From 7b0ea1f524f62c6ba68c15f2990cdc817a85c286 Mon Sep 17 00:00:00 2001 From: Dylan Roy <249851642+d6g8k5htny-coder@users.noreply.github.com> Date: Sat, 17 Oct 2026 20:27:23 -0500 Subject: [PATCH] math: derive compact Morse torus trap at literal elder death Conventional source-bound proof and fresh nonauthor AI review for one ordinary H0 interval. New adapter is not formalized; stochastic and all-bars obligations remain open. Delegated AI work for Dylan Roy: OpenAI Codex /root/adapter_feasibility_c292 (author), /root/c292_geometry_review (nonauthor review), /root (coordination and internal checks). Organizational independence 0; scientific effect NONE. --- .../PROOF.md | 268 ++++++ .../README.md | 28 + .../ROOT_ACCEPTANCE.json | 29 + .../SOURCE_IDENTITIES.json | 238 ++++++ .../evidence/ROOT_MATHEMATICAL_CHECK.md | 19 + .../inputs/C291/NEXT_TASK.md | 15 + .../inputs/C291/SourceLifetimeField.lean | 92 ++ .../inputs/C291/SourceLifetimeGeometry.lean | 98 +++ .../C291/evidence/final/NONAUTHOR_REVIEW.md | 87 ++ .../inputs/C291/inputs/Lifetime.lean | 634 ++++++++++++++ .../torus_filtration_20261011/OrdinaryH0.lean | 339 ++++++++ .../SuperlevelBar.lean | 299 +++++++ .../SuperlevelBarField.lean | 205 +++++ .../TorusTopology.lean | 247 ++++++ .../ElderBridge.lean | 83 ++ .../ElderClearance.lean | 129 +++ .../ElderInterval.lean | 392 +++++++++ .../provenance/INTERFACE_BOUNDARIES.md | 43 + .../DEPENDENCIES.json | 502 +++++++++++ .../README.md | 74 ++ .../SOURCE_MAP.json | 454 ++++++++++ .../asymmetric_elder/PROOF.md | 301 +++++++ .../asymmetric_elder/REVIEW.md | 155 ++++ .../PROOF.md | 739 ++++++++++++++++ .../Specifications/Field.lean | 380 +++++++++ .../universality/finite_h0_lifetime/PROOF.md | 526 ++++++++++++ .../QUANTITATIVE_FOLD_ISOLATION_PROPOSAL.md | 293 +++++++ .../fold_structural_universality/PROOF.md | 804 ++++++++++++++++++ .../reviews/NONAUTHOR_REVIEW.md | 124 +++ 29 files changed, 7597 insertions(+) create mode 100644 frontiers/compact_morse_elder_adapter_20261018/PROOF.md create mode 100644 frontiers/compact_morse_elder_adapter_20261018/README.md create mode 100644 frontiers/compact_morse_elder_adapter_20261018/ROOT_ACCEPTANCE.json create mode 100644 frontiers/compact_morse_elder_adapter_20261018/SOURCE_IDENTITIES.json create mode 100644 frontiers/compact_morse_elder_adapter_20261018/evidence/ROOT_MATHEMATICAL_CHECK.md create mode 100644 frontiers/compact_morse_elder_adapter_20261018/inputs/C291/NEXT_TASK.md create mode 100644 frontiers/compact_morse_elder_adapter_20261018/inputs/C291/SourceLifetimeField.lean create mode 100644 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a/frontiers/compact_morse_elder_adapter_20261018/PROOF.md b/frontiers/compact_morse_elder_adapter_20261018/PROOF.md new file mode 100644 index 000000000..bbfa40426 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/PROOF.md @@ -0,0 +1,268 @@ +# Compact-Morse torus geometry at the literal C263 death + +Author composition, 18 October 2026 UTC. OpenAI/Codex `/root/adapter_feasibility_c292`, delegated by the active C292 owner for Dylan Roy. Status: **CONVENTIONAL PROOF DRAFT, PENDING SEPARATE NONAUTHOR REVIEW**. This document extracts and proves the geometric adapter needed by the existing ordinary-H0 interval construction. It does not claim a novel general Morse theorem, a new stochastic admission theorem, or a Lean implementation. The accepted conventional source leaves are reused, not reopened or replaced. Same-provider organizational-independence credit is zero; scientific effect NONE. + +The source authority and exact byte identities are in [SOURCE_IDENTITIES.json](SOURCE_IDENTITIES.json). The parent owns current claim, review, custody and publication; this author writes only this proof, the exact source copies under `inputs/`, and the identity file. The separate elder-module registration study and source #468 ownership are unchanged. + +## 1. What is new in this handoff, and what is already conventional + +The conventional finite-H0 source, §§3–4, already identifies the intrinsic older-point threshold with the finite Morse merger level and proves once-counting through finite component/basis induction. The dimension-independent structural source, §2, states the same deterministic Morse/elder result on compact connected boundaryless smooth manifolds, with explicit classical Morse imports. Its §4 supplies a sufficient vector-ridge elder-transfer argument. The asymmetric source, §§2–6, derives geometric boundary/path inputs for its explicit two-dimensional chart and constructs an actual one-interval split conventionally. The existing Lean `ElderBridge.lean` then proves that arbitrary topological boundary/ceiling/path data instantiate the actual singular-H0 split. + +C292 does not reprove their stochastic results or claim to discover their elder theorem. Its bounded new composition is the explicit choice + +```text +D = closure(path component of M in the strict superlevel at C263 death), +``` + +and a complete derivation of that D's frontier and height bounds and a closed death-level path, directly from the literal smooth-periodic compact-Morse hypotheses. This produces the exact geometric inputs consumed by `elderBoundaryDirectSummand` without importing an elder pairing, a bar count, a splitting conclusion or a path attaining the desired death as an assumption. The argument includes the critical-level path-connectivity step that compact connected limits alone do not supply. + +The current formal gap is still real: no local-Morse chart, finite quotient critical-set theorem, critical-level local path-connectivity theorem, or the new adapter has been implemented or compiled here. This proof is a conventional source for that later work, not a claim that those APIs already exist. + +Principal immutable source routes: + +- [ST: dimension-independent structural source, §§2 and 4](https://github.com/d6g8k5htny-coder/main/blob/df590a438f1252309270ca625c6fe8531ddadc3b/experiments/universality/fold_structural_universality/PROOF.md), 43,927 bytes, SHA-256 `3d952a906553d578876456539a8f0dcd64aedd45099e8394cbe9db99866ec531`. +- [FH: finite-H0 source, §§3–4](https://github.com/d6g8k5htny-coder/main/blob/85b928b326ff0586715a56ae91f51d3fbd89557a/experiments/universality/finite_h0_lifetime/PROOF.md), 27,064 bytes, SHA-256 `5ea91111fa5f3d78d7c4188ebe84b3bc2c452b05dfdfc62c74b03123cb2696d1`. +- [AE: asymmetric source, §§2–6](https://github.com/d6g8k5htny-coder/Math-/blob/c8ec07034608741585111603f1b506393c340908/frontiers/gaussian_short_bar_theory_20261017/asymmetric_elder/PROOF.md), 28,291 bytes, SHA-256 `6d07631c068019bf9f081ad0b6f50325679e3ab554f221265e002d5c047f17ac`; [actual integrated review](https://github.com/d6g8k5htny-coder/Math-/blob/c8ec07034608741585111603f1b506393c340908/frontiers/gaussian_short_bar_theory_20261017/asymmetric_elder/REVIEW.md), 25,687 bytes, SHA-256 `16678abc7de50986b0fc6d54303449bf1b4c5ba8b10f6a3ccc7f9d03a97a4e9f`. +- [Actual geometric formal consumer](https://github.com/d6g8k5htny-coder/Math-/blob/c8ec07034608741585111603f1b506393c340908/frontiers/asymmetric_elder_bridge_20261017/ElderBridge.lean), 3,683 bytes, SHA-256 `6b0be1116f6b357b0ffbda202c572c27c2788b98bee639e228375a5ebeea1fa9`. + +Source wording matters. ST fixes a compact connected smooth Riemannian manifold without boundary of dimension d>=1, smooth real samples, ordinary superlevel H0 over any fixed coefficient field, increasing parameter minus the height, and the exclusion of the essential class. Its Morse facts are classical imports; a source-level proof is not a Lean theorem. AE's quantitative square/ridge hypotheses are specifically two-dimensional and include its embedded chart, half-chart endpoint and strict gap bounds. We do not extend that quantitative chart to higher dimensions. The present proof instead uses the genuine dimension-independent local Morse form. No Gaussian assumption or limit is needed for its deterministic result. + +## 2. Exact inputs, literal transport and external imports + +Fix an integer n>=1, a real L>0, and + +```text +g : EuclideanSpace R (Fin n) -> R, +hsmooth : UniversalLaw.Spec.Lifetime.IsSmooth n g, +hperiodic : UniversalLaw.Spec.Lifetime.IsLPeriodic n L g, +hMorse : UniversalLaw.Spec.Lifetime.IsMorseDistinct n L g, +M : EuclideanSpace R (Fin n), +hmaximum : UniversalLaw.Spec.Lifetime.IsNondegMax n g M, +hfinite : not TorusFiltration.superlevelBar_isEssential g M. +``` + +There is no finite-Fourier cutoff, coefficient law, chosen saddle, assumed elder event or interval among these inputs. The lower dimension bound is retained from the compact-manifold conventional sources. C291's allowance of n=0 does not widen this theorem. + +The unchanged C291 target `Lifetime.lean` and the frozen `Specifications/Field.lean` use the same literal formulas for the following objects, with their own namespaces: + +| Object | Literal formula to transport | +|---|---| +| `E n` | `EuclideanSpace R (Fin n)` | +| `latticeVec n L z` | sum over i of `EuclideanSpace.single i (L * (z i : R))` | +| `IsSmooth n g` | for every finite natural j, `ContDiff R j g` | +| `IsLPeriodic n L g` | `g(x+latticeVec n L z)=g x` for every x,z | +| `hessian n g x v w` | `iteratedFDeriv R 2 g x ![v,w]` | +| `IsCriticalPoint` | `fderiv R g x=0` | +| `HessianHasInertia` | complementary subspaces of the specified dimensions, negative and positive definiteness on the respective subspaces | +| `IsNondegMax` | criticality and inertia n | +| `IsMorseDistinct` | every critical point has some such inertia, together with the stated distinct-positive-torus-distance critical-height clause | + +Thus the Field-to-Lifetime transport used here is pointwise unfolding of these exact formulas, with no modification of the height, period, norm, derivatives or probability measure. This is a conventional identity check, not an executed Lean `rfl`/elaboration receipt. The local `Lifetime.lean` copy has SHA-256 `90f4892608e953ecc696bb32ea159d1c178ceb8a1c6edfd11acfceb14af462c6`; the Field file has SHA-256 `f22c91e082cd7e738ba47c9c3534e47ac687d066f0b78586d15103cf1132535e`. + +Let E be this Euclidean space, X=E/(L Z^n), pi:E->X the literal `torus_proj`, and F:X->R the literal `torus_descend`. Periodicity is transported to `torus_isPeriodic` by the identical lattice-vector formula. Therefore + +```text +F(pi x)=g(x), Mbar=pi M, m=g(M)=F(Mbar). +``` + +The existing `TorusTopology.lean` constructs the compact quotient, its continuous surjective projection and covering map, descent, and strict/closed higher-endpoint path equivalences. We additionally use the standard smooth quotient atlas of a Euclidean space by a discrete translation lattice: sufficiently small projected Euclidean balls are charts, with transition maps translations. In these charts F is literally a translate of g. In particular coordinate criticality and Hessians are exactly those of g; there is no Hessian rescaling. This smooth-atlas construction is an explicit conventional import, not a statement that the topology companion already formalizes it. + +For clarity the proof's remaining classical inputs are: + +1. **Finite-dimensional smooth calculus and inverse function theorem.** The Hessian of a smooth real function is symmetric and is the derivative of its coordinate gradient. A nonsingular Hessian makes a critical point isolated by the inverse function theorem. Negative definite Hessian gives a strict local maximum, for example by Taylor's formula. +2. **The local Morse lemma.** At a critical point with nonsingular Hessian, a smooth local coordinate change makes F-F(q) equal to a sum of positive coordinate squares minus a sum of negative coordinate squares. This is the actual local theorem for a smooth function, not a supplied finite-event or elder certificate. +3. **Elementary compact topology.