 such bounds. With the empty product equal to one when m=1, (5.3) gives + + |det H_S| <= r(h/2) [lambda_1+(3/2)rh] + product_{j=2}^m(lambda_j+rh). + +Hence the actual typed weight satisfies + + W_r <= (r^2 h^2/4) lambda_1[lambda_1+(3/2)rh] + product_{j=2}^m lambda_j(lambda_j+rh). (6.2) + +The second soft factor includes the saddle mixed-square term. Omitting it, or replacing it by an unrelated order-one factor, changes the radius power. No Gaussian-entry independence is needed for (6.2). + +## 7. Full matrix boundary-layer integration, including corank intersections + +By the predecessor's deterministic marked-cylinder theorem, pairing and the one-branch conclusion hold on + + G_r={lambda_min(-A_M)>[4/(3k)]rM3^2, + rM4<=3k/10}. + +On typed support A_M<0 automatically. It suffices to control G_r^c directly, not infer a converse from a local path-event bound. + +First suppose m>=2. In (4.3) absorb sqrt(m) into K and set U=J_r+Lambda>=1. Then h<=KU. Failure of the depth condition is contained in + + 00 and use h<=K(J_r+lambda). Depth failure implies + + lambda<=Dr(J_r+lambda)^2 + <=2DrJ_r^2+2Dr lambda^2. + +It is contained in the union + + 01/(4Dr). (7.5) + +The large-lambda branch is retained. On the near branch, h<=K(1+4D)J_r^2 for r<=1, J_r>=1. The bounded scalar density, (6.2), and direct integration in lambda give a numerator <=C r^5 E J_r^10. On the far branch, direct fourth-moment Markov with the actual weight gives + + E_Q[(W_r/r^2)1_far] + <=(4Dr)^4 E_Q[(W_r/r^2)lambda^4] <=C r^4. (7.6) + +Thus the scalar numerator is <=C r^5+C' r^6. This separate argument is why the all-dimension statement does not silently drop the scalar far branch. + +Finally, for every dimension, the fourth-derivative exception is bounded directly by + + E_Q[(W_r/r^2)1{rM4>3k_-/10}] + <=[10r/(3k_-)]^4 E_Q[(W_r/r^2)M4^4] <=C r^4. (7.7) + +This is a pointwise Markov bound followed by a joint moment estimate. It is not Cauchy-Schwarz with a missing square root. All joint moments in (7.6),(7.7) are finite by Sections 4-5. + +Divide the numerators by the FULL lower bound (5.5). The result is + + Q^W(G_r^c)<=C_3 r^3+C_4 r^4<=C r^3. (7.8) + +## 8. Pinned genericity and global event measurability + +The following argument applies at every fixed positive r and parameter, not on an asserted common probability-one set for uncountably many laws. + +Away from the pins, consider (grad f(x),Hf(x)v), with ||v||=1. Its range has dimension 2d; its parameter manifold has dimension 2d-1. Modulo the pin observations, these functionals have full covariance rank: gradient and Hessian orders are distinct, and the map H -> Hv from symmetric matrices onto R^d is surjective. For two distinct unpinned points, the map + + (grad f(x),grad f(y),f(x)-f(y)) + +has range dimension 2d+1 and parameter dimension 2d. For a tie to a pinned height use (grad f(x),f(x)-b), or the analogous s map, with dimensions d+1 and d. + +Exhaust the relevant domains by compact sets separated from pins and from coincident points; use finitely many sphere charts when v occurs. Covariance positivity from Section 2 and compactness give bounded Gaussian densities on each such set. An elementary overdetermined-zero lemma applies: on a mesh of size epsilon, a random C^1 map from dimension q to dimension p>q, with derivative norm bounded by N, could have a zero only if one of O(epsilon^-q) mesh values has norm O(N epsilon). The density bound makes the probability O(N^p epsilon^(p-q)), tending to zero. First fix N, then let N increase; the random derivative norm is almost surely finite. This proves absence of all these zeros. + +Pinned Hessians have a full density on their symmetric entries for fixed distinct pins, so their determinants are nonzero almost surely. The pinned heights differ because k,r>0. Countable exhaustion now proves that Q is almost surely Morse with distinct critical values. The transfer to Q^W follows by absolute continuity and 0f(M)} min_t f(gamma(t)), + +with supremum of an empty set equal to -infinity. The event d_f(M)>h can be represented by a countable union of polygonal-path tests, with rational interior vertices and endpoints, all staying strictly above h and with endpoint value above f(M). Strict clearances allow approximation by such paths. For variable M, use countably many coordinate charts and an initial segment depending continuously on M. These are Borel tests in (M,f). Thus equality d_f(M)=f(S) is Borel. On the Morse distinct-value locus it expresses ordinary elder death at S; an essential maximum has no older endpoint. + +On G_r the unchanged deterministic cap theorem controls every boundary exit and supplies a connection through S to a point higher than b. Therefore the GLOBAL ordinary elder partner is S, not merely a local adjacency candidate. Equations (7.8) and the full-probability locus prove Theorem A. The cap theorem also gives exactly one actual ascending branch to M. It does not identify that branch with the transverse ridge. + +## 9. A source-bound weighted Kac-Rice interface + +For clarity, this section specifies why a global elder mark is admissible; a formula for local Hessian index alone would not suffice. + +Apply Kac-Rice first on compact spatial pair domains separated from the diagonal, to the Gaussian vector field (grad f(x),grad f(y)). It is C^1 and has positive definite point covariance for x!=y by Section 2. Its derivative is block diagonal with Jacobian |det H_x det H_y|. Conditional laws of the whole smooth field depend continuously on the imposed gradient values by finite-dimensional Gaussian regression. Derivative moments are locally bounded. + +The expected-integral extension in Armentano-Azais-Leon, *On a general Kac-Rice formula for the measure of a level set*, arXiv:2304.07424v3, Theorem 7.1, gives the identity for bounded nonnegative continuous weights in the location and an additional Gaussian field mark. It is enough first to use cylinder weights depending on finitely many fixed evaluations of f and its derivatives through order two. Their conditional distributions depend continuously on the imposed gradients by Gaussian regression, so the paper's continuity requirements hold. The field is a random element of the separable C^2 space on the torus. On each compact pair domain the two measures on location x C^2 are finite, by the unweighted formula. Cylinder evaluations on a countable dense set, together with location coordinates, generate this Borel sigma-algebra. Equality on their bounded continuous test-function algebra gives equality of the Borel measures by the monotone-class theorem. This extends the identity to bounded Borel global marks, rather than pretending the elder indicator is continuous or lower semicontinuous. No independence between the mark and gradient field is assumed. + +Use the Borel elder mark from Section 8 and the type indicators. Disintegrate the two heights using their nondegenerate joint Gaussian density conditional on the gradients. This gives the full-pin density times E_Q[W] for candidates, and the same quantity times p_r for selected pairs. The continuous Gaussian regression kernel defines the canonical density version; the underlying measure identity is unaffected by height-null-set choices. + +Exhaust the off-diagonal spatial domain toward r=0. The r dr bound proved in the next section makes the expectations locally integrable, and nonnegative monotone convergence completes the near-diagonal formula. No finite-grid approximation of the continuum field is substituted. + +## 10. Exact radial ledger: the dimension cancels + +Let pi_r(R;v_r) denote the density of U_r at (3.3). Since U_r=T_r O_r, + + density_{O_r}(specified pins) + =12 r^(-(d+3)) pi_r(R;v_r). (10.1) + +Use midpoint z, directed separation h=r u, birth b, and gap mark k. The change from the ordered pair (M,S) to (z,h) has absolute determinant one. Polar separation contributes r^(d-1) dr d sigma(u), where sigma is ordinary, NOT probability-normalized, surface area on S^(d-1). The height change (f(M),f(S))=(b,b-k r^3), at fixed r, contributes r^3 db dk. There is no factor 1/2: maximum/saddle pairs are ordered by their different roles. + +Consequently the candidate intensity per unit midpoint volume is + + r A_r(b,k,u) dr db dk d sigma(u), + A_r=12 pi_r(R;v_r) (Z_r/r^2). (10.2) + +The powers are + + spatial (d-1) + height-mark 3 + pin -(d+3) + + determinant-normalizer 2 = 1. + +By Sections 3 and 5, + + A_r -> A_0=12 pi_0(R;v_0) z_0(b,k,R)>0 (10.3) + +uniformly and with a uniform finite upper bound. Although frames were used in the proof, (10.2) is invariant under a rotation of the transverse frame: the original zero-gradient law and determinant weight are unchanged by that orthogonal coordinate change. Thus it is a function of u alone. Uniformity was proved on the compact orthogonal group and descends to the sphere; no nonexistent global continuous transverse frame on the sphere is assumed. + +The selected intensity is r A_r p_r with the same measures. This supplies the domination needed below, rather than assuming a fixed-axis statement can be integrated. + +## 11. Lifetime pushforward and the exact compact coefficient + +For ell=k_->0. This proves (1.2). It is an O(ell^(2/3)) difference, not a claim that either density separately has a second-order expansion with that remainder. Convergence of A_r to A_0 was not assigned a numerical rate. + +## 12. Short bars, moments, and the exact remaining boundary + +Integrating the density versions gives the following expected per-unit-volume results for the same spatial/mark-restricted populations: + + E N_eld(0,t] ~ (3/2)c_{B,K} t^(2/3), + 0<=E[N_cand(0,t]-N_eld(0,t]]<=C t^(5/3). (12.1) + +For any real q>-2/3, + + E sum_{selected, ell<=t} ell^q + ~ c_{B,K} t^(q+2/3)/(q+2/3). (12.2) + +The same leading expression holds for candidates. For q>-5/3 the nonselected difference alone satisfies + + E sum_{candidate but not selected, ell<=t} ell^q + <= C_q t^(q+5/3). (12.3) + +For q<=-2/3 the two individual expectations in (12.2) are infinite for every sufficiently small positive t, because their densities have a positive ell^(-1/3) leading term. Equation (12.3) concerns the nonselected counting measure directly, not subtraction of two infinities. In particular the compact-window inverse-lifetime threshold is p<2/3 for integrability of ell^(-p), with logarithmic divergence at p=2/3. This is unrelated to the inverse-Hessian/Schur thresholds in issue #56. + +The corresponding unrestricted cumulative and individual-moment corollaries follow from Theorem C with c_{d,L} in place of c_{B,K}, by the same integrable-power argument. The compact-window nonselected difference rates (12.1),(12.3) are not automatically unrestricted rates. + +## 13. Removing all birth and gap cutoffs by a direct intensity majorant + +The covariance Sigma_r of U_r depends on radius and frame, NOT on the imposed target. Section 3 therefore supplies a fixed r_0>0, independent of b and k, on which its eigenvalues are bounded above and away from zero. Reduce r_0 below the torus injectivity scale and below one. Do NOT require a globally uniform lower bound for Z_r or a globally uniform constant in (1.1). + +For every b in R, k>0 and 0= c(b^2+k^2), (13.1) + +with fixed c,C>0. For the second inequality it suffices to use its two coordinates b-k r^3/2 and 12k: if r<=1, then b^2<=2(b-k r^3/2)^2+k^2/2. Thus the invertible covariance bound yields + + pi_r(R;v_r) <= C exp[-c(b^2+k^2)]. (13.2) + +Conditional Fourier-coefficient means are bounded by C|v_r| and their centered variances by one. The same summability/Minkowski proof as Section 4 now gives the TARGET-GROWTH bound + + E_Q (1+||f||_{C^3})^p <= C_p(1+|b|+k)^p (13.3) + +for every fixed finite p, uniformly over r and frames on this r_0 band. This is not the compact-target bound (4.1) applied outside its scope; it is its explicit polynomial-growth version. + +By (5.1),(5.3), for r<=1, + + |det H_i|/r <= C (1+||f||_{C^3})^d, + 0<=W_r/r^2<=C(1+||f||_{C^3})^(2d). + +These inequalities remain true when k is arbitrarily small; no division by k or by Z_r has occurred. Combining with (13.2),(13.3) gives the full unnormalized-intensity majorant + + 0<=A_r(b,k,u)=12 pi_r(R;v_r) Z_r/r^2 + <= C(1+|b|+k)^(2d) exp[-c(b^2+k^2)] =: H(b,k). (13.4) + +For every fixed positive r and every b,k>0, Z_r is positive and finite: append BOTH endpoint Hessians to the distinct-site pins; the full residual Gaussian density is positive on a product of an open maximum cone and an open index-(d-1) cone. Thus p_r is defined, even though no globally uniform lower floor has been used. For every fixed b,k>0, Theorem A on a compact neighborhood implies p_r->1 as r->0. + +At a fixed lifetime ell, the near-pair restriction r<=r_0 is exactly k>=ell/r_0^3. After the exact change (11.1), + + ell^(1/3) nu_cand^near(ell) + = integral_{R x (0,infinity) x S^(d-1)} + 1{k>=ell/r_0^3} A_{(ell/k)^(1/3)}(b,k,u) + /(3 k^(2/3)) db dk d sigma, (13.5) + +with the integrand defined to be zero where the indicator is zero. The selected near density has the additional factor p_r. Both integrands are dominated by H(b,k)/(3 k^(2/3)). This majorant is integrable: the k exponent -2/3 is strictly greater than -1 at zero, and Gaussian tails absorb the polynomial at infinity and in b. This explicitly handles the shrinking-k boundary where a compact-window selection constant can blow up. + +For each fixed b,k>0,u, the indicator tends to one, A_r->A_0 and p_r->1. Dominated convergence therefore gives the same limit for the two near densities: + + c_{d,L}=4 integral_{R x (0,infinity) x S^(d-1)} + k^(-2/3) pi_0(R;v_0) z_0(b,k,R) + db dk d sigma(u), (13.6) + 00, + D_u=E[(det A_u)^2 1{A_u<0} | V_u=0]. (15.1) + +Here A_u conditional on V_u=0 is an actual centered nondegenerate Gaussian symmetric matrix, not automatically GOE. Finite-jet rank proves tau_u>0, positive densities p_G(0),p_Vu(0), and D_u>0. All are continuous in the frame and invariant under transverse-frame changes. + +The contact density and cone moment factor as + + pi_0(R;v_0)=p_(f,V_u)(b,0) p_G(0) phi_tau_u(12k), + z_0=(6k)^2 E[(det A_u)^2 1{A_u<0} | f=b,V_u=0], + +where phi_tau is the centered scalar Gaussian density of standard deviation tau. Integrating b by ordinary Gaussian disintegration gives p_Vu(0) D_u. Thus (13.6) becomes + + c_{d,L}=144 integral_{S^(d-1)} p_G(0) p_Vu(0) D_u + [integral_0^infinity k^(4/3) + phi_tau_u(12k) dk] d sigma. + +The scalar integral can be evaluated exactly. With t=12k and the elementary gamma integral, + + 144 integral_0^infinity k^(4/3) phi_tau(12k) dk + = Gamma(7/6) tau^(4/3) / [24^(1/3) sqrt(pi)]. + +Consequently the full leading coefficient is + + c_{d,L}= Gamma(7/6) / [24^(1/3) sqrt(pi)] + * integral_{S^(d-1)} p_G(0) p_Vu(0) tau_u^(4/3) + D_u d sigma(u). (15.2) + +This eliminates both unbounded mark integrals. The remaining angular and negative-definite-cone expectation are finite, strictly positive and specified by derivatives of the EXACT periodized covariance. Formula (15.2) is an explicit finite-dimensional Gaussian representation, not a reported elementary closed form or numerical enclosure. The factor24 is a numerical Jacobian/gamma factor, NOT the side length L; it is the same factor for every L. Changing the field amplitude by a>0 scales the coefficient by a^(-2/3), consistent with nu_(af)(ell)=a^(-1)nu_f(ell/a); this is an additional normalization check. + +## 16. Supplied interfaces, retained limits, and review + +**Interfaces supplied by this candidate:** the actual transverse matrix boundary argument (including d=3); a uniform compact-mark asymptotic full normalizer; all orientations without false isotropy; the global Borel marked Kac-Rice interface; exact pin/radial/lifetime factors; compact-window leading density and quantitative selection difference; a separate integrable unnormalized Gaussian majorant removing ALL birth/gap cutoffs; an off-diagonal bound completing ALL spatial separations; the unrestricted finite-bar leading density and its cumulative/moment consequences; and analytic elimination of birth and gap in its coefficient. + +**Not supplied:** a numerical r_* or C on a prescribed band; a globally uniform O(r^3) probability constant over unbounded b,k; the compact O(ell^(2/3)) difference as an unrestricted remainder; an elementary closed form or numerical enclosure of (15.2); a rigorously evaluated lower coefficient for pairing failure; the original numerical H3/LPW replay; a shrinking-annulus RN certificate; a24-jet displacement enclosure; formal proof verification; nonauthor mathematical acceptance; or unrestricted P15 prize closure. No historical scientific status is changed by this author-side proof. + +The exact-arithmetic program checks pin transforms/targets, block determinants, finite typed-matrix envelopes, spectral Jacobians, radius/lifetime exponents, explicit small-eigenvalue integral models, global target coercivity and coefficient scaling. Semantic mutations challenge the load-bearing factors. These are finite algebra/implementation controls, NOT Gaussian simulation or proof of continuum estimates. + +Priority nonauthor review: (i) desingularized full residual covariance; (ii) uniform all-field moments and independent-matrix regression; (iii) both soft determinant factors; (iv) eigenvalue integration retaining the random largest-eigenvalue width and all corank strata; (v) scalar far branch; (vi) full-normalizer type convergence; (vii) continuous-to-Borel marked Kac-Rice extension; (viii) every factor in the pin/lifetime ledger; (ix) target-growth bound (13.3) and integrability at k=0 in (13.5); (x) distinct-site compact covariance in the far domain; and (xi) parity, b-disintegration and the gamma factor in (15.2). + +### Primary-source attribution + +Armentano, Azais and Leon, arXiv:2304.07424v3, Sections 2 and 7, supply established Gaussian Kac-Rice/expected-integral background. The precise full-field/Borel and pair application is explained in Section 9, not attributed wholesale to that paper. Curry, *The Fiber of the Persistence Map for Functions on the Interval*, arXiv:1706.06059v2, Section 3, supplies elder-rule background; the all-dimensional separating cap is the explicitly identified in-project predecessor. Beliaev, Cammarota and Wigman, *No repulsion between critical points for planar Gaussian random fields*, arXiv:1911.03455, is neighboring two-point/index-asymptotic work, not a source of our elder-selection conclusion. No priority or novelty claim follows from this bounded reconnaissance. diff --git a/specifications/lean_statements_20261010/sources/lifetime-remainder/LIFETIME_REMAINDER.md b/specifications/lean_statements_20261010/sources/lifetime-remainder/LIFETIME_REMAINDER.md new file mode 100644 index 000000000..247b3ecf8 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/lifetime-remainder/LIFETIME_REMAINDER.md @@ -0,0 +1,267 @@ +# A bounded unrestricted short-lifetime remainder + +**Object:** LIFETIME-BOUNDED-REMAINDER-20260924-v1. +**Author:** OpenAI / ChatGPT. **Disposition:** author-side derivation; nonauthor analytic review open. + +## 1. Exact dependency and new conclusion + +This is an additive successor to `UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1`, main issue 63, Drive `1foDgiDi4XIKOfbrZEE8dWKV_BkZU8LIb`, SHA256 `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. The unchanged deterministic import is `MARKED-CYLINDER-CAP-20260924-v1`, SHA256 `0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc`. + +Use the parent's exact centered variance-one Gaussian field on the fixed torus X=R^d/(L Z^d), for each fixed d>=2 and L>0: + + K_L(z)=sum_n exp(-|z+Ln|^2/2) / sum_n exp(-|Ln|^2/2). + +Every constant below may depend on d,L. It need not be uniform as L or d varies. No infinite-volume theorem is asserted. Pins are M=-ru/2, S=ru/2, heights b and b-k r^3, and zero gradients at both points, with b in R, k>0 and u a unit vector. Q is the continuous Gaussian regression law, W=|det H_M det H_S| times the maximum/index-(d-1)-saddle indicator, and Z=E_Q W is the FULL normalizer. Let p denote the Q^W=(W/Z)Q probability of global ordinary elder pairing. + +The marked Kac-Rice identity, pinned genericity, measurable elder convention and separating-cylinder implication are used exactly as in the parent. This note supplies quantitative estimates missing there; it is not independent acceptance of those interfaces. The coefficient c=c_{d,L} is exactly the parent's equation (15.2). + +**Theorem R (candidate).** There are ell_*>0 and C=1 and each finite derivative order q, + + ||F_r-F_0||_{L^p(C^q)} <= C_{p,q} r^2 P, + ||1+||F_r||_{C^q}+||F_0||_{C^q}||_p <= C_{p,q} P, + P=1+|b|+k. (R4) + +This is a coupling of laws, not an assertion of equality of two conditioned fields or independence of endpoint variables. + +Let pi_r be the density of U_r. For all b,k and r<=r_0, + + pi_r(v_r) <= C exp[-c(b^2+k^2)], + |pi_r(v_r)-pi_0(v_0)| + <= C r^2 P^2 exp[-c(b^2+k^2)]. (R5) + +To prove the difference estimate, interpolate the covariance and target linearly between r and 0. The covariance stays uniformly positive. The target keeps its 12k coordinate; its other nonzero height coordinate is b-theta*k*r^3/2, so its squared norm is uniformly coercive in b^2+k^2. Differentiating log Gaussian density gives a trace term O(r^2), a quadratic term O(r^2 P^2), and a target term O(r^2 kP). Integrating the derivative yields (R5), after reducing c. This explicitly handles arbitrarily large targets. + +## 3. An index-filtered determinant is more stable than its inertia indicator + +For a real symmetric n by n matrix H define + + F_j(H)=|det H| 1{H has exactly j negative eigenvalues}, + +and give it value zero on singular matrices. Two elementary lemmas suffice. + +**Lemma R3.1.** For symmetric X,Y, + + |F_j(X)-F_j(Y)| <= n max(||X||,||Y||)^(n-1)||X-Y||. (R6) + +If their indices agree, this follows from the determinant difference bound (or both terms are zero). If their indices differ and one term is nonzero, the straight segment reaches a singular matrix Z. The same determinant bound from that endpoint to Z applies, with norm and distance bounded by those of the full segment. When an endpoint is singular the same reasoning applies directly. The determinant bound follows by telescoping columns and Hadamard's inequality. In particular the discontinuous inertia indicator itself was not assumed Lipschitz. + +**Lemma R3.2 (quadratic off-diagonal cancellation).** For a symmetric (n-1) by (n-1) matrix A, scalar alpha, vector beta and t>=0, set + + K_t=[[alpha,sqrt(t) beta^T],[sqrt(t) beta,A]]. + +Then for every j and every r>=0, + + |F_j(K_r)-F_j(K_0)| + <= r |beta^T adj(A) beta|. (R7) + +Indeed det K_t=alpha det A-t beta^T adj(A) beta, INCLUDING singular A. If the endpoint indices agree, use the difference of absolute determinants. If only one endpoint contributes, a singular K_t lies in between by continuity of eigenvalues, and the affine determinant bounds the contributing endpoint by r times its slope. If the determinant is identically zero, both sides' filtered determinants are zero. The proof requires no inverse matrix and no lower eigenvalue margin. + +A naive application of (R6) to the sqrt(r) entries would lose a square root. Equation (R7) is the stronger fact that prevents this loss, also as k tends to zero. + +## 4. A refined all-mark intensity and contact error + +Under (R3), put A_0=D_y^2 F_0(0), A_i=D_y^2 F_r(i), +alpha_i=F_r,xx(i)/r and beta_i=grad_y F_r,x(i)/r. Let D_r=diag(sqrt(r),I). Then + + K_i=D_r^(-1) H_i D_r^(-1) + =[[alpha_i,sqrt(r) beta_i^T],[sqrt(r) beta_i,A_i]], + |det H_i|/r=|det K_i|. + +The exact gradient/height pin identities give + + |alpha_M+6k|+|alpha_S-6k| <= C r M4, + ||beta_i|| <= M3/2. (R8) + +Equation (R4) and the endpoint displacement give a nonnegative random T>=1+k with all finite moments bounded by C_p P^p such that + + ||A_i-A_0||<=r T, + ||A_i||+||A_0||+||beta_i||+M3+M4<=C T, + |alpha_M+6k|+|alpha_S-6k|<=C r T. (R9) + +For example T can include the C^4 norms in (R4) and r^(-2)||F_r-F_0||_{C^2}; the latter has the asserted uniform moments by (R4). No independence of T and A_0 is required here. + +Use j=d at M and j=d-1 at S. First remove the off-diagonal block by (R7); then compare diag(alpha_i,A_i) with diag(+-6k,A_0) by (R6). Equations (R8)-(R9) yield an O(r T^d) filtered determinant error at each endpoint. The exact block determinant also gives + + F_j(K_i) <= C(k+r) T^d, + F_d(diag(-6k,A_0)) <= C k T^(d-1), + F_(d-1)(diag(6k,A_0)) <= C k T^(d-1). + +The reference product is (6k)^2 det(A_0)^2 1{A_0<0}; no other transverse inertia can give both specified full indices. Multiplying the two estimates and taking expectations proves + + Z_r/r^2 <= C(k+r)^2 P^N, + |Z_r/r^2-z_0(b,k,u)| <= C r(k+r) P^N, (R10) + +for a fixed finite integer N depending only on d. Thus, defining exactly as in the parent + + A_r(b,k,u)=12 pi_r(v_r) Z_r/r^2, + A_0=12 pi_0(v_0) z_0, + +there is a majorant H(b,k)=C P^N exp[-c(b^2+k^2)] (increasing N when necessary) with + + 0<=A_r<=(k+r)^2 H, + 0<=A_0<=k^2 H, + |A_r-A_0|<=r(k+r) H. (R11) + +For the density difference term use (R5) and z_0<=C k^2 P^N; its r^2 k^2 factor is absorbed by r(k+r) times a further polynomial in P. These bounds hold for ALL b in R,k>0 and 0 (4/(3k)) r M3^2, + r M4 <= 3k/10. + +On typed support the cap theorem gives elder pairing on G_r. Define + + B_r=12 pi_r(v_r) E_Q[(W/r^2) 1{G_r^c}]. + +Always A_r(1-p_r)<=B_r<=A_r. We prove that with delta=r/k, + + B_r <= delta^3 H(b,k), whenever delta<=1. (R12) + +The constants in (R12) are independent of b,k; their target growth is inside H. This is NOT a globally uniform normalized probability estimate. + +Append the independent symmetric entries of A_M to U_r. The joint covariance is uniformly positive at contact and for r<=r_0, by distinct finite jets and compactness. Consequently its JOINT density obeys + + pi_r(v_r) density(A_M=A | U_r=v_r) + <= C exp[-c(b^2+k^2+||A||_F^2)]. (R13) + +Regress the whole field on this appended vector. Its centered residual g is independent of that entire vector and has uniform C^4 moments of every finite order. Its mean has C^4 norm bounded by C(P+||A||). Set J=1+||g||_{C^4}, T=P+J. On A_M<0 write B=-A_M with eigenvalues 0=2, failure of the depth condition implies + + 0 [lambda<=4D delta T^2] OR [lambda>1/(4D delta)]. (R15) + +On the near branch delta<=1 gives h<=C T^2. Direct scalar integration of lambda(lambda+CrT^2), with width 4D delta T^2 and (R13), again gives delta^3 H, since r delta^2=k delta^3. For the FAR branch, fourth-moment Markov gives + + pi_r E_Q[(W/r^2) 1_far] + <=(4D delta)^4 pi_r E_Q[(W/r^2)lambda^4] + <=delta^4 H<=delta^3 H. + +Joint moments here are bounded by polynomial Gaussian integrals from (R13), or directly by the target-growth regression bound. The scalar far branch was not discarded. + +Finally the fourth-derivative exception M4>3/(10delta) gives + + pi_r E_Q[(W/r^2)1_exception] + <=(10delta/3)^4 pi_r E_Q[(W/r^2)M4^4] + <=delta^4 H<=delta^3 H. + +These are weighted pointwise Markov inequalities, not Cauchy-Schwarz without a square root. Increasing the common polynomial degree and constants proves (R12). + +## 6. Integrate candidate error over the entire mark domain + +The exact parent polar/height/pin/determinant accounting gives r A_r dr db dk d sigma, with ordinary sphere area and no role-order factor 1/2. At lifetime ell set r=(ell/k)^(1/3). The near restriction r<=r_0 is k>=a, a=ell/r_0^3, and + + ell^(1/3) nu_cand^near(ell) + = integral_(k>=a) A_(ell/k)^(1/3)/(3 k^(2/3)) db dk d sigma. + +The full contact coefficient is integral A_0/(3 k^(2/3)) over all b,k>0,u. From (R11), the missing k=eta, delta=ell^(1/3)k^(-4/3)<=ell^(1/9)<=1. The loss estimate (R12) is therefore applicable, including large b,k. The scaled integrand is bounded by + + C ell k^(-14/3) H(b,k). + +Its integral is at most C ell eta^(-11/3)=C ell^(7/18), plus a smaller Gaussian-tail term. Combining with (R17), since 7/18>1/3, gives + + 0<=ell^(1/3)(nu_cand^near-nu_eld^near)<=C ell^(1/3). (R18) + +No normalized lower floor in the small-k region was used. The lower cutoff a must be retained: replacing it by zero in the r^2 k^(-2/3) term would create a divergent integral and lose the valid bound. + +On torus separations at least r_0 the original full-pin covariance is uniformly positive, as in parent Section 14. The direct unconditioned height-density estimate there gives 0<=nu_eld^far<=nu_cand^far<=C for 0-2/3, the individual candidate and finite-bar q-moments obey + + E sum_(ell_i<=t) ell_i^q / Vol(X) + = c t^(q+2/3)/(q+2/3)+O(t^(q+1)). (R20) + +For q>-1 the NONSELECTED counting measure alone has weighted mass at most C_q t^(q+1). This remains meaningful for -1=2 and use coordinates (x,y) in R x R^(d-1) on an embedded product cylinder + + D=[-2r,2r] x closed_ball(0,2r), r>0. + +Suppose f is C4 on a neighborhood of D, has critical pins M=(-r/2,0), S=(r/2,0), with f(M)=b and f(S)=b-kappa*r^3, kappa>0. For j=3,4 define the partial-block norm + + M_j=max_{a+c=j} sup_D || partial_x^a D_y^c f ||_op. + +For c=0 this is absolute value. For c>=1 the operator norm is the supremum on c transverse Euclidean unit vectors, independently chosen. In dimension two this is exactly the maximum of the coordinate partial derivatives used by #57, not the larger full-tensor operator norm. Put + + lambda=lambda_min(-D_y^2 f(M)). + +The explicit sufficient conditions are + + lambda > (4/(3*kappa))*r*M_3^2, + r*M_4 <= 3*kappa/10. (1) + +They give a unique transverse-critical ridge on D and a closed cap through S separating M from every older point. On a compact manifold, for a global Morse extension with distinct critical values, the ordinary superlevel elder death partner of M is S. Exactly one ascending unstable branch of S ends at M, for a smooth Riemannian gradient. No Morse-Smale assumption and no identification of the ridge with a flow trajectory are used. + +**Probabilistic corollary, narrower scope.** Retain EXACTLY the fixed-x-axis 2D normalized SIDE24 six-pin law of #57/#58: b=6/5, kappa=1/6, 08*r*m^2, r*n<=1/20. (4) + +The exact critical pins give the Hermite identity + + f(S)-f(M)=-(1/2) integral_{-r/2}^{r/2} + (x+r/2)(r/2-x) f_xxx(x,0) dx. + +The positive kernel has integral r^3/6, so its weighted average of f_xxx is 2. In particular m>=2 and some x0 in the pin interval satisfies f_xxx(x0,0)=2. This forced lower bound on m is useful; replacing m by the larger 1+m is unnecessary. + +The partial-block norms control increments by |Delta x|+||Delta y||, with no factor depending on dimension. Every point of D is within this product distance <5r of (x0,0), and likewise <5r from M. Thus + + f_xxx >= 2-5rn >= 7/4, + D_y^2 f <= -(lambda-5rm) I = -delta I, (5) + delta>rm(8m-5), 8m-5>=11. + +The Hessian inequality is in the quadratic-form order throughout D. A scalar lower bound on a diagonal entry cannot substitute for this matrix inequality when d>2. + +## 3. A vector ridge exists globally across the cylinder + +Let w(x)=grad_y f(x,0). It vanishes at the two pins and ||w''||<=m. Applying the scalar two-node interpolation remainder to every unit-vector projection of w gives, including outside the pin interval, + + ||w(x)|| <= (m/2)|x^2-r^2/4| <= (15/8)mr^2 <2mr^2. (6) + +The bound outside the nodes follows from the interpolation remainder on the convex hull of the three points, not from an unjustified inside-only estimate. + +Also integral_{-r/2}^{r/2} w'(t)dt=0. Subtract this average from w'(x), use the Lipschitz constant m, and integrate |x-t| to obtain + + ||w'(x)|| <= (m/r) integral_{-r/2}^{r/2}|x-t|dt + <=2mr, |x|<=2r. (7) + +This vector average argument does not posit a common point at which every component of w' vanishes. + +For fixed x, f(x,.) is delta-strongly concave on the transverse ball. On its boundary ||y||=2r, + + grad_y f(x,y).y <= ||w(x)|| ||y|| -delta||y||^2 <0. + +The maximum on the compact ball is therefore interior (moving inward increases the value at a boundary point). Strict concavity makes it unique. Call it h(x). It solves grad_y f(x,h(x))=0, has h(+-r/2)=0 and is C3 by the implicit function theorem. Strong monotonicity gives + + ||h(x)|| <= ||w(x)||/delta <2r/k, k=8m-5>=11. (8) + +Its derivative satisfies + + h'=-(D_y^2 f)^(-1) partial_x grad_y f. + +Equations (7),(8) and the mixed derivative bound give + + ||h'|| < (2rm+m||h||)/delta + <2/k+2/k^2=:u, u<=24/121. (9) + +Since m=(k+5)/8, + + m*u=(1+6/k+5/k^2)/4 <=48/121. (10) + +All these inequalities use transverse operator norms; there is no hidden dimension-dependent coordinate sum. + +## 4. The correct ridge derivative and all cap boundary faces + +Set g(x)=f(x,h(x)), F=g'=f_x(x,h(x)). With v=(1,h'), differentiating the transverse critical equation gives (H_f v)_y=0. In the third derivative of g, every term involving h'' is therefore zero. The exact identity is + + F''=f_xxx +3 f_xxy[h'] +3 f_xyy[h',h'] + + f_yyy[h',h',h']. (11) + +This is valid for a vector h. It does not assert that the ridge is an integral curve of the gradient. + +Using (9),(10), the adverse part is at most + + (48/121)[3+3*(24/121)+(24/121)^2] + =2554128/1771561. + +Consequently + + F'' > 7/4-2554128/1771561 + =2184415/7086244 >1/4. (12) + +The bound is weaker than #58's F''>1, but it is already sufficient for an older connecting endpoint. This is the second source of the improved depth condition. + +F has the prescribed zeros a=-r/2,c=r/2. Strict convexity makes them its only zeros, with F negative between and positive outside. Every critical point in D lies on the ridge, hence D contains precisely M and S. The Schur complement of D_y^2 f at these points is F'(a)<0 and F'(c)>0, proving that M is a maximum and S has exactly one positive Hessian direction. + +Define + + C=[-2r,r/2] x closed_ball(0,2r). + +Strong concavity gives f(x,y)<=g(x) and g(x)<=b on C. To control the longitudinal faces, subtract (x-a)(x-c)/8 from F. The difference is convex with zero values at a,c, hence nonnegative outside [a,c]. Integrating on either exterior interval gives + + g(-2r) <= b-(9/32)r^3 = s-(11/96)r^3, + g( 2r) >= s+(9/32)r^3 = b+(11/96)r^3, (13) + +where s=b-r^3/6. These are lower/upper inequalities at the actual ridge endpoints, not Taylor fits. + +For a transverse boundary point ||y||=2r, (5),(8) imply + + f(x,y) <=g(x)-(delta/2)||y-h(x)||^2 + rm(8m-5)>=22r and ||y-h||>20r/11. This covers the entire curved side of the cylinder, including its intersections with the end faces; in dimension two it covers both horizontal edges and their corners. + +On the right cap face x=c, h(c)=0, so + + f(c,y) <= s-(delta/2)||y||^2, + +with equality at S only. On the left face use f<=g and (13). Thus every boundary exit from C has value <=s. + +Finally, the ridge segment a<=x<=2r starts at M, passes through S and ends at z=(2r,h(2r)), where f(z)>b by (13). Its minimum is exactly s. A path through S at level s is not a preemption path staying strictly above s. + +## 5. Global elder pairing, actual branches, and mark scaling + +Every point with value >b lies outside C. Every continuous path from M to such a point must first intersect the boundary of C, so its minimum is <=s regardless of later remote excursions or reentry. The preceding ridge path has minimum exactly s and endpoint >b. Hence the maximin connection level to an older point is s. + +For a compact Morse function with distinct critical values, ordinary superlevel-component persistence identifies this maximin value with M's elder death level; the older endpoint excludes the essential/global-maximum case. The unique critical value s identifies S. This is the same separating-cap criterion established in #58, now with all its boundary premises re-proved under the weaker event. + +At S the Hessian is negative definite on the transverse hyperplane. The unstable eigenvector for gradient ascent has nonzero x component: its Hessian quadratic form is positive, whereas that hyperplane is strictly negative. This argument also holds for a smooth positive-definite Riemannian metric. One half-branch enters the cap at height >s; the other enters outside. The first cannot exit because its height increases and every boundary value is <=s. Its omega-limit is a critical point in the compact cap. Gradient monotonicity and finiteness of the critical set make this limit a single point; it cannot be S, so it is M. The other branch cannot enter C or tend to its interior point M. Exactly one branch ends at M. + +For general kappa>0 apply the normalized theorem to f/(6*kappa). Its pins have gap r^3/6 and arbitrary birth b/(6*kappa). Its derivative norms and transverse curvature are divided by 6*kappa. Condition (4) becomes exactly (1). The geometric ridge is unchanged. The reduced third derivative is >3*kappa/2, the longitudinal drop/rise is at least (27/16)*kappa*r^3, and the excess over the specified gap is (11/16)*kappa*r^3. The radial drop exceeds (26400/121)*kappa*r^3. + +An arbitrary orthonormal frame can be used in this pathwise argument as long as its product chart is embedded. Scaling or rotating the function here is not a proof that SIDE24's conditional covariance, pin energy or normalizer are invariant under that operation. + +## 6. What is imported for the 2D probability calculation + +Use the fixed-axis Q,Q^W law from Section 1 and the same random-variable coupling as #57 Sections 5 and 7. The following are retained premises, not newly rerun numerical programs: + +* Z>=(17/10)r^2, six-pin regression energy<8, and ||det H_M det H_S||_2/Z<23/5. +* q=f_yy(M) has density <=1/3 and ||q||_8<5. On typed support q<0. +* There is T>=1, JOINTLY independent of q, with ||T||_8<320 and M_3<=T+8|q|; also ||M_4||_8<340. +* In the SAME coupling, at q=-ru, the full typed weight satisfies the pointwise bound + + W<=r^4[T^2*u^2+3*T^3*u], u>=0. (15) + + Internal correlations among the residual field, axial/mixed variables and transverse innovation are retained. Independence is only from q. The coefficient 3 includes the saddle's mixed-square contribution. +* The pinned-genericity candidate #49 supplies the full-probability Morse/distinct-value locus for each fixed r. No simultaneous null-set statement over all parameters is imported. + +These facts are read from exact source bytes and their explicit chain. Hash verification does not establish their truth; H3 and LPW interval programs were not rerun by this delivery. The stronger geometry does not depend on #56's endpoint spectral candidate. + +## 7. The smaller exceptional layer and its retained far branch + +Set G_new={-q>8r M_3^2, rM_4<=1/20}. Section 2-5 give global pairing on G_new. Unlike an inference from a local-event probability, this uses a direct bound on G_new's complement. + +Put lambda=-q>0 on typed support. Failure of the depth condition implies + + lambda<=8r(T+8lambda)^2 + <=16rT^2+1024r lambda^2. + +Thus the failure lies in + + {01/(2048r)}. (16) + +The large-lambda branch is essential to the quadratic inequality and has NOT been discarded. + +For the near event, disintegrate on q independent of T. Use its density cap 1/3, dq=r du, and (15): + + E[W 1_near] <= (r^5/3) E integral_0^(32T^2) + [T^2 u^2+3T^3 u]du + = (r^5/3)[(32^3/3)E T^8+(3*32^2/2)E T^7]. (17) + +Division by the full Z and the moment bounds gives C3_new*r^3. Weighted Cauchy-Schwarz, INCLUDING its square root, and eighth-moment Markov give + + Q^W{lambda>1/(2048r)} < (23/5)(10240r)^4, + Q^W{M_4>1/(20r)} < (23/5)(6800r)^4. (18) + +Together (17),(18) prove (2),(3) at the imported premises. The same single complement controls nonadjacency and the event that both unstable branches end at M; no independence or equality of these events off G_new is asserted. + +The earlier LPW lower premise, when separately consumed, is unaffected: c_LPW=13/468574870385996070912000 and r<=1/356352 still compose with this upper bound for two-sided cubic order in the one fixed-axis law. This delivery does not re-establish that lower premise or give a limiting coefficient. + +## 8. Exact comparison and remaining work + +The predecessor constants from #57/#58 are + + C3_old=(10/51)[(256^3/3)320^8+(3*256^2/2)320^7], + C4_old=(23/5)[81920^4+6800^4]. + +Exact arithmetic checks C3_old>500*C3_new, C4_old>500*C4_new and + + C3_new+C4_new/20 <2.4*10^23. + +Their absorbed-constant ratio is about 511.803 (display only; the comparison >500 is rational). The tighter good event is strictly weaker than the old sufficient event; neither claims to characterize all successful pairs. + +For the three-dimensional probability program the deterministic matrix/ridge obstacle now has a concrete candidate solution, but the probability of a small transverse eigenvalue and intersections of matrix boundary strata remains to be proved under the correct eight-pin law. Full-annulus expected counts, 24-jet modulus certificates, all-angle/all-mark estimates and lifetime integration remain distinct targets. Small failure probability alone does not bound a count on the failure event. + +Source identities and external reconnaissance are in SOURCE_OBSERVATIONS.json and RECONNAISSANCE.md. Tests check exact algebra, finite examples and deliberate mutations, not arbitrary continuous fields or the analytic imported chain. Review should focus on the vector average in (7), all partial-block norms, the h'' cancellation, entire radial side, marked scaling, and the independence/density/normalizer powers in (17). diff --git a/specifications/lean_statements_20261010/sources/p15-full-price/PROOF.md b/specifications/lean_statements_20261010/sources/p15-full-price/PROOF.md new file mode 100644 index 000000000..582180e41 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/p15-full-price/PROOF.md @@ -0,0 +1,153 @@ +# Full transformed-price budgets without a probability cutoff + +**Object:** P15-FULL-TRANSFORMED-PRICE-20260924-v1. +**Author:** OpenAI / ChatGPT. **Disposition:** author-side mathematical derivation; nonauthor review open. + +[Research home](https://github.com/d6g8k5htny-coder/main) · [Math index](../../README.md) · [Original realization](../three_fronts_20260924/P15_REALIZED_COVERS.md) · [Demand-one counterexample](../three_fronts_20260924/P15_PRICE_BOUNDARY.md) + +## 1. Exact scope and result + +Use the original realized family, not arbitrary P15 inputs. H is a finite clutter of block supports, every edge has size at least two (empty H is allowed). The original coordinate blocks are disjoint, with + + |X_i|=a_i d_i+1, a_i>=1, d_i>=2 integers. + +The decreasing family D consists of U with |U intersect X_i|<=a_i in each block and with occupied-block support containing no H-edge. Its complete minimal original forbidders and its actual local restrictions are proved in P15-REALIZED-COVERS-20260924-v1, SHA256 `c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9`. + +Write O_K(D) for sets not partitionable into at most K members of D. Let K_H(d) be the least palette size with |P_i|>=d_i and empty intersection over every H-edge. For K>=K_H(d), the specified family of full-block generators {X_i} covers O_K(D), regardless of probabilities or prices. A generator g costs product_(v in g)c_v. The empty generator costs one; an empty family costs zero. + +Put phi(p)=min(1,-log(1-p)), with phi(1)=1. Define + + p_star=1-exp(-1), + h_star=3-log(3e-2), + rho_star=1/h_star. (F1) + +**Theorem F.** For EVERY independent original probability vector p in [0,1]^X, every price vector 0<=c_v<=phi(p_v), and the SAME palette K>=K_H(d), + + covercost_c(O_K(D)) <= min(1,rho_star[-log mu_p(D)]). (F2) + +Before capping, the union of full-block generators has price at most rho_star times the global hazard. The constant rho_star is optimal uniformly over this entire realized class, even when alternative generator covers are allowed. Explicitly, + + 0.84547981724898672067 < rho_star + < 0.84547981724898672068 < 6/7. (F3) + +Thus 6/7 is a simpler valid rational factor. The probability ceilings 1/4 and 3/7 are no longer required. Demand d_i>=2 is still essential to this uniform family guarantee. This is not an unrestricted downset/prize theorem; palette size remains K_H(d), which can depend on H and d. + +The prior probability<=1/4 theorem, SHA `3b79d2de60d77df9dd0d81cea60935d3dbceeb26fcf2d38d62667a425180a535`, gives the stronger factor16/27 on its smaller domain and should still be used there. The local-only3/7 note is a historical intermediate result, not a prerequisite here. No prior proof or counterexample is overwritten. + +## 2. A general hazard interpolation lemma + +Let A be any proper decreasing family on n>=1 coordinates containing the empty set. Put + + p_i(t)=1-exp(-t_i), t in [0,1]^n, + F_A(t)=-log mu_(p(t))(A), + H_A=F_A(1,...,1)>0. + +The probability is positive on this compact cube because the empty set is admitted and every p_i(t)<1. In one coordinate, after averaging over all others, its good-event probability is exactly + + mu(A)=A0+B0 exp(-t_i), A0>=0, B0>=0. (F4) + +Here A0 is the probability of membership when that coordinate is present, and A0+B0 is the probability when it is absent. The inequality B0>=0 follows from decreasingness. Their sum is positive. Direct differentiation yields + + partial_i F_A = B0 exp(-t_i)/(A0+B0 exp(-t_i)), + partial_i^2 F_A = -A0 B0 exp(-t_i) + /(A0+B0 exp(-t_i))^2 <= 0. (F5) + +Only SEPARATE coordinatewise concavity is asserted; joint concavity is not needed or claimed. Also F_A>=0. The chord inequality in coordinate i therefore gives + + F_A(t) >= t_i F_A(t with t_i=1). + +Iterating across coordinates, keeping all factors nonnegative, proves + + F_A(t) >= (product_i t_i) H_A. (F6) + +Now take arbitrary original probabilities p_i in [0,1], set t_i=phi(p_i), and let p'_i=1-exp(-t_i). Then p'_i<=p_i, including p_i=1. Decreasingness gives mu_p(A)<=mu_(p')(A), so + + -log mu_p(A) >= H_A product_i phi(p_i). (F7) + +This holds as an extended inequality if the left side is infinite. At a zero probability the product is zero; no division by it is performed. For any transformed prices, + + product_i c_i <= min(1, [-log mu_p(A)]/H_A). (F8) + +This is a PRICE estimate for the full-ground generator. It does not assert that this generator alone covers an arbitrary obstruction. That setwise property is supplied separately for the capacity construction below. + +The constant1/H_A in the uncapped hazard comparison is exact: at every p_i=p_star and every c_i=1, the two sides of (F7) are equal. In particular a full-ground generator universally meets the capped hazard budget precisely when H_A>=1. The necessity uses that same reference vector; it does not infer coverage of arbitrary A. + +## 3. Capacity families and the sharp worst case + +For a capacity restriction A={S:|S|<=a} on n coordinates, + + H_(n,a)=-log P(Bin(n,p_star)<=a). (F9) + +Our n=ad+1 and d>=2 imply n>=2a+1. Adding independent Bernoulli trials without raising the threshold can only decrease the good probability. It remains to compare odd majority tails. Let S~Bin(2a+1,p), with p>1/2. Directly adjoining two trials gives + + P(Bin(2a+3,p)<=a+1)-P(S<=a) + =(1-p)^2 P(S=a+1)-p^2 P(S=a) + =p(1-2p)P(S=a)<0. (F10) + +The ratio P(S=a+1)/P(S=a)=p/(1-p) proves the second equality. Since a>=1, the maximal odd-tail probability occurs at a=1,n=3. Consequently + + H_(n,a) >= H_(3,1) + =-log[3exp(-2)-2exp(-3)] + =3-log(3e-2)=h_star>1. (F11) + +For the last inequality, e>2 and + + e^2-(3e-2)=(e-1)(e-2)>0, + +so 3exp(-2)-2exp(-3)=K_H(d); (F13) gives its budget. If its union price exceeds one, use the empty generator. This proves (F2). If a local good probability is zero, global goodness has probability zero and the price-one cap proves the assertion. If global goodness is zero for crossing reasons alone, the same cap applies. Zero-price generators are kept for setwise completeness. + +For the established six blocks of409 coordinates (a_i=1,d_i=408, all15 four-block supports), the SAME816-label cover now works for every independent p_v in [0,1] and every c_v<=phi(p_v). Its whole-ground chromatic number remains818. No coordinates are cloned, and the full crossing family is retained. At sufficiently low probabilities the earlier16/27 budget is better; on the whole cube (F2) is valid. + +## 5. Sharpness and the exact demand boundary + +Take one block, a=1,d=2,n=3, no macro edges and K=2. Every proper subset is two-decomposable and the full triple is not. At p_i=p_star and c_i=1, every possible generator has price one, including the empty generator. Any cover of the nonempty obstruction therefore costs at least one, and one generator attains it. The available hazard is h_star. Replacing rho_star by any smaller constant in (F2) would make its right side strictly less than one. This proves sharpness even among alternative covers, not only for our chosen full-block cover. + +If d=1, n=a+1 and K=1, the sole obstruction is again the full block. At p_i=p_star,c_i=1 every cover costs at least one, but + + mu_p(D)=1-p_star^(a+1)>1-p_star=exp(-1), + +hence its hazard is less than one. The universal same-palette transformed-price guarantee is false in this demand-one family. This preserves and generalizes the earlier two-coordinate counterexample; it does not retract original P15-B, whose local-budget premise is missing in that case. + +Thus d>=2 is sufficient for the realized capacity family on the full probability cube, while a uniform statement admitting all d=1 examples would be false. This is not a necessity claim for every multi-block instance with some demand one; other constraints may change its budget. + +## 6. Exact numerical certificate and validation limits + +The proof of the simpler bound rho_star<6/7 can be made with short rational certificates. The exponential series and its positive geometric tail give + + e < 31967/11760 < 87/32, + 87/32-31967/11760=11/23520. + +The positive Taylor polynomial through degree5 satisfies + + sum_(j=0)^5 (11/6)^j/j! -197/32=26081/933120>0. + +Therefore 3e-2<197/327/6 and rho_star<6/7. + +For (F3), full_price.py bounds e by its positive series plus a geometric tail. It encloses logarithms using binary reduction and + + log x = 2 sum_(j>=0) t^(2j+1)/(2j+1), t=(x-1)/(x+1), + +on 1<=x<=2, with remainder bounded by2 t^(2m+1)/[(2m+1)(1-t^2)]. Exact Fraction endpoints are propagated monotonically through the binomial tail and logarithm, then reciprocated and rounded outward to20 decimal places. No floating-point value determines a certified endpoint. + +The finite suite checks the recurrence against subset enumeration, the odd-majority identity, the coordinatewise derivative sign, exact constants and endpoint conventions, heterogeneous probability grids, and preservation of the demand-one counterexample. These checks support algebra and implementation; the continuum hazard inequality is the written proof, not a conclusion from the grid. No independent reviewer, formal checker, Gaussian/RN closure or unrestricted prize acceptance is claimed. + +## 7. Source and reconnaissance + +Project inputs are the unchanged realized-cover and price-boundary sources linked above, original P15-B (Drive19D-eHQAIXMGGy2ThZUfZ0GGjIKWm5C2j; SHA9b18b6e9abc90d18deef06ab12e3aa7794dad40e1618d99daa88e369c300e8c3), and the published restricted-price note (Math-PR4). This proof derives (F4)-(F13) directly rather than importing a theorem from an abstract. + +Primary-source reconnaissance on24September2026 read the arXiv abstracts of Gunby–He–Narayanan, Down-set thresholds,2112.08525v2, and Warnke, Note on down-set thresholds,2310.11662v2. Their downset/expectation-threshold results provide neighboring context, not the hazard-transfer theorem above, and do not license arbitrary-downset closure. The bounded search did not establish novelty or historical priority. Attribution to classical separate concavity, Bernoulli conditioning, and binomial identities is not replaced by a project-specific invention claim. diff --git a/specifications/lean_statements_20261010/sources/p15-price-boundary/P15_PRICE_BOUNDARY.md b/specifications/lean_statements_20261010/sources/p15-price-boundary/P15_PRICE_BOUNDARY.md new file mode 100644 index 000000000..1d6946658 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/p15-price-boundary/P15_PRICE_BOUNDARY.md @@ -0,0 +1,43 @@ +# Why the realized-cover theorem cannot simply use every transformed price + +**Object:** P15-PRICE-BOUNDARY-20260924-v1. Author: OpenAI / ChatGPT. +This is an exact counterexample to one proposed extension, not to P15-B or to +an unrestricted large-palette theorem. + +The realized-cover theorem deliberately assumes c_v<=p_v. Consider its smallest +one-block member: a_1=d_1=1, X={x,y}, no macro edges. Its good sets are the empty +set and singletons, its palette optimum is K_H(d)=1, and its one-piece obstruction +is exactly {X}. The prescribed cover is {{x,y}}. + +Take independent original probabilities p_x=p_y=1/2 and prices c_x=c_y=2/3. +These are admissible TRANSFORMED prices: phi(1/2)=log 2>2/3. For example, + + log 2=2 integral_0^(1/3) 1/(1-t^2) dt > 2/3. + +Nevertheless mu_p(D)=3/4, and + + min(1,-log mu_p(D))=log(4/3)<1/3, + +using log(1+x)0. Every cover of the obstruction {X} must contain a +generator g subset X. The possible generator costs are1,2/3,2/3,4/9 for the empty +set, the two singletons, and X respectively. Additional nonnegative-price +generators cannot reduce the minimum. Therefore the EXACT optimal cover price is + + covercost_c(O_1(D))=4/9 > 1/3 > log(4/3). + +Thus replacing c<=p by c<=phi(p) in the realized-family theorem while keeping its +same palette K_H(d) is FALSE in general, even if arbitrary alternative covers +rather than the prescribed cover are permitted. This is stronger than merely +saying that the existing proof does not establish the extension. + +It does not contradict the original P15-B amalgamation: that source separately +requires each local cover to meet its local phi(q_i) budget, which fails here. +It also does not refute a theorem allowing more colors: with two colors this +particular whole ground set is properly colorable, the obstruction is empty, +and the empty family has price zero. No conclusion about the original408-scale +or unrestricted prize follows from a one-color counterexample. + +The accompanying three finite tests enumerate all generator families on the +actual two-coordinate ground set, compute the product-measure probability exactly, +and check the rational comparisons used above. The two elementary logarithm +inequalities are proved in this note, not replaced by floating-point evaluations. diff --git a/specifications/lean_statements_20261010/sources/p15-price-budget/PROOF.md b/specifications/lean_statements_20261010/sources/p15-price-budget/PROOF.md new file mode 100644 index 000000000..9b17d179d --- /dev/null +++ b/specifications/lean_statements_20261010/sources/p15-price-budget/PROOF.md @@ -0,0 +1,152 @@ +# Restoring the transformed-price budget under explicit hypotheses + +**Object:** P15-TRANSFORMED-PRICE-BUDGET-20260924-v1. +**Author:** OpenAI / ChatGPT. **Disposition:** author-side derivation; nonauthor review open. + +[Research home](https://github.com/d6g8k5htny-coder/main) · [Math index](../../README.md) · [Prior construction](../three_fronts_20260924/P15_REALIZED_COVERS.md) · [Unchanged counterexample](../three_fronts_20260924/P15_PRICE_BOUNDARY.md) + +## 1. What changes and what does not + +The exact original-coordinate construction in `P15_REALIZED_COVERS.md`, SHA256 +`c0dbb821fb57b685a20cc321e4074456f39a9697f3104afd1f728e4732179bb9`, proves a +cover theorem for prices c_v<=p_v. The separate two-coordinate counterexample, +SHA256 `498af650ef7139c39d2630561dbfd0822c495ce6308a6cac8b767aefe0c2a40b`, +refutes extending the SAME palette to every c_v<=phi(p_v) with no additional +hypotheses. Both sources remain unchanged. Here phi(p)=min(1,-log(1-p)), with +phi(1)=1. This note supplies additional sufficient hypotheses; it does not retract +that counterexample, accept an unrestricted prize, or claim new classical coloring theory. + +Use the prior construction exactly. H is a clutter of block supports, with every +edge of size at least two; the original disjoint blocks satisfy |X_i|=a_i*d_i+1. +The downset D consists of sets using at most a_i vertices of each block and whose +occupied-block support contains no H-edge. The complete crossing witnesses are +ALL H-edge transversals. K_H(d) is the integer demand-palette optimum. The specified +cover consists of the full blocks X_i; an obstructed set contains a generator. +A generator g costs product_(v in g)c_v, the empty generator costs one, and an +empty family costs zero. No random coordinates are copied. + +**Theorem (restricted transformed-price extension).