** Continuous functions attain extrema on nonempty compact sets; connected components are closed; nested nonempty compact connected sets in a compact Hausdorff space have connected nonempty intersection; an infinite subset of a compact metric space has an accumulation point. Open subsets of manifolds are locally path connected, and in a locally path-connected space connected components are path connected and open. + +These are the explicit geometric/topological imports. The first two are the classical local-Morse/finite-dimensional inputs already admitted in scope by FH/ST. No new primary-book audit is claimed. No critical-free flow or handle-attachment theorem is needed by the argument below; their earlier use in FH/ST is not turned into a premise here. We do not assume a barcode theorem, an elder basis, critical-level attainment, an H0 comparison field or a component-counting certificate. + +Only the nondegeneracy conjunct of `IsMorseDistinct` is needed for this one-summand geometric theorem. The full source hypothesis is retained. In particular no unproved equivalence between the literal `torusDist` formula and quotient inequality is needed in this proof, and no uniqueness of a saddle is concluded from that unused second conjunct. + +## 3. The theorem and exact acceptance boundary + +Define the *literal existing* C263 set and number + +```text +Clear(M) = {a : there is y with g(y)>m and a path M->y + all of whose heights are strictly greater than a}, +delta = superlevelBar_deathHeight g M. +``` + +C263 uses a real supremum of the nonempty clearance set and a default zero if that set is empty. Its proved nonemptiness theorem applies here, so the empty default will not occur. + +Set + +```text +U = {x in X : F(x)>delta}, +C = {q in X : JoinedIn U Mbar q}, +D = closure_X C. +``` + +**Theorem (compact-Morse boundary data at literal death).** Under exactly the inputs of §2, + +```text +delta < m, +D is closed, Mbar is in D, +F(q) <= m for all q in D, +F(q) <= delta for all q in frontier_X D, +there exist Q in X and a continuous path gamma:Mbar->Q such that + F(Q)>m and F(gamma(t))>=delta for all t in [0,1]. +``` + +The endpoint Q can be chosen to be a nondegenerate local maximum. The path attains height delta somewhere. Consequently these data instantiate `elderTrap_of_boundary` and, over any fixed coefficient field k, the existing `elderBoundaryDirectSummand` gives one actual ordinary singular-H0 interval direct summand on [-m,-delta). + +The conclusion provides one split for this supplied maximum. It does not construct a simultaneous split for all maxima, identify a uniquely labeled quotient saddle, or prove an exhaustive once-counted barcode. No existence of D, Q or gamma, no relation delta=F(S), and no target H0 equality is an input. + +## 4. Genuine Morse geometry follows from the literal hypotheses + +Because g is smooth, its Hessian bilinear form B is symmetric. At a critical point the inertia predicate supplies complementary negative and positive subspaces V,W with E=V+W. This does imply nondegeneracy, even though no cross-term orthogonality was included in the predicate. If z=v+w lies in the radical of B, then + +```text +B(v,v)=-B(w,v)=-B(v,w)=B(w,w). +``` + +Strict negative and strict positive definiteness force v=w=0. Thus z=0. This verifies the nonsingularity needed by the classical Morse lemma; it is not assumed through a different Morse interface. At M the negative subspace has dimension n and hence equals E, so the Hessian is negative definite. + +The quotient charts described in §2 transport these facts to F. Its critical set is closed because the derivative is continuous in local charts. Every critical point is isolated by the inverse function theorem applied to the coordinate gradient. A closed discrete critical set in compact metric X is finite: otherwise an accumulating subsequence would limit to another critical point, contrary to that point's isolation. Denote this finite set by Crit(F). + +For every real a, the closed superlevel A_a={F>=a} is compact and locally path connected. Here is the critical-level detail. At a point with F>a a small neighborhood lies inside A_a. At a point with F=a and nonzero derivative, the submersion coordinate form identifies a local piece with a closed half-space. At a critical point with F=a, the Morse lemma identifies it near the origin with + +```text +{(u,v) : ||u||^2-||v||^2 >= 0}. +``` + +Its intersection with any centered sufficiently small ball is star shaped about the origin, hence path connected. This supplies arbitrarily small relatively open path-connected neighborhoods at the cone vertex. Away from the vertex its boundary is regular, so the preceding half-space argument applies. At a local maximum the cone may be the single point, which is also path connected. Thus all cases give local path connectivity of A_a. It follows that every connected component of A_a is path connected. + +This uses the actual Morse quadratic cone, not the invalid general assertion that compact connected sets are path connected. + +## 5. Strict death drop below the nondegenerate maximum + +By continuity and hfinite, C263 proves Clear(M) nonempty, bounded above, and + +```text +a < delta if and only if a is in Clear(M). (1) +``` + +Use the negative definite Hessian at the Euclidean representative M. Taylor's formula provides a radius rho>0 such that g(x)=1. Let c0 be the maximum of g on that sphere. Then c0m lies outside that ball. Every continuous path from M to y meets its sphere by the intermediate value theorem applied to ||gamma(t)-M||. Hence no such path stays strictly above c0. More generally every clearance member a satisfies aa} to some point y with F(y)>m. + (3) +``` + +At a=delta this strict quotient connection is impossible. + +## 6. Fixing an older maximum across levels approaching death + +For any aa}. By (2), Mbar belongs to O_a; by (3), U_a contains some y with F(y)>m. Since O_a is an open subset of the manifold X, it is locally path connected. Its path components are therefore both open and closed relative to O_a. In particular U_a is open in X and + +```text +closure_X(U_a) intersect O_a = U_a. (4) +``` + +The nonempty compact closure of U_a has a maximizer Q_a of F. Its height is at least F(y)>m>a. Equation (4) places Q_a back in U_a. Since U_a is open in X, Q_a is a local maximum of F on X, not just a constrained maximum of a closed set. It is critical, and the Morse condition makes its Hessian negative definite. Thus it is one of the finitely many members of + +```text +Older = {Q in Crit(F) : Q is a local maximum and F(Q)>m}. (5) +``` + +Moreover Mbar and Q_a have an actual path inside O_a, by the defining property of U_a. We have proved nonemptiness of Older and this path statement, rather than assuming that a higher endpoint was already a higher critical maximum. + +Choose a_j=delta-1/(j+1), j=0,1,... . For each j choose Q_j as above. Since Older is finite, some fixed Q in Older occurs for infinitely many j. Enumerate those indices as a strictly increasing sequence j_l. It is unbounded, so a_(j_l) increases to delta. Write b_l=a_(j_l). For every l there is a path from Mbar to this one Q in {F>b_l}. + +The endpoint Q is not prescribed in advance and need not be the global maximum. This finite-subsequence step is what prevents a sequence of merely higher endpoints from approaching height m and losing the strict inequality needed by the interval constructor. + +## 7. Closed death-level attainment + +Let K_l be the connected component of Mbar in the compact closed superlevel A_(b_l)={F>=b_l}. Each K_l is closed in A_(b_l), hence compact in X, and is connected. The path just constructed places Q in K_l. + +Since b_(l+1)>=b_l, K_(l+1) is a connected subset of A_(b_l) containing Mbar. By maximality of the connected component, K_(l+1) is contained in K_l. The nested compact connected intersection + +```text +K = intersection over l of K_l +``` + +is nonempty and connected; it contains both Mbar and Q. Every x in K satisfies F(x)>=b_l for all l, hence F(x)>=delta. Thus K is a connected subset of A_delta containing both endpoints. They belong to the same connected component of A_delta. + +Section 4 proves that A_delta is locally path connected even if delta is a critical value. Its connected components are therefore path connected. There exists an actual continuous path gamma:Mbar->Q with F(gamma(t))>=delta for all t. Since Q belongs to Older, F(Q)>m. + +This is the required attained closed path. The proof did not take a limit in a space of paths and did not infer path connectivity directly from connectedness of K. + +For completeness, its minimum is exactly delta. The compact parameter interval and continuity give an attained minimum. If the minimum were strictly greater than delta, the same quotient path would be a strict higher-endpoint path at delta, contradicting (3). Thus some parameter has F(gamma(t))=delta. Closed attainment is consistent with the exclusion of strict clearance at delta. + +## 8. Constructing the exact closed domain and frontier bound + +Recall U={F>delta}, C the path component of Mbar in U, and D=closure_X C. By (2), Mbar is in C. Local path connectivity of U makes C open in X and closed relative to U. Therefore + +```text +D intersect U = C, C subset interior_X D. (6) +``` + +For q in C, if F(q)>m then its defining path from Mbar inside U would be a strict higher-endpoint path at delta, contradicting (3). Hence F<=m throughout C. The set {F<=m} is closed, so + +```text +F<=m throughout D. (7) +``` + +The domain D is closed by construction and contains Mbar. If q is in frontier_X D and F(q)>delta, then q lies in D intersect U=C by (6), and consequently in the interior of D. This contradicts q being on the frontier of the closed set D. Thus + +```text +F(q)<=delta for every q in frontier_X D. (8) +``` + +The older maximum Q lies outside D by (7) and F(Q)>m. Equations (2), (7), (8), F(Mbar)=m, Mbar in D and the actual path of §7 are exactly the geometric hypotheses of `elderTrap_of_boundary`. No desired separation, elder relation, death equality or H0 equation was supplied among the theorem's inputs. + +Unlike the special asymmetric chart, this D can contain multiple younger branches. Its active intersection need not be a single path component. That does not invalidate the boundary constructor: its hypotheses require clopen separation and a height ceiling, not a unique interior component. + +## 9. Consumption by the existing ordinary-H0 constructor + +Fix any field k. Its Field structure is nonzero; characteristic two is allowed. Apply the unchanged `elderTrap_of_boundary` with + +```text +X, F, D, m, delta, Mbar, Q, +IsClosed D, frontier bound (8), delta0. The carrier remains `SmoothSample n L`, the measure remains `smoothLaw n L`, and the spatial field remains the literal `smoothSourceField` through the canonical Euclidean coordinate equivalence. Nothing in this proof changes the covariance, normalization, Gaussian law or field. + +However C291's `actualGaussianFieldInterface` has no Morse-genericity field. The predicate hMorse on this omega remains an explicit additional deterministic hypothesis. This proof establishes neither its probability-one validity under the literal law nor a measurable choice of D, Q, gamma or the splitting. There is no stochastic conclusion without that further work. + +The dimension-independent result here derives from compact Morse geometry. It does not transplant AE's quantitative two-dimensional chart, the finite full-square Gaussian source, or its factorial estimates to arbitrary n. The local sphere proof and Morse cone argument also cover n=1; there are no transverse variables to invent in that case. n=0 remains outside the stated scope. + +No limit is taken in (9). The auxiliary sequence of levels b_l approaches the fixed deterministic death of one fixed function; it is not a random-field approximation, a small-lifetime limit, a thermodynamic limit, or a cutoff limit. The existing fixed-law small-lifetime theorems, fixed-target conditional couplings, and varying-L/cutoff process families keep their respective parameter orders and full normalizers. + +We conclude exactly one actual interval summand for each supplied nonessential maximum on the source Morse locus. We do not conclude a coherent simultaneous decomposition, an endpoint-preserving bijection, exhaustive once-counted bars, a unique