** Suppose every a_i>=1, +every d_i>=2, and all independent original probabilities satisfy 0<=p_v<=1/4. +Then for EVERY set of prices 0<=c_v<=phi(p_v), at the SAME K>=K_H(d), + + covercost_c(O_K(D)) <= min(1,(16/27)*[-log mu_p(D)]). (T1) + +The uncapped full-block union has cost at most the expression inside the second +argument of this minimum. Cap with the empty generator when needed. The palette +and the original-coordinate support family are not enlarged. The constants in +this theorem are explicit; unlike the Gaussian lifetime remainder, there is no +unevaluated radius or error constant here. + +## 2. Local budget proof + +For 0<=p<1, elementary integration gives + + -log(1-p)=integral_0^p dt/(1-t) <= p/(1-p). (T2) + +Consider one block with capacity a>=1, demand d>=2 and n=ad+1. Its local failure +probability is q=P(number of selected original coordinates > a). If any p_v=0, +the full-block generator price is zero and the desired inequality is immediate; +the zero-price generator remains in the cover for setwise completeness. + +Otherwise choose any subset S of a+1 original coordinates. The event that all +members of S are selected is a local failure, regardless of the other coordinates. +By independence, q>=product_(v in S)p_v. Using (T2), p_v<=1/4, and n>=2a+1, + + product_(v in X)c_v / product_(v in S)p_v + <= product_(v notin S)p_v / product_(v in X)(1-p_v) + <= (1/4)^(n-a-1)/(3/4)^n + = (4^(a+1))/(3^n) + <= (4/3)*(4/9)^a <=16/27. (T3) + +Consequently the full-block generator satisfies + + price({X}) <= (16/27)q <= phi(q). (T4) + +This proof does not require equal probabilities. No independence of different +crossing-witness events is used. The extra demand hypothesis d>=2 is load-bearing: +when a=d=1, n=2, the ratio calculation does not supply (T3), and the earlier +counterexample applies. The bound p_v<=1/4 is a sufficient range, not an asserted +sharp threshold. + +A more general sufficient ratio, for any common upper bound p_*<1, is + + rho(a,d,p_*)=p_*^(ad-a)/(1-p_*)^(ad+1). (T5) + +If rho<=1 the same argument proves price({X})<=q for that block. This is only a +sufficient test; rho>1 is not a counterexample and must not be labeled infeasible. + +## 3. Global budget and unchanged palette + +Write q_i=P(U intersect X_i notin D_i) for the ACTUAL capacity restriction. +Disjoint original blocks make their local-good events independent. Global goodness +implies all local restrictions are good, even though crossing constraints may +create further failures. Therefore + + mu_p(D)<=product_i(1-q_i), + sum_i q_i<=-log(product_i(1-q_i))<=-log mu_p(D). (T6) + +Sum (T4), then use (T6). The exact setwise argument of the prior realization +shows that the union of full-block generators covers O_K(D) for K>=K_H(d), +independently of prices. This proves (T1). If mu_p(D)=0 the price-one cap applies; +if mu_p(D)=1 the union price is zero by the same estimates. This is the same +local-cover/budget interface as original P15-B, source SHA256 +`9b18b6e9abc90d18deef06ab12e3aa7794dad40e1618d99daa88e369c300e8c3`. + +## 4. Exact original-coordinate consequence for the 816-label example + +Use the prior six blocks of 409 original coordinates, a_i=1, d_i=408, and all +15 four-block forbidden supports. At p_v=1/40900, every hypothesis above holds. +Thus the SAME 816-label obstruction cover is valid for the FULL transformed range + + 0<=c_v<=log(40900/40899), + +not merely c_v<=1/40900. At maximal equal prices its nonzero union price is + + 6*log(40900/40899)^409 <=6/(40899^409). (T7) + +An explicit rational price strictly larger than p is + + c=p+p^2/2=81801/3345620000. + +It is admissible since integral_0^p 1/(1-t)dt > integral_0^p(1+t)dt for p>0. +Its cover price is exactly 6*c^409. The palette remains 816; the entire ground +set still requires 818 colors, and there are 2454 original coordinates. No large +witness enumeration or new claim about arbitrary old P15 instances is involved. + +## 5. A useful exact checker beyond the simple uniform hypothesis + +The supplied standard-library program computes q exactly for rational inputs via +a truncated Poisson-binomial recursion. If b_j denotes P(exactly j successes) for +j<=a, one new coordinate updates b_j to (1-p)b_j+p b_(j-1); discarded higher +counts never return. Hence q=1-sum_(j=0)^a b_j after all coordinates. + +It bounds every transformed generator price by + + U=product_v min(1,p_v/(1-p_v)), + +with the p_v=1 factor defined as one. It certifies a sufficient local budget if +U<=q, or if q>=2/3 (then phi(q)=1, since log3>1). The low-probability theorem +additionally verifies U<=(16/27)q. It rejects inexact inputs and explicitly limits +problem size; its arithmetic cost is O(n*a) rational operations, not a polynomial +bit-complexity or arbitrary-hypergraph optimization claim. + +A non-certified output means only these sufficient tests did not prove the budget. +The program is not a full cover optimizer, a logarithmic interval decision procedure, +a test of global scientific status, or an independent proof reviewer. The earlier +two-coordinate counterexample is correctly not certified. The proof of log3>1 +is 2*integral_0^(1/2)1/(1-t^2)dt>1, so the cap test is also exact. + +## 6. Review and attribution + +Review (T3)'s ratio exponents, the demand and probability conditions, (T6)'s +inequality direction, zero-price completeness, and the distinction between the +specified cover threshold and whole-ground colorability. Finite tests compare +the local recursion with subset enumeration and preserve the earlier counterexample. +They do not establish independent analytic acceptance. + +Primary-source reconnaissance on 24 September 2026 inspected Park–Pham, +*A Proof of the Kahn–Kalai Conjecture*, arXiv:2203.17207 abstract, and +Gunby–He–Narayanan, *Down-set thresholds*, arXiv:2112.08525 abstract. +These are neighboring threshold results, not inputs to (T1) and not evidence of +novelty. The proof here uses the named in-project sources and the explicitly derived +elementary inequalities. No theorem from those abstracts is imported. diff --git a/specifications/lean_statements_20261010/sources/p15-realized-covers/P15_REALIZED_COVERS.md b/specifications/lean_statements_20261010/sources/p15-realized-covers/P15_REALIZED_COVERS.md new file mode 100644 index 000000000..173881916 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/p15-realized-covers/P15_REALIZED_COVERS.md @@ -0,0 +1,140 @@ +# P15: actual-coordinate realizations with nonempty covers and exact palette demand + +**Object:** P15-REALIZED-COVERS-20260924-v1. +**Author:** OpenAI / ChatGPT. **Disposition:** author-side mathematical derivation; no unrestricted prize closure or independent review. + +## 1. The source interface, not a substitute hypergraph + +The original source is P15-B, `P15-B_PALETTE_SEPARATED_LOCALIZATION.md`, Drive `19D-eHQAIXMGGy2ThZUfZ0GGjIKWm5C2j`, 6286 bytes, SHA256 `9b18b6e9abc90d18deef06ab12e3aa7794dad40e1618d99daa88e369c300e8c3`. It was read in full. P15-B requires ACTUAL local restrictions, compatible local covers at the same original-coordinate prices, and palette separation on EVERY crossing minimal original forbidden set. Main issues59/61 and the later consecutive-palette construction left that application check open in general. + +This note supplies an explicit, nontrivial family for which all those inputs are proved, including NONEMPTY local generator covers. It does not claim that every existing P15 downset belongs to this family or that the previous abstract examples already specified these original coordinates. The price range is deliberately stated as 0<=c_v<=p_v, a subset of P15-B's possible transformed-price range c_v<=phi(p_v). No assertion is made here for the additional prices p_v=1; removing a vertex leaves a proper subset of e, which contains no H-edge by the clutter property. Thus it is minimal. Conversely, any bad set either exceeds a local capacity and contains a set in (a), or has support containing an H-edge and contains a transversal from (b). This proves completeness, not just existence of some witnesses. + +In particular the crossing-support hypergraph is EXACTLY H, and the actual local restriction is + + D_i=D intersect 2^(X_i)={S subset X_i: |S|<=a_i}. (P2) + +The restriction does not inherit an artificial stronger condition. Neither a sampled list of transversals nor an incomplete lifting can replace (b). + +## 3. Real, positive-price local obstruction covers + +The k-fold decomposability of a capacity family is exact: + + I_k(D_i)={S:|S|<=a_i k}. + +Necessity follows by summing the k capacities; sufficiency partitions S into successive groups of at most a_i elements. For k=d_i and n_i=a_i d_i+1, the only obstructed subset of X_i is X_i itself. Hence + + G_i={X_i} (P3) + +is an exact local obstruction cover. It is not the empty family arising from proper colorability of the entire block with d_i colors. + +Select the ORIGINAL coordinates independently with arbitrary probabilities p_v in [0,1], and let 0<=c_v<=p_v. With q_i=P(U_i notin D_i), + + price(G_i)=product_(v in X_i)c_v + <=product_(v in X_i)p_v + =P(U_i=X_i)<=q_i<=phi(q_i), (P4) + +where phi(t)=min(1,-log(1-t)), phi(1)=1. When all c_v>0 the local price is positive. When a zero price occurs the generator is still retained for setwise coverage. + +The independent-block calculation uses (P2), so + + mu_p(D)<=product_i(1-q_i), + sum_i price(G_i)<=sum_i q_i + <=-log(product_i(1-q_i))<=-log mu_p(D). (P5) + +If some q_i=1 the trivial price-one cover handles the endpoint. When the summed price exceeds1 use {empty set}; otherwise retain union_i G_i. This produces the capped hazard bound whenever the palette condition below holds. Crossing events are not declared independent, and no coordinates are cloned. + +## 4. The exact number of colors needed for this cover + +Let K_H(d) be the minimum size K of an integer palette universe admitting subsets P_i with |P_i|>=d_i and + + intersection_(i in e) P_i = empty, for every e in H. (P6) + +Equivalently each label's incident macro-indices form an independent set of H. By discarding excess labels within blocks, one may take |P_i|=d_i. This is the actual macro-demand coloring problem; fractional palettes are not used. + +**Theorem P2.** For D in (P1), the fixed family G={X_1,...,X_b} covers O_K(D) if and only if + + K>=K_H(d). (P7) + +For sufficiency take a set U avoiding all generators. Then |U_i|<=a_i d_i. Split U_i into at most d_i groups of size<=a_i and assign them distinct labels from P_i. A global label's block support is H-independent by (P6), and it uses at most a_i vertices in each block. Each resulting color class is in D. Thus U is K-decomposable. + +For necessity choose U with |U_i|=a_i d_i in EVERY block, omitting one original vertex per block. It avoids every generator X_i. Any D-coloring of U needs at least d_i distinct colors in block i because of its capacity a_i. If a label occurs in every block of an H-edge, completeness of the transversal forbidders makes that label class bad. Thus its used-color sets form palettes satisfying (P6); at least K_H(d) colors are necessary. For K=K_H(d), + + covercost_c(O_K(D)) <= min(1,-log mu_p(D)). (P8) + +Optimality in (P7) is for the SPECIFIED union-of-block-generators cover. It does not prove that no different cover with the same budget works at fewer colors. It is not unrestricted prize optimality. + +For comparison, the chromatic number of the ENTIRE ground set X is K_H(d+1), because ceil(n_i/a_i)=d_i+1. The same palette argument gives both inequalities. Therefore cover demand and whole-ground colorability are distinct quantities. + +## 5. Realized structural subclasses + +For any H whose independent sets form a loopless matroid, the established matroid palette formula in main issue59 can be applied to this actual family. It is not necessary to establish a new matroid theorem here. For a consecutive-capacity presentation, the corresponding consecutive-palette theorem can similarly be consumed only when its exact complete-support hypotheses hold. + +A fully elementary case is H consisting of all (s+1)-subsets of b indices, 1<=s<=b. Each label may occur in at most s blocks, and + + K_H(d)=max(max_i d_i, ceil(sum_i d_i/s)). (P9) + +The two lower bounds follow from one block and from total label incidences. For the matching construction, concatenate segments of lengths d_i along the cyclic word 0,1,...,K-1. Each block has distinct labels because d_i<=K. Each label occurs at most ceil(sum d_i/K)<=s times. This proves (P9) without enumerating individual labels when demands are large. + +The macro triangle at unit demands needs3 colors, although a formula considering only size-two edge loads would give2. The theorem uses K_H itself or a proved structural method, not an unrestricted edge-load formula. + +## 6. A nonempty-cover realization of the 816-label benchmark + +Take six blocks, a_i=1 and d_i=408. Thus EACH original block has409 vertices and the original ground set has2454 vertices. H consists of all15 four-block subsets. Every D-color class contains at most one vertex per block and occupies at most three blocks. + +The original minimal forbidders are exactly + + 6*binom(409,2)=500616 internal pairs, + 15*409^4=419743994415 crossing transversal quadruples. (P10) + +These are counts obtained from a proved complete structural formula, not a claim that the program enumerated more than four hundred billion sets. + +Use408 labels on macro-indices {0,2,4} and408 different labels on {1,3,5}, giving + + K_H(408,...,408)=816. + +The actual cover has six generators, the six FULL409-vertex blocks. At816 colors it covers the obstruction; at815 it does not, because omitting one vertex from each block leaves2448 vertices, each color holds at most3, and815*3<2448. Pairwise-disjoint local palettes would have used2448 labels. The whole ground set, unlike the cover-avoiding sets, needs818 colors, since2454/3=818. + +This is a NEW specified original-coordinate family instantiating the previous abstract benchmark; it is not a statement that every earlier six-block application had409 vertices per block or these complete transversals. + +A concrete low-failure, positive-price instance is p_v=c_v=1/40900. The cover price is exactly + + 6/(40900^409)>0. + +A union bound on internal pairs and four-block occupancy gives + + Q=1-mu_p(D) + <=500616/(40900^2)+15/(100^4) + =2449227/8180000000 < 3/10000. (P11) + +For the second term each block's occupancy probability is at most409p=1/100, and occupancies of distinct blocks are independent. This is only a union bound, not an equality of the global bad probability. Thus the hazard budget is not being satisfied solely in the trivial high-failure regime; the actual-good probability exceeds0.9997. The cost comparison (P5) supplies (P8), rather than comparing this positive price to an unproved numerical hazard lower bound. + +## 7. Exact verification and boundaries + +The executable finite checker constructs actual block coordinates for small cases. It independently enumerates minimal bad subsets, tests equality with (a)-(b), computes actual restrictions, compares original-coordinate colorability with residual-demand dynamic programming, and checks every cover-avoiding subset in declared bounded examples. A removed transversal is a negative control: its now-good original set still violates the macro support condition, so incomplete lifting is detected. The large816 example uses the proved formulas (P9)-(P11), not an unbounded enumeration. + +The original P15-D source (Drive `1z-vrL-f5tDBbqybco-ZiS7X9vJq70ifA`) was also read. It treats complete GRAPH-transversal coupling, arbitrary macro chromatic number and a different scalar/hazard budget. This note does not improve its256 factor or extend it to arbitrary hypergraphs at the same fixed palette. It supplies a separate actual-realization theorem whose palette remains K_H(d). + +The unrestricted P15 prize, full transformed-price range, arbitrary input support identification, and nonauthor review remain open. Classical coloring, hypergraph demand coloring, union bounds and the source P15-B amalgamation are not claimed as inventions. The contribution supplied here is the exact compatible original-coordinate family, nonempty covers, and source-hypothesis verification that were absent from a merely abstract palette output. diff --git a/specifications/lean_statements_20261010/sources/remote-collision/PROOF.md b/specifications/lean_statements_20261010/sources/remote-collision/PROOF.md new file mode 100644 index 000000000..7b48a88e2 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/remote-collision/PROOF.md @@ -0,0 +1,328 @@ +# Witness collisions in the fixed-remote height window: a second factorial moment and matching probability asymptotics + +**Object:** CL-D5-REMOTE-COLLISION-20260928-v1. +**Author:** Anthropic Claude (Claude Code session `session_017Mi3hxjaxV45x6zo6o1ee3`). +**Disposition:** author-side proof candidate. **Nonauthor review is required.** +**Scientific effect:** NONE. No `STATUS`, `PROOF_INDEX` verdict, `GRAPH`, claim, prize or source body changes. + +## 1. Results + +### Setting + +Use the exact model, observations and weight of the reviewed fixed-remote theorem, `frontiers/remote_window_20260924/PROOF.md` (RN-FIXED-REMOTE-WINDOW-20260924-v1, Git blob `b383bfcc88ec4ad497dff01fb6640e429ba24a84`, "Theorem A" there). + +- **Field.** Fix `d>=2` and `L>0`. `f` is the centered variance-one Gaussian field on `X=R^d/(LZ^d)` with the exact normalized periodized Gaussian covariance. +- **Marks.** Compact `B=[b_-,b_+]` and `K=[k_-,k_+]` with `0=rho}` and `I_r=(b-kr^3,b)`. +- **Count.** For Borel `E` contained in `D_rho` and an index `j`, + + N_j(E) = #{x in E: grad f(x)=0, index H_x=j, f(x) in I_r}, + N(E) = sum_j N_j(E). + +The reviewed Theorem A gives + + | E_(Q^W) N_j(E) - k r^3 integral_E Lambda_j | <= C r^4 |E|, (A) + +with a continuous kernel `Lambda_j` that is bounded above and away from zero on `D_rho`. It states explicitly that (A) is **not** a matching lower bound on the event probability. Its §6 handles pairs only at a **fixed** separation `eta>0` and records that shrinking separation is uncontrolled. This is the open "witness-collision" region of the D5 graph (`math.rn-region.witness-collision`, "eta->0 mutual witness separation"). The present note treats that region **only when both witnesses lie in the fixed remote region `D_rho`**; see §8. + +### Theorem C (near-diagonal witness pairs) + +Fix `00`, depending on `d,L,rho,eta_0,B,K`, such that for `0= 2 } <= C r^5 |E|. (D) + +### Corollary E (matching probability asymptotics, fixed remote region) + +For every index `j` and Borel `E` contained in `D_rho`, + + | Q_r^W{ N_j(E) >= 1 } - k r^3 integral_E Lambda_j(x) dx | <= C r^4 |E|. (E) + +The same holds for `N(E)` with `Lambda=sum_j Lambda_j`. In particular, for `E` of positive volume the `Q_r^W`-probability that the remote region contains an additional window critical point is asymptotic to `k r^3 integral_E Lambda`. + +### Corollary F (lower bound on the whole torus in every fixed `d`; matching upper bound only for `d=2`, conditional on one pending import) + +For all small `r`, + + Q_r^W{ some critical point x != M,S has f(x) in I_r } >= c r^3, (F-) + +with `c=(k_-/2) inf integral_(D_rho) Lambda_j > 0`, for a fixed index `j` (or with `Lambda=sum_j Lambda_j` throughout, applying (E) to the total count). The infimum is over marks and frames. `r_*` may need to be reduced. The factor `1/2` is needed: an asymptotic `A r^3 + O(r^4)` with a possibly negative `O(r^4)` term only licenses constants strictly below `A`. + +**Upper half, planar only.** For `d=2`, consume the global single-witness first moment (I5) of `frontiers/intermediate_window_20260928`. That source is Math-#107, still pending integration; its nonauthor review record is merged as `reviews/d5_intermediate_window_claude_20260928/` via Math-#109. Markov then gives the matching upper bound `<= C r^3`, so the order of the event probability is exactly `r^3`. This half is **conditional** on that import. + +For `d>2` the upper half stays conditional on a dimension-matched global first moment. #107 does not supply one, because (I5) is planar. + +The lower half (F-) uses only (A) and (D) and holds in every fixed `d>=2`. Theorem C and Corollaries D and E also keep their fixed-`d` scope. + +## 2. Dependencies + +| Consumed | Where | Status | +|---|---|---| +| Theorem A (A), the kernel `Lambda_j`, and the uniform remote nondegeneracy argument | `frontiers/remote_window_20260924/PROOF.md` §§1–5 | ACCEPT on the fixed-rho interfaces (main#76 review and reconciliation) | +| Uniform `Q_r` moments `sup E_Q||f||_(C^m)^p= z_* r^2` | same parent, (5.5) | A3 ACCEPT (Math-#106) | +| Weighted Kac–Rice, general Borel sets | Armentano–Azaïs–León, arXiv:2304.07424v3, Thms 2.1, 2.2, 7.1, Remarks 7–8 | primary source; same use as parent §9 and the remote §4 | + +Nothing else is imported. In particular the cap theorem, elder selection and #107 are **not** used for Theorem C or Corollaries D–E. + +## 3. The pair Kac–Rice formula + +Let `D'` be an open neighbourhood of `D_rho` at distance `>=rho/2` from `0`. Let `T` be the open off-diagonal set of `D' x D'`, and define `G(x,x')=(grad f(x), grad f(x'))`, which maps into `R^(2d)`. + +Under `Q_r` (a Gaussian law with bounded mean), the covariance of `G(x,x')` is positive definite at every point of `T`: + +- the functionals are gradients at two distinct sites, both separated from the pins; +- by positivity of every Fourier weight of the periodized kernel (parent §2), distinct derivative functionals at distinct sites are linearly independent modulo the `2(d+1)` pin functionals. + +On compact subsets of `T`, continuity and compactness bound the density of `G` above. The derivative of `G` is block diagonal, `diag(H_x, H_x')`, so its Jacobian is `|det H_x||det H_x'|`. + +For the Theorem 7.1 mark, use the truncated weight + + g_m = min(W_r, m) 1{H_x in O_i} 1{H_x' in O_j} 1{f(x) in I_r} 1{f(x') in I_r}. + +Here `O_j` is the open set of nonsingular symmetric matrices with `j` negative eigenvalues. + +The witness determinants are **not** part of the mark. Theorem 7.1's Jacobian factor `Delta=|det H_x||det H_x'|` supplies them, and `Delta 1{H_x in O_i}1{H_x' in O_j} = F_i(H_x)F_j(H_x')`, so each witness determinant appears exactly once. The mark satisfies hypotheses (a)–(b) of Theorem 7.1: + +- `W_r` is a continuous nonnegative functional of the field; +- indicators of the open sets `O_i`, `O_j` and `I_r` are lower semicontinuous. + +So Theorem 7.1 of arXiv:2304.07424v3 applies. Its hypothesis (c) holds by Remark 8, and Theorem 2.1's hypotheses hold by Remark 7. Monotone convergence in `m`, together with the nondegenerate joint density of the four heights and gradients (below), then gives the ordered-pair formula for every Borel `B` contained in `T`: + + E_Q[ sum_{(x,x') in B: G=0} W_r 1{H_x in O_i} 1{H_x' in O_j} 1{f(x),f(x') in I_r} ] + = integral_B integral_(I_r) integral_(I_r) + p_(grad f(x),grad f(x'),f(x),f(x'))(0,0,y,y') + E_Q[ W_r F_i(H_x) F_j(H_x') | grad f(x)=grad f(x')=0, f(x)=y, f(x')=y' ] + dy dy' dx dx'. (3.1) + +Divide by `Z_r` to obtain the `Q_r^W` count. The endpoint weight appears once, and each witness determinant appears once. This is the `m=2` case of the remote §6 formula, now on the whole off-diagonal set rather than `|x-x'|>=eta`. + +## 4. Divided-difference coordinates, uniform through collision + +Write `x'=x+delta e`, with `0= 0, integral w_delta = 1. (4.2) + +**Limits.** As `delta` tends to 0, `V_delta` tends to + + V_0 = (grad f(x), H_x e, f(x), -(1/12) d_e^3 f(x)), + +in every `L^p`, uniformly in `x` and `e`. + +**Change of variables.** The map from `(grad f(x), grad f(x'), f(x), f(x'))` to `V_delta` is linear and block lower triangular. Its diagonal blocks are `I_d`, `delta^(-1) I_d`, `1` and `delta^(-3)`. So + + p_(grad f(x),grad f(x'),f(x),f(x'))(0,0,y,y') = delta^(-d-3) p_(V_delta)(0,0,y,(y'-y)/delta^3), (4.3) + +and on `{grad f(x)=grad f(x')=0}` we have `G_delta=0` and `D3_delta=(f(x')-f(x))/delta^3`. + +**Lemma 1 (uniform floor).** There are `c_1>0`, `C_1` and `r_1>0` such that + + c_1 I <= Cov_(Q_r)(V_delta) <= C_1 I, |E_(Q_r) V_delta| <= C_1, + +for all `00`, and its `r=0` limit is the contact jet `U_0` (parent §3). So the joint covariance is a continuous function on the compact set + `[0,r_1] x [0,eta_0] x D_rho x S^(d-1) x O(d)`. +2. **Positive definiteness for `delta>0`.** For `r>0`, `U_r` is an invertible transform of the actual endpoint observations `(f, grad f)` at `M` and `S`. Only at `r=0` is it the contact jet `U_0` at `0`. In both cases the pin functionals are supported within distance `r/2<=rho/4` of `0`, whereas `x` and `x'` are at distance `>=rho/2` from `0`. So the functionals are, up to an invertible map, distinct derivative functionals at distinct sites: `M`, `S` (or `0` at `r=0`), `x` and `x'`. +3. **Positive definiteness at `delta=0`.** At `x`, the functionals `f`, `grad f`, `H e` and `d_e^3 f` have distinct derivative orders 0, 1, 2 and 3. `H -> He` maps `Sym_d` onto `R^d` for every unit `e` (checked exactly in `collision_exact_check.py`), so the order-2 block has full rank. Together with the pin functionals, which are the endpoint observations at `M`, `S` for `r>0` and the contact jets at `0` for `r=0`, supported at distinct sites, the positive-Fourier-weight argument makes the covariance positive definite. +4. **Conclusion.** Continuity of `lambda_min` and compactness give the floor. The `Q_r` covariance is the Schur complement on `U_r`, which inherits the floor. The target `v_r` is bounded, so the means are bounded. ∎ + +**Lemma 2 (conditional moments and density).** For every `p` there is `C_p` such that, for all parameters above, `y` in `I_r` and real `t`, + + E_(Q_r)[ K^p | V_delta=(0,0,y,t) ] <= C_p (1+|t|)^p, + p_(V_delta)(0,0,y,t) <= C_p (1+|t|)^(-p), (4.4) + +where `K = C_d(1+||f||_(C^3(X)))`. + +*Proof.* Under `Q_r`, the pair `(f, V_delta)` is jointly Gaussian. + +- `Var V_delta<=C_1`, so every cross-covariance `|Cov_Q(D^alpha f(z), V_delta)|` is bounded by Cauchy–Schwarz and (4.1) of the parent. +- Regress `f` on `V_delta`. The conditional mean moves by at most `C(1+|y|+|t|)` in `C^3`. +- The centered residual has moments bounded by those of `f - E f` plus `C` times those of `V_delta`. +- The Gaussian density bound follows from Lemma 1. ∎ + +## 5. Pathwise bounds + +**Lemma 3 (endpoint weight).** Pathwise, `W_r <= r^2 K^(2d)` for `r<=1`. + +*Proof.* By parent (5.1), `|alpha_i|` and `||beta_i||` are at most `M3/2<=K`, and `||A_i||<=K`. By (5.3), +`|det H_i| = r|alpha_i det A_i - r beta_i^T adj(A_i) beta_i| <= r K^d`. Multiply the two endpoints. ∎ + +**Lemma 4 (a critical pair forces two small Hessian directions).** If `grad f(x)=grad f(x+delta e)=0`, then + + |H_x e|, |H_(x') e| <= K delta / 2, + |det H_x|, |det H_(x')| <= (delta/2) K^d. (5.1) + +*Proof.* + +1. `0 = grad f(x+delta e)-grad f(x) = integral_0^delta H(x+te) e dt`, so + `delta H_x e = -integral_0^delta (H(x+te)-H_x) e dt`, and `|H_x e| <= delta^(-1) integral_0^delta K t dt = K delta/2`. +2. The same argument from `x'` in direction `-e` gives the bound for `H_(x') e`. +3. For symmetric `H` and unit `e`: `|det H| = product sigma_i <= sigma_min ||H||^(d-1) <= |He| ||H||^(d-1)`, and `||H||<=K`. ∎ + +Both witnesses of a colliding pair therefore carry a small Hessian direction. This is what compensates the `delta^(-d)` singularity of the gradient density. + +## 6. Proof of Theorem C + +**Rewrite the near pairs.** Restrict (3.1) to `B_near={(x,x') in E x E: 0<|x-x'|=z_* r^2`, + + E_(Q^W) T_near(E) <= C r^3 |E| integral_0^infinity delta min(1, k r^3/delta^3) d delta + = C r^3 |E| (3/2) (k r^3)^(2/3) + <= C' r^5 |E|. + +The last integral is evaluated exactly by splitting at `delta_0=(k r^3)^(1/3)`: it equals `delta_0^2/2 + k r^3/delta_0 = (3/2) delta_0^2`. This proves (C). ∎ + +**Remark (where the gain comes from).** Pairs closer than about `r` cost only one window factor `r^3`, because the second height is then automatically in the window. They are rare because of their small area, `O(r^2)`. For pairs farther apart than `r`, the second window supplies the extra factor `r^3/delta^3`. Both effects give `r^3 * r^2`. + +## 7. Separated pairs and the corollaries + +**Lemma 5 (separated pairs).** On `B_sep={(x,x') in E x E: |x-x'|>=eta_0}`, + + E_(Q^W) T_sep(E) <= C r^6 |E|^2. + +*Proof.* The site pairs lie in a compact set of distinct, separated sites. As in Lemma 1, the joint density of `(grad f(x), grad f(x'), f(x), f(x'))` under `Q_r` is bounded, and conditional `K`-moments are bounded. By Lemma 3, and since `|det H|<=K^d`, the integrand in (3.1) is at most `C r^2`. Integrating the two windows gives `(k r^3)^2`, and dividing by `Z_r` gives `r^6`. This is the `m=2`, `eta=eta_0` estimate of the remote §6, (15)–(16), re-derived here for mixed indices. ∎ + +**Proof of Corollary D.** Sum Theorem C and Lemma 5 over the finitely many index pairs. This gives `E N(E)(N(E)-1) <= C r^5|E| + C r^6|E|^2`. + +The second term is absorbed explicitly: `r^6|E|^2 = r^5|E| * (r|E|) <= r_* L^d * r^5|E|`, using `r<=r_*` and `|E|<=L^d`. So `E N(E)(N(E)-1) <= C' r^5|E|`, uniformly in `E`, including sets of tiny volume. For integers `N>=0`, `1{N>=2} <= N(N-1)/2`. ∎ + +**Proof of Corollary E.** For integers `N>=0`, `N - N(N-1)/2 <= 1{N>=1} <= N`. + +Apply this to `N_j(E)`. Since `0<=N_j(E)<=N(E)` and `n -> n(n-1)` is nondecreasing on the nonnegative integers, `N_j(E)(N_j(E)-1) <= N(E)(N(E)-1)`, so (D) for the total count also bounds the index-`j` factorial moment. Take expectations and use (A) and (D): + + k r^3 integral_E Lambda_j - C r^4|E| - C r^5|E| <= Q^W{N_j(E)>=1} <= k r^3 integral_E Lambda_j + C r^4|E|. ∎ + +**Proof of Corollary F.** The event contains `{N_j(D_rho)>=1}`. By (E), its probability is at least `k r^3 integral_(D_rho) Lambda_j - C r^4 >= r^3 (2c - C r)`, with `c` as above. After reducing `r_*` so that `C r_* <= c`, this is at least `c r^3`. For `d=2`, the upper half is Markov applied to the (I5) first moment and is conditional on that import. For `d>2` no upper half is claimed. ∎ + +## 8. Scope and non-claims + +**Established:** the witness-collision region **inside the fixed remote region `D_rho`**, meaning pairs at all separations below `eta_0`, including coincidence limits, at first-order factorial-moment level. Also a **matching lower bound** on the window-event probability, which the remote theorem explicitly did not supply. + +**Not established:** + +- **A torus-wide second factorial moment, or `math.rn-region.witness-collision` in its full torus-wide sense.** Pairs with either witness in the local, collar or intermediate regions are not estimated here. Corollary F's lower bound is global only because the global event contains the remote event. + +- **Pairs with a point near the pins or at intermediate scales.** One or both witnesses in `|x| < rho` would need the collision estimate combined with the #105 and #107 machinery. The remote nondegeneracy of Lemma 1 is not uniform as `rho` tends to 0. +- **All-height pairs.** Only window pairs are counted. +- **Higher factorial moments, a Poisson limit, or an explicit numerical constant.** +- **Anything about elder selection.** D1 is separate. +- **Uniformity** as `k -> 0`, as marks grow, or as `d` or `L` vary. +- Any status transition. + +## 9. Literature and method + +The divided-difference change of variables near the diagonal is the standard tool for two-point Kac–Rice formulas at short range. See: + +- S. Ladgham, R. Lachièze-Rey, *Local repulsion of planar Gaussian critical points*, SPA 2023, which gives the exact second factorial moment in small balls; +- D. Beliaev, V. Cammarota, I. Wigman, *No repulsion between critical points for planar Gaussian random fields*, ECP 2020; +- L. Gass, M. Stecconi, *The number of critical points of a Gaussian field: finiteness of moments*, PTRF 2024; +- S. Muirhead, *A second moment bound for critical points of planar Gaussian fields in shrinking height windows*, SPL 160 (2020), arXiv:1901.11336, which treats unconditioned stationary fields in large domains. + +None of these treats the pinned, determinant-weighted law `Q_r^W`, and none extracts the two-window factor `min(1,r^3/delta^3)`. Their theorems are not imported. Only the method, divided differences plus Kac–Rice, is shared. No priority claim is made from this bounded search. It used Consensus and the arXiv records, and a Drive search found the program's July 2026 literature pass on three-point machinery, which is consistent with this assessment. + +## 10. Finite controls + +`collision_exact_check.py` uses the Python standard library only and exact arithmetic. It checks: + +- the trapezoid identity (4.2), including the constant `1/12` and the kernel mass; +- the divided-difference average for `G_delta`; +- the Jacobian `delta^(-d-3)` of (4.3), for `d=2,3`; +- surjectivity of `H -> He` for exact rational unit vectors, `d=2,3`; +- the determinant inequality `det(H)^2 <= |He|^2 ||H||_F^(2(d-1))` on 3,000 exact rational matrices, and the equality `|det H| = |He| ||H||_op^(d-1)` on diagonal families with `e` along the smallest eigenvalue (so Lemma 4 is sharp); +- the fold example `f=t^3/3 - eps t`: its two critical points `x=-sqrt(eps)`, `x'=sqrt(eps)` are at distance `delta=2 sqrt(eps)`, with `f(x')-f(x) = -delta^3/6 = -(1/12) f''' delta^3`, which is (4.2) with equality; +- the `delta` and `r` power ledgers; +- the exact integral `(3/2) a^(2/3)` for `a=q^3`; +- the Bonferroni inequalities for `N<=60`. + +Five mutants, altering the Jacobian exponent, the trapezoid constant, a dropped determinant factor, the window factor, and the Bonferroni sign, must fail. These checks do not prove the continuum or Gaussian steps. + +## Appendix A. Details for the continuum steps held in review 5342481786 + +These spell out standard steps; none changes a statement. + +**A1. Theorem 7.1 hypotheses for the pair process.** +- Take `T` (the open off-diagonal set of `D' x D'`) as parameter set, `X(x,x') = G(x,x') = (grad f(x), grad f(x'))` (in AAL's notation both the parameter dimension and the value dimension equal `2d`), and the auxiliary process `Z = (f, Df, D^2 f)` with continuous paths on the torus. +- Theorem 2.1's (i): `f` has smooth paths, so `X` is `C^1`. +- Theorem 2.1's (ii): under `Q_r`, `X(t)` is Gaussian with covariance continuous in `t` and positive definite on `T` (§3), so its density is continuous and locally uniformly bounded on compact subsets of `T`. +- Theorem 2.1's (iii) is Remark 7: the conditional law given `X(t)=v` is a Gaussian regression, continuous in `v` for the `C^1` topology. +- Theorem 7.1's (a)–(b): `g_m(t,Z)` is a product of the continuous nonnegative `min(W_r,m)`, which depends only on `Z` at the fixed pins, with indicators of the open sets `{H_x in O_i}`, `{H_x' in O_j}`, `{f(x), f(x') in I_r}`. `(t,Z) -> H_x, f(x)` is jointly continuous, so each indicator is lower semicontinuous in `t` and in `Z` (uniform convergence on compacts), and so is the product. +- Theorem 7.1's (c) is Remark 8, since `(X, Z)` is jointly Gaussian under `Q_r`. + +**A2. Removing the truncation.** +- `g_m` increases to `g = W_r 1{O_i}1{O_j}1{I_r}` pointwise. +- The left side of (3.1) is an expected sum of nonnegative terms, each increasing in `m`. +- The right side is `integral_B p(0) E[Delta g_m | X=0] dt`, whose integrand increases in `m`, first inside the conditional expectation and then in `t`. +- Monotone convergence applies on both sides, so (3.1) holds for `g`, as an equality of extended nonnegative numbers. The subsequent estimates show it is finite. +- Height disintegration uses the nondegenerate joint density of `(X(t), f(x), f(x'))`, again from §3/Lemma 1. + +**A3. Uniform `L^p` continuity of `V_delta` through `delta=0` (Lemma 1).** +- Each coordinate of `V_delta` has the form `integral D^alpha f(x+te)[...] mu_delta(dt)`, with `mu_delta` a probability measure on `[0,delta]` (a point mass, the uniform law, or the trapezoid weight `w_delta`). As `delta` tends to 0, `mu_delta` tends to the point mass `delta_0`. +- Covariances of such coordinates are `integral integral D^alpha D^beta K_L((x+te)-(y+se)) mu(dt) nu(ds)`. `K_L` is smooth (periodized Gaussian), so these are jointly continuous in `(delta, x, e)`, including `delta=0`. The same holds for cross-covariances with the pin functionals `U_r`, jointly with `r` in `[0,r_1]` (parent §3 / A2 of Math-#106). +- `V_delta - V_0` is centered Gaussian, so `L^2` continuity gives `L^p` continuity for every `p`. +- Positive definiteness at every parameter point is parent §2 (A1 of Math-#106, accepted): distinct derivative functionals at distinct sites have positive-definite covariance because every Fourier weight `a_n>0`. At `delta=0` the order-2 functionals `(He)_i` are distinct because `H -> He` is onto. + +**A4. Lemma 2.** +- Write `Sigma = Cov_Q(V_delta)`, with `c_1 I <= Sigma <= C_1 I`, and `A(z) = Cov_Q(D^alpha f(z), V_delta) Sigma^(-1)`. +- `|Cov_Q(D^alpha f(z), V_delta)| <= (Var_Q D^alpha f(z))^(1/2) C_1^(1/2)`, which is uniformly bounded by parent (4.1). So `sup_z |A(z)| <= C`. +- Conditioning on `V_delta = w` shifts the mean of `D^alpha f` by `A(z)(w - E_Q V_delta)`, of size at most `C(1+|w|)`. +- The residual `f - E_Q[f | V_delta]` is Gaussian and independent of `V_delta`. Its `C^3` sup norm has moments bounded by those of `f - E_Q f`, plus `C` times those of `V_delta`. +- Hence `E_Q[K^p | V_delta = w] <= C_p (1+|w|)^p`. The density bound is the Gaussian density with `Sigma >= c_1 I` and bounded mean. + +## Revision history + +- **v1**, `a4d4f74`: initial candidate. +- **v2**: clarifications from the OpenAI/ChatGPT review [5342280263](https://github.com/d6g8k5htny-coder/Math-/pull/110#pullrequestreview-5342280263) at `a4d4f74`. That review accepts the fixed-remote near-diagonal mechanism, its `O(r^5)` order, and Corollaries E and F at their stated scopes. It is source-exposed on the remote dependency. The clarifications: + - Corollary D now states the absorption `r^6|E|^2 <= r_* L^d r^5|E|` explicitly. + - Corollary E now states `N_j(N_j-1) <= N(N-1)`. + - §1 and §8 now say that this does not close a torus-wide second factorial moment or the torus-wide witness-collision region. + - No mathematical change was made. +- **v3**: adds Appendix A, which spells out the continuum steps held in the xAI/Harper review [5342481786](https://github.com/d6g8k5htny-coder/Math-/pull/110#pullrequestreview-5342481786): the Theorem 7.1 hypotheses for the pair process, truncation removal, `L^p` continuity through `delta=0`, and Lemma 2. That review accepts the finite identities, the (3.1) mark and the Lemma 1 mechanism. No statement changes. +- **v4**: amendments from the OpenAI foreground review ([comment 5876143293](https://github.com/d6g8k5htny-coder/Math-/pull/110#issuecomment-5876143293)): + - Corollary F's explicit constant is now `c=(k_-/2) inf integral Lambda_j`, not the infimum itself. + - The matching upper half of F is restricted to `d=2`, because #107's (I5) is planar. For `d>2` it remains conditional on an unsupplied dimension-matched first moment. + - Lemma 1 step 2 now distinguishes the endpoint observations (for `r>0`) from the contact jet (at `r=0`). + - Theorem C, Corollaries D and E, and (F-) are unchanged in scope. diff --git a/specifications/lean_statements_20261010/sources/rn-annulus-bridge/PROOF.md b/specifications/lean_statements_20261010/sources/rn-annulus-bridge/PROOF.md new file mode 100644 index 000000000..6f317515b --- /dev/null +++ b/specifications/lean_statements_20261010/sources/rn-annulus-bridge/PROOF.md @@ -0,0 +1,245 @@ +# D5 all-height fixed-annulus bridge candidate + +**Object:** D5-ALL-HEIGHT-ANNULUS-BRIDGE-20260925-v1. +**Author:** OpenAI / ChatGPT, session OA-D5-ANNULUS-BRIDGE-20260925. +**Disposition:** AUTHOR-SIDE ANALYTIC CANDIDATE; independent review pending. +**Scientific effect:** NONE. No theorem/claim-status register is changed. + +## 1. Statement and exact scope + +Fix L>0 and the centered variance-one Gaussian field on the two-dimensional torus with covariance + + K_L(h) = sum_(n in Z^2) exp(-|h+Ln|^2/2) + / sum_(n in Z^2) exp(-|Ln|^2/2). + +Fix a compact birth interval, 00, uniform in the stated marks, frames, Borel E and j=0,1,2, such that + + E_(Q_r^W) N_j(rE) <= C r^3 area(E), 0=k_->0, (2) can also be written <=C' k r^3 area(E), but that rewriting uses k_-: **there is no height-window factor in the proof and no assertion uniform as k approaches zero**. + +This addresses the fixed scaled-annulus region in d=2. It does not include pin neighborhoods, A approaching 1/2, B growing with r, intermediate physical distances r<=3, uniform-L estimates, numerical all-cell certificates, elder selection or unrestricted lifetime acceptance. Choose r_* small enough that all local segments used below embed in the torus. + +## 2. What was missing, and what is added + +PR22's two-scale estimate controlled the conditional product of three Hessians by C delta^-6, leading to the physical intensity C r^-5 delta^-8 exp(-c/delta^2). That estimate alone retains an adverse r power on a strip of fixed positive scaled width. + +The additional fact is deterministic: **three noncollinear zero gradients force all three Hessians to be small**. The Gaussian input must control the FULL conditional C3 norm after the witness gradient condition, not only endpoint-only moments. Combining the two facts supplies the factor + + min(1,(r/|v|)^6), (3) + +with value 1 at v=0. Splitting at |v|=r, not sending a fixed-chart constant to infinity, absorbs the remaining inverse powers into the Gaussian penalty. No new height conditioning or shrinking polynomial/logarithmic cutoff is needed. + +All precursor sources remain unreviewed same-provider candidates. Sections 3-8 give the required arguments explicitly rather than treating those sources as accepted premises. The separate reviewer must verify these arguments, not infer their validity from matching implementations. + +## 3. Uniform endpoint regression, moments and original Z + +Let a=r/2, g(x)=f(x,0), h(x)=f_z(x,0). Let H_r be the cubic interpolating g and g' at +/-a, and L_r the affine interpolant of h at +/-a. Their midpoint coefficients form an invertible observation vector + + U_r=(H_r(0),H'_r(0),H''_r(0),H'''_r(0),L_r(0),L'_r(0)). + +For arbitrary data define s0=(g(-a)+g(a))/2, d0=(g(a)-g(-a))/(2a), s1=(g'(-a)+g'(a))/2 and d1=(g'(a)-g'(-a))/(2a). Then + + U_r=(s0-a^2 d1/2,(3d0-s1)/2,d1,3(s1-d0)/a^2,L_r(0),L'_r(0)). + +The exact target under (1) is + + v_r=(b-k r^3/2,-3k r^2/2,0,12k,0,0), + H_r(x)=b-k r^3/2-3k r^2 x/2+2k x^3, L_r=0. (4) + +The finite-r targets are retained BEFORE rescaling. U_r converges in every fixed Lp at rate O(r^2) to U_0=(f,f_x,f_xx,f_xxx,f_z,f_xz)(0). + +The covariance has strictly positive, Gaussian-decaying Fourier variances at every lattice mode. The sum of their square roots times any polynomial frequency weight is finite. Minkowski applied to the Fourier series therefore bounds every fixed C^m supremum in each finite Lp, uniformly over frames. For any distinct one-site derivative monomials, a zero-variance combination gives a polynomial symbol zero on the rotated lattice. Undo rotation and successively fix integer coordinates: that polynomial is identically zero. Thus the jet covariance is positive definite; compactness of O(2) gives uniform positive/finite bounds. At distinct sites the analogous statement follows from uniqueness of Fourier coefficients of distributions and isolation of the supports by test functions. + +Taylor formulas justify the U_r convergence, including mixed covariances. Its inverse covariance and its C^m field cross-covariances are bounded. The Gaussian coupling + + F_Q = F + Cov(F,U_r) Cov(U_r)^-1 (v_r-U_r) + +shows that all fixed C^m/Lp moments under Q_r remain uniformly bounded. This uses the whole local Taylor domain, including pins, midpoint and segments, rather than an annulus-only derivative bound. + +For completeness the full endpoint-only normalizer has a uniform lower bound + + Z_r >= c_Z r^2. (5) + +Indeed, with A_r=f_zz(0), the exact finite-pin Taylor relations give, under Q_r, + + det H_M/r = -6k A_r+O_Lp(r), + det H_S/r = 6k A_r+O_Lp(r), + f_xx(M)/r = -6k+O_Lp(r). + +For example f_xx(0)=-(r^2/24)f_xxxx(0)+O_Lp(r^4), f_xxx(0)=12k+O_Lp(r^2), and f_xz at the endpoints is O_Lp(r). Products of remainders are controlled with Holder. Appending f_zz to U_0 leaves distinct derivative monomials. Thus A_r has conditional variance bounded above and below and bounded mean, so Q_r(-2<=A_r<=-1)>=p0>0. Remove the events where any displayed error exceeds 3k_-; their union has probability O(r^p). On the remainder H_M is negative definite, H_S is indefinite, and both absolute determinants are >=3k_-r. This proves (5). No witness-conditioned quantity replaces Z_r. + +## 4. Deterministic three-Hessian lemma + +Assume grad f(M)=grad f(S)=grad f(X)=0, X=(ru,rv), |u|,|v|<=B and v!=0. Let M_3 bound the trilinear operator norm of D^3 f on a convex neighborhood of M,S,X. + +Taylor's integral formula from M to S gives + + ||H_M e1|| <= r M_3/2. + +Put d=(u+1/2,v). The formula from M to X gives + + ||H_M d|| <= r |d|^2 M_3/2. + +Consequently + + |v| ||H_M e2|| <= r M_3 (|u+1/2|+|d|^2)/2. + +Hessian Lipschitz control along the remaining segments yields + + max(||H_M||,||H_S||,||H_X||) <= C_B r M_3/|v|, (6) + +where, for example, a sufficiently generous choice is C_B=16(B+1)^2. Only a finite constant depending on B is used. Since |det H|<=||H||^2 in d=2, + + W_r F_j(H_X) <= C_B^6 r^6 |v|^-6 M_3^6. (7) + +Dropping types only increases the bound. Also, without noncollinearity, + + W_r F_j(H_X) <= M_2^6, (8) + +if M_2 bounds the Hessian norm everywhere on the local domain. Estimate (8) is the valid fallback on the axis. No division by v is made there. The lemma constrains all three Hessians; an endpoint-only smallness estimate would not suffice. + +## 5. The stable two-scale Gaussian block near the axis + +Fix 1=0, + + det(M M^T) >= min(A_u^2 B_u^2,u^2)/2 > 0 (12) + +uniformly on the compact longitudinal chart. The trace is bounded. The nine jets (U_0,Q,T,S) have uniformly positive covariance by Section 3. Hence the joint covariance of (U_r,Y), and the Schur complement Sigma=Cov_Qr(Y), have positive/finite eigenvalue bounds for delta<=delta0. The conditional mean m_Y is bounded. This proves uniformity also as alpha->0; using only the axial minor would not. + +Let d(u)=6k(u^2-1/4)>=d_*>0. At zero gradient the Y target is tau=(-d(u)/delta,0). For small delta, + + |tau-m_Y|>=d_*/(2delta), |tau-m_Y|<=C/delta. + +The exact Gaussian density and det diag(r^2 delta,r delta)=r^3 delta^2 give + + p_(grad f(X)|pins)(0) <= C r^-3 delta^-2 exp(-c/delta^2). (13) + +This is a JOINT two-dimensional density, not a marginal small-ball probability. + +## 6. The added conditional-field moment interface + +The essential extension beyond a finite Hessian-vector estimate is + + E_Qr[M_2^6+M_3^6 | grad f(X)=0] <= C delta^-6. (14) + +Here M_2,M_3 may be bounded by a fixed multiple of ||F||_C3 on the whole torus. Under Q_r these supremum moments are uniformly finite. The conditional Y moments are uniformly bounded by (11) and its stable regression on U_r. For C(x)=Cov_Qr(F(x),Y), Cauchy-Schwarz bounds each derivative through degree three uniformly in x, so ||C||_C3<=C0. + +Under Q_r define the Gaussian field residual + + R=F_Q-E_QF_Q-C Sigma^-1 (Y-m_Y). + +It is independent of Y and has uniformly bounded C3/L6 norm: apply the triangle inequality to this explicit expression, bounded ||C Sigma^-1||_C3, and the uniform moments of F_Q and Y. Under the additional condition Y=tau the field has law + + E_QF_Q+C Sigma^-1(tau-m_Y)+R. + +Its C3 sixth moment is <=C(1+|tau-m_Y|^6)<=C delta^-6. This proves (14) after *all* gradient pins. It does not assume independence between endpoint and witness Hessians. The regression construction retains the original pins and gives the exact additional zero gradient, so (6)-(8) hold under this conditional law. + +Combining (7),(8),(14) gives + + E_Qr[W_r F_j(H_X) | grad f(X)=0] + <= C delta^-6 min(1,(r/|v|)^6), (15) + +where the minimum is defined to be 1 at v=0. Different constants in the two bounds are absorbed into C. This is the missing factor, not a new power inferred from unconditioned Hessians. + +## 7. Weighted intensity, crossover and Gaussian absorption + +For each fixed r>0 the annulus excludes M,S. Distinct-site derivative nondegeneracy and smooth Gaussian regression justify weighted Kac-Rice there. In detail, use the gradient as the zero-counted field and take the nonnegative mark W_r times the indicator of a nonsingular index-j witness Hessian. The endpoint filtered determinants are continuous through singular matrices (their determinant factors vanish); the witness type set is open. This mark is lower semicontinuous, and joint Gaussian regression supplies the requisite continuity. Truncation removes polynomial-growth restrictions. Height is not a mark in this application. + +The exact intensity per physical area is + + rho_j^W(X)=Z_r^-1 p_(grad f(X)|pins)(0) + E_Qr[W_r F_j(H_X) | grad f(X)=0]. (16) + +This formula is under the underlying Gaussian Q_r; the tilted Q_r^W is not declared Gaussian. Applying (5),(13),(15) proves + + rho_j^W(ru,rv) + <= C r^-5 delta^-8 exp(-c/delta^2) + min(1,(r/|v|)^6). (17) + +Now split at the actual crossover |v|=r. + +* If |v|<=r then r<=delta<=sqrt(2)r, so (17) is <=C r^-13 exp(-c/(2r^2)). +* If r<=|v| and delta<=delta0 then |v|<=delta<=sqrt(2)|v|, and (17) is <=C r |v|^-14 exp(-c/(2v^2)). + +For every s,c0>0, the seventh positive term in exp(c0/s^2) gives + + exp(-c0/s^2) <= 7! c0^-7 s^14. (18) + +Use (18) with s=r in the first case and s=|v| in the second. Both bounds become <=Cr. Thus there is a FIXED near-axis strip, not merely a shrinking-width region, on which + + rho_j^W(ru,rv) <= Cr (19) + +uniformly whenever a0<=|u|<=B and delta<=delta0. The constants are qualitative and may be large; no optimal power prefactor or numerical bound is claimed. + +## 8. Off-axis chart and complete fixed-annulus cover + +Choose a0=(A+1)/2>1 and epsilon>0 so small that sqrt(A^2-epsilon^2)>a0 and epsilon<=delta0/2. Then choose r_* small enough that r_*<=delta0/2. Every annulus point with |v|<=epsilon satisfies a0<|u|<=B and delta=epsilon, retain the finite-r pin corrections and use only + + J=(f_x(ru,rv)/r^2, f_z(ru,rv)/r). + +With a=f_zz(0), q=f_xxz(0), c1=f_xzz(0), the expansion under Q_r is + + J = (6k(u^2-1/4)+q u v+c1 v^2/2, a v)+O_Lp(r). (20) + +The coefficient matrix on genuinely random jets (a,q,c1) is + + [0,u v,v^2/2; v,0,0]. + +Its minor in the a,c1 columns is -v^3/2, uniformly separated from zero on this chart. Appending these jets to U_0 preserves covariance positivity. Therefore J has uniformly bounded mean, positive covariance and bounded C3 field cross-covariance; conditioning J=0 yields a uniformly bounded C3 sixth moment. The raw gradient Jacobian is r^3, hence its density is <=Cr^-3. Equation (7) gives conditional triple weight <=Cr^6 since |v|>=epsilon. Divide by (5) to obtain rho_j^W<=Cr on the off-axis chart as well. + +The two charts cover K_AB without a gap. Physical area is r^2 du dv, so (16),(19),(20) give (2) for every Borel E. Null boundaries can be assigned to either chart; the fixed-r intensity is absolutely continuous. There is exactly one endpoint normalizer and one witness determinant. No height-window Jacobian is multiplied into this all-height count. + +## 9. Scope and independent-review requirements + +This is a stronger local count candidate than the between-pin-height statement in PR16, but it has the SAME fixed d2/compact-mark/fixed-annulus restrictions. The estimate does not give a matching lower bound, a numerical constant/radius, a fixed-remote all-height bound, an all-scales RN theorem, a persistence partner statement or a 3D result. Its k_->0 dependence is essential to the Gaussian penalty. + +The decisive review interfaces are: R1 exact unconditional subtraction and finite-r remainders in (9)-(11); R2 full-rank compactification (12); R3 conditional full-field moments after the large target (14); R4 deterministic three-Hessian bound (6)-(8); R5 ORIGINAL Z (5), not a witness-conditioned replacement; R6 weighted Kac-Rice and the absence of a height factor; R7 both crossover powers, absorption and complete annulus cover. Any unsupported interface prevents acceptance of (2). + +## 10. Source relationships and external framework + +Read source PR22 at b2e1652f1374c3b45759324a1ad1fd4177458500, `frontiers/rn_thin_tube_20260925/TWO_SCALE_ADDENDUM.md` (SHA256079f9399aef401d58749b3684f3acbb7f49ffdcbcac9f501b79e06c28a7f4e7d); PR16 at32b80ee085dc6a40113d1e46e333cda50d57ba21, `reviews/downstream_boundary_20260925/TRANSVERSE_BOUND_CANDIDATE.md` (blob024d927779f79fadaf932541ea6474f2995f8c50); and PR19 at e93eade078cac2e8b14aa39d402a3043b7cdb1c7. These are exposed same-provider author sources, not outside confirmation. Their files and review dispositions are unchanged by this package. + +External framework inspected: Armentano, Azais and Leon, arXiv:2304.07424v3, Theorems 2.2 and 7.1, Remark 8 and section 8.1, https://arxiv.org/html/2304.07424v3. These supply the Gaussian/weighted expected-count framework and its hypotheses, not estimates (13)-(20). NIST DLMF 3.3, https://dlmf.nist.gov/3.3, is interpolation context only. No comprehensive novelty or priority claim is made. + +The finite program verifies exact row/minor identities, raw six-pin interpolation, polynomial triple-gradient fixtures and exponent accounting, including deliberate omission controls. It is not a proof checker for the continuum argument, nor independent organizational review. A separate analyst must review the exact candidate source before it is consumed as a closed prerequisite. diff --git a/specifications/lean_statements_20261010/sources/rn-count-interface/RN_COUNT_INTERFACE.md b/specifications/lean_statements_20261010/sources/rn-count-interface/RN_COUNT_INTERFACE.md new file mode 100644 index 000000000..371fd6d17 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/rn-count-interface/RN_COUNT_INTERFACE.md @@ -0,0 +1,113 @@ +# From rare pairing failure to expected counts: exact interface and sharp limits + +**Object:** RN-COUNT-INTERFACE-20260924-v1. +**Author:** OpenAI / ChatGPT. **Disposition:** author-side mathematical derivation, not an RN numerical certificate. + +## 1. What is, and is not, supplied + +Campaign task RN-04 asks whether a pairing-failure probability bound can supply the unresolved RN expected-critical-count estimate. The answer depends on an extra count-weighted estimate. This note proves the exact implication, its limitations, and the corresponding full-pin Kac-Rice expression. It does not invent an absent RN driver, claim a 24-jet enclosure, or relabel a probability theorem as an expected-count theorem. + +At the observed main hardening source `docs/math_status/STATUS_RN_UNIF.md`, Git blob `ccd542fb174e5a242ee343fad213aa40b35b733d`, the original lane remains OPEN with separate ENV-RESCOV, ALLCELL-FDZ-Q4, LOGQ-TAIL, SYM-Fw-jet and CH-LIFT obligations. Those source-specific predicates are not modified by this new logical interface. + +## 2. General probability-to-count estimates + +Let E_r be an event, N_r>=0 an integer-valued random variable, and assume P(E_r)<=C r^3. A count of interest may be supported on E_r; otherwise the results here apply to N_r 1_Er, not its entire expectation. + +Markov gives P(N_r>=1)<=E N_r. It cannot be reversed. For every p>1, Holder gives + + E[N_r 1_Er] <= (E N_r^p)^(1/p) P(E_r)^(1-1/p). (N1) + +If E N_r^p<=A r^(-beta), then + + E[N_r 1_Er] <= A^(1/p) C^(1-1/p) + r^([3(p-1)-beta]/p). (N2) + +In particular a uniformly bounded p-th moment supplies r^(3-3/p), not r^3. The powers in (N2) refer to the p-th MOMENT itself, not its L^p norm; conflating those conventions changes beta's contribution. + +For a probability q=P(E_r)>0 there is an exact identity + + E[N_r 1_Er]=q E[N_r | E_r]. (N3) + +Thus a uniform conditional mean bound E[N_r|E_r]<=A, plus P(E_r)<=C r^3, is enough for the desired cubic rate. When additionally q>=c r^3, a cubic expectation bound is equivalent to a bounded conditional mean. Without this lower comparison, bounded conditional mean is sufficient but not necessary. + +Another exact route is to establish the weighted quantity E[N_r 1_Er] directly. Merely knowing that all derivative suprema have all moments is not a count-moment estimate without a separate theorem tying the number of zeros to those quantities. + +## 