quotient saddle partner, or identification with a pre-assumed Field/Lifetime elder relation. Such further work must preserve quotient points versus multiple periodic lifts and elder death versus branch adjacency. The domain and endpoint constructed here can depend on M; no measurable global family of choices is asserted. + +The result does not prove that all small bars are local regular folds, an all-small-bars converse, singular-mark or remote complements, field-first stochastic transfer, regional shrinking-window multiple-witness collision estimates, a quantitative remainder, or universality. Global factorial bounds do not supply regional collision estimates. Source #468, the elder-registration study, the numerical owners and their explicit stops retain their scopes. No scientific or institutional alignment status changes. + +## 11. Evidence scope + +Exact source bodies already fetched and verified during read-only reconnaissance were copied without network rereads to `inputs/`. The identity file binds repository, immutable cut, source path, Git blob, byte length and SHA-256. C291 inputs were copied from the unchanged publication package after matching their previously verified hashes. The source map's historical PENDING/HOLD labels remain unedited history. + +This document is an author proof composition awaiting substantive nonauthor review. No simulated numerical test can certify its Morse/topological steps, and none is presented as such. The isolated ball, fixed-older-maximum subsequence and critical-level cone arguments are the mathematical controls against the precise failure modes identified in §§5–7. Source review is not human or provider-distinct acceptance, compilation, kernel checking, scientific promotion, or a transfer of another owner's scope. diff --git a/frontiers/compact_morse_elder_adapter_20261018/README.md b/frontiers/compact_morse_elder_adapter_20261018/README.md new file mode 100644 index 000000000..1ddbfcfed --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/README.md @@ -0,0 +1,28 @@ +# Compact Morse torus geometry for one ordinary H0 interval + +This packet derives the geometric premises of the existing ordinary-H0 interval construction for a supplied nonessential nondegenerate maximum of a fixed smooth periodic Morse function. It constructs the closed domain as the closure of the maximum's strict-superlevel path component at the literal C263 death, proves its height and frontier bounds, and obtains an actual path to a strictly older maximum at that closed death level. + +The conclusion is one interval direct summand on `[-g(M), -death(M))`, over any fixed coefficient field, for each supplied maximum. The proof uses finite critical-point geometry to retain one older endpoint while approaching death; compact connected intersections and the local Morse form then establish closed-level path attainment. It consumes the existing `elderBoundaryDirectSummand` instead of changing or reimplementing it. + +## Evidence and status + +- [Complete frozen author proof](PROOF.md): conventional mathematical composition with explicitly named classical imports. Its draft frontmatter records its author-time status; the separately bound review and root acceptance determine the later delivery status. +- [Fresh nonauthor mathematical review](reviews/NONAUTHOR_REVIEW.md): substantive reconstruction and failure-mode analysis of the exact proof, with source exposure disclosed. +- [Root acceptance](ROOT_ACCEPTANCE.json): exact proof/review bindings and retained limits. +- [Exact source identities](SOURCE_IDENTITIES.json): 23 immutable input bodies, with repository, cut, path, length, SHA-256 and Git blob identity. `inputs/` contains those unchanged public source bytes, including the literal Field/Lifetime definitions and the formal consumer. + +Actual proof author: OpenAI Codex `/root/adapter_feasibility_c292`. Actual nonauthor reviewer: OpenAI Codex `/root/c292_geometry_review`. Coordinator/internal mathematical check and custody: OpenAI Codex `/root`, chat `01a0bbb5-2fcb-77f0-b78b-4d220ddd7ab2`. All work is delegated AI work for Dylan Roy. Organizational independence 0; human review 0; scientific effect NONE. Neither source identity nor agreement proves the result by itself. + +## Exact scope + +Inputs retain `n >= 1`, fixed `L > 0`, literal smoothness, periodicity, `IsMorseDistinct`, a supplied `IsNondegMax` and the literal nonessentiality predicate. No death equality, selected saddle, elder relation, attained closed path, bar count or interval decomposition is an input. The local Morse lemma, inverse function theorem, smooth lattice-quotient atlas and elementary compact topology are explicit conventional imports. No novelty or new audit of those classical theorems is claimed. + +This packet contains no new Lean implementation or compiler/kernel execution. The copied formal modules retain only their existing historical evidence. Hosted repository checks, if observed on this PR, verify their own registered targets and repository contracts; they do not formalize this new conventional proof. + +C291 supplies the literal smooth periodic Gaussian field interface. Applying the result almost surely to its `smoothLaw` still requires Morse admission for that exact law. Simultaneous coherent splitting, exhaustive once-counting, a unique saddle partner, measurability of choices, all-small-bars converse, regional shrinking-collision estimates and finite-window/sampler/count/confidence transfer remain separate. The auxiliary level sequence approaches one fixed deterministic death; it is not a field-law, small-lifetime, period or cutoff limit. + +## Next useful task + +Under a fresh claim and current owner check, formalize the first missing deterministic geometric prerequisites on the existing quotient: transport the literal complementary-sign Hessian predicate to a genuine Morse local form, derive finite critical points and local path connectivity of closed superlevels, then implement the source-bound trap construction in this proof. Keep companion, component-kernel and hosted-registration owners intact. A record assuming finite Morse events or the target closed path would move the gap into a premise and does not complete this task. + +No merge, Release, permission change, source deletion, scientific-status update or transfer of another owner's claim is part of this delivery. The broader C261 program and project-wide audit remain active and incomplete. diff --git a/frontiers/compact_morse_elder_adapter_20261018/ROOT_ACCEPTANCE.json b/frontiers/compact_morse_elder_adapter_20261018/ROOT_ACCEPTANCE.json new file mode 100644 index 000000000..3139c6102 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/ROOT_ACCEPTANCE.json @@ -0,0 +1,29 @@ +{ + "accepted_boundary": "Fixed n>=1,L>0 smooth periodic literal MorseDistinct nonessential maximum -> closed trap and older closed path at literal death -> one actual ordinary H0 interval", + "accepted_utc": "2026-10-18T01:26:25.819299+00:00", + "author": "OpenAI Codex /root/adapter_feasibility_c292", + "exposure": "Fresh nonauthor; task/design and exact source exposure; actual full proof reconstruction; root internal check is not another peer vote", + "human_review_credit": 0, + "new_formal_execution": "NOT_IMPLEMENTED_NOT_RUN", + "nonauthor_reviewer": "OpenAI Codex /root/c292_geometry_review", + "organizational_independence_credit": 0, + "proof_bytes": 25662, + "proof_sha256": "39f92734da35aa4c3ad97ea9d8e194cea4ee3a0b23054a8a412f331294d74762", + "retained_obligations": [ + "formal local Morse quotient geometry and adapter", + "almost-sure Morse under literal smoothLaw", + "measurable choices", + "simultaneous exhaustive barcode and once-counting", + "unique saddle pairing", + "all-small-bars converse", + "regional shrinking collisions", + "finite-window sampler count and confidence transfer" + ], + "review_bytes": 16983, + "review_sha256": "ab8ab0084449e676f0466510e31f8e0a84b36616d1d4f7488f470c47a30a4ede", + "root_internal_check": "evidence/ROOT_MATHEMATICAL_CHECK.md", + "scientific_effect": "NONE", + "source_count": 23, + "source_manifest_sha256": "8b0fc8892a1936ad6a37c49b45e5fcf987154068fb7107e07b0b48d55c9f5d44", + "verdict": "ACCEPT_SCOPED_CONVENTIONAL" +} diff --git a/frontiers/compact_morse_elder_adapter_20261018/SOURCE_IDENTITIES.json b/frontiers/compact_morse_elder_adapter_20261018/SOURCE_IDENTITIES.json new file mode 100644 index 000000000..b105e4972 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/SOURCE_IDENTITIES.json @@ -0,0 +1,238 @@ +{ + "schema": "c292-source-identities-v1", + "role": "SOURCE_BOUND_CONVENTIONAL_COMPOSITION_NOT_NEW_SOURCE_REVIEW", + "created_utc": "2026-10-18T01:17:40.267291+00:00", + "sources": [ + { + "repo": "d6g8k5htny-coder/Math-", + "cut": "c8ec07034608741585111603f1b506393c340908", + "path": "frontiers/asymmetric_elder_bridge_20261017/provenance/INTERFACE_BOUNDARIES.md", + "blob": "c1f0ae9cb4b7da1bdba68bf1d678b3c39ff476e2", + "local": "inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/provenance/INTERFACE_BOUNDARIES.md", + "bytes": 5080, + "sha256": "7ec0f6f8a7a5f58e2c233d0aae9f69f18f13a2a0e340259c6931504ae6843cf9", + "source_method": "Previously fetched immutable connector body, previously raw-verified where listed; 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This is an internal author-side reconstruction, not an additional nonauthor vote. Proof checked in full: 25,662 bytes, SHA-256 `39f92734da35aa4c3ad97ea9d8e194cea4ee3a0b23054a8a412f331294d74762`. + +I checked the literal `SuperlevelBar.lean` definitions: essentiality is global maximality, clearance quantifies actual Euclidean paths with strict height inequalities, and death is the conditional real supremum with an empty-set default. Nonessentiality and continuity activate the existing nonempty strict-cut theorem; the proof does not interpret the empty default as a finite bar. + +The complementary-sign Hessian argument is exact: if z=v+w lies in the radical, symmetry yields B(v,v)=-B(v,w)=B(w,w). Negative and positive definiteness on the complementary subspaces force v=w=0, even with nonzero cross terms. At a maximum the negative subspace has full dimension. This justifies the classical local Morse and inverse-function imports without adding a nonsingular-Hessian premise. + +The strict drop uses the first exit from an ordinary Euclidean ball around M, so no quotient-chart injectivity is silently required for that step. The compact sphere is nonempty because n>=1. A path to a higher endpoint must exit; the uniform sphere ceiling bounds every member of the clearance set and hence the nonempty supremum. + +The endpoint step maximizes F over the compact closure of an open path component of {F>a}. The maximum is strictly above a, so relative closedness returns it to the original open component. It is an unconstrained local maximum. Finiteness of the quotient critical set then supplies one fixed higher maximum along a cofinal subsequence; no limit of merely higher endpoint heights is used. + +The nested sets are closed connected components of closed superlevels, not a purported compact space of paths. Their intersection gives only connectedness. The Morse quadratic cone supplies local path connectivity of the ambient closed superlevel, whose connected component then supplies an actual path. This is the missing justification that prevents an invalid connected-implies-path-connected inference. + +For C the strict-death path component and D its closure, D intersect {F>death}=C and C is open. A frontier point of D above death would lie in C and thus in the interior of D. A point of C above the birth height would contradict the strict cut at death. Continuity extends the birth ceiling to D. The older endpoint is outside D by its strictly larger value. These match every input of the unchanged `elderTrap_of_boundary` and `elderBoundaryDirectSummand` read in full. + +Boundary controls are mathematical, not numerical experiments: (1) the essential maximum is excluded before the empty-default death could be used; (2) strict clearance at death is impossible while closed