3. Sharp counterexamples: all finite moments still do not preserve the exponent + +These are abstract probability counterexamples to an inference. They are NOT asserted to be realizations of the Gaussian field. + +On one space take U uniform on [0,1]. For 0 infinity. (N4) + +Nevertheless for EACH finite p>=1, + + sup_r E N_r^p <= sup_(t>=1) e^(-3t)(t+1)^p < infinity. + +The last supremum is finite because its logarithmic derivative is -3+p/(t+1); beyond its possible finite maximum it decreases exponentially. The constants may depend on p. Hence even a statement giving every finite moment, with no controlled dependence on the order, does not establish the cubic expectation rate. + +For sharpness at one specified p, take N_r=ceil(r^(-3/p))1_Er. Its p-th moment is bounded by 2^p for r<=1, while E N_r is of order r^(3-3/p). Integer rounding does not remove the exponent loss. + +There is a useful exponential-tail version. If P(N_r>t)<=A exp(-t/B), A>=1, then layer cake gives, for q=P(E_r)>0, + + E[N_r 1_Er] <= integral_0^infinity min(q,Ae^(-t/B))dt + = B q[1+log(A/q)]. (N5) + +A cubic event bound therefore gives a cubic rate with a logarithmic loss for sufficiently small r. The example (N4) even satisfies a uniform exponential tail with suitable fixed A,B, so that loss cannot be removed from the tail assumption alone. It does not become a cubic estimate merely by optimizing a large Holder exponent. + +## 4. Apply the interface to the actual pinned weighted law + +Use exactly the parent main issue63 Gaussian regression law Q conditioned on O_r=v, with + + W_r=|det H_M det H_S| 1{M maximum,S index-(d-1) saddle}, + Z_r=E_Q W_r, dQ^W=(W_r/Z_r)dQ. + +The parent source SHA is `9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`. For a Borel region B not containing M,S and a Borel selector 0<=chi(x,f)<=1, define + + N_B(f)=sum_(x in B: grad f(x)=0) chi(x,f). + +For instance chi may select a critical-point type, a height window, or another exact witness property. If it is not 0/1 this is a weighted count; all formulas still apply. Start on compact B separated from both pins. Residual finite-jet rank gives a nonsingular conditional gradient law there, and Gaussian regression gives locally finite conditional moments. Apply the marked Kac-Rice identity under Q, retaining the ENDPOINT weight as a field mark. The result is + + E_Q^W N_B + = 1/Z_r integral_B p_(grad f(x)|O_r=v)(0) + E[W_r |det H_x| chi(x,f) + | O_r=v, grad f(x)=0] dx. (N6) + +For E_r insert 1_Er INSIDE the conditional expectation. It must not be factored out. The result is E_Q^W[N_B 1_Er]. Endpoint Hessians, the witness Hessian and E_r can be strongly dependent under these pins. + +The Borel-mark passage is the parent Section9 convention: establish equality on bounded continuous cylinder marks on a compact domain and extend the resulting finite measures, then truncate W_r if needed and use monotone convergence. For regions approaching pins, exhaust by compacts separated from pins. Equation (N6) then holds as an equality of nonnegative extended quantities, possibly +infinity. Uniform finiteness or a rate cannot be concluded simply from this identity. + +With a lower normalizer bound Z_r>=z_*r^2, an actual enclosure + + integral_B p_(grad f(x)|O_r=v)(0) + E[W_r |det H_x| chi(x,f) | all pins] dx <= A r^5 (N7) + +would imply E_Q^W N_B<=(A/z_*)r^3. If the event indicator is included, the same statement controls the event-weighted count instead. The r^5 numerator and its three determinant factors are explicit. A cell decomposition must cover the WHOLE declared B and all stated parameter faces before its cell enclosures sum to (N7). + +The cap probability argument controls E_Q[W_r 1_Gc], not the integral in (N7). That is the missing mathematical distinction. An original RN spatial parameterization may carry additional change-of-variable factors; they must be kept when translating (N7) to that parameterization rather than copied from the probability ledger. + +## 5. Where support on the rare event is actually valid + +The exact deterministic marked-cylinder source (Drive `1BnPods7Lf-ECdD34noQihZcEfcqpy7R5`, SHA `0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc`) proves that its full cylinder contains precisely M,S on G_r. Therefore an additional-critical count in a region contained in THAT cylinder, excluding the pins, is zero on G_r. For that count, N=N1_Gc and (N1)-(N7) apply to its entire expectation. + +For a remote region outside that cylinder, this support statement is not supplied. A remote critical point can exist while the prescribed maximum/saddle pair is correctly paired. Even before a moment estimate, a proposed RN transfer must identify its count region and prove the required support inclusion, or work with N1_Gc only. + +This location distinction is independent of the parent's unconditional fixed-off-diagonal O(1) height-density bound. That bound is not a shrinking-annulus triple-count enclosure. + +## 6. Completion test for this interface + +A genuine cubic expected-count completion must provide at least one of: + +1. the source-specific full integrated numerator (N7), with all chart/Jacobian and conditioning controls; or +2. a uniform conditional count mean on the actual rare event and the required support relation. + +A finite p-th moment gives the precise weaker rate (N2). An exponential tail gives (N5). Neither is silently rounded up to exponent3. A collection of finite-radius samples does not prove any all-small-r version of these claims. + +The supplied program checks exact Bernoulli examples, rational rate exponents and the correct direction of the count/event inequalities. It does not evaluate the Gaussian triple integral or discharge ENV-RESCOV, the 24-jet roster or any historical missing carrier. + +External context inspected: Armentano-Azais-Leon, arXiv:2304.07424v3 Theorem7.1 for expected integrals and Section8.1 for critical points. Armentano et al., arXiv:1909.10243v2, abstract, identifies separate differentiability, derivative-moment and density hypotheses for scalar level-set moment results. Only its abstract was inspected; no uniform vector-gradient count-moment theorem is imported from it. Holder, layer cake and the counterexamples are derived directly above. diff --git a/specifications/lean_statements_20261010/sources/rn-fixed-annulus-window/FIXED_ANNULUS_CANDIDATE.md b/specifications/lean_statements_20261010/sources/rn-fixed-annulus-window/FIXED_ANNULUS_CANDIDATE.md new file mode 100644 index 000000000..081abc13c --- /dev/null +++ b/specifications/lean_statements_20261010/sources/rn-fixed-annulus-window/FIXED_ANNULUS_CANDIDATE.md @@ -0,0 +1,230 @@ +# D5 fixed scaled annulus: a height-window stitching candidate + +**Object:** D5-FIXED-ANNULUS-STITCH-20260925-v1. **Author:** OpenAI / ChatGPT. +**Disposition:** AUTHOR-SIDE ANALYTIC CANDIDATE; independent analytic review OPEN. +**Scientific effect:** NONE. No global RN, elder-selection, lifetime, or historical numerical gate is closed by this file. The earlier PROOF.md and TWO_SCALE_ADDENDUM.md are unchanged. + +## 1. Exact target, and what is stronger and weaker + +Fix L>0 and the centered variance-one Gaussian field on R^2/(L Z^2) with FULL covariance + + K_L(h)=sum_{n in Z^2} exp(-|h+Ln|^2/2) + /sum_{n in Z^2} exp(-|Ln|^2/2). + +For any orthonormal frame E write f(x,z)=F(x e1+z e2). Let b range over a fixed compact interval and 00, uniform in the frame, compact marks, E0 and j=0,1,2, such that + + E_Qr^W N_j(r E0) <= C*k*r^3*area(E0), 0=3, pin-collision, r<=p0>0. On this event, excluding determinant errors exceeding 3k_- and longitudinal-diagonal errors exceeding 3k_- removes only O(r^p) probability. On the remaining positive-probability event H_M is negative definite, det H_S<0, and both absolute determinants are >=3k_- r. Consequently + + Z_r>=c_Z*r^2. (A8) + +This is the FULL normalizer under (A1), not after witness conditioning. Its upper finiteness follows from the same moment bounds. No numerical eigenfloor or prior acceptance is imported. + +## 3. Retain the height: the three-row contact vector + +All variables in this section are under Q_r; no zero pin target is substituted into an UNCONDITIONED covariance. At X=(ru,rt) set + + J_r=( f_x(X)/r^2, f_z(X)/r, + [f(X)-b-(r*t/2)f_z(X)]/r^3 ). (A9) + +For bounded u,t, (A6) and multivariate Taylor imply uniform L^p expansions + + f_x(X)/r^2=6k(u^2-1/4)+u*t*T+(t^2/2)*C+O_Lp(r), + f_z(X)=r*t*S+r^2[(u^2-1/4)T/2+u*t*C+t^2*D/2]+O_Lp(r^3), + f(X)-b=r^2*t^2*S/2+r^3[k(2u^3-3u/2-1/2) + +(u^2-1/4)t*T/2+u*t^2*C/2+t^3*D/6]+O_Lp(r^4), + +where S=f_zz(0), T=f_xxz(0), C=f_xzz(0), D=f_zzz(0). The quadratic height cancels in (A9), as does the C contribution to its third row. In the coupling (A5), + + J_r = d(u,k)+M(u,t) V_0+O_L2(r), (A10) + d=(6k(u^2-1/4),0,k(2u^3-3u/2-1/2)), + V_0=(S,T,C,D) under U_0=v_0, + M=[ 0 u*t t^2/2 0 ] + [ t 0 0 0 ] + [ 0 (u^2-1/4)*t/4 0 -t^3/12 ]. + +The four V_0 jets are independent modulo the six U_0 derivatives; their conditional covariance G obeys lambda I<=G<=Lambda I uniformly in frame. Means are uniformly bounded. Hence J_r has uniformly bounded L^p norms, and + + ||Cov_Qr(J_r)-M G M^T|| <= C*r (A11) + +uniformly on the entire bounded scaled region. The O(r) comparison is absolute, not falsely asserted relative to a vanishing covariance. + +The minor of M on columns (S,C,D) is exactly t^6/24. Cauchy-Binet gives det(M M^T)>=t^12/576. Its largest eigenvalue is uniformly bounded for |u|,|t|<=B0. Therefore, with changed constants, + + lambda_min(M G M^T)>=c*|t|^12. (A12) + +The exponent 12 is deliberately conservative. Optimizing it is unnecessary for this proof. It is positive for every nonzero t, including u=0; it does NOT cover t=0. + +## 4. An explicit shrinking cutoff makes perturbation uniform + +Choose q=1/24 and h_r=r^q. For |t|>=h_r and r<=1, + + r/|t|^12 <= r^(1-12q)=r^(1/2) -> 0. (A13) + +Combining (A11)-(A13) and shrinking r_* to absorb the constants gives + + Sigma_r=Cov_Qr(J_r)>=c1*|t|^12 I, + ||Sigma_r^-1||<=C*|t|^-12, det(Sigma_r)^-1/2<=C*|t|^-18. (A14) + +These estimates also hold away from a fixed small axis neighborhood by compactness. There is no substitution of a shrinking eta into unspecified PR16 constants. The cutoff is selected AFTER deriving the explicit eigenfloor and O(r) perturbation. A q with 12q>=1 would not justify this perturbation step. + +## 5. The joint height density keeps the rare-gradient exponential + +Choose A1=(A0+1)/2>1. Fix epsilon>0 sufficiently small that epsilon<1, epsilon^2=A1. Write d0=6k_-(A1^2-1/4)>0. From (A10) and bounded jet means, + + E_Qr J_r1 >= d0-C(|t|+r) >= d0/2, + Var_Qr J_r1 <= C*(t^2+r^2). (A15) + +The variance estimate uses the O_L2(r) remainder and the t factors in the first random contact row. On h_r<=|t|<=epsilon, r<=|t|, so Var J_r1<=C*t^2. + +At witness gradient zero and height f(X)=b+r^3*tau, the J_r target is y=(0,0,tau), tau in [-k,0], a fixed compact range. For any positive covariance Sigma and vector z, + + z^T Sigma^-1 z >= z_1^2/Sigma_11 (A16) + +by Cauchy-Schwarz applied to Sigma^(1/2)e1 and Sigma^(-1/2)z. Thus (A15) supplies exp(-c/t^2) in the FULL three-dimensional Gaussian density at y, regardless of correlations with height. Together with (A14) and the exact determinant r^-6 of the affine map (f_x,f_z,f)->J_r, + + p_(grad,height)|pins(0,b+r^3*tau) + <= C*r^-6*|t|^-18*exp(-c/t^2) (A17) + +on the outer part of the small axis neighborhood. Away from |t|<=epsilon, the same argument without the exponential gives p<=C*r^-6 with constants depending on epsilon. We never infer a joint density bound from a marginal alone: the joint determinant is bounded separately by (A14). + +## 6. Account for all three Hessians after gradient AND height pins + +Let M3 be the supremum norm of third derivatives on a local convex ball containing M,S,X. Under the three zero-gradient constraints, deterministic Taylor identities give + + ||H_M e1||<=C*r*M3, + ||H_M (u+1/2,t)^T||<=C*r*M3. + +The two-direction matrix has determinant t. For t!=0, + + ||H_M||+||H_S||+||H_X|| <= C*r*|t|^-1*M3, (A18) + +using Hessian Lipschitz control between these points. The constant is uniform for bounded u,t. Therefore the typed determinant product is bounded, pointwise under the constrained law, by + + W_r F_j(H_X) <= C*r^6*|t|^-6*M3^6. (A19) + +Smallness of an endpoint-only Hessian is NOT simply reused after extra conditioning. + +To control M3 after the additional pins, start with the Q_r field in (A5). Its C3 moments and J_r moments are uniformly bounded. For every derivative through order three, the supremum of its covariance with J_r is bounded by Cauchy-Schwarz. Gaussian regression at J_r=y represents the additional conditional field as + + f^*=f^(r)+Cov_Qr(f^(r),J_r) Sigma_r^-1 (y-J_r). + +This coupling retains all original pins and imposes the actual witness pins. By (A14), Minkowski, and bounded sixth moments of y-J_r, + + E[M3^6 | original pins, grad=0, height=b+r^3*tau] + <= C*|t|^-72 (A20) + +for h_r<=|t|<=epsilon. This is intentionally crude and includes possible large conditional means. Combining (A19)-(A20) gives a single, correlated triple-determinant expectation <=C*r^6*|t|^-78. No independence or factorization of endpoint and witness Hessians is assumed. On |t|>=epsilon the same reasoning gives <=C*r^6. + +## 7. Weighted Kac-Rice yields a bounded angular envelope + +At each fixed r>0 the annulus stays away from the pinned sites. Distinct-site derivative distributions are independent for this periodic field: if a finite distribution annihilates every Fourier mode, it is zero, and local test functions isolate its coefficients. This proves the needed positive finite-r gradient/value/Hessian covariance off the pins. Smooth Gaussian regression provides continuous conditional laws. The joint gradient and symmetric-Hessian jet is the relevant subspace, not an unconstrained vector derivative space. + +Apply the weighted Kac-Rice formula to the gradient under Q_r with nonnegative weight W_r times the nonsingular index indicator and the open height-window indicator. Equivalently augment the auxiliary Gaussian field with f, its Hessians, and the endpoint Hessians. These weights are lower semicontinuous on the nonsingular index sets and can be truncated and approximated from below; continuous Gaussian regression checks the conditional-law requirement. Disintegrating the witness height then gives + + rho_win,j^W(X)=Z_r^-1 int_(b-k*r^3)^b + p_(grad,height)|pins(0,s) + E_Qr[W_r F_j(H_X) | grad=0,height=s] ds. (A21) + +One may use Armentano--Azais--Leon, arXiv:2304.07424v3, Theorem 7.1 with Theorem 2.2, and the critical-point example in Section 8.1. The paper supplies the integration framework; the actual r,t-uniform bounds are proved here. No formula for almost every gradient level is substituted for the required zero-gradient level. + +For h_r<=|t|<=epsilon, (A8),(A17),(A19)-(A20), and the ACTUAL height interval of length k*r^3 give + + rho_win,j^W(ru,rt) + <= C*k*r * |t|^-96 * exp(-c/t^2). (A22) + +The angular exponent is 18+72+6=96. Although this polynomial cost is large, its product with the exponential is uniformly bounded for 0<|t|<=epsilon. For example, e^y>=y^48/48! gives + + |t|^-96 exp(-c/t^2) <= 48!*c^-48. (A23) + +Consequently rho_win,j^W<=C*k*r on this entire shrinking-cutoff outer region. For |t|>=epsilon, fixed-angle covariance and (A18)-(A20) yield the same bound directly. The r ledger is density -6, triple determinants +6, height +3, original normalizer -2: physical-area intensity +1. + +## 8. Inner strip: all-height suppression is enough + +It remains to handle |t|=A1 for small r. This is exactly the two-scale regime; the following summarizes its proved-in-package candidate ingredients without assuming outside review acceptance. + +Put delta=sqrt(r^2+t^2), alpha=r/delta, beta=t/delta. Before unconditional covariance analysis subtract H'_r(ru)+r*t*L'_r(0) from f_x and L_r(ru) from f_z, then divide by (r^2*delta,r*delta). The joint contact rows on the genuinely random jets (f_xxxx,f_xxz,f_zz) are + + [u(u^2-1/4)*alpha/6, u*beta, 0], + [0,(u^2-1/4)*alpha/2,beta]. (A24) + +Taylor remainders are O_L2(delta) uniformly in the aspect ratio: the first component's error is bounded by C(r^2+r|t|+t^2)/delta, the second by C(r^2+r|t|+r*t^2)/delta. The three minors, with a=u(u^2-1/4)/6 and b=(u^2-1/4)/2, are ab*alpha^2,a*alpha*beta,u*beta^2. Their squared sum is bounded below by c(alpha^4+beta^4)>=c/2. Thus the conditional covariance is uniformly positive, while its Gaussian target has size at least c/delta, after allowing a bounded mean. The raw gradient density is <=C*r^-3*delta^-2*exp(-c/delta^2). + +For all three Hessians, conditional regression under this rare gradient target gives O(delta^-1) means and bounded covariance, so the single determinant-product expectation is <=C*delta^-6. With the original normalizer (A8), the ALL-HEIGHT physical-area intensity satisfies + + rho_all,j^W(ru,rt)<=C*r^-5*delta^-8*exp(-c/delta^2). (A25) + +Detailed derivations are in unchanged TWO_SCALE_ADDENDUM.md Sections 2--7. In this inner strip, r<=delta and delta^2<=2r^(1/12), hence + + rho_all,j^W<=C*r^-13*exp[-c/(2*r^(1/12))]<=C*r. (A26) + +The last step follows explicitly from e^y>=y^168/168!, producing a factor C*r^14. Constants may be huge but finite; no usable numerical radius is claimed. A height restriction only decreases this intensity. Since k>=k_->0, C*r can be written C'*k*r with C'=C/k_-. This does NOT give uniformity as k tends to zero. + +## 9. Conclusion, remaining review, and non-extrapolation + +Equations (A22)-(A26) cover every point in the fixed scaled annulus. Integrating rho_win,j^W<=C*k*r over r E0, whose physical area is r^2 area(E0), proves the candidate (A2). Boundary lines can be assigned to either region; the count measure is absolutely continuous by Kac-Rice, so fixed area-null boundaries have zero expected count. + +The proof uses TWO different strengths: a height-restricted, three-Hessian cancellation outside h_r, and an all-height exponential bound inside h_r. It does not give all-height O(r^3) over the full annulus, a positive leading coefficient, or a matching probability lower bound. + +Scope covered at author-candidate level: one witness in each fixed scaled annulus A0>1,B0infinity, r<=3, k_->0, uniform torus-size/field-class limits, numerical24-jet constants, or persistence elder pairing. This is downstream D5 candidate work, not activation of main#94. + +Independent review must check: R1 full six-pin regression and original normalizer; R2 conditioned-law expansion (A10) and O(r) covariance perturbation; R3 contact eigenfloor and cutoff constants; R4 full joint density with retained exponential; R5 deterministic three-gradient Hessian lemma and extra-conditioned C3 moments; R6 legitimate weighted/height-disintegrated Kac-Rice; R7 pointwise inner-strip estimate and union of regions; R8 exact scope. Test success, same-provider agreement, and publication provide no independent acceptance of these interfaces. + +Source context inspected: PR22 head b2e1652f1374c3b45759324a1ad1fd4177458500; PR16 head32b80ee085dc6a40113d1e46e333cda50d57ba21, TRANSVERSE_BOUND_CANDIDATE.md blob024d927779f79fadaf932541ea6474f2995f8c50; current Math#24 review assignment. This addendum quantifies the fixed-transverse constants and supplies a new stitching argument; it neither edits nor claims independent review of those same-provider sources. The exact algebra tests verify finite pin completions, the contact matrix/minor, and exponent budgets only, not this analytic proof. diff --git a/specifications/lean_statements_20261010/sources/rn-fixed-remote-window/PROOF.md b/specifications/lean_statements_20261010/sources/rn-fixed-remote-window/PROOF.md new file mode 100644 index 000000000..b383bfcc8 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/rn-fixed-remote-window/PROOF.md @@ -0,0 +1,217 @@ +# Remote critical points in a shrinking height window + +**Object:** RN-FIXED-REMOTE-WINDOW-20260924-v1. +**Author:** OpenAI / ChatGPT. **Disposition:** author-side proof candidate; nonauthor analytic review open. + +[Research home](https://github.com/d6g8k5htny-coder/main) · [Math index](../../README.md) · [Earlier count interface](../three_fronts_20260924/RN_COUNT_INTERFACE.md) + +## 1. The count and the new scope + +The earlier RN count interface identified a three-determinant integral whose unnormalized numerator must be O(r^5), but did not bound it. This note supplies that bound for a precise part of the problem: **a fixed remote spatial region and the shrinking height window between the pinned maximum and saddle**. The spatial exclusion radius must not shrink with r. + +Fix d>=2 and L>0. On X=R^d/(L Z^d), use the centered variance-one Gaussian field with the exact normalized periodic covariance + + K_L(z)=sum_(n in Z^d) exp(-|z+Ln|^2/2) + / sum_(n in Z^d) exp(-|Ln|^2/2). + +Fix compact B=[b_-,b_+] and K=[k_-,k_+] with 00 the pins are + + M=-(r/2)u, S=(r/2)u, + f(M)=b, f(S)=b-k r^3, grad f(M)=grad f(S)=0. + +Q_r denotes continuous Gaussian regression on these 2(d+1) observations. For a symmetric matrix H define + + F_j(H)=|det H| 1{H has j negative eigenvalues}, + +with value zero on singular matrices. Define the actual endpoint weight and FULL normalizer + + W_r=F_d(H_M) F_(d-1)(H_S), + Z_r=E_(Q_r) W_r, dQ_r^W=(W_r/Z_r)dQ_r. + +There is no adjacency condition, no additional pin-density factor, and no Jacobian inside Z_r. + +Fix 0=rho}. For a Borel E contained in D_rho and j in {0,...,d}, let + + N_(r,j)(E)=#{x in E: grad f(x)=0, index H_x=j, + b-k r^30 and C=1} <= C k r^3 |E|. (3) + +For fixed E of positive volume the expected count is asymptotic to a strictly positive constant times r^3. This is NOT a matching lower bound on the event probability. Sum over the finitely many indices to count all critical types. + +The normalized mean measures converge in total variation on D_rho at rate O(r): a density version differs uniformly from Lambda_j by O(r). No numerical value of C or r_* is supplied. No constant is claimed uniform as rho tends to zero, marks become unbounded, or d,L vary. + +## 2. Exact observations and uniform remote nondegeneracy + +Order the original axial observations as f(M),f_x(M),f(S),f_x(S), followed by pairs f_yj(M),f_yj(S). Use the centered transform + + U_r=[(f(M)+f(S))/2, + (f(S)-f(M))/r, + (f_x(S)-f_x(M))/r, + (6/r^2){f_x(M)+f_x(S)-2(f(S)-f(M))/r}, + (f_yj(M)+f_yj(S))/2, (f_yj(S)-f_yj(M))/r for j]. + +It has absolute determinant 12 r^(-(d+3)) and target + + v_r=(b-k r^3/2,-k r^2,0,12k,0,...,0). + +Its contact limit and target are + + U_0=(f,f_x,f_xx,f_xxx,f_yj,f_xyj for j) at 0, + v_0=(b,0,0,12k,0,...,0). (4) + +These are the exact conventions of the parent matrix-cap source, not a new normalization. Since U_r is an invertible re-expression of the original pins, conditioning on U_r=v_r gives the SAME Q_r. Its determinant does not get multiplied into the conditional count formula again. + +Poisson summation gives strictly positive Fourier variances proportional to exp(-2pi^2|n|^2/L^2). All sums of their square roots times any fixed polynomial in |n| converge. The real Fourier series therefore has smooth versions and finite moments of every fixed derivative supremum. + +Any finite collection of distinct derivative-evaluation functionals at distinct torus sites has positive covariance. A zero-variance linear combination would annihilate every Fourier mode, hence would be the zero distribution. Derivatives of point masses at distinct sites are independent, as tested by smooth functions with individually prescribed finite jets. At one site the rotated derivative monomials are independent; only independent symmetric Hessian entries are listed. This proves the assertion without a finite-mode or sampled-eigenvalue substitution. + +For x in D_rho write Y_x=(grad f(x),f(x)). The joint vector (U_0,Y_x) has no repeated jet functionals and positive covariance. Its covariance varies continuously on the compact set O(d) x D_rho. It consequently has a uniform positive smallest eigenvalue and finite largest eigenvalue. The same holds after adjoining the independent entries of A_0=D_y^2 f(0) and H_x. These extra entries are distinct from the observations. + +Centered differences imply U_r-U_0=O(r^2) in every finite L^p norm. More precisely, modewise Taylor remainder bounds and the Fourier summability above give second-order convergence of the observation covariance and its cross-covariances with any fixed derivative field, in every fixed C^q norm. For example the fourth axial row is f_xxx(0)+(r^2/40)f_xxxxx(0)+O(r^4). The bounds are uniform in frame and remote position. + +Choose r_*<=rho and below the torus injectivity scale. Uniform convergence shows that the covariances of (U_r,Y_x), their inverses, and the relevant Schur complements are uniformly bounded and positive for r<=r_*. Every x stays at least rho/2 from both endpoints. The appended endpoint-transverse and witness-Hessian conditional covariances are also positive by their contact limits. These arguments use compactness, not rotational invariance of the torus. + +## 3. A common regression coupling, with the extra height pin retained + +For t in [b-k r^3,b], let V_r=(U_r,Y_x) have target a_r=(v_r,0,t). At contact take V_0=(U_0,Y_x) with target a_0=(v_0,0,b). If F is one unconditioned field copy, define + + F_(r,x,t)(z)=F(z)+Cov(F(z),V_r) Cov(V_r)^(-1)(a_r-V_r), + F_(0,x,b)(z)=F(z)+Cov(F(z),V_0) Cov(V_0)^(-1)(a_0-V_0). + +Each marginal has precisely the indicated conditional Gaussian law. All targets are bounded on the stated compact sets, and |a_r-a_0|<=C(r^2+|t-b|). Section 2 gives, for every fixed finite p>=1 and derivative order q, + + ||F_(r,x,t)-F_(0,x,b)||_(L^p(C^q)) <= C_(p,q) r^2, + E[(1+||F_(r,x,t)||_(C^q)+||F_(0,x,b)||_(C^q))^p]<=C. (5) + +Here |t-b|<=k_+ r^3 is included. The same construction without Y_x applies to the endpoint-only laws. The fields, their Hessians and remote observations are NOT declared independent; this is a coupling, not a factorization. + +The conditional density p_(Y_x|U_r=v_r)(0,t) differs uniformly by O(r^2) from p_(Y_x|U_0=v_0)(0,b). To see this, the conditional means and covariances are Schur expressions in the uniformly positive joint covariance, and differ by O(r^2); t-b=O(r^3). On the bounded target/positive covariance domain the Gaussian density is continuously differentiable with bounded derivatives. In particular all these densities are uniformly bounded. No actual mean value is replaced by a zero mean unless the regression supplies it. + +## 4. The endpoint weights vanish to second order even after remote conditioning + +All extra conditional fields retain both original gradient pins and the exact height gap. Put + + alpha_i=f_xx(i)/r, beta_i=grad_y f_x(i)/r, A_i=D_y^2 f(i), + H_i=[[r alpha_i,r beta_i^T],[r beta_i,A_i]]. + +The gradient integral identities give |alpha_i|,||beta_i||<=M3/2. The Hermite height identity forces an averaged f_xxx equal to 12k, whence + + alpha_M=-6k+O(r M4), alpha_S=6k+O(r M4). (6) + +All derivative moments in these bounds remain uniform under the EXTRA remote value and gradient pins by (5). This extra-conditioning verification is essential: an estimate established only under Q_r cannot simply be presumed uniform after Y_x has been fixed. + +Let A_0 be the transverse block of F_(0,x,b) at 0. Equation (5), displacement of endpoints by r/2, and (6) show, in every fixed finite L^p norm, + + A_i-A_0=O(r), alpha_M+6k=O(r), alpha_S-6k=O(r), + beta_i=O(1), H_x(F_(r,x,t))-H_x(F_(0,x,b))=O(r^2). (7) + +For clarity two purely deterministic facts are reproved. For symmetric X,Y, + + |F_j(X)-F_j(Y)| + <= n max(||X||,||Y||)^(n-1)||X-Y||. (8) + +If one filtered determinant contributes and the indices differ, the segment from X to Y meets a singular matrix. The determinant difference bound between that endpoint and the singular matrix proves (8). Equal-index cases follow by the same determinant bound; if neither contributes the difference is zero. Thus the index indicator itself is not asserted Lipschitz. + +For K_s=[[alpha,sqrt(s) beta^T],[sqrt(s) beta,A]], + + det K_s=alpha det A-s beta^T adj(A)beta, + |F_j(K_r)-F_j(K_0)|<=r |beta^T adj(A)beta|. (9) + +The determinant is affine in s, including singular A. If an index changes, the continuous matrix path has a singular intermediate point and its affine determinant controls the contributing endpoint. If the determinant is identically zero both filtered determinants vanish. No inverse Hessian or lower eigenvalue bound occurs in (9). + +With D_r=diag(sqrt(r),I), H_i=D_r K_i D_r. Congruence preserves inertia and det(D_r)^2=r. Remove the sqrt(r) block by (9), then compare each endpoint to its own filtered block-diagonal determinant F_d(diag(-6k,A_0)) and F_(d-1)(diag(6k,A_0)) by (8) and (7). Since k>0, each reference factor is 6k |det A_0|1{A_0<0}. Uniform higher moments and Holder control the product error, giving + + W_r/r^2 = w_0+O_(L^p)(r), + w_0=(6k)^2(det A_0)^2 1{A_0<0}. (10) + +No equality of the separate finite-r type indicators, and no independence of endpoint weights, is asserted. + +Under the endpoint-only contact law, z_0=E[w_0|U_0=v_0] is continuous and strictly positive: the conditional law of A_0 has full density on an open negative-definite matrix ball. Compactness, k_->0, and the endpoint-only version of (10) give + + Z_r/r^2=z_0+O(r), inf z_0>0, + (Z_r/r^2)^(-1)=z_0^(-1)+O(r). (11) + +This is the FULL normalizer, not a restriction to a geometric good event. The remote conditioning changes the law inside (10), not the denominator in (11). + +## 5. Kac-Rice yields the actual bound and the limiting density + +Apply Kac-Rice under Q_r to grad f(x) on the compact remote domain. Its conditional gradient covariance is uniformly nonsingular there. Use the endpoint determinant product as a field mark, and F_j(H_x) to incorporate the witness Jacobian and index. The endpoint and witness filtered determinants are continuous functions of finite jets by (8). Truncation and uniform Gaussian moments handle their polynomial growth. Approximate the height-window indicator by continuous functions and then use height disintegration; location restrictions extend to Borel sets as equality of the finite measures. The standard Gaussian weighted Kac-Rice formula therefore gives + + E_(Q_r^W) N_(r,j)(E) + = (r^2/Z_r) integral_E integral_(b-k r^3)^b + p_(Y_x|U_r=v_r)(0,t) + E[(W_r/r^2) F_j(H_x)|U_r=v_r,Y_x=(0,t)] dt dx. (12) + +An underlying critical-point Jacobian is counted exactly once through F_j(H_x). No separate |det H_x| should be multiplied into F_j. Endpoint weight is counted exactly once. The height disintegration uses dt, not a radial/pin Jacobian. Equality of (12) also establishes zero expected count at either single boundary height. + +The contact kernel is + + Lambda_j(x;b,k,u) + = p_(Y_x|U_0=v_0)(0,b)/z_0 + * E[w_0 F_j(H_x)|U_0=v_0,Y_x=(0,b)]. (13) + +The dependence between A_0 and H_x is retained INSIDE this expectation. The contact field has a forced degenerate critical jet at 0 and is not a globally Morse field; no such claim is needed. Only its separated remote jets and their Gaussian density are used. + +By (5),(8),(10), Holder and uniform moments, the expectation in (12) differs from its contact counterpart by O(r), uniformly in x,b,k,R,t. Multiply by the density estimate from Section 3 and (11); the complete integrand after r^2/Z_r differs uniformly from Lambda_j by O(r). Integrating over a window of exact length k r^3 proves (2). The unnormalized version of (12) has the explicit bound + + integral_E integral_window p_(Y_x|U_r)(0,t) + E[W_r F_j(H_x)|U_r,Y_x] dt dx + <= C k r^5 |E|. (14) + +This is the required three-determinant numerator for THIS declared spatial/height region. It is not deduced from a cap-failure probability. Markov is used only in the valid direction after the expectation is bounded, giving (3). + +Continuity of Lambda_j follows from continuous nondegenerate Gaussian conditional parameters and polynomial moment domination, since the filtered determinants are continuous. Strict positivity follows from full conditional support of (A_0,H_x): use an open ball around A_0=-I and a separate open ball around a nonsingular H_x of index j. Their joint density is positive; it need not factor. Compactness supplies common positive lower and finite upper bounds. Invariance under changing the transverse frame follows from determinant/inertia invariance, so Lambda_j is a function of u. No global continuous choice of transverse frame over the sphere is required. + +## 6. Several mutually separated witnesses + +Fix an integer m>=2 and eta>0. Restrict the ordered location tuples to + + D_(rho,eta,m)={ (x1,...,xm): xi in D_rho, + dist_X(xi,xj)>=eta for i!=j }. + +The domain may be empty; then all corresponding quantities are zero. Count ordered distinct m-tuples of index-j critical points in the height window on this domain. Use the SAME endpoint weight W_r once, not its m-th power, and normalize by Z_r once. Append all Y_xi to U_r. Distinct-site jet rank and compact separation give the same uniform covariance, coupling and moment argument. Thus, writing T_(r,j,m,eta) for this ordered count, + + E_(Q_r^W) T_(r,j,m,eta) + = (k r^3)^m integral_(D_(rho,eta,m)) Lambda_(j,m)(x1,...,xm) dx + + O((k r^3)^m r). (15) + +The coefficient uses the JOINT conditional density of all remote gradients/heights and the conditional expectation of w_0 times the product of the m witness filtered determinants, divided by z_0. It is not a product of one-point kernels. If the tuple domain has positive measure, its integral is strictly positive. + +There are m+2 Hessian determinant factors, one O(r^2) endpoint product and m height-window integrations. The radius exponent is 2+3m-2=3m. Since each unordered qualifying m-set contributes m! ordered tuples, + + P(exists m pairwise-eta-separated qualifying points) + <= E T_(r,j,m,eta)/m! = O(r^(3m)). (16) + +For m=2 this supplies an O(r^6) bound on the probability of a separated pair. It does not control two nearly coincident witnesses. There is no uniformity as eta decreases to zero, no full factorial-moment bound, no matching probability lower bound, and no Poisson-limit claim. + +## 7. RN crosswalk and remaining work + +The earlier RN_COUNT_INTERFACE.md, SHA256 aa993f5217bd70e6d1060402d121ab659f6582a098e29042c67585a345a8e3ab, required an actual O(r^5) count numerator before division by Z~r^2. Equation (14) now supplies it for every fixed remote region and the entire BETWEEN-PIN HEIGHT WINDOW. Any additional 0/1 witness selector is bounded by this count when its type/location/value conditions imply (1). Arbitrary global selectors are not asserted to share the explicit positive leading kernel (13). + +What remains outside this result: + +- Spatial regions with distance from the pins tending to zero, including the intermediate annulus between a multiple of r and fixed rho. Compactness constants here may diverge there. +- Witness tuples whose mutual separation tends to zero; (15) does not control that collision. +- All remote critical points with NO shrinking height restriction. The r^3 window factor is absent for that different count. +- A numerical value of C or r_*, the original RN all-cell/24-jet enclosures, and a proof that a particular legacy witness definition lies in this note's count region. +- Independent analytic review of this note or any parent persistence theorem. A source checksum or green test cannot supply that review. + +The Gaussian model and exact observation conventions are those of UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1, SHA256 9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7. The weight stability algebra is reused, and rederived above, from LIFETIME-BOUNDED-REMAINDER-20260924-v1, SHA256 380b7d0abdb0fe2de5a6564565af9560d3f1b1ce22fd5538927db0c52f4f3a4a. This proof does NOT consume their global elder-selection conclusion or the deterministic cap theorem. It treats the actual pinned law and remote critical-point count directly. + +## 8. Attribution and test scope + +Armentano, Azais and Leon, *On a general Kac-Rice formula for the measure of a level set*, arXiv:2304.07424v3, Theorem 7.1 and Section 8.1 supply the established Gaussian expected-integral/critical-point framework. The application to the stated pinned law, extra height disintegration, uniformity and endpoint scaling is argued here. This note claims no historical priority for Kac-Rice, Gaussian regression, rare-window scaling or factorial moment methods. + +The standard-library exact tests check observation transforms, singular typed-determinant cases, conditional covariance calculations, dependence controls, height-window integration and ordered-count radius factors. They do not compute Lambda_j for the actual field, certify continuum bounds, or substitute for a mathematical review. diff --git a/specifications/lean_statements_20261010/sources/side24-enclosure/ENCLOSURE.json b/specifications/lean_statements_20261010/sources/side24-enclosure/ENCLOSURE.json new file mode 100644 index 000000000..57af39a05 --- /dev/null +++ b/specifications/lean_statements_20261010/sources/side24-enclosure/ENCLOSURE.json @@ -0,0 +1,29 @@ +{ + "cone_moments": { + "2": "4/3", + "3": "29/6-sqrt(6)" + }, + "dimensions": { + "2": { + "lower": "0.07340691930603427103", + "upper": "0.07340691930603427104" + }, + "3": { + "lower": "0.04177593184059834334", + "upper": "0.04177593184059834335" + } + }, + "image_ledger": { + "covariance_relative_bound": "31763607879717/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "density_comparison_below_reported": true, + "derivative_bound": "10587869293239/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "exp_lower_gt_10": true, + "image_constant": 21175738586478, + "relative_below_eps": true + }, + "method": "outward rational arithmetic; analytic image and Stirling remainder bounds", + "object": "SIDE24-COEFFICIENT-D23-20260924-v1", + "relative_periodization_bound": "1e-106", + "scientific_acceptance": false, + "scope": "coefficient of issue63 Eq15.2; parent theorem unreviewed" +} diff --git a/specifications/lean_statements_20261010/sources/side24-proof/PROOF.md b/specifications/lean_statements_20261010/sources/side24-proof/PROOF.md new file mode 100644 index 000000000..44b66f04f --- /dev/null +++ b/specifications/lean_statements_20261010/sources/side24-proof/PROOF.md @@ -0,0 +1,216 @@ +# SIDE24 coefficient in dimensions two and three + +Object: SIDE24-COEFFICIENT-D23-20260924-v1. Author: OpenAI / ChatGPT. +Disposition: author-side coefficient-evaluation candidate; nonauthor review open. + +## Scope and exact parent + +This note evaluates the coefficient defined in equation (15.2) of +UNIFORM-MATRIX-CAP-LIFETIME-20260924-v1, SHA256 +`9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7`, +Drive ID `1foDgiDi4XIKOfbrZEE8dWKV_BkZU8LIb`, main issue63. +It does not independently accept that parent's global elder-selection or Kac-Rice +proof. Interpreting this coefficient as the finite-bar asymptotic constant remains +conditional on that proof. The parent source is not modified. + +Use its centered variance-one field with the exact normalized covariance + + K_24(z) = sum_{n in Z^d} exp(-|z+24n|^2/2) + / sum_{n in Z^d} exp(-|24n|^2/2), d in {2,3}. + +Let c_d,24 be precisely its equation (15.2), using ordinary sphere area, ordered +maximum/saddle roles and the full height/gradient pin Jacobian. Then the bounds +proved below, with endpoints rational decimal numbers, are + + 0.07340691930603427103 < c_2,24 < 0.07340691930603427104, + 0.04177593184059834334 < c_3,24 < 0.04177593184059834335. + +The uncertainty in these intervals is numerical/arithmetic uncertainty in this +specified expression, not a probability of correctness of the parent theorem. +No finite-radius error estimate, unrestricted remainder or RN/24-jet claim follows. + +## 1. A nonperiodic reference coefficient with exact cone moments + +Use K_infty(z)=exp(-|z|^2/2) only to define a reference CONTACT covariance and its +coefficient expression. No infinite-volume persistence theorem is asserted. +In an orthonormal frame with u=e1, covariance differentiation gives + + Cov(G)=I_d, + Cov(H_ij,H_kl)=delta_ij delta_kl+delta_ik delta_jl+delta_il delta_jk, + Var(t)=15, Cov(t,G)=(-3,0,...,0), t=partial_1^3 f. + +Thus V=Hu has covariance diag(3,1,...,1), tau^2=Var(t|G=0)=6, and + + p_G(0) p_V(0) = (2pi)^(-d) / sqrt(3). + +Writing m=d-1 and A for the transverse Hessian, Gaussian conditioning on V=0 gives + + Cov(A_ij,A_kl | V=0) + = (2/3)delta_ij delta_kl + delta_ik delta_jl + delta_il delta_jk. + +Equivalently A=Q+sqrt(2/3) Z I_m, with Q's diagonal variances2, off-diagonal +variances1, and independent upper-triangular entries, and Z an independent standard +normal. This explicitly specified reference law is not substituted for the actual +periodic law without an error bound. + +For m=1 the variance of A is8/3. Centered Gaussian symmetry gives + + D_1=E[A^2 1{A<0}]=4/3. + +For m=2 write A=[[s+x,y],[y,s-x]]. Then s,x,y are independent, +Var(s)=5/3 and Var(x)=Var(y)=1. The negative-definite cone is s<-R, +R=sqrt(x^2+y^2), and det A=s^2-R^2. The variable R^2 has density e^(-z/2)/2. +For a>=0, elementary integration gives + + integral_0^a (a-z)^2 e^(-z/2) dz/2 = a^2-4a+8-8e^(-a/2). + +Using symmetry in s, E s^2=5/3, E s^4=25/3 and +E exp(-s^2/2)=sqrt(3/8), one obtains + + D_2 = (1/2)[25/3-20/3+8-8sqrt(3/8)] + = 29/6-sqrt(6). + +The truncation R<|s| is essential: half the UNTRUNCATED determinant-squared moment +would instead give29/6, an incorrect value. The shared scalar shift of the matrix +is likewise essential. + +Substituting these moments and |S^1|=2pi, |S^2|=4pi into the exact parent coefficient +and simplifying 6^(2/3)/24^(1/3)=(3/2)^(1/3) gives + + c_d,ref = Gamma(7/6)*(3/2)^(1/3)*D_(d-1) + / [2sqrt(3)*pi^(d-1)*sqrt(pi)], d=2,3. (1) + +This eliminates BOTH the angular and cone integrals for the reference expression. + +## 2. A uniform, deterministic bound for every omitted periodic image + +For q<=6 unit directional differentiations of phi(x)=exp(-|x|^2/2), product-rule +pairings bound the absolute derivative by + + e^(-|x|^2/2) sum_{j=0}^{floor(q/2)} + q! |x|^(q-2j) / [2^j j! (q-2j)!]. + +At |x|>=1 this is at most76 |x|^6 e^(-|x|^2/2). The coefficient sum for q=6 is +1+15+45+15=76 and it bounds the smaller orders as well. At zero, every such +contraction has absolute value at most15. + +For an integer point with maximum coordinate magnitude j>=1, there are at most +27j^3 possibilities for d<=3, |n|^6<=27j^6, and |n|^2>=j^2. Consequently + + sum_{n!=0} |n|^6 e^(-288|n|^2) + <=729 sum_{j>=1} j^9 e^(-288j^2) + <=1458 e^(-288). + +The last step follows because successive terms have ratio at most512e^(-864)<1/2. +The code proves e^(288/125)>10 using the positive rational Taylor sum through20; +hence e^(-288)<10^(-125) and the geometric-ratio inequality follows too. + +Normalizing by S=sum_n exp(-288|n|^2)>=1 costs the subtraction of +(S-1)D^q phi(0), which is included. Thus for EVERY unit-direction contraction, +including mixed directions, through total order6, + + |D^q K_24(0)-D^q K_infty(0)| + < E := 1458*(76*24^6+15)*10^(-125) + = 21175738586478 *10^(-125). (2) + +This is a bound on the infinite image sum, not a sampled orientation or truncated +floating-point comparison. It applies to the exact variance-one normalization. + +## 3. Full joint covariance and all orientations + +Use the joint vector (G,t,svec H), where svec lists diagonal Hessian entries and +sqrt(2) times independent off-diagonal entries. Its dimension is at most10. +Each covariance entry differs from the reference by at most2E, by (2). +Hence the spectral norm of the difference is at most20E. + +The reference Hessian block has eigenvalues d+2 on trace and2 on traceless matrices; +it is independent of the odd derivative block. The only nontrivial odd block is +[[1,-3],[-3,15]], whose eigenvalues are8 +/- sqrt(58). Its smaller eigenvalue exceeds +1/3: subtract I/3 and use first principal minor2/3 and determinant7/9. +All remaining gradient variances are1. Thus C_ref >= I/3 and + + (1-epsilon) C_ref <= C_24 <= (1+epsilon) C_ref, + epsilon=10^(-108), since60E0 its expectation scales by a^m, since h has degree2m and the cone is scale +invariant. Thus the exact conditional cone moment has the same bounds, multiplied +by (1-epsilon)^m D0 and (1+epsilon)^m D0 respectively. No continuity claim about +an indicator at a singular matrix is used. + +Combining this with the two d-dimensional densities and tau^(4/3) yields an +integrand ratio between + + (1-epsilon)^a/(1+epsilon)^b and (1+epsilon)^a/(1-epsilon)^b, + a=m+2/3+n/2, b=d+n/2. + +For d=2,3, a+2b<14 and2a+b<13. Since epsilon is far below1/28, elementary bounds +on log(1+epsilon), -log(1-epsilon) and exp give ratio between1-13epsilon and +1+28epsilon, hence safely between1-32epsilon and1+32epsilon. Angular integration +preserves these multiplicative bounds because every reference integrand is positive. +It follows that + + |c_d,24/c_d,ref -1| < 10^(-106), d=2,3. (4) + +The exact periodic constant is NOT claimed equal to the reference constant. +Equation (4) bounds it and justifies the numerical interval in this note. + +## 5. Outward arithmetic and special-function remainder + +`coefficient.py` uses only integer and Fraction arithmetic to obtain (1)-(4). +Every interval operation rounds outward to the rational grid10^(-80); negative +values use mathematical floor/ceiling rather than rounding toward zero. Floats and +booleans are rejected as inputs. Roots use exact integer root brackets. + +Pi uses Machin's identity16atan(1/5)-4atan(1/239) with alternating-series remainder. +Log uses range reduction and the positive atanh series, with its geometric tail. +Exp uses reduction to[0,1/4], its positive Taylor series and a ratio-bounded tail; +negative arguments use reciprocal intervals. + +Gamma(7/6) is obtained by shifting to z=199/6, using Stirling's LOG-gamma expansion +through B20, enclosing the remainder by the positive first omitted B22 term, then +subtracting the32 recurrence logarithms and exponentiating. The rigorous positive- +real remainder rule is NIST DLMF5.11(ii), and recurrence is DLMF5.5(i). +The periodization allowance in (4) is applied to exact rational endpoints BEFORE +the final outward rounding; it is not silently rounded to zero. +The displayed endpoints are outward rational decimal bounds, not rounded guesses. + +The supplied unit tests and mutation runs are finite implementation/algebra checks. +They do not constitute formal verification or nonauthor analytic acceptance. +`ENCLOSURE.json` records the computed bounds and explicit scope. + +## Review and remaining work + +Nonauthor review should independently check the shared trace variance5/3, Rayleigh +cone integration, ordered-pair/gamma normalization inherited from the parent, +all-direction derivative bound, covariance and Schur-complement comparison, and +rational interval implementation. No frozen proof, review grade, RN/24-jet obligation, +P15 assertion or project completion flag is changed. + +### External reconnaissance, 24 September 2026 + +Primary sources read: NIST DLMF5.11, especially equation5.11.1 and positive-real +remainder statement in5.11(ii); DLMF5.5 for the gamma recurrence. These are established +numerical-analysis inputs. The Gaussian covariance/cone and image comparison above +are derived here; no priority or novelty claim follows. Main issue63 and its exact +Drive source provide the project coefficient and conventions. diff --git a/specifications/lean_statements_20261010/test_check.py b/specifications/lean_statements_20261010/test_check.py new file mode 100644 index 000000000..39a2449ea --- /dev/null +++ b/specifications/lean_statements_20261010/test_check.py @@ -0,0 +1,1307 @@ +#!/usr/bin/env python3 +"""Positive and negative controls for check.py (standard library only). + +Synthetic controls build a disposable repository with a miniature packet (five modules, two +pinned sources, a register) so that every refusal is exercised in isolation. Real-tree controls +check the committed packet's pins and module scans. Lean is required only by the one test that +is guarded by `skipUnless(lake)`; it builds a dependency-free fixture package, never Mathlib. +No test here is evidence about any mathematical claim; scientific effect NONE. +""" +import contextlib +import hashlib +import importlib.util +import io +import json +import os +from pathlib import Path +import re +import shutil +import subprocess +import sys +import tempfile +import unittest +from unittest import mock + +HERE = Path(__file__).resolve().parent +spec = importlib.util.spec_from_file_location('lean_spec_check', HERE / 'check.py') +check = importlib.util.module_from_spec(spec) +spec.loader.exec_module(check) + +COMMIT = '760340e921ac4ceda296b8118da936f1133e956e' +SOURCE_TEXT = 'Section 1. In the source, two and two make four; a given object is named here.\n' +LANDING_TEXT = json.dumps({'claims': [{'claim_id': 'demo-claim', 'disposition': 'HOLD_WITH_DOMAIN'}]}) + '\n' +LOCK = {'version': '1.2.0', 'packagesDir': '.lake/packages', 'packages': [ + {'name': 'batteries', 'url': 'https://github.com/leanprover-community/batteries', 'type': 'git', 'subDir': None, + 'scope': 'leanprover-community', 'rev': 'f2effa3d803fda822b1f97b806c47cf2adfbcbc2', 'manifestFile': 'lake-manifest.json', + 'inputRev': 'main', 'inherited': True, 'configFile': 'lakefile.toml'}, + {'name': 'mathlib', 'url': 'https://github.com/leanprover-community/mathlib4.git', 'type': 'git', 'subDir': None, + 'scope': '', 'rev': check.MATHLIB_REV, 'manifestFile': 'lake-manifest.json', 'inputRev': check.MATHLIB_REV, + 'inherited': False, 'configFile': 'lakefile.lean'}], + 'name': 'universal_law_formal', 'lakeDir': '.lake', 'fixedToolchain': True} +LAKEFILE = ('name = "universal_law_specifications"\nversion = "0.1.0"\ndefaultTargets = ["Specifications"]\n\n[[require]]\n' + 'name = "mathlib"\ngit = "https://github.com/leanprover-community/mathlib4.git"\nrev = "' + check.MATHLIB_REV + + '"\n\n[[lean_lib]]\nname = "Specifications"\n' + check.LEAN_OPTIONS_LINE + '\n') + + +def sha(data): + return hashlib.sha256(data).hexdigest() + + +def blob(data): + return hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest() + + +def labels(anchor='two and two make four', text=SOURCE_TEXT, path='docs/DEMO.md', interface='none.'): + return (f'Source: d6g8k5htny-coder/Math- commit {COMMIT} path {path} sha256 {sha(text.encode())}\n' + f'Anchor: "{anchor}"\nInterface: {interface}\nDoes not claim: anything about the source theorem.') + + +FIELD = f'''import Mathlib + +/-! Demo module: statements only. The word theorem may appear in a module docstring. -/ + +namespace UniversalLaw.Spec.Field + +open MeasureTheory + +/-- Ambient space (auxiliary; not registered). -/ +abbrev E (d : ℕ) : Type := EuclideanSpace ℝ (Fin d) + +/-- A helper constant (auxiliary; not registered). -/ +noncomputable def helper (x : ℝ) : ℝ := x + 1 + +/-- Two and two. + +{labels()} -/ +def TwoAndTwo (n : ℕ) (h : n = 2 := by rfl) : Prop := n + n = 4 + +/-- An interface. + +{labels('a given object is named here', interface='the object is given.')} -/ +structure DemoInterface (d : ℕ) where + x : E d + pos : 0 < d + +/-- A multi-line header with a Prop-valued binder. + +{labels()} -/ +def MultiLine (d : ℕ) (I : DemoInterface d) + (P : (E d → ℝ) → Prop) : + Prop := + P (fun _ => 0) ∧ 0 < d + +end UniversalLaw.Spec.Field +''' + + +def module(stem): + return (f'import Mathlib\n\nnamespace UniversalLaw.Spec.{stem}\n\n/-- Demo statement.\n\n{labels()} -/\n' + f'def Demo{stem} : Prop := (2 : ℕ) + 2 = 4\n\nend UniversalLaw.Spec.{stem}\n') + + +def statement(module_path, declaration, kind='def_prop', package=check.PACKAGE, progress='specified', anchor='two and two make four'): + return {'package': package, 'module': module_path, 'declaration': declaration, 'kind': kind, 'formal_progress': progress, + 'anchor': anchor, 'does_not_claim': 'the source theorem; review; acceptance'} + + +class Fixture: + """A disposable repository with a miniature packet that passes the source check.""" + + def __init__(self): + self.tmp = tempfile.mkdtemp() + self.repo = Path(self.tmp) / 'repo' + self.packet = self.repo / check.PACKAGE + (self.packet / 'Specifications').mkdir(parents=True) + self.write('formal/lake-manifest.json', json.dumps(LOCK, indent=2) + '\n', repo=True) + self.write(check.WORKFLOW, 'name: placeholder\n', repo=True) + self.write('.gitignore', '.lake/\n__pycache__/\n') + for name in ('README.md', 'SCOPE.md', 'check.py', 'test_check.py', 'replay.sh'): + self.write(name, '# bound copy\n') + self.write('lakefile.toml', LAKEFILE) + self.write('lean-toolchain', check.TOOLCHAIN + '\n') + self.write('lake-manifest.json', json.dumps(dict(LOCK, name=check.LOCK_NAME), indent=2) + '\n') + self.write('Specifications.lean', ''.join('import ' + m[:-5].replace('/', '.') + '\n' for m in check.MODULES)) + self.write('Specifications/Field.lean', FIELD) + for m in check.MODULES[1:]: + self.write(m, module(Path(m).stem)) + self.write('sources/demo/DEMO.md', SOURCE_TEXT) + self.write('sources/landing-claims/LANDING_CLAIMS.json', LANDING_TEXT) + sources = [self.source('demo', 'docs/DEMO.md', SOURCE_TEXT, 'sources/demo/DEMO.md'), + self.source('landing-claims', 'claims/LANDING_CLAIMS.json', LANDING_TEXT, 'sources/landing-claims/LANDING_CLAIMS.json')] + self.write('sources/SOURCES.json', json.dumps({'schema_version': 1, 'scientific_effect': 'NONE', 'meaning': 'pins', + 'sources': sources}, indent=2) + '\n') + register = { + 'schema_version': 1, 'object': check.REGISTER_OBJECT, 'scientific_effect': 'NONE', 'scientific_status_authority': False, + 'alignment_status': 'PENDING_INDEPENDENT_REVIEW', 'meaning': 'inventory of formal statements; labels only', + 'toolchain': {'lean': check.TOOLCHAIN, 'lean_commit': check.LEAN_COMMIT, 'mathlib': check.MATHLIB_REV}, + 'vocabulary': {'formal_progress': list(check.FORMAL_PROGRESS)}, + 'sources': [{k: s[k] for k in check.SOURCE_KEYS} for s in sources], + 'claims': [ + {'claim_id': 'demo-claim', 'kind': 'landing_claim', 'title': 'Demo claim', 'layer0': {'source_id': 'demo', 'heading': 'Theorem'}, + 'disposition_transcribed': 'HOLD_WITH_DOMAIN', 'formal_progress': 'specified', + 'statements': [statement('Specifications/Field.lean', 'UniversalLaw.Spec.Field.TwoAndTwo'), + statement('Specifications/Field.lean', 'UniversalLaw.Spec.Field.DemoInterface', 'structure', anchor='a given object is named here'), + statement('Specifications/Field.lean', 'UniversalLaw.Spec.Field.MultiLine'), + statement('UniversalLaw/Demo.lean', 'UniversalLaw.Demo.two_add', 'theorem', 'formal', 'proved', 'two and two make four')], + 'kernel_evidence': [{'package': 'formal', 'receipt_or_workflow': 'formal-lean.yml', 'note': 'arithmetic skeleton only'}], + 'not_established': 'the claim itself', 'next_step': 'independent alignment review'}, + {'claim_id': 'demo-graph-node', 'kind': 'graph_node', 'title': 'Demo node', 'layer0': {'source_id': 'demo', 'heading': None}, + 'disposition_transcribed': None, 'formal_progress': 'specified', + 'statements': [statement(m, f'UniversalLaw.Spec.