attainment is allowed; (3) connectedness is never promoted to path connectivity without local Morse geometry; (4) the older endpoint is fixed before compact intersection; (5) the region may contain younger branches, so the theorem does not replace elder death by first adjacency; (6) a split for each supplied M is not a simultaneous exhaustive barcode. The review must independently assess these points. + +I found no must-fix mathematical defect in this frozen draft. This internal conclusion is subject to the separate actual nonauthor review. The proof remains conventional and deterministic; formalization, exact-law almost-sure Morse admission, all-bars/counting and regional collision claims are absent. diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/NEXT_TASK.md b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/NEXT_TASK.md new file mode 100644 index 000000000..36bd4d61f --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/NEXT_TASK.md @@ -0,0 +1,15 @@ +# Next mathematical join — offered, not claimed + +The new constructor supplies the literal smooth Gaussian field and its exact law/covariance interface. A useful next formal unit is a source-bound adapter from the existing compact-Morse/torus geometric hypotheses to the existing ordinary-H0 interval construction. Check current source owners and claims before pickup. This offer does not take over the separately owned elder-module registration study, reopen accepted conventional proofs, or assert that the adapter is already proved. + +The boundary-map source cut is Math main `c8ec07034608741585111603f1b506393c340908`, where source packet #475 is merged. Its [interface-boundary map](https://github.com/d6g8k5htny-coder/Math-/blob/c8ec07034608741585111603f1b506393c340908/frontiers/asymmetric_elder_bridge_20261017/provenance/INTERFACE_BOUNDARIES.md) (Git blob `c1f0ae9cb4b7da1bdba68bf1d678b3c39ff476e2`) distinguishes the proved geometric adapter from remaining analytic, compact-Morse, torus and all-bar joins. The map's original no-branch/no-PR wording is historical; current merged source availability is a separate observation, not new execution or alignment credit. + +In that source slice, `elderBoundaryDirectSummand` constructs one interval direct summand from explicit closed-domain/frontier-height/inside-ceiling/actual-higher-endpoint-path data. `elderBoundary_death_eq` relates the same Euclidean geometry to the literal C263 death. These are useful proved targets. They do not provide an exhaustive simultaneous barcode decomposition or establish once-counted bar multiplicities merely from one witness. + +An eligible next author should first freeze the exact conventional chart and compact-Morse source statements already carried by [the Gaussian source packet](https://github.com/d6g8k5htny-coder/Math-/blob/c8ec07034608741585111603f1b506393c340908/frontiers/gaussian_short_bar_theory_20261017/README.md) and identify the minimum missing formal adapter. Retain their dimension, compactness, distinct-critical-value, coefficient-field, period and source-law hypotheses literally; the present constructor's inclusion of dimension zero does not extend downstream theorems to that dimension. The adapter must derive geometric inputs from those source hypotheses rather than assume a desired death value, interval split, bar count or pairing conclusion. + +The finite acceptance boundary should state exactly whether it yields one actual ordinary-H0 summand, an endpoint-preserving bijection, or an exhaustive once-counted barcode statement. Keep elder pairing distinct from branch adjacency and quotient representatives distinct from periodic lifts. A subsequent all-small-bars converse, stochastic field-first sampling transfer or shrinking-region multiple-witness collision estimate needs its own hypotheses and evidence. Global factorial bounds do not supply the regional collision estimate. Parameter/limit order and full normalizers remain source-bound. + +No new review, dispatch, lease, institutional alignment or scientific promotion is created by this routing note. Existing source/integration owners and explicit stops remain effective; source-exposed same-provider AI work earns organizational-independence credit zero. + +The bounded architecture read through 2026-10-18 00:39:28 UTC records a pause on additional toy-control work after current integrations and directs future finite-window/count comparisons through existing Math193 owners. The actual NUM owner has conditionally adopted the qualitative ideal-law identification while retaining the finite-window, sampler, count-transfer and confidence obligations; its certified full-spectrum moment kernel is separately claimed and must not be duplicated. The proposed original-observable effective remainder is input only, with no proved remainder, assigned adapter or quantitative acceptance. See [owner adoption and retained limits](https://github.com/d6g8k5htny-coder/main/issues/275#issuecomment-6243599816) and [current owner/implementation boundaries](https://github.com/d6g8k5htny-coder/main/issues/275#issuecomment-6243819291). This proposed formal geometric join should serve that same original observable and preserve those owners and stops; it is not an additional toy experiment or a transfer of their scope. diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/SourceLifetimeField.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/SourceLifetimeField.lean new file mode 100644 index 000000000..a07036a15 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/SourceLifetimeField.lean @@ -0,0 +1,92 @@ +import SourceLifetimeGeometry +import SourceThetaBridge + +/-! The actual smooth source fills the unchanged Lifetime Gaussian field interface. +The only final parameter hypothesis is `0 < L`; theta, Gaussian, covariance and +smoothness properties are proved from the frozen literal source. +-/ + +noncomputable section +open MeasureTheory ProbabilityTheory +open UniversalLaw.Spec.Lifetime I4SmoothCarrier I4ThetaBridge +open scoped BigOperators ENNReal NNReal + +namespace I4LifetimeField + +theorem actualField_covariance (d : ℕ) {L : ℝ} (hL : 0 < L) + (x y : E d) : + cov[fun ω => actualField d L ω x, fun ω => actualField d L ω y; smoothLaw d L] = + periodicCovariance d L (x - y) := by + rw [actualField_covariance_spectral d hL] + exact sourceSpectralKernel_eq_periodicCovariance d hL (x - y) + +theorem actualField_raw_second_moment (d : ℕ) {L : ℝ} (hL : 0 < L) + (x y : E d) : + ∫ ω, actualField d L ω x * actualField d L ω y ∂smoothLaw d L = + periodicCovariance d L (x - y) := by + simpa only [ProbabilityTheory.covariance, actualField_integral_zero d hL, sub_zero] + using actualField_covariance d hL x y + +theorem actualField_linearCombination_variance (d : ℕ) {L : ℝ} (hL : 0 < L) + {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ) : + Var[fun ω => ∑ i, a i * actualField d L ω (x i); smoothLaw d L] = + covQuadForm d L x a := by + let : IsProbabilityMeasure (smoothLaw d L) := smoothLaw_isProbabilityMeasure d hL.ne' + rw [variance_fun_sum (fun i => + (actualField_eval_hasGaussianLaw d hL (x i)).memLp_two.const_mul (a i))] + simp only [covariance_const_mul_left, covariance_const_mul_right, + actualField_covariance d hL, covQuadForm] + apply Finset.sum_congr rfl + intro i _ + apply Finset.sum_congr rfl + intro j _ + ring + +theorem covQuadForm_nonneg (d : ℕ) {L : ℝ} (hL : 0 < L) + {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ) : + 0 ≤ covQuadForm d L x a := by + rw [← actualField_linearCombination_variance d hL x a] + exact variance_nonneg _ _ + +theorem actualField_linearCombination_map (d : ℕ) {L : ℝ} (hL : 0 < L) + {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ) : + (smoothLaw d L).map (fun ω => ∑ i, a i * actualField d L ω (x i)) = + gaussianReal 0 ⟨covQuadForm d L x a, covQuadForm_nonneg d hL x a⟩ := by + rw [(actualField_linearCombination_hasGaussianLaw d hL x a).map_eq_gaussianReal, + actualField_linearCombination_integral_zero d hL, + actualField_linearCombination_variance d hL] + congr 1 + apply Subtype.ext + exact Real.coe_toNNReal _ (covQuadForm_nonneg d hL x a) + +/-- The original target on the unchanged coefficient carrier and preserved source law. -/ +def actualGaussianFieldInterface (d : ℕ) (L : ℝ) (hL : 0 < L) : + GaussianFieldInterface d L where + Ω := SmoothSample d L + mΩ := inferInstance + P := smoothLaw d L + isProb := smoothLaw_isProbabilityMeasure d hL.ne' + F := actualField d L + smooth := actualField_smooth d hL + periodic := actualField_periodic d hL + summable := theta_summable d hL + quad_nonneg := covQuadForm_nonneg d hL + gaussian := actualField_linearCombination_map d hL + covariance := actualField_raw_second_moment d hL + +theorem actualGaussianFieldInterface_exists (d : ℕ) (L : ℝ) (hL : 0 < L) : + Nonempty (GaussianFieldInterface d L) := + ⟨actualGaussianFieldInterface d L hL⟩ + +@[simp] theorem actualGaussianFieldInterface_carrier (d : ℕ) (L : ℝ) (hL : 0 < L) : + (actualGaussianFieldInterface d L hL).Ω = SmoothSample d L := rfl + +@[simp] theorem actualGaussianFieldInterface_law (d : ℕ) (L : ℝ) (hL : 0 < L) : + (actualGaussianFieldInterface d L hL).P = smoothLaw d L := rfl + +@[simp] theorem actualGaussianFieldInterface_field (d : ℕ) (L : ℝ) (hL : 0 < L) + (ω : SmoothSample d L) (x : E d) : + (actualGaussianFieldInterface d L hL).F ω x = + smoothSourceField d L ω (coordinates d x) := rfl + +end I4LifetimeField diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/SourceLifetimeGeometry.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/SourceLifetimeGeometry.lean new file mode 100644 index 000000000..e83eb8e13 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/SourceLifetimeGeometry.lean @@ -0,0 +1,98 @@ +import Lifetime +import SourceSmoothCarrier + +/-! Coordinate and probability transport for the literal smooth Fourier source. +No theta identity or target field-law premise is introduced in this module. +-/ + +noncomputable section +open MeasureTheory ProbabilityTheory +open UniversalLaw.Spec.Lifetime I4SmoothCarrier +open scoped BigOperators ENNReal NNReal + +namespace I4LifetimeField + +def coordinates (d : ℕ) : E d ≃L[ℝ] (Fin d → ℝ) := + EuclideanSpace.equiv (Fin d) ℝ + +@[simp] theorem coordinates_apply (d : ℕ) (x : E d) (i : Fin d) : + coordinates d x i = x i := rfl + +theorem coordinates_latticeVec (d : ℕ) (L : ℝ) (n : Fin d → ℤ) : + coordinates d (latticeVec d L n) = fun i => L * (n i : ℝ) := by + ext i + simp [coordinates, latticeVec] + +/-- The same carrier, same law, and same literal source; only spatial coordinates change. -/ +def actualField (d : ℕ) (L : ℝ) (ω : SmoothSample d L) (x : E d) : ℝ := + smoothSourceField d L ω (coordinates d x) + +theorem actualField_smooth (d : ℕ) {L : ℝ} (hL : 0 < L) + (ω : SmoothSample d L) : IsSmooth d (actualField d L ω) := by + intro n + exact (smoothSourceField_contDiff d hL.ne' ω n).comp + (coordinates d).contDiff + +theorem actualField_periodic (d : ℕ) {L : ℝ} (hL : 0 < L) + (ω : SmoothSample d L) : IsLPeriodic d L (actualField d L ω) := by + intro x n + unfold actualField smoothSourceField + rw [map_add, coordinates_latticeVec] + exact I4Lattice.sourceField_lattice_shift d hL.ne' (sampleInclusion d L ω) n + (coordinates d x) + +theorem actualField_eval_measurable (d : ℕ) (L : ℝ) (x : E d) : + Measurable (fun ω => actualField d L ω x) := + (I4Source.sourceField_eval_measurable d L (coordinates d x)).comp + (sampleInclusion_measurable d L) + +theorem actualField_eval_hasGaussianLaw (d : ℕ) {L : ℝ} (hL : 0 < L) + (x : E d) : HasGaussianLaw (fun ω => actualField d L ω x) (smoothLaw d L) := by + exact (smoothSourceField_finiteVector_hasGaussianLaw d hL.ne' 1 + (fun _ => coordinates d x)).eval 0 + +theorem actualField_linearCombination_measurable (d : ℕ) (L : ℝ) {m : ℕ} + (x : Fin m → E d) (a : Fin m → ℝ) : + Measurable (fun ω => ∑ i, a i * actualField d L ω (x i)) := by + exact Finset.measurable_sum _ (fun i _ => + measurable_const.mul (actualField_eval_measurable d L (x i))) + +theorem actualField_linearCombination_hasGaussianLaw (d : ℕ) {L : ℝ} + (hL : 0 < L) {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ) : + HasGaussianLaw (fun ω => ∑ i, a i * actualField d L ω (x i)) (smoothLaw d L) := by + have hX : Measurable (fun η : I4Source.SourceMode d → ℝ => + ∑ i, a i * I4Source.sourceField d L η (coordinates d (x i))) := + Finset.measurable_sum _ (fun i _ => measurable_const.mul + (I4Source.sourceField_eval_measurable d L (coordinates d (x i)))) + refine ⟨(actualField_linearCombination_measurable d L x a).aemeasurable, ?