{Path(m).stem}.Demo{Path(m).stem}') for m in check.MODULES[1:]], + 'kernel_evidence': [], 'not_established': 'everything beyond the statement', 'next_step': 'none planned'}]} + self.write('REGISTER.json', json.dumps(register, indent=2, ensure_ascii=False) + '\n') + + @staticmethod + def source(sid, path, text, local): + data = text.encode() + return {'id': sid, 'repository': 'd6g8k5htny-coder/Math-', 'commit': COMMIT, 'path': path, 'blob': blob(data), + 'bytes': len(data), 'sha256': sha(data), 'local_copy': local} + + def path(self, rel, repo=False): + return (self.repo if repo else self.packet) / rel + + def write(self, rel, text, repo=False): + p = self.path(rel, repo) + p.parent.mkdir(parents=True, exist_ok=True) + p.write_text(text, encoding='utf-8') + + def read(self, rel, repo=False): + return self.path(rel, repo).read_text(encoding='utf-8') + + def mutate(self, rel, old, new, repo=False, count=1): + text = self.read(rel, repo) + if old not in text: + raise AssertionError(f'{old!r} not in {rel}') + self.write(rel, text.replace(old, new, count), repo) + + def inject(self, rel, line): + """Insert a column-0 line before the closing `end` of a module.""" + text = self.read(rel) + head, _, tail = text.rpartition('\nend ') + self.write(rel, head + '\n' + line + '\n\nend ' + tail) + + def register(self): + return json.loads(self.read('REGISTER.json')) + + def save_register(self, reg): + self.write('REGISTER.json', json.dumps(reg, indent=2, ensure_ascii=False) + '\n') + + def run(self, allow_dirty=True): + return check.source_check(self.packet, self.repo, allow_dirty) + + def cleanup(self): + shutil.rmtree(self.tmp, ignore_errors=True) + + +class Base(unittest.TestCase): + def setUp(self): + # Fixture packets are synthetic: their pinned commits exist in no git history and their + # worktree is not a CI checkout, so the CI-only requirements that check.py applies when + # GITHUB_ACTIONS is set (GITHUB_SHA == HEAD, repository name, every pinned source verified + # against git history) must not apply to them. Identity.test_clean_worktree_and_ci re-enables + # CI mode explicitly inside its own patch to test exactly those refusals. + patcher = mock.patch.dict(os.environ) + patcher.start() + self.addCleanup(patcher.stop) + for key in ('GITHUB_ACTIONS', 'GITHUB_SHA', 'GITHUB_REPOSITORY', 'GITHUB_RUN_ID', 'GITHUB_RUN_ATTEMPT'): + os.environ.pop(key, None) + self.f = Fixture() + self.addCleanup(self.f.cleanup) + + def assertRefused(self, fragment, allow_dirty=True): + with self.assertRaises(ValueError) as ctx: + self.f.run(allow_dirty) + self.assertIn(fragment, str(ctx.exception)) + return str(ctx.exception) + + +class Positive(Base): + def test_fixture_passes(self): + info = self.f.run() + self.assertEqual(info['prop_definitions'], 6) + self.assertEqual(info['structures'], 1) + self.assertEqual(info['auxiliary_definitions'], 2) + self.assertEqual((info['claims'], info['statements'], info['registered_local_statements']), (2, 8, 7)) + self.assertEqual(info['formalization_status'], 'specified') + self.assertEqual(info['alignment_status'], 'PENDING_INDEPENDENT_REVIEW') + self.assertEqual((info['checked_commit'], info['worktree']), (None, 'UNVERIFIED')) + self.assertEqual(info['dependency_revisions']['mathlib'], check.MATHLIB_REV) + names = {d['name'] for d in info['scans']['Specifications/Field.lean']} + self.assertEqual(names, {'UniversalLaw.Spec.Field.E', 'UniversalLaw.Spec.Field.helper', 'UniversalLaw.Spec.Field.TwoAndTwo', + 'UniversalLaw.Spec.Field.DemoInterface', 'UniversalLaw.Spec.Field.MultiLine'}) + + def test_cli(self): + cmd = [sys.executable, '-B', '-S', str(HERE / 'check.py'), '--root', str(self.f.packet), '--repo', str(self.f.repo), '--allow-dirty'] + proc = subprocess.run(cmd, capture_output=True, text=True) + self.assertEqual(proc.returncode, 0, proc.stderr) + self.assertIn('SOURCE_CHECK_PASS', proc.stdout) + self.assertIn('WORKTREE_UNVERIFIED', proc.stderr) + proc = subprocess.run(cmd + ['source'], capture_output=True, text=True) + self.assertEqual(proc.returncode, 0, 'the DESIGN 3b spelling `check.py source` is the default mode: ' + proc.stderr) + self.assertIn('SOURCE_CHECK_PASS', proc.stdout) + proc = subprocess.run(cmd + ['execute'], capture_output=True, text=True) + self.assertEqual(proc.returncode, 2, 'only `source` is a positional mode') + self.assertNotIn('SOURCE_CHECK_PASS', proc.stdout) + self.f.inject('Specifications/RN.lean', 'theorem injected : True := trivial') + proc = subprocess.run(cmd, capture_output=True, text=True) + self.assertEqual(proc.returncode, 1) + self.assertIn('LEAN_SPEC_CHECK_FAIL', proc.stderr) + self.assertIn('refused token', proc.stderr) + self.assertNotIn('SOURCE_CHECK_PASS', proc.stdout) + + def test_comments_may_mention_refused_words(self): + self.f.inject('Specifications/RN.lean', '-- theorem sorry axiom native_decide set_option') + self.f.mutate('Specifications/RN.lean', 'Demo statement.', 'Demo statement; the source says theorem and the proof uses sorry nowhere.') + self.f.run() + + def test_unnamed_end_is_refused(self): + self.f.mutate('Specifications/RN.lean', '\nend UniversalLaw.Spec.RN', '\nend\n\nend UniversalLaw.Spec.RN') + self.assertRefused('does not close') + + +class Identity(Base): + def test_requires_git_without_allow_dirty(self): + self.assertRefused('not a git repository', allow_dirty=False) + + @unittest.skipUnless(shutil.which('git'), 'git not available') + def test_clean_and_dirty_worktrees(self): + def g(*args): + return subprocess.run(['git', '-C', str(self.f.repo), '-c', 'user.name=t', '-c', 'user.email=t@t', '-c', 'commit.gpgsign=false', + *args], capture_output=True, text=True, check=True).stdout + g('init', '-q') + g('add', '-A') + g('commit', '-q', '-m', 'fixture') + with mock.patch.dict(os.environ): + os.environ.pop('GITHUB_ACTIONS', None) + info = self.f.run(allow_dirty=False) + self.assertEqual(info['worktree'], 'CLEAN') + self.assertRegex(info['checked_commit'], '^[0-9a-f]{40}$') + self.f.mutate('Specifications/RN.lean', '(2 : ℕ) + 2 = 4', '(2 : ℕ) + 2 = 5') + self.assertRefused('uncommitted or untracked', allow_dirty=False) + g('checkout', '--', '.') + self.f.write('sources/demo/EXTRA.md', 'untracked\n') + self.assertRefused('uncommitted or untracked', allow_dirty=False) + self.f.path('sources/demo/EXTRA.md').unlink() + self.f.mutate('formal/lake-manifest.json', '"version": "1.2.0"', '"version": "1.2.1"', repo=True) + self.assertRefused('uncommitted or untracked', allow_dirty=False) + g('checkout', '--', '.') + self.f.run(allow_dirty=False) + os.environ['GITHUB_ACTIONS'] = 'true' + os.environ['GITHUB_SHA'] = '0' * 40 + os.environ['GITHUB_REPOSITORY'] = check.REPOSITORY + self.assertRefused('GITHUB_SHA', allow_dirty=False) + os.environ['GITHUB_SHA'] = info['checked_commit'] + os.environ['GITHUB_REPOSITORY'] = 'someone/fork' + self.assertRefused('GITHUB_REPOSITORY', allow_dirty=False) + os.environ['GITHUB_REPOSITORY'] = check.REPOSITORY + # The fixture history does not contain the pinned commit, so CI must refuse (fetch-depth 0 is required). + self.assertRefused('verified against its commit in git history', allow_dirty=False) + + +class Lexical(Base): + def test_refused_tokens(self): + cases = {'theorem': 'theorem injected : True := trivial', 'lemma': 'lemma injected : True := trivial', + 'example': 'example : True := trivial', 'instance': 'instance : Inhabited ℕ := ⟨0⟩', + 'axiom': 'axiom hidden : False', 'sorry': 'def Injected : Prop := sorry', + 'admit': 'def Injected : Prop := by admit', 'native_decide': 'def Injected : Prop := (by native_decide : 2 + 2 = 4)', + 'unsafe': 'unsafe def Injected : Prop := True', 'opaque': 'opaque Injected : Prop', + 'partial': 'partial def Injected (n : ℕ) : ℕ := Injected n', 'macro': 'macro "tt" : term => `(True)', + 'notation': 'notation "⊤⊤" => True', 'set_option': 'set_option autoImplicit true', + 'syntax': 'syntax "tt" : term', 'elab': 'elab "tt" : term => pure (mkConst ``True)', + 'implemented_by': '@[implemented_by helper] def Injected : Prop := True', 'extern': '@[extern "x"] def Injected : Prop := True', + 'deriving': 'deriving instance Repr for Nat', 'attribute': 'attribute [simp] helper', + 'inductive': 'inductive Injected | a', 'class': 'class Injected where\n x : ℕ', + 'local': 'local notation "x" => True', 'include': 'include x', 'mutual': 'mutual\nend'} + for token, line in cases.items(): + with self.subTest(token=token): + f = Fixture() + self.addCleanup(f.cleanup) + f.inject('Specifications/RN.lean', line) + with self.assertRaises(ValueError) as ctx: + f.run() + self.assertIn('refused token', str(ctx.exception)) + self.assertIn(token, str(ctx.exception)) + + def test_refused_token_detection_unit(self): + self.assertEqual(check.refused_tokens('theorem t : False := by sorry\n'), ['sorry', 'theorem']) + self.assertEqual(check.refused_tokens('def partialSups : ℕ := 0\n'), []) + self.assertEqual(check.refused_tokens('def x := Foo.sorry\n'), ['sorry']) + + def test_column0_commands(self): + for line, fragment in (('@[simp] def Injected : Prop := True', 'refused character'), + ('private def Injected : Prop := True', 'off column 0'), + ('protected def Injected : Prop := True', 'off column 0'), + ('universe u', 'column-0 command'), + ('noncomputable section', 'off column 0'), + ('| injected => True', 'column-0 command')): + with self.subTest(line=line): + f = Fixture() + self.addCleanup(f.cleanup) + f.inject('Specifications/RN.lean', line) + with self.assertRaises(ValueError) as ctx: + f.run() + self.assertIn(fragment, str(ctx.exception)) + + def test_section_and_variable_are_refused(self): + """`variable (h : False)` mentioned in a body adds a vacuating hypothesis the scan cannot see, + so both commands are refused until a module needs them (DESIGN 3b lists them; none uses them).""" + for line in ('section\nend', 'section Named\nend Named', 'variable (h : False)', 'variable (d : ℕ)', 'noncomputable section\nend'): + with self.subTest(line=line): + f = Fixture() + self.addCleanup(f.cleanup) + f.inject('Specifications/RN.lean', line) + with self.assertRaises(ValueError) as ctx: + f.run() + self.assertRegex(str(ctx.exception), 'column-0 command not allowed|off column 0') + self.f.inject('Specifications/RN.lean', 'variable (h : False)\n\n/-- s\n\n' + labels() + ' -/\ndef Vacuous : Prop := h.elim') + reg = self.f.register() + reg['claims'][1]['statements'].append(statement('Specifications/RN.lean', 'UniversalLaw.Spec.RN.Vacuous')) + self.f.save_register(reg) + self.assertIn("'variable'", self.assertRefused('column-0 command not allowed')) + + def test_imports_must_be_mathlib(self): + self.f.mutate('Specifications/RN.lean', 'import Mathlib\n', 'import Mathlib.Data.Real.Basic\n') + self.assertRefused('imports must be exactly') + self.f.mutate('Specifications/RN.lean', 'import Mathlib.Data.Real.Basic\n', 'import Mathlib\nimport Specifications.Field\n') + self.assertRefused('imports must be exactly') + + +class Escapes(Base): + """The escape families found by the adversarial review of the first checker: constructs that + Lean 4.34.1 accepts and a regex scanner over a naive comment stripper let through. Each must + be refused by the source scan; the environment audit of --execute is the second, independent + refusal point (LeanHelpers.test_fixture_package).""" + + def register_rn(self, *names): + reg = self.f.register() + for n in names: + reg['claims'][1]['statements'].append(statement('Specifications/RN.lean', 'UniversalLaw.Spec.RN.' + n)) + self.f.save_register(reg) + + def test_theorem_axiom_sorry_between_string_literals_with_opener_registered(self): + self.f.inject('Specifications/RN.lean', f'/-- s\n\n{labels()} -/\ndef Opener : Prop := "/-" = "x"\naxiom hidden : False\n' + 'theorem smuggled : False := hidden\ndef Sorried : Prop := sorry\ndef Closer : Prop := "-/" = "x"') + self.register_rn('Opener') + msg = self.assertRefused('refused token') + for tok in ('axiom', 'sorry', 'theorem'): + self.assertIn(tok, msg, 'string literals no longer open a fake comment') + + def test_string_literal_alone_is_refused(self): + self.f.inject('Specifications/RN.lean', f'/-- s\n\n{labels()} -/\ndef S : Prop := "x" = "x"') + self.register_rn('S') + self.assertRefused('refused character') + + def test_line_comment_desync_with_statement_registered(self): + self.f.inject('Specifications/RN.lean', f'/-- s\n\n{labels()} -/\ndef S2 : Prop := "--" = "" ∧ sorry') + self.register_rn('S2') + self.assertIn('sorry', self.assertRefused('refused token')) + + def test_guillemet_desync(self): + self.f.inject('Specifications/RN.lean', 'def «/-» : ℕ := 0\ntheorem smuggled : True := trivial\ndef «-/» : ℕ := 0') + self.assertIn('theorem', self.assertRefused('refused token')) + self.f = Fixture() + self.addCleanup(self.f.cleanup) + self.f.inject('Specifications/RN.lean', 'def «x y» : ℕ := 0') + self.assertRefused('refused character') + + def test_indented_commands(self): + for line, fragment in ((' def Hidden : Prop := False', 'off column 0'), + (' def proofHidden : (2 : ℕ) + 2 = 4 := rfl', 'off column 0'), + (' @[simp] def Hidden : Prop := False', 'refused character'), + (' #eval IO.println "escaped"', 'refused character'), + (' #check Nat', 'refused character'), + (' namespace Other\n structure Hidden where\n n : ℕ\n end Other', 'off column 0'), + (' private def Hidden : Prop := False', 'off column 0'), + (' open Classical', 'off column 0'), + (' section\n end', 'off column 0'), + (' abbrev Hidden : Prop := True', 'off column 0')): + with self.subTest(line=line): + f = Fixture() + self.addCleanup(f.cleanup) + f.inject('Specifications/RN.lean', line) + with self.assertRaises(ValueError) as ctx: + f.run() + self.assertIn(fragment, str(ctx.exception)) + + def test_in_chains(self): + for line, fragment in (('open Classical in def Hidden : Prop := True', 'off column 0'), + ('variable (n : ℕ) in def Hidden : Prop := n = n', 'off column 0'), + ('open Classical in', '`open … in`'), + ('open Classical\n Real in', '`open … in`'), + ('variable (n : ℕ) in', 'column-0 command not allowed'), + ('variable (n : ℕ)\n (m : ℕ) in', 'column-0 command not allowed')): + with self.subTest(line=line): + f = Fixture() + self.addCleanup(f.cleanup) + f.inject('Specifications/RN.lean', line) + with self.assertRaises(ValueError) as ctx: + f.run() + self.assertIn(fragment, str(ctx.exception)) + + def test_in_inside_binders_is_fine(self): + self.f.inject('Specifications/RN.lean', 'open Finset\n (sum)\n\ndef inBinders (s : Finset ℕ) : ℕ := ∑ i in s, i') + info = self.f.run() + self.assertEqual(info['auxiliary_definitions'], 3) + + def test_two_commands_on_one_line(self): + self.f.inject('Specifications/RN.lean', f'/-- a\n\n{labels()} -/\ndef A : Prop := True def B : Prop := False') + self.register_rn('A') + self.assertRefused('off column 0') + + def test_proof_written_as_definition(self): + for line in ('def proofAsDef : (2 : ℕ) + 2 = 4 := rfl', 'abbrev proofAsAbbrev : True := trivial', + 'def proofImp : (1 : ℕ) = 1 → True := fun _ => trivial', 'def proofForall : ∀ n : ℕ, n = n := fun _ => rfl', + 'noncomputable def proofLe : (1 : ℝ) ≤ 2 := by norm_num', 'def proofMem (s : Set ℕ) (h : 0 ∈ s) : 0 ∈ s := h', + 'def proofNot : ¬ False := id', 'def proofFalse : False → True := fun h => h.elim'): + with self.subTest(line=line): + f = Fixture() + self.addCleanup(f.cleanup) + f.inject('Specifications/RN.lean', line) + with self.assertRaises(ValueError) as ctx: + f.run() + self.assertIn('proof-shaped', str(ctx.exception)) + + def test_relations_inside_brackets_are_types_not_proofs(self): + self.f.inject('Specifications/RN.lean', 'def dec (n : ℕ) : Decidable (n = 2) := inferInstance\n' + 'def two : {x : ℕ // x = 2} := ⟨2, rfl⟩\ndef pred : Set (ℕ × ℕ) := {p | p.1 ≤ p.2}') + info = self.f.run() + self.assertEqual(info['auxiliary_definitions'], 5) + + def test_untyped_definition(self): + for line in ('def Untyped := (2 : ℕ) + 2 = 4', 'abbrev U := True', 'noncomputable def c := (1 : ℝ)'): + with self.subTest(line=line): + f = Fixture() + self.addCleanup(f.cleanup) + f.inject('Specifications/RN.lean', line) + with self.assertRaises(ValueError) as ctx: + f.run() + self.assertIn('explicit result type', str(ctx.exception)) + + def test_sort_and_root(self): + self.f.inject('Specifications/RN.lean', 'def SortZero : Sort 0 := True') + self.assertIn('Sort', self.assertRefused('refused token')) + self.f = Fixture() + self.addCleanup(self.f.cleanup) + self.f.inject('Specifications/RN.lean', 'def _root_.Escaped : ℕ := 0') + self.assertIn('_root_', self.assertRefused('refused token')) + + def test_underscore_name_components(self): + for name in ('_hidden', 'Inner._hidden'): + with self.subTest(name=name): + f = Fixture() + self.addCleanup(f.cleanup) + f.inject('Specifications/RN.lean', f'def {name} : ℕ := 0') + with self.assertRaises(ValueError) as ctx: + f.run() + self.assertIn('may not start with `_`', str(ctx.exception)) + + def test_commands_that_add_declarations_under_other_names(self): + for token, line in (('export', 'export Nat (succ)'), ('alias', 'alias succ := Nat.succ'), + ('irreducible_def', 'irreducible_def x : ℕ := 0'), ('universe', 'universe u')): + with self.subTest(token=token): + f = Fixture() + self.addCleanup(f.cleanup) + f.inject('Specifications/RN.lean', line) + with self.assertRaises(ValueError) as ctx: + f.run() + self.assertIn('refused token' if token != 'universe' else 'column-0 command', str(ctx.exception)) + self.assertIn(token, str(ctx.exception)) + + +class Root(Base): + def test_order(self): + self.f.write('Specifications.lean', 'import Specifications.Lifetime\nimport Specifications.Field\nimport Specifications.Side24\n' + 'import Specifications.RN\nimport Specifications.P15\n') + self.assertRefused('root module must import') + + def test_missing_and_extra(self): + self.f.write('Specifications.lean', ''.join('import ' + m[:-5].replace('/', '.') + '\n' for m in check.MODULES[:-1])) + self.assertRefused('root module must import') + self.f.write('Specifications.lean', ''.join('import ' + m[:-5].replace('/', '.') + '\n' for m in check.MODULES) + 'import Mathlib\n') + self.assertRefused('root module must import') + self.f.write('Specifications.lean', ''.join('import ' + m[:-5].replace('/', '.') + '\n' for m in check.MODULES) + 'def x : Prop := True\n') + self.assertRefused('root module must import') + + def test_comments_in_root_ignored(self): + self.f.write('Specifications.lean', '/-! root -/\n' + ''.join('import ' + m[:-5].replace('/', '.') + '\n' for m in check.MODULES)) + self.f.run() + + +class Namespaces(Base): + def test_wrong_namespace(self): + self.f.mutate('Specifications/RN.lean', 'UniversalLaw.Spec.RN', 'UniversalLaw.Spec.Other', count=2) + self.assertRefused('outside namespace UniversalLaw.Spec.RN') + self.f.write('Specifications/RN.lean', 'import Mathlib\n\nnamespace UniversalLaw.Spec.Other\n\nend UniversalLaw.Spec.Other\n') + self.assertRefused('exactly one namespace') + + def test_two_namespaces(self): + self.f.inject('Specifications/RN.lean', 'namespace UniversalLaw.Spec.RN.Inner\nend UniversalLaw.Spec.RN.Inner') + self.assertRefused('exactly one namespace') + + def test_unclosed(self): + self.f.mutate('Specifications/RN.lean', '\nend UniversalLaw.Spec.RN\n', '\n') + self.assertRefused('unclosed namespace') + + def test_mismatched_end(self): + self.f.mutate('Specifications/RN.lean', '\nend UniversalLaw.Spec.RN\n', '\nend UniversalLaw.Spec.Field\n') + self.assertRefused('does not close') + + def test_declaration_outside_namespace(self): + self.f.mutate('Specifications/RN.lean', 'namespace UniversalLaw.Spec.RN\n', 'def Early : Prop := True\n\nnamespace UniversalLaw.Spec.RN\n') + self.assertRefused('declaration outside namespace') + + +class Scanner(unittest.TestCase): + def sources(self): + return {'demo': {'id': 'demo', 'repository': 'd6g8k5htny-coder/Math-', 'commit': COMMIT, 'path': 'docs/DEMO.md', + 'sha256': sha(SOURCE_TEXT.encode()), 'text': SOURCE_TEXT}} + + def scan(self, body, stem='Field'): + return check.scan_module(f'import Mathlib\n\nnamespace UniversalLaw.Spec.{stem}\n\n{body}\n\nend UniversalLaw.Spec.{stem}\n', stem, self.sources()) + + def test_fixture_module_shapes(self): + decls = {d['name']: d for d in check.scan_module(FIELD, 'Field', self.sources())} + self.assertEqual(decls['UniversalLaw.Spec.Field.E']['kind'], None) + self.assertEqual(decls['UniversalLaw.Spec.Field.helper'], {'name': 'UniversalLaw.Spec.Field.helper', 'lean': 'def', 'kind': None, + 'result': 'ℝ', 'line': 13}) + self.assertEqual(decls['UniversalLaw.Spec.Field.TwoAndTwo']['kind'], 'def_prop') + self.assertEqual(decls['UniversalLaw.Spec.Field.MultiLine']['result'], 'Prop') + self.assertEqual(decls['UniversalLaw.Spec.Field.DemoInterface']['kind'], 'structure') + self.assertEqual(decls['UniversalLaw.Spec.Field.DemoInterface']['doc_anchor'], 'a given object is named here') + self.assertEqual(decls['UniversalLaw.Spec.Field.TwoAndTwo']['source_id'], 'demo') + + def test_where_definition_is_auxiliary(self): + decls = self.scan('structure Data where\n n : ℕ\n\ndef sample : Data where\n n := 2\n'.replace('structure Data', f'/-- s\n\n{labels()} -/\nstructure Data')) + self.assertEqual([d['kind'] for d in decls], ['structure', None]) + + def test_arrow_prop_result_refused(self): + with self.assertRaises(ValueError) as ctx: + self.scan(f'/-- p\n\n{labels()} -/\ndef Pred : ℕ → Prop := fun n => n = 0') + self.assertIn('exactly as `: Prop`', str(ctx.exception)) + with self.assertRaises(ValueError): + self.scan(f'/-- p\n\n{labels()} -/\ndef Pred (n : ℕ) : (Prop) := n = 0') + + def test_prop_abbrev_is_a_registered_kind(self): + decls = self.scan(f'/-- p\n\n{labels()} -/\nabbrev Short : Prop := True') + self.assertEqual(decls[0]['kind'], 'def_prop') + self.assertEqual(decls[0]['lean'], 'abbrev') + + def test_pattern_arms_end_header(self): + decls = self.scan('def f : ℕ → ℕ\n | 0 => 1\n | n + 1 => n') + self.assertEqual(decls[0]['result'], 'ℕ → ℕ') + + def test_docstring_requirements(self): + with self.assertRaises(ValueError) as ctx: + self.scan('def Bare : Prop := True') + self.assertIn('has no docstring', str(ctx.exception)) + with self.assertRaises(ValueError) as ctx: + self.scan('/-! module doc, not a docstring -/\ndef Bare : Prop := True') + self.assertIn('has no docstring', str(ctx.exception)) + with self.assertRaises(ValueError) as ctx: + self.scan(f'/-- p\n\n{labels()} -/\n\ndef Gap : ℕ := 0\n\ndef Bare : Prop := True') + self.assertIn('has no docstring', str(ctx.exception)) + for label in check.LABELS: + with self.subTest(label=label), self.assertRaises(ValueError) as ctx: + self.scan(f'/-- p\n\n{labels().replace(label, label[:-1] + ".")} -/\ndef P : Prop := True') + self.assertIn('exactly one', str(ctx.exception)) + with self.assertRaises(ValueError) as ctx: + self.scan(f'/-- p\n\n{labels(anchor="two and three make four")} -/\ndef P : Prop := True') + self.assertIn('docstring anchor is not a verbatim', str(ctx.exception)) + with self.assertRaises(ValueError) as ctx: + self.scan(f'/-- p\n\n{labels(anchor="two and two")} -/\ndef P : Prop := True') + self.assertIn('shorter than 16', str(ctx.exception), 'a short verbatim anchor locates nothing') + self.scan(f'/-- p\n\n{labels(anchor="two and two make")} -/\ndef P : Prop := True') + with self.assertRaises(ValueError) as ctx: + self.scan(f'/-- p\n\n{labels(text="other bytes")} -/\ndef P : Prop := True') + self.assertIn('names no registered', str(ctx.exception)) + with self.assertRaises(ValueError) as ctx: + self.scan(f'/-- p\n\n{labels()} -/\ndef P : Prop := True'.replace('Anchor: "two and two make four"', 'Anchor: two and two make four')) + self.assertIn('double-quoted', str(ctx.exception)) + self.scan(f'/-- p\n\n{labels()} -/\ndef P : Prop := True'.replace('Does not claim: anything', 'Does not claim:\n anything')) + + def test_lex(self): + code, docs = check.lex('/- a /- nested -/ b -/\n-- line /- not opened\n/-- doc -/\ndef x := 1 -- tail\n') + self.assertEqual([line.strip() for line in code.split('\n')], ['', '', '', 'def x := 1', '']) + self.assertEqual(len(code.split('\n')), 5, 'newlines are preserved so line numbers stay stable') + self.assertEqual(docs, [(2, 2, '/-- doc -/')]) + with self.assertRaises(ValueError): + check.lex('/- open\n') + + def test_lex_strings_and_guillemets_do_not_open_comments(self): + code, _ = check.lex('def a := "/-" -- c\ntheorem t : True := trivial\ndef b := "-/"\n') + self.assertIn('theorem t', code, 'a string literal "/-" must not hide the next lines') + self.assertIn('"/-"', code, 'the literal stays visible so the scan can refuse it') + self.assertNotIn('-- c', code) + code, _ = check.lex('def c := "--" ∧ sorry\n') + self.assertIn('sorry', code) + code, _ = check.lex('def «/-» := 0\ntheorem t : True := trivial\n') + self.assertIn('theorem t', code) + code, _ = check.lex('def d := "a\\"/-" -- x\ndef e := 1\n') + self.assertIn('def e', code, 'escaped quotes stay inside the literal') + code, _ = check.lex('def f := "two\nlines"\ndef g := 1\n') + self.assertEqual(len(code.split('\n')), 4, 'newlines inside literals keep line numbers stable') + for bad in ('def x := "open\n', 'def «open := 0\n'): + with self.assertRaises(ValueError): + check.lex(bad) + self.assertEqual(check.lex('/- "unterminated inside a comment is fine -/\ndef x := 1\n')[0].split('\n')[1], 'def x := 1') + + def test_depth_zero(self): + self.assertEqual(check.depth_zero('Decidable (n = 2)'), 'Decidable' + ' ' * 8) + self.assertEqual(check.depth_zero('a = b'), 'a = b') + self.assertEqual(check.depth_zero('{x : ℕ // x = 2} → ℕ'), ' ' * 17 + '→ ℕ') + self.assertEqual(check.depth_zero('∀ n, (n : ℕ) = n'), '∀ n, = n') + + def test_header_and_result_type(self): + self.assertEqual(check.result_type(' (x : ℕ) (h : x = 1 := rfl) : Prop '), 'Prop') + self.assertEqual(check.result_type(' {X : Type} [Fintype X] (F : Set X) :\n Set (X × X) '), 'Set (X × X)') + self.assertIsNone(check.result_type(' (x : ℕ) ')) + self.assertEqual(check.header(['def f (x : ℕ) : ℕ := x', 'next'], 0, ' (x : ℕ) : ℕ := x'), ' (x : ℕ) : ℕ ') + self.assertEqual(check.header(['structure S (d : ℕ) where', ' x : ℕ'], 0, ' (d : ℕ) where'), ' (d : ℕ) ') + + +class Inventory(Base): + def test_short_anchors_are_refused(self): + reg = self.f.register() + reg['claims'][0]['statements'][0]['anchor'] = 'two and two' + self.f.save_register(reg) + self.assertIn('TwoAndTwo', self.assertRefused('shorter than 16')) + reg = self.f.register() + reg['claims'][0]['statements'][0]['anchor'] = 'two and two make four' + reg['claims'][0]['statements'][3]['anchor'] = 'four;' + self.f.save_register(reg) + self.assertIn('two_add', self.assertRefused('shorter than 16')) + reg = self.f.register() + reg['claims'][0]['statements'][3]['anchor'] = 'two and two make four' + self.f.save_register(reg) + self.f.run() + + def test_unregistered_declaration(self): + self.f.inject('Specifications/RN.lean', f'/-- extra\n\n{labels()} -/\ndef Extra : Prop := True') + msg = self.assertRefused('unregistered declaration') + self.assertIn('UniversalLaw.Spec.RN.Extra', msg) + + def test_unregistered_structure(self): + self.f.inject('Specifications/RN.lean', f'/-- extra\n\n{labels()} -/\nstructure Extra where\n n : ℕ') + self.assertRefused('unregistered declaration') + + def test_missing_declaration(self): + reg = self.f.register() + reg['claims'][0]['statements'][0]['declaration'] = 'UniversalLaw.Spec.Field.DoesNotExist' + self.f.save_register(reg) + self.assertRefused('is not declared') + reg['claims'][0]['statements'][0]['declaration'] = 'UniversalLaw.Spec.Field.helper' + self.f.save_register(reg) + self.assertRefused('is not declared') + + def test_kind_and_module_mismatch(self): + reg = self.f.register() + reg['claims'][0]['statements'][0]['kind'] = 'structure' + self.f.save_register(reg) + self.assertRefused('registered as structure') + reg = self.f.register() + reg['claims'][0]['statements'][0]['module'] = 'Specifications/RN.lean' + self.f.save_register(reg) + self.assertRefused('is not declared') + + def test_control_row_helper(self): + reg = self.f.register() + reg['claims'].append(check.control_row('Specifications/Field.lean', 'UniversalLaw.Spec.Field.DoesNotExistControl', 'two and two make four')) + self.f.save_register(reg) + self.assertRefused('is not declared') + with self.assertRaises(ValueError): + check.control_row('Specifications/Field.lean', 'UniversalLaw.Spec.Field.DoesNotExistControl', 'x') + + def test_source_side_negative_controls(self): + """The four source-side controls that --execute runs, exercised on the fixture without Lean.""" + info = self.f.run() + reg = self.f.register() + sources, _ = check.load_sources(self.f.packet, None, reg) + ctrl = Path(self.f.tmp) / 'controls' + ctrl.mkdir() + outcomes, missing, injected = check.source_controls(self.f.packet, reg, info['scans'], sources, check.MODULES, ctrl) + self.assertEqual(outcomes, {k: 'REFUSED_BY_SOURCE_CHECK' for k in + ('injected_theorem', 'missing_declaration', 'unregistered_declaration', 'proved_label')}) + self.assertEqual(missing, 'UniversalLaw.Spec.Field.DoesNotExistControl') + self.assertIn('theorem injected : True := trivial', injected) + self.assertIn('def UnregisteredControl : Prop := True', (ctrl / 'Unregistered.lean.txt').read_text()) + mutations = json.loads((ctrl / 'register_mutations.json').read_text()) + self.assertEqual(mutations['proved_label']['formal_progress'], 'proved') + self.assertEqual(self.f.read('Specifications/Field.lean'), FIELD, 'the packet module is never modified by a control') + with self.assertRaises(ValueError) as ctx: + check.expect_refusal(lambda: None, 'anything', 'escape') + self.assertIn('escaped', str(ctx.exception)) + with self.assertRaises(ValueError) as ctx: + check.expect_refusal(lambda: check.require(False, 'other reason'), 'expected reason', 'wrong') + self.assertIn('wrong reason', str(ctx.exception)) + + +class