_⟩ + change IsGaussian ((smoothLaw d L).map (fun ω => + ∑ i, a i * I4Source.sourceField d L (sampleInclusion d L ω) (coordinates d (x i)))) + rw [observation_map_eq d hL.ne' _ hX] + exact (I4SourceGaussian.source_linearCombination_hasGaussianLaw d hL.ne' m + (fun i => coordinates d (x i)) a).isGaussian_map + +theorem actualField_integral_zero (d : ℕ) {L : ℝ} (hL : 0 < L) (x : E d) : + ∫ ω, actualField d L ω x ∂smoothLaw d L = 0 := by + change (∫ ω, I4Source.sourceField d L (sampleInclusion d L ω) (coordinates d x) + ∂smoothLaw d L) = 0 + rw [(sampleInclusion_measurePreserving d hL.ne').integral_comp + (MeasurableEmbedding.subtype_coe (goodSet_measurable d L)) + (fun η => I4Source.sourceField d L η (coordinates d x))] + exact I4SourceCovariance.sourceField_integral_zero d hL.ne' (coordinates d x) + +theorem actualField_linearCombination_integral_zero (d : ℕ) {L : ℝ} + (hL : 0 < L) {m : ℕ} (x : Fin m → E d) (a : Fin m → ℝ) : + ∫ ω, (∑ i, a i * actualField d L ω (x i)) ∂smoothLaw d L = 0 := by + rw [integral_finsetSum _ (fun i _ => + (actualField_eval_hasGaussianLaw d hL (x i)).integrable.const_mul (a i))] + simp only [integral_const_mul, actualField_integral_zero d hL, mul_zero, + Finset.sum_const_zero] + +theorem actualField_covariance_spectral (d : ℕ) {L : ℝ} (hL : 0 < L) + (x y : E d) : + cov[fun ω => actualField d L ω x, fun ω => actualField d L ω y; smoothLaw d L] = + I4Kernel.sourceSpectralKernel d L (coordinates d (x - y)) := by + simpa only [actualField, map_sub] using + smoothSourceField_covariance d hL.ne' (coordinates d x) (coordinates d y) + +end I4LifetimeField diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/evidence/final/NONAUTHOR_REVIEW.md b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/evidence/final/NONAUTHOR_REVIEW.md new file mode 100644 index 000000000..e918bfdc1 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/evidence/final/NONAUTHOR_REVIEW.md @@ -0,0 +1,87 @@ +**C291 final nonauthor source/result review — ACCEPT_SCOPED_FORMAL_CONSTRUCTOR** + +Reviewed at 2026-10-18T00:42Z by **OpenAI Codex `/root/c291_nonauthor_review`**, an actual source-exposed nonauthor AI reviewer delegated within Dylan Roy's project. I did not author or edit the production Lean sources and did not execute Lean. I reviewed source, exact identities, actual receipts and full relevant logs, and ran the two read-only evidence verifiers described below. This is same-provider work: **organizational-independence credit 0; no human review or provider-distinct review; scientific effect NONE**. + +The scoped verdict is acceptance of the exact literal-source construction of the unchanged `UniversalLaw.Spec.Lifetime.GaussianFieldInterface d L`, for every natural dimension `d` and real `L` with `0 < L`, together with the observed local formal execution evidence. No unresolved must-fix finding remains for that scope. Institutional statement alignment, scientific acceptance and downstream theorem closure are separate obligations. + +**Exact source identity and mathematical correspondence.** I read the governing PLAN/SCOPE, original `Lifetime.lean`, the supplied `THETA_BRIDGE.md` and `FIELD_CONSTRUCTOR.md`, the new implementation/control/audit sources, and their verification setup. Final readback confirmed all 19 working, frozen and publication-package Lean files are byte-identical to the reviewed inventory. All 13 inherited source hashes match the pinned original-source record; the target interface is unchanged. The decisive SHA-256 identities are: + +| Source | SHA-256 | +|---|---| +| `Lifetime.lean` | `90f4892608e953ecc696bb32ea159d1c178ceb8a1c6edfd11acfceb14af462c6` | +| `SourceSmoothCarrier.lean` | `5defdf223e85e7ccd23b5a903a1a306fe7f723727eb6aabf62eef523ef8639ed` | +| `SourceThetaBridge.lean` | `c03df303f47ac056af332ad421a6720966464c474ee77e671f85242d30f6cc9b` | +| `SourceLifetimeGeometry.lean` | `8ec4b18959055f20d849337e44a70d0c5b39b7959b54cb4d7b6c2a473ee0612d` | +| `SourceLifetimeField.lean` | `daf0949a14d4ed8bd19fbdc564f0f8c3fad72ed243de73e39d65b2951576919a` | +| `SourceLifetimeControls.lean` | `f50da7cb374c3e75280ae465696b657902f9070cff1256784dc28050017cd171` | +| `SourceLifetimeAxiomAudit.lean` | `fea8df29106f295238b3a039bb2aae350167a9ada2326d6d00ba6f6663be52c3` | + +`I4LifetimeField.actualGaussianFieldInterface` assigns all eleven original fields. Its only endpoint hypotheses are the original dimension/period data and `0 < L`. The carrier and measure are literally repaired `SmoothSample` and `smoothLaw`; the field is the original `smoothSourceField` evaluated through `EuclideanSpace.equiv (Fin d) Real`. The law is not replaced or renormalized. The source carrier's inclusion preserves the full original coefficient law, and positive controls explicitly retain this property. + +The geometry bridge uses the canonical continuous linear equivalence. It does not identify the ordinary Pi sup norm with Euclidean norm. The theta bridge uses `EuclideanSpace.real_norm_sq_eq` where the sum-of-squares identity is needed; the `(1,1)` control distinguishes the Euclidean geometry. + +The theta proof performs the one-dimensional Poisson substitution with alpha `L^2/(2*pi)` and beta `-t*L/(2*pi)`. It forms the absolutely convergent complex product before taking its real part. The factor `(sqrt(2*pi)/L)^d`, full-lattice `sourceZ`, DC contribution and paired-coefficient factor 2 remain present. The proof does not use a product-of-cosines replacement or silently omit normalization terms. The zero-lag identity cancels only a proved nonzero common scale. + +Centered covariance is transported under the literal measure. The raw product integral follows by substituting both proved zero means. Gaussian L2 integrability supports the finite double-sum variance calculation; variance nonnegativity supplies the exact quadratic-form parameter. The finite-combination pushforward is the required `gaussianReal` law, including zero variance, without division by variance or a positive-variance premise. Dimension zero, empty families, repeated points and cancelling coefficients remain admissible. There is no new production `axiom`/`sorry`, hidden positive-dimension restriction or endpoint strengthening. + +**Observed final local execution.** The final build directory was created empty. Nineteen exact project sources subsequently produced nineteen successful build receipts in dependency order. Five new modules then passed standard named-module `leanchecker -v` replay. The final transcript contains **26 actual attempts: 24 successes and two preserved failures**, with explicit recovery lineage. No successful build or successful kernel phase was rerun. + +| Preserved attempt or recovery | Actual observation | +|---|---| +| Initial final audit, `20261018T000515Z-build-SourceLifetimeAxiomAudit-93b1c912` | Return -15, timed out at the 420-second bound; 425.671 seconds including shutdown. Empty logs, unchanged 19 sources and 18 imported objects, no audit object. Original sequence remains `HOLD_FAILED_PHASE`. | +| Audit-only recovery, `20261018T001437Z-build-SourceLifetimeAxiomAudit-20f1f279` | Return 0, no timeout, 458.502 seconds under the explicit 900-second bound. Same sources and imported objects; full positive census completed. | +| Initial kernel attempt, `20261018T002219Z-kernel-SourceThetaBridge-9ba2eb35` | Return 1 after 3.943 seconds, no timeout. Exact diagnostic: `could not execute external process 'lean'`. This was child-executable discovery failure, not an observed kernel rejection. All 19 objects/sources remained unchanged. The intermediate recovery result remains `HOLD_FAILED_RECOVERY_PHASE`. | +| Kernel-only recovery | Uses the pinned runtime `bin` followed by `/usr/bin:/bin:/usr/sbin:/sbin`, checks child discovery and records the exact resolved Lean hash. Reuses all 19 successful builds and runs the five unresolved named-module phases. All five return 0 without timeout. | + +The five successful kernel durations were 159.308, 166.171, 162.787, 166.490 and 201.561 seconds for ThetaBridge, Geometry, Field, Controls and AxiomAudit respectively. I read each actual stdout/stderr: stdout contains only `replaying` followed by the correct module name; stderr is empty. Receipts bind the pinned checker, pinned child Lean, exact commands/import paths, pre/post object maps and source identities. + +This is **standard named-module replay against the imported environment**, using the same Lean kernel. It is not `--fresh`, an external verifier, or a source rebuild/replay of all Mathlib. **`lake build` was not run.** The publication package's Lake mapping was separately reviewed, but that does not manufacture a Lake execution receipt. + +The dependency inventory contains **83,897 artifacts, 8,418,477,755 bytes**, including relevant standard-library/package compiled artifacts and runtime companions. Those dependencies were pre-existing. Initial full hashes, per-phase path-set/metadata checks and a terminal full byte/path-set rehash bound their use. The actual terminal rehash completed at 2026-10-18T00:41:16.298861Z, after the terminal sequence, with exact path-set equality and no mismatches. I inspected the rehash receipt and verifier binding; I did not independently rerun the 8.4 GB scan. This explicit shared-cache trust boundary remains. + +**Positive axiom census.** The final recovered audit output contains exactly 409 unique declarations across the 18 specified source/target/control modules, including 40 private declarations and all 27 required names. Every reported transitive axiom is in `{propext, Classical.choice, Quot.sound}`. Module counts and actual declaration counts agree, exactly one `LIFETIME_AUDIT_STANDARD_ONLY_PASS COUNT 409` occurs, and no warning/error/sorryAx occurs. Multiline axiom lists were parsed completely. Final output is byte-identical to the independently reviewed development census; the successful final audit belongs to the new source/object chain. The audit module itself is also one of the five named-module replay targets. + +Earlier development audit attempts remain historical: one 240-second timeout, and one deliberate parent stop after a native sample localized execution to import initialization. The sample did not establish a theorem or census-loop failure. Narrower imports preserved audit coverage; their later success is not proof that import selection alone explains timing variation. + +**Meaningful controls and actual refusal evidence.** Positive Lean controls retain the literal carrier/law/field and coefficient-law pushforward, dimension zero, zero-lag variance, empty/cancelled/reinforced families, Euclidean norm, all-order smoothness on one carrier, lattice translations and the shifted mean. + +The source-mutation harness clones the actual constructor proof body into isolated namespaces without importing the production constructor. A matched zero-edit baseline compiled successfully. Its initial wrapper reported `control_pass: false` because proof references were elided in printing; that original observation is preserved. A separate exact-source probe enabled proof printing and showed all 21 local helper references in the seven constructors' quadratic-form, Gaussian-law and raw-covariance fields. Two-part acceptance binds the original compiled object and both actual receipts. + +All seven single-edit mutants then returned actual intended errors, without timeout or import/unknown-identifier failure: + +| Mutation | Intended refusal observed | +|---|---| +| Zero field | Gaussian pushforward and raw-covariance fields cannot reuse the literal source proofs. | +| Field shifted by 1 | Zero-mean Gaussian law and raw product integral no longer match. | +| Zero measure | Probability and raw-covariance obligations no longer match. | +| Added positive-dimension premise | The unrestricted dimension-zero consumer fails. | +| Doubled raw covariance | The covariance theorem's required scale differs. This is an output-scale control, not a direct mutation of the theta denominator implementation. | +| Covariance at `x+y` | The required stationary displacement is `x-y`. | +| Negated quadratic form | Variance nonnegativity does not prove nonnegativity of its negative. | + +The raw mutant output contains additional cascading projection errors; acceptance uses the intended diagnostic locations. Its unused-hd warning is not the refusal mechanism. A global sign reversal inside the fully summed cosine kernel would be equivalent and is not credited as a rejected control. + +The separate private-unused-axiom carrier compiled, then the copied whole-module audit rejected exactly the unreferenced private custom axiom. This checks module-wide enumeration, not merely endpoint dependency reachability. Its negative environment lists module counts totaling 421 but emits 419 declarations before the intended refusal; it is not a completed census. The emitted-set difference from the positive environment includes the generated `UniversalLaw.Spec.Lifetime.covQuadForm.eq_1`, with the production field count 11 instead of 12 in that import context. That negative run is not substituted for the complete positive 409-declaration census. + +**Reviewer verification and resolved findings.** I invoked `verify_mutations.verify(ROOT)` once on actual terminal negative evidence: **265 checks passed**, at 2026-10-17T23:43:28Z. I invoked the final recovery-aware `verify_execution.verify(ROOT)` once after terminal execution and dependency rehash: **579 checks passed**, at 2026-10-18T00:42:08Z. Neither invocation runs Lean or edits original evidence. The final verifier hashes were `8d9ec220f9db776fccf9ec6cf71c924217913700ee3d20ce8e47119ff99d0a32` and `85a809843d79dc8d68684c0b53b672d27ff4f86e436c3f903eaa280c4316c7c4` respectively. + +Early findings concerned imported-object binding, one-shot orchestration, exact phase/module coverage, stale rehash acceptance, mutation receipt/probe identity and historical schema handling. Source repairs and actual final records resolve those findings. Exact original successes/failures remain pinned; both recoveries have exclusive START records and distinct result files; all object transitions and complete receipt sets are checked. Synthetic corruption tests were kept distinct from actual execution evidence. Chronological draft review and interim findings are preserved in `REVIEW_C291_DRAFT_HISTORY.json` and the review-specific evidence records. + +The final acceptance anchors are: + +| Artifact | SHA-256 | +|---|---| +| `FINAL_SOURCE_INVENTORY.json` | `82f685a5d71671d9e682ca3df87485e97e8321e16f89c094965a49c2f8a100fe` | +| `FINAL_SEQUENCE_TERMINAL_RESULT.json` | `d2eb35bbf4afe56015ca65a64636690313c05e60253fa530e5633757ca18ba4e` | +| `FINAL_DEPENDENCY_REHASH.json` | `b67936f4f98bdd84d33eff5ea57d3c6aafcf8a5c6112afb75cbef501b1f3e884` | +| `NEGATIVE_FINAL_INDEX.json` | `3dc4f214c87fcf90096b67bb9bbc49fc57d823d1b96c91e0f79ef56c641f57ae` | +| Reviewer actual execution-verifier record | `b5a2e2982846f749500875318ecbec5429d17742824a05b7764cb3f7c1f551b3` | +| Reviewer actual mutation-verifier record | `4a2a8b85c2ac8f9fd9501e7c58d008a09956f7919a05b2a28aefe744b56930fb` | + +`REVIEW_C291_FINAL_DISPOSITION.json` records every final source and all 26 receipt identities. The original failed sequence and audit-recovery intermediate result retain hashes `a8dddfaf3d46287a573b70fae91d71ba000b219ea4d4b0bed148341e0fe4825d` and `e28e03d16b3df144317b5f1fe1bbec48379665b3dd39db4422d52209c549dc07`. + +**Publication, custody and downstream scope.** I reviewed the public evidence selector, full-text three-prefix projection, exact-byte readback/master mechanisms, and limited static-validator agent-token allowance. The projected public scripts are reviewable evidence, not a standalone executable substitute for the private packet's compiled objects and full dependency inventory. Actual final projection/static checks, current PR/raw-byte readback, hosted CI, private upload/download custody and final master execution are not observed or accepted by this report. They remain the parent's separate delivery gates. The new package's hosted Lean build is **NOT_REGISTERED_NOT_CLAIMED**; existing repository hosted checks cannot be credited as building it. + +The `NEXT_TASK.md` routing wording was checked against the supplied saved source readback and exact interface-map blob. It offers a compact-Morse/torus geometric adapter to the existing ordinary-H0 interval construction; it does not claim to prove that adapter, take over another owner, reopen accepted conventional proofs, or turn one interval summand into an exhaustive once-counted barcode. That wording check is not a new mathematical review of those downstream sources. + +This acceptance does not close Morse admission, canonical elder pairing, once-counting, all-small-bars converse, exhaustive barcode identification, shrinking-region multiple-witness collision estimates, stochastic transfer, universality, institutional alignment, or the broader research-system audit. The current constructor's inclusion of dimension zero does not extend downstream theorems to that dimension. Scientific effect remains **NONE**, and the broader project remains active/incomplete. diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/inputs/Lifetime.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/inputs/Lifetime.lean new file mode 100644 index 000000000..f2508f7f2 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/C291/inputs/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/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/OrdinaryH0.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/OrdinaryH0.lean new file mode 100644 index 000000000..7bc5f8002 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/OrdinaryH0.lean @@ -0,0 +1,339 @@ +import Mathlib +import ClosedSuperlevel + +/-! +# Actual ordinary singular H0 and path components, naturally + +C264 isolated source candidate. Compilation and kernel audit: NOT_RUN. +The homology objects and maps below are Mathlib's singular homology functor. +The comparison is its existing objectwise `TopCat.singularHomology₀Iso`; +its naturality is proved here from actual zero-cycle generators. + +The coefficient object belongs to an arbitrary preadditive category with +the required coproducts and homology. Nonzero/distinction controls explicitly +require `𝟙 R ≠ 0`; no nonzero claim is made for a zero coefficient object. + +This supplies the component/H0 comparison for the existing closed-superlevel +filtration. It supplies no Morse chart, critical-level attainment theorem, +interval decomposition, elder correspondence, Field adapter, count identity, +stochastic law, or scientific-status promotion. + +Owner repairs at the pin (Anthropic/Claude lane, companion owner, recorded as the +actual repair author; handoff bytes SHA-256 +261dd0526d42890d8a4ac1225fcf47228eb3c9799af4106cb643a528bcbf968e did not elaborate +with `-DwarningAsError=true`, Math- #428 comments 6113300903 and 6113372506): the +homology instances `[Preadditive C] [CategoryWithHomology C]` enter the `variable` +scope only after `ordinaryH0_componentCoproductMap_iota`, `ordinaryH0_componentFiltration` +and `ordinaryH0_componentFiltration_map_iota`, which do not use them (the two filtration +declarations therefore precede the homology ones; no statement changed); +`ordinaryH0_iso_naturality` is proved by explicit `rw`/`erw` reassociation of both sides +(the source author's five-step `calc` elaborated with an unresolved `Trans` instance +at the pin) instead of a `simp only` that left the right side unreassociated; `isoWhiskerLeft` is referred to as `Functor.isoWhiskerLeft`; the +three declaration names ending in `ι` are spelled `…_iota` / `…Iota…` for the +companion's ASCII identifier rule. No statement changed. +-/ + +noncomputable section + +open CategoryTheory Limits AlgebraicTopology +open scoped Simplicial + +universe w v u + +namespace TorusFiltration + +/-- Proves: actual path-component quotients form a functor on topological spaces. +Method: use the existing continuous-map quotient map and its identity/composition laws. +Does not establish: equality of connected and path-connected components. -/ +def ordinaryH0_componentFunctor : TopCat.{w} ⥤ Type w where + obj X := ZerothHomotopy X + map f := TypeCat.ofHom (superlevel_componentMap f.hom) + map_id X := by + apply ConcreteCategory.hom_ext + intro q + exact congrFun superlevel_componentMap_id q + map_comp f g := by + apply ConcreteCategory.hom_ext + intro q + exact congrFun (superlevel_componentMap_comp f.hom g.hom) q + +variable {C : Type u} [Category.{v} C] [HasCoproducts.{w} C] + +/-- Proves: the component-indexed coproduct construction is a functor. +Method: compose the actual component functor with the categorical coproduct functor. +Does not establish: an identification with homology until naturality is proved below. -/ +def ordinaryH0_componentCoproductFunctor (R : C) : TopCat.{w} ⥤ C := + ordinaryH0_componentFunctor ⋙ sigmaConst.obj R + +/-- Proves: a coproduct generator maps to the component of its continuous image. +Method: the universal coproduct map sends each inclusion to its indexed image. +Does not establish: that distinct source components stay distinct. -/ +theorem ordinaryH0_componentCoproductMap_iota (R : C) {X Y : TopCat.{w}} + (f : X ⟶ Y) (q : ZerothHomotopy X) : + Sigma.ι (fun _ : ZerothHomotopy X => R) q ≫ + (ordinaryH0_componentCoproductFunctor R).map f = + Sigma.ι (fun _ : ZerothHomotopy Y => R) (superlevel_componentMap f.hom q) := by + simp [ordinaryH0_componentCoproductFunctor, ordinaryH0_componentFunctor, sigmaConst] + +/-- Proves: the existing closed-superlevel family has a component-indexed coproduct filtration. +Method: compose its actual continuous inclusions with the proved component coproduct functor. +Does not establish: regularity, finite dimensionality, or Morse hypotheses for the height function. -/ +def ordinaryH0_componentFiltration {X : Type w} [TopologicalSpace X] + (g : X → ℝ) (R : C) : ℝ ⥤ C := + superlevel_filtration g ⋙ ordinaryH0_componentCoproductFunctor R + +/-- Proves: a filtration component generator maps by the existing actual component inclusion. +Method: specialize the coproduct generator map formula to a closed-superlevel inclusion. +Does not establish: a reverse inclusion or endpoint deletion. -/ +theorem ordinaryH0_componentFiltration_map_iota {X : Type w} [TopologicalSpace X] + (g : X → ℝ) (R : C) {s t : ℝ} (hst : s ≤ t) + (q : ZerothHomotopy (superlevel_set g s)) : + Sigma.ι (fun _ : ZerothHomotopy (superlevel_set g s) => R) q ≫ + (ordinaryH0_componentFiltration g R).map (homOfLE hst) = + Sigma.ι (fun _ : ZerothHomotopy (superlevel_set g t) => R) + (superlevel_componentInclusion g hst q) := by + exact ordinaryH0_componentCoproductMap_iota R + (TopCat.ofHom (superlevel_inclusion g hst)) q + +variable [Preadditive C] [CategoryWithHomology C] + +/-- Proves: a zero-simplex defines a morphism into actual simplicial homology. +Method: include its chain generator, lift through the zero differential, then take its class. +Does not establish: a replacement definition of homology or any naturality premise. -/ +def ordinaryH0_simplicialPointClass (S : SSet.{w}) (R : C) (x : S _⦋0⦌) : + R ⟶ S.homology R 0 := + (S.chainComplex R).liftCycles (S.ιChainComplex x) 0 (by simp) (by simp) ≫ + (S.chainComplex R).homologyπ 0 + +/-- Proves: the existing simplicial H0 isomorphism sends a point class to its component. +Method: apply Mathlib's zero-cycle generator formula. +Does not establish: naturality merely from the existence of the objectwise isomorphism. -/ +theorem ordinaryH0_simplicialPointClass_iso (S : SSet.{w}) (R : C) + (x : S _⦋0⦌) : + ordinaryH0_simplicialPointClass S R x ≫ (S.homology₀Iso R).hom = + Sigma.ι (fun _ : S.π₀ => R) (SSet.