Vocabulary(Base): + def set_statement(self, **changes): + reg = self.f.register() + reg['claims'][0]['statements'][0].update(changes) + self.f.save_register(reg) + + def set_top(self, key, value): + reg = self.f.register() + reg[key] = value + self.f.save_register(reg) + + def set_row(self, index, **changes): + reg = self.f.register() + reg['claims'][index].update(changes) + self.f.save_register(reg) + + def test_local_statement_never_proved(self): + self.set_statement(formal_progress='proved') + self.assertRefused('never proved') + + def test_local_statement_never_kernel_checked_or_none(self): + for value in ('kernel-checked', 'none', 'verified'): + with self.subTest(value=value): + self.set_statement(formal_progress=value) + self.assertRefused('kernel-checked appears only in a run receipt') + + def test_local_statement_never_theorem(self): + self.set_statement(kind='theorem') + self.assertRefused('never theorem') + + def test_external_theorem_label(self): + reg = self.f.register() + reg['claims'][0]['statements'][3]['formal_progress'] = 'specified' + self.f.save_register(reg) + self.assertRefused('outside this package a theorem is proved') + + def test_claim_level_progress(self): + for value in ('proved', 'kernel-checked', 'verified'): + with self.subTest(value=value): + self.set_row(0, formal_progress=value) + self.assertRefused('claim-level formal_progress') + + def test_specified_requires_def_prop(self): + reg = self.f.register() + reg['claims'][0]['statements'] = [reg['claims'][0]['statements'][1], reg['claims'][0]['statements'][3]] + reg['claims'].append({**reg['claims'][1], 'claim_id': 'rest', 'statements': [ + statement('Specifications/Field.lean', 'UniversalLaw.Spec.Field.TwoAndTwo'), + statement('Specifications/Field.lean', 'UniversalLaw.Spec.Field.MultiLine')]}) + self.f.save_register(reg) + self.assertRefused('specified requires a def_prop statement') + + def test_top_level_vocabulary(self): + for key, value, fragment in (('alignment_status', 'ACCEPTED', 'alignment_status'), ('alignment_status', 'ALIGNED', 'alignment_status'), + ('scientific_effect', 'LOW', 'scientific_effect'), ('scientific_status_authority', True, 'scientific_status_authority'), + ('vocabulary', {'formal_progress': ['none', 'specified', 'proved', 'kernel-checked', 'verified']}, 'vocabulary'), + ('object', 'LEAN-STATEMENT-REGISTER-20261010-v2', 'object must be'), ('schema_version', 2, 'schema_version'), + ('toolchain', {'lean': check.TOOLCHAIN, 'lean_commit': check.LEAN_COMMIT, 'mathlib': 'f' * 40}, 'toolchain'), + ('meaning', '', 'meaning')): + with self.subTest(key=key, value=value): + self.set_top(key, value) + self.assertRefused(fragment) + self.f = Fixture() + self.addCleanup(self.f.cleanup) + + def test_unknown_keys(self): + self.set_top('formalization_status', 'kernel-checked') + self.assertRefused('top-level keys') + self.f = Fixture() + self.addCleanup(self.f.cleanup) + self.set_row(0, review_verdict='ALIGNED') + self.assertRefused('fixed schema') + self.f = Fixture() + self.addCleanup(self.f.cleanup) + self.set_statement(reviewed=True) + self.assertRefused('fixed schema') + + def test_row_fields(self): + for index, changes, fragment in ((0, {'kind': 'theorem_row'}, 'kind must be one of'), (1, {'claim_id': 'demo-claim'}, 'unique'), + (0, {'not_established': ''}, 'not_established'), (0, {'next_step': ' '}, 'next_step'), + (0, {'layer0': {'source_id': 'nope', 'heading': None}}, 'layer0.source_id'), + (0, {'layer0': {'source_id': 'demo'}}, 'layer0 must be'), + (0, {'kernel_evidence': [{'package': 'formal'}]}, 'kernel_evidence rows'), + (0, {'title': ''}, 'title')): + with self.subTest(changes=changes): + self.f = Fixture() + self.addCleanup(self.f.cleanup) + self.set_row(index, **changes) + self.assertRefused(fragment) + + def test_dispositions_are_transcriptions(self): + self.set_row(0, disposition_transcribed='REVIEWED_SCOPED') + self.assertRefused('verbatim') + self.set_row(0, disposition_transcribed='ACCEPT') + self.assertRefused('LANDING_CLAIMS disposition') + self.set_row(0, disposition_transcribed='HOLD_WITH_DOMAIN', claim_id='not-a-landing-claim') + self.assertRefused('landing_claim rows must carry') + self.f = Fixture() + self.addCleanup(self.f.cleanup) + self.set_row(1, disposition_transcribed='HOLD_WITH_DOMAIN') + self.assertRefused('must be null or the verbatim') + self.set_row(1, disposition_transcribed=None) + self.f.run() + + +class Anchors(Base): + def test_statement_anchor_verbatim(self): + reg = self.f.register() + reg['claims'][0]['statements'][0]['anchor'] = 'two and two make five' + self.f.save_register(reg) + self.assertRefused('verbatim substring') + + def test_external_anchor_verbatim(self): + reg = self.f.register() + reg['claims'][0]['statements'][3]['anchor'] = 'absent words of sufficient length' + self.f.save_register(reg) + self.assertRefused('verbatim substring') + reg['claims'][0]['statements'][3]['anchor'] = 'two and two make five' + reg['claims'][0]['layer0']['source_id'] = None + self.f.save_register(reg) + self.assertRefused('need a layer0 source') + + def test_docstring_anchor_verbatim(self): + self.f.mutate('Specifications/RN.lean', 'Anchor: "two and two make four"', 'Anchor: "two and two make four"') + self.assertRefused('docstring anchor is not a verbatim') + + +class Sources(Base): + def test_local_copy_bytes(self): + self.f.write('sources/demo/DEMO.md', SOURCE_TEXT + '\n') + self.assertRefused('local copy differs') + + def test_register_differs_from_sources_json(self): + def mutate_bytes(reg): + reg['sources'][0]['bytes'] += 1 + + def mutate_blob(reg): + reg['sources'][0]['blob'] = 'x' + for mutation, fragment in ((mutate_bytes, 'differs from sources/SOURCES.json'), + (lambda reg: reg['sources'].reverse(), 'exactly the sources/SOURCES.json ids'), + (mutate_blob, 'fixed schema')): + with self.subTest(fragment=fragment): + self.f = Fixture() + self.addCleanup(self.f.cleanup) + reg = self.f.register() + mutation(reg) + self.f.save_register(reg) + self.assertRefused(fragment) + + def test_sources_json_blob(self): + self.f.mutate('sources/SOURCES.json', blob(SOURCE_TEXT.encode()), '0' * 40) + self.assertRefused('local copy differs') + + def test_nonpublic_repository(self): + self.f.mutate('sources/SOURCES.json', 'd6g8k5htny-coder/Math-', 'someone/private', count=1) + reg = self.f.register() + reg['sources'][0]['repository'] = 'someone/private' + self.f.save_register(reg) + self.assertRefused('nonpublic') + + def test_local_copy_location(self): + self.f.mutate('sources/SOURCES.json', '"local_copy": "sources/demo/DEMO.md"', '"local_copy": "DEMO.md"') + reg = self.f.register() + reg['sources'][0]['local_copy'] = 'DEMO.md' + self.f.save_register(reg) + shutil.move(self.f.path('sources/demo/DEMO.md'), self.f.path('DEMO.md')) + self.assertRefused('live under sources/') + + +class Pins(Base): + def test_lock_name(self): + self.f.mutate('lake-manifest.json', check.LOCK_NAME, 'other_name') + self.assertRefused('name must be') + + def test_lock_drift(self): + self.f.mutate('lake-manifest.json', 'f2effa3d803fda822b1f97b806c47cf2adfbcbc2', 'f2effa3d803fda822b1f97b806c47cf2adfbcbc3') + self.assertRefused('differs from formal/lake-manifest.json') + + def test_mathlib_rev_pinned_even_if_reference_moves(self): + other = 'a' * 40 + self.f.mutate('lake-manifest.json', check.MATHLIB_REV, other, count=2) + self.f.mutate('formal/lake-manifest.json', check.MATHLIB_REV, other, repo=True, count=2) + self.f.mutate('lakefile.toml', check.MATHLIB_REV, other) + self.assertRefused('lakefile.toml lacks') + + def test_toolchain(self): + self.f.write('lean-toolchain', 'leanprover/lean4:v4.34.2\n') + self.assertRefused('lean-toolchain must read') + self.f.write('lean-toolchain', check.TOOLCHAIN) + self.assertRefused('lean-toolchain must read') + + def test_lakefile(self): + self.f.mutate('lakefile.toml', 'name = "universal_law_specifications"', 'name = "other"') + self.assertRefused('lakefile.toml lacks') + self.f = Fixture() + self.addCleanup(self.f.cleanup) + self.f.write('lakefile.toml', LAKEFILE + '\n[[require]]\nname = "extra"\ngit = "https://example.invalid/x.git"\nrev = "' + 'b' * 40 + '"\n') + self.assertRefused('exactly one dependency') + + def test_auto_implicit_must_be_off(self): + for replacement in ('', 'leanOptions = { autoImplicit = true }\n', 'leanOptions = { relaxedAutoImplicit = false }\n'): + with self.subTest(replacement=replacement): + self.f = Fixture() + self.addCleanup(self.f.cleanup) + self.f.mutate('lakefile.toml', check.LEAN_OPTIONS_LINE + '\n', replacement) + self.assertRefused('lakefile.toml lacks') + + +class Docs(Base): + def test_placeholders_refused(self): + for name in ('README.md', 'SCOPE.md'): + with self.subTest(name=name): + self.f = Fixture() + self.addCleanup(self.f.cleanup) + self.f.write(name, '# bound copy\n\n## Declarations\n\n<>\n') + self.assertRefused('placeholder') + + def test_placeholder_check_runs_after_the_inventory(self): + self.f.write('README.md', '<>\n') + self.f.inject('Specifications/RN.lean', f'/-- extra\n\n{labels()} -/\ndef Extra : Prop := True') + self.assertRefused('unregistered declaration') + + +class Tree(Base): + def test_extra_file(self): + self.f.write('NOTES.md', 'stray\n') + self.assertRefused('packet tree differs') + + def test_extra_lean_module(self): + self.f.write('Specifications/Extra.lean', 'import Mathlib\n') + self.assertRefused('packet tree differs') + + def test_missing_fixed_file(self): + self.f.path('SCOPE.md').unlink() + self.assertRefused('packet tree differs') + + def test_gitignore(self): + self.f.write('.gitignore', '__pycache__/\n') + self.assertRefused('.gitignore must ignore') + + def test_lake_directory_ignored(self): + (self.f.packet / '.lake' / 'build').mkdir(parents=True) + self.f.write('.lake/build/junk.olean', 'x') + self.f.run() + + +class Audit(unittest.TestCase): + def test_allowed(self): + text = "'A.x' depends on axioms: [propext, Classical.choice, Quot.sound]\n'A.y' does not depend on any axioms\n" + self.assertEqual(check.audit_axioms(text, ['A.x', 'A.y']), {'A.x': ['Classical.choice', 'Quot.sound', 'propext'], 'A.y': []}) + + def test_forbidden(self): + for bad in ('sorryAx', 'Lean.ofReduceBool', 'hiddenPremise'): + with self.assertRaises(ValueError) as ctx: + check.audit_axioms("'A.x' depends on axioms: [propext, " + bad + "]\n", ['A.x']) + self.assertIn('forbidden transitive axiom', str(ctx.exception)) + + def test_missing_duplicate_unrecognized(self): + with self.assertRaises(ValueError): + check.audit_axioms("'A.x' does not depend on any axioms\n", ['A.x', 'A.y']) + with self.assertRaises(ValueError): + check.audit_axioms("'A.x' does not depend on any axioms\n" * 2, ['A.x']) + with self.assertRaises(ValueError): + check.audit_axioms("'A.z' does not depend on any axioms\n", ['A.x']) + with self.assertRaises(ValueError): + check.audit_axioms("warning: x\n'A.x' does not depend on any axioms\n", ['A.x']) + + def test_lean_version(self): + good = f'Lean (version 4.34.1, x86_64-unknown-linux-gnu, commit {check.LEAN_COMMIT}, Release)\n' + self.assertTrue(check.check_lean_version(good).startswith('Lean (version 4.34.1,')) + self.assertEqual(check.check_lean_version('info: toolchain not updated; already up-to-date\n' + good), good.strip()) + for bad in (good.replace('4.34.1', '4.34.2'), good.replace(check.LEAN_COMMIT, check.LEAN_COMMIT[:12]), + good.replace(check.LEAN_COMMIT, 'f' * 40), good.replace('Release', 'Debug'), '', + 'warning: something\n' + good, good + 'trailing line\n'): + with self.assertRaises(ValueError): + check.check_lean_version(bad) + + def test_strict_json(self): + self.assertEqual(check.load_json('{"a": 1}'), {'a': 1}) + for raw in ('{"a": 1, "a": 2}', '{"a": 1.5}', '{"a": NaN}', '[Infinity]'): + with self.assertRaises(ValueError): + check.load_json(raw) + + def test_signatures(self): + log = 'A.x (n : ℕ) : Prop\nA.y (long : ℕ) :\n Prop\nA.z : Type\n' + self.assertEqual(check.signatures(log, ['A.x', 'A.y']), {'A.x': 'A.x (n : ℕ) : Prop', 'A.y': 'A.y (long : ℕ) :\n Prop'}) + with self.assertRaises(ValueError): + check.signatures(log, ['A.w']) + with self.assertRaises(ValueError): + check.signatures(log + 'A.x : Prop\n', ['A.x']) + with self.assertRaises(ValueError): + check.signatures('A.xy : Prop\n', ['A.x']) + + def test_parse_audit(self): + log = ('Environment.lean:9:0: info: AUDIT statement A.x\nEnvironment.lean:9:0: info: AUDIT structure A.S\n' + 'Environment.lean:9:0: info: AUDIT generated A.S.mk\nEnvironment.lean:9:0: info: AUDIT auxiliary A.h\n' + 'Environment.lean:9:0: info: AUDIT END 4 0\n') + self.assertEqual(check.parse_audit(log), {'statement': {'A.x'}, 'structure': {'A.S'}, 'generated': {'A.S.mk'}, 'auxiliary': {'A.h'}}) + refused = log.replace('AUDIT auxiliary A.h', 'AUDIT REFUSED:theorem A.h').replace('END 4 0', 'END 4 1') + self.assertEqual(check.parse_audit(refused)['REFUSED:theorem'], {'A.h'}) + for bad in (log.replace('END 4 0', 'END 5 0'), log.replace('END 4 0', 'END 4 1'), log.replace('Environment.lean:9:0: info: AUDIT END 4 0\n', ''), + log + 'Environment.lean:9:0: info: AUDIT END 4 0\n', log.replace('AUDIT auxiliary A.h', 'AUDIT auxiliary A.x').replace('END 4', 'END 3')): + with self.assertRaises(ValueError): + check.parse_audit(bad) + + def test_audit_source(self): + src = check.audit_source('Specifications', ['Specifications.Field', 'Specifications.RN']) + self.assertIn('import Specifications\n', src) + self.assertIn('def auditModules : Array Name := #[`Specifications.Field, `Specifications.RN]', src) + self.assertIn('run_cmd do', src) + self.assertIn('"REFUSED:theorem"', src) + self.assertIn('forallTelescope ci.type fun _ body => pure body.isProp', src) + self.assertNotIn('isPropFormerType', src, 'the reducing classification called Set-valued definitions statements') + ctrl = check.audit_source('Specifications', [], check.HIDDEN_CONTROL) + self.assertIn('def auditModules : Array Name := #[]', ctrl) + self.assertLess(ctrl.index('theorem smuggledBetweenStrings'), ctrl.index('run_cmd do')) + self.assertIn('def auditPrefix : Name := `UniversalLaw.Spec.Control', ctrl) + + +LAKE = check.lake_binary() + + +FIXTURE_FIELD = '''namespace UniversalLaw.Spec.Field + +/-- two and two, parametrized -/ +def TwoAndTwo (n : Nat) : Prop := n + n = 4 + +/-- interface with a Prop field: its projection is a generated theorem -/ +structure Iface (d : Nat) where + x : Nat + pos : 0 < x + +/-- extending interface: `toIface` is a generated projection -/ +structure Ext (d : Nat) extends Iface d where + y : Nat + hy : y = x + +/-- auxiliary definition with a nested proof: `_proof_1` is a generated, range-less theorem -/ +def toFin3 (x : Nat) : Fin 3 := ⟨x % 3, Nat.mod_lt _ (by decide)⟩ + +/-- auxiliary abbreviation -/ +abbrev E (d : Nat) : Type := Fin d → Nat + +/-- stand-in for Mathlib's `def Set (α : Type u) := α → Prop` (Mathlib/Data/Set/Defs.lean): a type +former whose unfolding is a Prop former -/ +def Fam (α : Type) : Type := α → Prop + +/-- auxiliary `Fam`-valued definition (the real modules have eleven `Set`-valued ones): its type +`Fam (Nat × Nat)` unfolds to `Nat × Nat → Prop`, so a reducing classification (`isPropFormerType`) +calls it a statement while the scan's `: Prop` rule has it auxiliary -/ +def pred : Fam (Nat × Nat) := fun p => p.1 ≤ p.2 + +end UniversalLaw.Spec.Field +''' +FIXTURE_EXPECTED = {'statement': ['UniversalLaw.Spec.Field.TwoAndTwo'], 'structure': ['UniversalLaw.Spec.Field.Iface', 'UniversalLaw.Spec.Field.Ext'], + 'auxiliary': ['UniversalLaw.Spec.Field.toFin3', 'UniversalLaw.Spec.Field.E', 'UniversalLaw.Spec.Field.Fam', 'UniversalLaw.Spec.Field.pred']} + + +@unittest.skipUnless(LAKE, 'lake not available; the Lean-driving helpers are exercised only with the toolchain') +class LeanHelpers(unittest.TestCase): + """A dependency-free fixture package: real `lake build`, `leanchecker`, `#check`, `#print axioms`, + the environment audit, its two controls, the Lean-side negative controls, and module copies + carrying each escape family that the environment audit must refuse. No Mathlib; nothing here + is about the real modules.""" + + def setUp(self): + self.tmp = Path(tempfile.mkdtemp()) + self.addCleanup(shutil.rmtree, self.tmp, True) + (self.tmp / 'Specifications').mkdir() + (self.tmp / 'lakefile.toml').write_text('name = "spec_fixture"\ndefaultTargets = ["Specifications"]\n\n[[lean_lib]]\nname = "Specifications"\n' + + check.LEAN_OPTIONS_LINE + '\n') + (self.tmp / 'lean-toolchain').write_text(check.TOOLCHAIN + '\n') + (self.tmp / 'Specifications.lean').write_text('import Specifications.Field\n') + (self.tmp / 'Specifications/Field.lean').write_text(FIXTURE_FIELD) + self.out = self.tmp / '.lake' / 'evidence' + (self.out / 'controls').mkdir(parents=True) + self.env = check.lean_env(LAKE) + + def build(self, module_text=None, expect_success=True): + if module_text is not None: + (self.tmp / 'Specifications/Field.lean').write_text(module_text) + if (self.tmp / '.lake' / 'build').exists(): + shutil.rmtree(self.tmp / '.lake' / 'build') + return check.run([LAKE, 'build'], 'build', self.out, self.tmp, self.env, expect_success=expect_success) + + def audit(self, expected=FIXTURE_EXPECTED): + return check.environment_audit(LAKE, self.tmp, self.env, self.out, ['Specifications.Field'], expected) + + def test_fixture_package(self): + names = ['UniversalLaw.Spec.Field.TwoAndTwo', 'UniversalLaw.Spec.Field.Iface', 'UniversalLaw.Spec.Field.Ext'] + with contextlib.redirect_stdout(io.StringIO()): + check.check_lean_version(check.run([LAKE, 'env', 'lean', '--version'], 'version', self.out, self.tmp, self.env)) + log = self.build() + self.assertNotIn('is a proposition; use', log) + check.run([LAKE, 'env', 'leanchecker', 'Specifications'], 'leanchecker', self.out, self.tmp, self.env) + axioms, sigs = check.kernel_audit(LAKE, self.tmp, self.env, self.out, names) + self.assertEqual(axioms, {n: [] for n in names}) + self.assertEqual(sigs['UniversalLaw.Spec.Field.TwoAndTwo'], 'UniversalLaw.Spec.Field.TwoAndTwo (n : Nat) : Prop') + self.assertEqual(sigs['UniversalLaw.Spec.Field.Ext'], 'UniversalLaw.Spec.Field.Ext (d : Nat) : Type') + counts = self.audit() + self.assertEqual((counts['statement'], counts['structure'], counts['auxiliary']), (1, 2, 4)) + self.assertGreater(counts['generated'], 20, 'projections, recursors, injEq/inj/sizeOf_spec, _proof_1 are tolerated as generated') + env_log = (self.out / 'environment.log').read_text() + self.assertIn('AUDIT generated UniversalLaw.Spec.Field.Iface.pos', env_log, 'a Prop-field projection is a generated theorem') + self.assertIn('AUDIT generated UniversalLaw.Spec.Field.toFin3._proof_1', env_log) + self.assertIn('AUDIT auxiliary UniversalLaw.Spec.Field.pred', env_log, + 'a definition whose type merely unfolds to `… → Prop` is auxiliary, as in the scan (statement = syntactically `∀ …, Prop`)') + self.assertIn('AUDIT auxiliary UniversalLaw.Spec.Field.Fam', env_log) + self.assertNotIn('REFUSED', env_log) + with self.assertRaises(ValueError) as ctx: + self.audit({**FIXTURE_EXPECTED, 'auxiliary': ['UniversalLaw.Spec.Field.E']}) + self.assertIn('unexpected', str(ctx.exception)) + with self.assertRaises(ValueError) as ctx: + self.audit({**FIXTURE_EXPECTED, 'statement': FIXTURE_EXPECTED['statement'] + ['UniversalLaw.Spec.Field.Ghost']}) + self.assertIn('missing', str(ctx.exception)) + with self.assertRaises(ValueError): + check.environment_audit(LAKE, self.tmp, self.env, self.out, ['Specifications.Missing'], FIXTURE_EXPECTED) + self.assertEqual(check.lean_control_type_error(LAKE, self.tmp, self.env, self.out), 'REJECTED_BY_LEAN') + self.assertEqual(check.lean_control_missing(LAKE, self.tmp, self.env, self.out, 'UniversalLaw.Spec.Field.Missing'), 'REJECTED_BY_LEAN') + with self.assertRaises(ValueError): + check.kernel_audit(LAKE, self.tmp, self.env, self.out, names + ['UniversalLaw.Spec.Field.Missing']) + self.assertIn('REFUSED_BY_ENVIRONMENT_AUDIT', check.lean_control_environment(LAKE, self.tmp, self.env, self.out)) + hidden_log = (self.out / 'control_environment_hidden.log').read_text() + for verdict in ('REFUSED:theorem UniversalLaw.Spec.Control.smuggledBetweenStrings', 'REFUSED:proof UniversalLaw.Spec.Control.proofAsDef', + 'REFUSED:axioms:sorryAx UniversalLaw.Spec.Control.usesSorry', 'REFUSED:axiom UniversalLaw.Spec.Control.hidden', + 'REFUSED:instance UniversalLaw.Spec.Control.controlInstance', 'statement UniversalLaw.Spec.Control.HiddenIndented'): + self.assertIn('AUDIT ' + verdict, hidden_log) + injected = self.out / 'Injected.lean' + injected.write_text('import Specifications\ntheorem injected : True := trivial\n#print axioms injected\n') + log = check.run([LAKE, 'env', 'lean', str(injected)], 'injected', self.out, self.tmp, self.env) + self.assertIn("'injected' does not depend on any axioms", log) + + def test_environment_audit_refuses_escapes_lean_accepts(self): + """Module copies Lean builds without error (the source scan never saw them) must be refused + by the environment audit, each for its own reason; the build log check catches the proof.""" + cases = { + 'indented Prop definition': (FIXTURE_FIELD.replace('end UniversalLaw', ' def HiddenIndented : Prop := False\nend UniversalLaw'), 'unexpected'), + 'theorem between string literals': (FIXTURE_FIELD.replace('end UniversalLaw', 'def Opener : Prop := "/-" = "x"\ntheorem smuggled : True := trivial\n' + 'def Closer : Prop := "-/" = "x"\nend UniversalLaw'), 'REFUSED:theorem'), + 'axiom': (FIXTURE_FIELD.replace('end UniversalLaw', 'axiom hidden : False\nend UniversalLaw'), 'REFUSED:axiom'), + 'instance': (FIXTURE_FIELD.replace('end UniversalLaw', 'instance hiddenInst : Inhabited Nat := ⟨0⟩\nend UniversalLaw'), 'REFUSED:instance'), + 'untyped Prop definition': (FIXTURE_FIELD.replace('end UniversalLaw', 'def Untyped := (2 : Nat) + 2 = 4\nend UniversalLaw'), 'unexpected'), + 'Sort 0': (FIXTURE_FIELD.replace('end UniversalLaw', 'def SortZero : Sort 0 := True\nend UniversalLaw'), 'unexpected'), + 'opaque': (FIXTURE_FIELD.replace('end UniversalLaw', 'opaque opq : Nat\nend UniversalLaw'), 'REFUSED:opaque'), + 'inductive': (FIXTURE_FIELD.replace('end UniversalLaw', 'inductive Two | a | b\nend UniversalLaw'), 'REFUSED:inductive'), + } + with contextlib.redirect_stdout(io.StringIO()): + for label, (text, fragment) in cases.items(): + with self.subTest(label=label): + self.build(text) + with self.assertRaises(ValueError) as ctx: + self.audit() + message = str(ctx.exception) + (self.out / 'environment.log').read_text() + self.assertIn(fragment, message) + text = FIXTURE_FIELD.replace('end UniversalLaw', 'def proofAsDef : (2 : Nat) + 2 = 4 := rfl\nabbrev proofAsAbbrev : True := trivial\nend UniversalLaw') + log = self.build(text) + self.assertIn('is a proposition; use', log, 'the build-log check of execute() refuses the defProp linter warning') + with self.assertRaises(ValueError): + self.audit() + self.assertIn('AUDIT REFUSED:proof UniversalLaw.Spec.Field.proofAsDef', (self.out / 'environment.log').read_text()) + self.assertIn('AUDIT REFUSED:proof UniversalLaw.Spec.Field.proofAsAbbrev', (self.out / 'environment.log').read_text()) + log = self.build(FIXTURE_FIELD.replace('end UniversalLaw', 'def usesSorry : Prop := sorry\nend UniversalLaw')) + self.assertIn('declaration uses', log) + with self.assertRaises(ValueError): + self.audit() + self.assertIn('AUDIT REFUSED:axioms:sorryAx UniversalLaw.Spec.Field.usesSorry', (self.out / 'environment.log').read_text()) + + def test_auto_implicit_is_off(self): + """With `autoImplicit = false` an unbound name in a header is an error instead of a silent + implicit parameter (under the default, `def X (f : Fin d → Nat) : Prop` gets `{d : Nat}`).""" + with contextlib.redirect_stdout(io.StringIO()): + log = self.build(FIXTURE_FIELD.replace('end UniversalLaw', 'def X (f : Fin d → Nat) : Prop := f = f\nend UniversalLaw'), expect_success=False) + self.assertIn('Unknown identifier `d`', log) + + +REAL_REPO = HERE.parents[1] + + +@unittest.skipUnless((REAL_REPO / check.REFERENCE_LOCK).is_file() and (HERE / 'REGISTER.json').is_file(), 'real tree not present') +class RealPacket(unittest.TestCase): + """The committed packet's pins and module scans, independent of the register inventory.""" + + def test_pins(self): + revs = check.check_lock(HERE, REAL_REPO) + self.assertEqual(revs['mathlib'], check.MATHLIB_REV) + self.assertEqual(len(revs), 9) + check.check_root(HERE, check.MODULES) + lock = json.loads((HERE / 'lake-manifest.json').read_text()) + ref = json.loads((REAL_REPO / check.REFERENCE_LOCK).read_text()) + self.assertEqual(lock['packages'], ref['packages']) + self.assertEqual(lock['name'], check.LOCK_NAME) + self.assertIn('.lake/', (HERE / '.gitignore').read_text().splitlines()) + self.assertIn(check.LEAN_OPTIONS_LINE, (HERE / 'lakefile.toml').read_text().splitlines()) + + def test_register_skeleton_and_sources(self): + reg = check.load_json((HERE / 'REGISTER.json').read_text(encoding='utf-8')) + self.assertEqual(set(reg), check.TOP_KEYS) + self.assertEqual(reg['vocabulary'], {'formal_progress': list(check.FORMAL_PROGRESS)}) + self.assertEqual(reg['alignment_status'], 'PENDING_INDEPENDENT_REVIEW') + self.assertIs(reg['scientific_status_authority'], False) + sources, _ = check.load_sources(HERE, None, reg) + self.assertGreaterEqual(len(sources), 20) + self.assertIn(check.LANDING_SOURCE, sources) + landing = check.landing_dispositions(sources) + self.assertTrue(set(landing.values()) <= set(check.DISPOSITIONS)) + + def test_modules_scan(self): + reg = check.load_json((HERE / 'REGISTER.json').read_text(encoding='utf-8')) + sources, _ = check.load_sources(HERE, None, reg) + for m in check.MODULES: + with self.subTest(module=m): + if not (HERE / m).is_file(): + self.skipTest(m + ' not written yet') + decls = check.scan_module((HERE / m).read_text(encoding='utf-8'), Path(m).stem, sources) + self.assertTrue(any(d['kind'] == 'def_prop' for d in decls), 'a statement module declares at least one Prop') + + @staticmethod + def chunks(module): + """Top-level declaration chunks of a module: {short name: (docstring, declaration text)}, each + the comment-stripped code from the declaration line to the next column-0 line.""" + text = (HERE / module).read_text(encoding='utf-8') + code, docs = check.lex(text) + lines = code.split('\n') + doc_end = {end: raw for _, end, raw in docs} + out = {} + for i, line in enumerate(lines): + m = re.match(r'^(?:noncomputable )?(?:def|abbrev|structure)\s+(\S+)', line) + if not m: + continue + j = i + 1 + while j < len(lines) and (not lines[j] or lines[j][0].isspace()): + j += 1 + k = i - 1 + while k >= 0 and k not in doc_end and not lines[k].strip(): + k -= 1 + out[m.group(1)] = (doc_end.get(k), '\n'.join(lines[i:j]).rstrip()) + return out + + def test_shared_declarations_are_byte_identical(self): + """Field.lean is duplicated into Lifetime.lean (26 declarations, docstrings and bodies) and + RN.lean (9 declarations; bodies identical, the `pinM`/`pinS` docstrings differ as the RN header + declares) because the scan forces `import Mathlib` exactly. The copies must not drift.""" + field = self.chunks('Specifications/Field.lean') + lifetime = self.chunks('Specifications/Lifetime.lean') + rn = self.chunks('Specifications/RN.lean') + shared_l = sorted(set(field) & set(lifetime)) + shared_r = sorted(set(field) & set(rn)) + self.assertEqual(len(shared_l), 26, shared_l) + self.assertEqual(len(shared_r), 9, shared_r) + for name in shared_l: + with self.subTest(module='Lifetime', name=name): + self.assertEqual(field[name], lifetime[name]) + for name in shared_r: + with self.subTest(module='RN', name=name): + self.assertEqual(field[name][1], rn[name][1], 'body drifted') + if name in ('pinM', 'pinS'): + self.assertNotEqual(field[name][0], rn[name][0], 'the RN header declares these two docstrings differ') + else: + self.assertEqual(field[name][0], rn[name][0], 'docstring drifted') + + def test_source_check(self): + reg = check.load_json((HERE / 'REGISTER.json').read_text(encoding='utf-8')) + if not reg['claims']: + self.skipTest('REGISTER.json has no claim rows yet; the register compiler fills them, after which this test runs the full source check') + info = check.source_check(HERE, REAL_REPO, allow_dirty=True) + self.assertEqual(info['formalization_status'], 'specified') + self.assertGreater(info['prop_definitions'], 0) + + +if __name__ == '__main__': + unittest.main(verbosity=2)