π₀.mk x) := by + simpa only [ordinaryH0_simplicialPointClass, Category.assoc] using + SSet.liftCycles_ιChainComplex_homologyπ_homology₀Iso_hom S R x + +/-- Proves: the actual simplicial homology map carries a point class to its image class. +Method: homology projection naturality, cycle-lift naturality, and the chain generator formula. +Does not establish: a transported or postulated substitute for the actual homology map. -/ +theorem ordinaryH0_simplicialPointClass_map (R : C) {S T : SSet.{w}} + (f : S ⟶ T) (x : S _⦋0⦌) : + ordinaryH0_simplicialPointClass S R x ≫ SSet.homologyMap f R 0 = + ordinaryH0_simplicialPointClass T R (f.app _ x) := by + simp only [ordinaryH0_simplicialPointClass, SSet.homologyMap, Category.assoc, + HomologicalComplex.homologyπ_naturality, + HomologicalComplex.liftCycles_comp_cyclesMap_assoc, SSet.ι_chainComplexMap_f] + +/-- Proves: the singular zero-simplex of a point maps to that of its continuous image. +Method: unfold the point/simplex correspondence and the singular-set map. +Does not establish: a chain or homology identification by itself. -/ +theorem ordinaryH0_toSSet_pointMap {X Y : TopCat.{w}} (f : X ⟶ Y) (x : X) : + (TopCat.toSSet.map f).app _ (TopCat.toSSetObj₀Equiv.symm x) = + TopCat.toSSetObj₀Equiv.symm (f x) := by + rfl + +/-- Proves: each point has its actual ordinary singular zero-homology class morphism. +Method: use the actual singular zero-simplex and the cycle/projection construction above. +Does not establish: a free-module definition standing in for singular H0. -/ +def ordinaryH0_pointClass (X : TopCat.{w}) (R : C) (x : X) : + R ⟶ ((singularHomologyFunctor C 0).obj R).obj X := + ordinaryH0_simplicialPointClass (TopCat.toSSet.obj X) R + (TopCat.toSSetObj₀Equiv.symm x) + +/-- Proves: actual continuous-map-induced singular H0 maps transport point classes. +Method: specialize the proved simplicial generator naturality to the singular-set functor. +Does not establish: injectivity of the induced map. -/ +theorem ordinaryH0_pointClass_map (R : C) {X Y : TopCat.{w}} + (f : X ⟶ Y) (x : X) : + ordinaryH0_pointClass X R x ≫ ((singularHomologyFunctor C 0).obj R).map f = + ordinaryH0_pointClass Y R (f x) := by + change ordinaryH0_simplicialPointClass (TopCat.toSSet.obj X) R + (TopCat.toSSetObj₀Equiv.symm x) ≫ + SSet.homologyMap (TopCat.toSSet.map f) R 0 = + ordinaryH0_simplicialPointClass (TopCat.toSSet.obj Y) R + (TopCat.toSSetObj₀Equiv.symm (f x)) + rw [ordinaryH0_simplicialPointClass_map, ordinaryH0_toSSet_pointMap] + +/-- Proves: Mathlib's existing topological comparison sends actual point classes to components. +Method: combine the simplicial generator formula with the actual path-component equivalence. +Does not establish: any connected-component substitution. -/ +theorem ordinaryH0_pointClass_iso (X : TopCat.{w}) (R : C) (x : X) : + ordinaryH0_pointClass X R x ≫ (TopCat.singularHomology₀Iso X R).hom = + Sigma.ι (fun _ : ZerothHomotopy X => R) (ZerothHomotopy.mk x) := by + change ordinaryH0_simplicialPointClass (TopCat.toSSet.obj X) R + (TopCat.toSSetObj₀Equiv.symm x) ≫ + ((SSet.homology₀Iso (TopCat.toSSet.obj X) R).hom ≫ + (sigmaConst.obj R).map TopCat.zerothHomotopyEquiv.toIso.inv) = _ + rw [← Category.assoc, ordinaryH0_simplicialPointClass_iso] + simp [sigmaConst] + +/-- Proves: inverse comparison takes each component generator to any representing point class. +Method: cancel the comparison isomorphism and use the point-class formula. +Does not establish: a choice of distinguished geometric representative. -/ +theorem ordinaryH0_componentIota_isoInv (X : TopCat.{w}) (R : C) (x : X) : + Sigma.ι (fun _ : ZerothHomotopy X => R) (ZerothHomotopy.mk x) ≫ + (TopCat.singularHomology₀Iso X R).inv = ordinaryH0_pointClass X R x := by + apply (cancel_mono (TopCat.singularHomology₀Iso X R).hom).1 + simp only [Category.assoc, Iso.inv_hom_id, Category.comp_id, ordinaryH0_pointClass_iso] + +/-- Proves: morphisms out of actual singular H0 agree if they agree on all point classes. +Method: cancel the inverse comparison, use coproduct extensionality and quotient surjectivity. +Does not establish: a basis of geometric maxima or finiteness of components. -/ +theorem ordinaryH0_hom_ext (X : TopCat.{w}) (R : C) {A : C} + {f g : ((singularHomologyFunctor C 0).obj R).obj X ⟶ A} + (h : ∀ x : X, ordinaryH0_pointClass X R x ≫ f = ordinaryH0_pointClass X R x ≫ g) : + f = g := by + apply (cancel_epi (TopCat.singularHomology₀Iso X R).inv).1 + apply Sigma.hom_ext + intro q + obtain ⟨x, rfl⟩ := ZerothHomotopy.mk_surjective q + simpa only [← Category.assoc, ordinaryH0_componentIota_isoInv] using h x + +/-- Proves: the existing actual singular H0 comparison commutes with every continuous map. +Method: test both composites on actual point classes, transport them, and compare component images. +Does not establish: naturality by adding an assumption or a compatibility record. -/ +theorem ordinaryH0_iso_naturality (R : C) {X Y : TopCat.{w}} (f : X ⟶ Y) : + ((singularHomologyFunctor C 0).obj R).map f ≫ + (TopCat.singularHomology₀Iso Y R).hom = + (TopCat.singularHomology₀Iso X R).hom ≫ + (ordinaryH0_componentCoproductFunctor R).map f := by + apply ordinaryH0_hom_ext X R + intro x + rw [← Category.assoc, ordinaryH0_pointClass_map, ordinaryH0_pointClass_iso] + -- the right side composes through `(ordinaryH0_componentCoproductFunctor R).obj X`, which is + -- `∐ fun _ => R` only after unfolding, so its reassociation is rewritten at default transparency + erw [← Category.assoc, ordinaryH0_pointClass_iso, ordinaryH0_componentCoproductMap_iota] + rfl + +/-- Proves: actual ordinary singular H0 is naturally isomorphic to component-indexed coproducts. +Method: assemble the existing objectwise isomorphisms using the commuting square proved above. +Does not establish: an interval decomposition or persistence-bar classification. -/ +def ordinaryH0_naturalIso (R : C) : + (singularHomologyFunctor C 0).obj R ≅ ordinaryH0_componentCoproductFunctor R := + NatIso.ofComponents (fun X => TopCat.singularHomology₀Iso X R) + (fun f => ordinaryH0_iso_naturality R f) + +/-- Proves: ordinary augmentation sends every actual point-class morphism to the coefficient identity. +Method: use the comparison's augmentation formula and the coproduct universal property. +Does not establish: reduced-H0 vanishing; this retains the ordinary point class. -/ +theorem ordinaryH0_pointClass_augmentation (X : TopCat.{w}) (R : C) (x : X) : + ordinaryH0_pointClass X R x ≫ TopCat.singularHomology₀ε X R = 𝟙 R := by + rw [← TopCat.singularHomology₀Iso_sigma_desc_id X R, ← Category.assoc, + ordinaryH0_pointClass_iso] + simp + +/-- Proves: an actual point class is nonzero for a coefficient object with nonzero identity. +Method: compose a proposed zero class with ordinary augmentation. +Does not establish: nonzero classes for a zero coefficient object. -/ +theorem ordinaryH0_pointClass_ne_zero (X : TopCat.{w}) (R : C) + (hR : (𝟙 R : R ⟶ R) ≠ 0) (x : X) : ordinaryH0_pointClass X R x ≠ 0 := by + intro h + have hε := ordinaryH0_pointClass_augmentation X R x + rw [h, zero_comp] at hε + exact hR hε.symm + +/-- Proves: points in the same actual path component give equal ordinary H0 class morphisms. +Method: cancel the objectwise comparison and compare their component indices. +Does not establish: a converse for zero coefficients. -/ +theorem ordinaryH0_pointClass_eq_of_component_eq (X : TopCat.{w}) (R : C) + (x y : X) (h : ZerothHomotopy.mk x = ZerothHomotopy.mk y) : + ordinaryH0_pointClass X R x = ordinaryH0_pointClass X R y := by + apply (cancel_mono (TopCat.singularHomology₀Iso X R).hom).1 + rw [ordinaryH0_pointClass_iso, ordinaryH0_pointClass_iso, h] + +/-- Proves: the difference of same-component point classes is zero for arbitrary coefficients. +Method: subtract the equality proved from actual path-component equality. +Does not establish: that either individual point class dies. -/ +theorem ordinaryH0_sameComponent_difference_zero (X : TopCat.{w}) (R : C) + (x y : X) (h : ZerothHomotopy.mk x = ZerothHomotopy.mk y) : + ordinaryH0_pointClass X R x - ordinaryH0_pointClass X R y = 0 := by + rw [ordinaryH0_pointClass_eq_of_component_eq X R x y h, sub_self] + +/-- Proves: distinct actual path components have distinct ordinary H0 class morphisms if coefficients are nonzero. +Method: separate the two coproduct injections by an identity/zero component projection. +Does not establish: distinction with zero coefficients or distinction of points within one component. -/ +theorem ordinaryH0_pointClass_ne_of_component_ne (X : TopCat.{w}) (R : C) + (hR : (𝟙 R : R ⟶ R) ≠ 0) (x y : X) + (hxy : ZerothHomotopy.mk x ≠ ZerothHomotopy.mk y) : + ordinaryH0_pointClass X R x ≠ ordinaryH0_pointClass X R y := by + classical + intro h + have hc : Sigma.ι (fun _ : ZerothHomotopy X => R) (ZerothHomotopy.mk x) = + Sigma.ι (fun _ : ZerothHomotopy X => R) (ZerothHomotopy.mk y) := by + simpa only [ordinaryH0_pointClass_iso] using + congrArg (fun k => k ≫ (TopCat.singularHomology₀Iso X R).hom) h + let p : (∐ fun _ : ZerothHomotopy X => R) ⟶ R := + Sigma.desc (fun q => if q = ZerothHomotopy.mk x then 𝟙 R else 0) + apply hR + simpa [p, Ne.symm hxy] using congrArg (fun k => k ≫ p) hc + +/-- Proves: with nonzero coefficients, actual point-class equality is exactly path-component equality. +Method: combine the same-component formula with the separating projection control. +Does not establish: equality of geometric points or connected components. -/ +theorem ordinaryH0_pointClass_eq_iff (X : TopCat.{w}) (R : C) + (hR : (𝟙 R : R ⟶ R) ≠ 0) (x y : X) : + ordinaryH0_pointClass X R x = ordinaryH0_pointClass X R y ↔ + ZerothHomotopy.mk x = ZerothHomotopy.mk y := by + constructor + · intro h + by_contra hn + exact ordinaryH0_pointClass_ne_of_component_ne X R hR x y hn h + · exact ordinaryH0_pointClass_eq_of_component_eq X R x y + +/-- Proves: the actual H0 map sends a difference of point classes to the difference of image classes. +Method: distribute composition over subtraction and use proved actual point-class transport. +Does not establish: death of the individual ordinary point classes. -/ +theorem ordinaryH0_difference_map (R : C) {X Y : TopCat.{w}} (f : X ⟶ Y) (x y : X) : + (ordinaryH0_pointClass X R x - ordinaryH0_pointClass X R y) ≫ + ((singularHomologyFunctor C 0).obj R).map f = + ordinaryH0_pointClass Y R (f x) - ordinaryH0_pointClass Y R (f y) := by + rw [Preadditive.sub_comp, ordinaryH0_pointClass_map, ordinaryH0_pointClass_map] + +/-- Proves: with nonzero coefficients, a mapped difference vanishes exactly when the image points merge in path components. +Method: actual homology-map transport, subtraction cancellation, and the component equality criterion. +Does not establish: a Morse merger event, elder selection, death height, or barcode. -/ +theorem ordinaryH0_difference_map_eq_zero_iff (R : C) (hR : (𝟙 R : R ⟶ R) ≠ 0) + {X Y : TopCat.{w}} (f : X ⟶ Y) (x y : X) : + (ordinaryH0_pointClass X R x - ordinaryH0_pointClass X R y) ≫ + ((singularHomologyFunctor C 0).obj R).map f = 0 ↔ + ZerothHomotopy.mk (f x) = ZerothHomotopy.mk (f y) := by + rw [ordinaryH0_difference_map, sub_eq_zero, ordinaryH0_pointClass_eq_iff Y R hR] + +/-- Proves: the existing actual singular H0 filtration is naturally isomorphic to its component coproduct filtration. +Method: restrict the proved natural comparison along the existing topological filtration. +Does not establish: the later interval, elder, maximum/saddle, count, Field, or stochastic bridges. -/ +def ordinaryH0_filtrationIso {X : Type w} [TopologicalSpace X] + (g : X → ℝ) (R : C) : + superlevel_h0Filtration g R ≅ ordinaryH0_componentFiltration g R := + Functor.isoWhiskerLeft (superlevel_filtration g) (ordinaryH0_naturalIso R) + +/-- Proves: all ordered closed-superlevel inclusion maps satisfy the actual H0/component square. +Method: read the naturality of the proved filtration isomorphism at the preorder morphism. +Does not establish: strict-level conventions or any unproved geometric persistence hypothesis. -/ +theorem ordinaryH0_filtration_naturality {X : Type w} [TopologicalSpace X] + (g : X → ℝ) (R : C) {s t : ℝ} (hst : s ≤ t) : + (superlevel_h0Filtration g R).map (homOfLE hst) ≫ + (ordinaryH0_filtrationIso g R).hom.app t = + (ordinaryH0_filtrationIso g R).hom.app s ≫ + (ordinaryH0_componentFiltration g R).map (homOfLE hst) := + (ordinaryH0_filtrationIso g R).hom.naturality (homOfLE hst) + +/-- Proves: under a closed-superlevel inclusion, a difference vanishes exactly when the two included points are joined in the target closed superlevel. +Method: apply the actual H0 difference control and the existing component/JoinedIn equivalence. +Does not establish: attainment of a critical death threshold or an elder/barcode theorem. -/ +theorem ordinaryH0_filtration_difference_zero_iff_joinedIn + {X : Type w} [TopologicalSpace X] (g : X → ℝ) (R : C) + (hR : (𝟙 R : R ⟶ R) ≠ 0) {s t : ℝ} (hst : s ≤ t) + (x y : superlevel_set g s) : + (ordinaryH0_pointClass (TopCat.of (superlevel_set g s)) R x - + ordinaryH0_pointClass (TopCat.of (superlevel_set g s)) R y) ≫ + (superlevel_h0Filtration g R).map (homOfLE hst) = 0 ↔ + JoinedIn (superlevel_set g t) x.val y.val := by + exact (ordinaryH0_difference_map_eq_zero_iff R hR + (TopCat.ofHom (superlevel_inclusion g hst)) x y).trans + (superlevel_component_eq_iff_joinedIn g t + (superlevel_inclusion g hst x) (superlevel_inclusion g hst y)) + +end TorusFiltration diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/SuperlevelBar.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/SuperlevelBar.lean new file mode 100644 index 000000000..b047384e2 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/SuperlevelBar.lean @@ -0,0 +1,299 @@ +import Mathlib + +/-! +# C263 total strict-clearance cut for continuous Euclidean heights + +Proves: total essentiality, strict-clearance set and real death value with an +explicit empty-set value zero; an unconditional bound for clearance members; +nonemptiness for continuous nonessential inputs; strict witness improvement; +the exact strict supremum cut, guarded death bound and cut uniqueness. + +Method: the starting endpoint bounds every clearance; a straight path and the +attained minimum on its compact unit-interval parameter supply a lower witness +and a uniform strict margin; conditional real supremum laws identify the cut. + +Does not establish: an inhabitant of the separately owned Field or Lifetime +record, an elder relation, Morse endpoint attainment, ordinary H0 bar pairing, +finite-bar counts, a Gaussian field or weighted law, or scientific acceptance. +The scalar zero on an empty set is a default, not an essential-bar death. + +Authored by OpenAI/Codex c262_lean_source_review, now acting as C263 implementer, +delegated for Dylan Roy. Source/design-exposed; organizational independence 0; +no human review. Consumes C260 prose SHA-256 +d2462639df32364e6fc8b0fea0703106c668cedb0aa7bb00e139aa81561bd845. +Target pin: Lean 4.34.1 / Mathlib d13f23b723b8a846827a245b89c10fc7d3f11612. +Elaboration, kernel audit and executed controls are recorded in separate +source-bound receipts; this text makes no execution claim. + +Compile fix at the pin (Anthropic/Claude lane, companion owner; handoff bytes +SHA-256 d74dc8b2c25eff5b90ce7133e51a0888c7885cae24ca6d5e648bc514089637a8 did not +elaborate with `-DwarningAsError=true`): in `superlevelBar_death_eq_sSup_of_nonempty` +the simp lemma `if_pos` (deprecated in Lean core since 2026-07-21) is replaced by +its successor `ite_eq_left`, the same statement; no other byte changes. +-/ + +noncomputable section + +namespace TorusFiltration + +/-- Proves: the ambient type is real Euclidean space in the stated dimension. +Method: a type abbreviation. Does not establish: a positive-dimensional or +periodic quotient-space hypothesis. -/ +abbrev superlevelBar_E (d : ℕ) := EuclideanSpace ℝ (Fin d) + +/-- Proves: the essentiality datum is total. Method: global order comparison. +Does not establish: an actual persistence interpretation off the guarded locus. -/ +def superlevelBar_isEssential {d : ℕ} (g : superlevelBar_E d → ℝ) + (M : superlevelBar_E d) : Prop := + ∀ y, g y ≤ g M + +/-- Proves: a total strict-clearance set with the source's existential endpoint. +Method: quantify over actual paths and every parameter, including endpoints. +Does not establish: a path attaining the supremum with closed inequalities. -/ +def superlevelBar_clearanceSet {d : ℕ} (g : superlevelBar_E d → ℝ) + (M : superlevelBar_E d) : Set ℝ := + {h | ∃ y, g M < g y ∧ ∃ γ : Path M y, ∀ t : unitInterval, h < g (γ t)} + +/-- Proves: the death datum is a total real number with explicit empty default. +Method: nonempty conditional supremum; boundedness is proved below. +Does not establish: a finite death for an essential class. -/ +def superlevelBar_deathHeight {d : ℕ} (g : superlevelBar_E d → ℝ) + (M : superlevelBar_E d) : ℝ := by + classical + exact if (superlevelBar_clearanceSet g M).Nonempty then + sSup (superlevelBar_clearanceSet g M) else 0 + +/-- Proves: essentiality is exactly global maximality for these total data. +Method: unfold the defining proposition. Does not establish: agreement with +an independently defined essential persistence class. -/ +theorem superlevelBar_essential_iff {d : ℕ} (g : superlevelBar_E d → ℝ) + (M : superlevelBar_E d) : + superlevelBar_isEssential g M ↔ ∀ y, g y ≤ g M := Iff.rfl + +/-- Proves: an unconditional strict bound on each clearance member. +Method: evaluate its path at zero. Does not establish: an unconditional bound +on the default death value when the clearance set is empty. -/ +theorem superlevelBar_mem_lt_start {d : ℕ} (g : superlevelBar_E d → ℝ) + (M : superlevelBar_E d) {h : ℝ} (hh : h ∈ superlevelBar_clearanceSet g M) : + h < g M := by + obtain ⟨y, hy, γ, hγ⟩ := hh + have hzero := hγ (0 : unitInterval) + simpa only [γ.source] using hzero + +/-- Proves: the clearance set is bounded above, including when empty. +Method: the initial height bounds each member. Does not establish: that the +empty-set default death value obeys the same bound. -/ +theorem superlevelBar_clearance_bddAbove {d : ℕ} (g : superlevelBar_E d → ℝ) + (M : superlevelBar_E d) : BddAbove (superlevelBar_clearanceSet g M) := by + exact ⟨g M, fun h hh => (superlevelBar_mem_lt_start g M hh).le⟩ + +/-- Proves: essentiality forces the clearance set to be empty. +Method: contradict the required strictly higher endpoint. +Does not establish: a converse for arbitrary discontinuous heights. -/ +theorem superlevelBar_clearance_empty_of_essential {d : ℕ} + (g : superlevelBar_E d → ℝ) (M : superlevelBar_E d) + (hM : superlevelBar_isEssential g M) : superlevelBar_clearanceSet g M = ∅ := by + ext h + constructor + · rintro ⟨y, hy, γ, hγ⟩ + exact False.elim ((not_lt_of_ge (hM y)) hy) + · intro hh + exact False.elim hh + +/-- Proves: an empty clearance set selects the declared real value zero. +Method: simplify the conditional definition. Does not establish: a geometric +death event or finite persistence interval at that zero. -/ +theorem superlevelBar_death_eq_zero_of_empty {d : ℕ} + (g : superlevelBar_E d → ℝ) (M : superlevelBar_E d) + (he : superlevelBar_clearanceSet g M = ∅) : superlevelBar_deathHeight g M = 0 := by + classical + simp [superlevelBar_deathHeight, he] + +/-- Proves: essential inputs have the declared zero default. +Method: essentiality implies empty clearance, then use the empty branch. +Does not establish: an essential-bar death height. -/ +theorem superlevelBar_death_eq_zero_of_essential {d : ℕ} + (g : superlevelBar_E d → ℝ) (M : superlevelBar_E d) + (hM : superlevelBar_isEssential g M) : superlevelBar_deathHeight g M = 0 := by + exact superlevelBar_death_eq_zero_of_empty g M + (superlevelBar_clearance_empty_of_essential g M hM) + +/-- Proves: nonempty clearance selects its real supremum in the total data. +Method: simplify the nonempty conditional branch. Does not establish: +nonemptiness without a witness or a regularity hypothesis. -/ +theorem superlevelBar_death_eq_sSup_of_nonempty {d : ℕ} + (g : superlevelBar_E d → ℝ) (M : superlevelBar_E d) + (hne : (superlevelBar_clearanceSet g M).Nonempty) : + superlevelBar_deathHeight g M = sSup (superlevelBar_clearanceSet g M) := by + classical + simp only [superlevelBar_deathHeight, ite_eq_left hne] + +/-- Proves: a continuous height attains a minimum along each actual path. +Method: compact unit interval and the extreme-value theorem. +Does not establish: a minimum over all paths or an optimal path. -/ +theorem superlevelBar_path_minimum {d : ℕ} (g : superlevelBar_E d → ℝ) + (hg : Continuous g) {M y : superlevelBar_E d} (γ : Path M y) : + ∃ t₀ : unitInterval, ∀ t : unitInterval, g (γ t₀) ≤ g (γ t) := by + obtain ⟨t₀, ht₀, hmin⟩ := + (isCompact_univ : IsCompact (Set.univ : Set unitInterval)).exists_isMinOn + (show (Set.univ : Set unitInterval).Nonempty from ⟨0, Set.mem_univ _⟩) + (hg.comp γ.continuous).continuousOn + exact ⟨t₀, fun t => hmin (Set.mem_univ t)⟩ + +/-- Proves: nonessential continuous inputs have a nonempty clearance set. +Method: a higher endpoint, Path.segment, and minimum minus one. +Does not establish: compactness of the whole Euclidean domain. -/ +theorem superlevelBar_clearance_nonempty {d : ℕ} (g : superlevelBar_E d → ℝ) + (hg : Continuous g) (M : superlevelBar_E d) + (hM : ¬ superlevelBar_isEssential g M) : + (superlevelBar_clearanceSet g M).Nonempty := by + classical + have hhigh : ∃ y, g M < g y := by + by_contra hn + apply hM + intro y + exact le_of_not_gt (fun hy => hn ⟨y, hy⟩) + obtain ⟨y, hy⟩ := hhigh + let γ : Path M y := Path.segment M y + obtain ⟨t₀, hmin⟩ := superlevelBar_path_minimum g hg γ + refine ⟨g (γ t₀) - 1, y, hy, γ, ?_⟩ + intro t + have ht := hmin t + linarith + +/-- Proves: for continuous heights, clearance is nonempty exactly off the +essential locus. Method: the higher witness endpoint and the straight-path +construction. Does not establish: this equivalence without continuity. -/ +theorem superlevelBar_clearance_nonempty_iff {d : ℕ} + (g : superlevelBar_E d → ℝ) (hg : Continuous g) (M : superlevelBar_E d) : + (superlevelBar_clearanceSet g M).Nonempty ↔ ¬ superlevelBar_isEssential g M := by + constructor + · rintro ⟨h, y, hy, γ, hγ⟩ hM + exact (not_lt_of_ge (hM y)) hy + · exact superlevelBar_clearance_nonempty g hg M + +/-- Proves: any smaller threshold than an existing clearance is a clearance. +Method: reuse the same endpoint and path and compose strict inequalities. +Does not establish: strict improvement to a larger threshold. -/ +theorem superlevelBar_clearance_downward {d : ℕ} (g : superlevelBar_E d → ℝ) + (M : superlevelBar_E d) {h c : ℝ} (hhc : h < c) + (hc : c ∈ superlevelBar_clearanceSet g M) : h ∈ superlevelBar_clearanceSet g M := by + obtain ⟨y, hy, γ, hγ⟩ := hc + exact ⟨y, hy, γ, fun t => hhc.trans (hγ t)⟩ + +/-- Proves: each strict witness has a larger strict witness on its same path. +Method: attained minimum and a midpoint below it. +Does not establish: a witness with clearance strictly above the supremum. -/ +theorem superlevelBar_clearance_improve {d : ℕ} (g : superlevelBar_E d → ℝ) + (hg : Continuous g) (M : superlevelBar_E d) {h : ℝ} + (hh : h ∈ superlevelBar_clearanceSet g M) : + ∃ c : ℝ, h < c ∧ c ∈ superlevelBar_clearanceSet g M := by + obtain ⟨y, hy, γ, hγ⟩ := hh + obtain ⟨t₀, hmin⟩ := superlevelBar_path_minimum g hg γ + have hmargin := hγ t₀ + refine ⟨(h + g (γ t₀)) / 2, ?_, y, hy, γ, ?_⟩ + · linarith + · intro t + have ht := hmin t + linarith + +/-- Proves: the exact strict cut for continuous nonessential inputs. +Method: least-upper-bound property, downward closure and strict improvement. +Does not establish: closed-threshold attainment at the death height. -/ +theorem superlevelBar_strict_cut {d : ℕ} (g : superlevelBar_E d → ℝ) + (hg : Continuous g) (M : superlevelBar_E d) + (hM : ¬ superlevelBar_isEssential g M) (h : ℝ) : + h < superlevelBar_deathHeight g M ↔ h ∈ superlevelBar_clearanceSet g M := by + classical + have hne := superlevelBar_clearance_nonempty g hg M hM + rw [superlevelBar_death_eq_sSup_of_nonempty g M hne] + constructor + · intro hlt + have hex : ∃ c ∈ superlevelBar_clearanceSet g M, h < c := by + by_contra hn + have hub : ∀ c ∈ superlevelBar_clearanceSet g M, c ≤ h := by + intro c hc + exact le_of_not_gt (fun hhc => hn ⟨c, hc, hhc⟩) + exact (not_le_of_gt hlt) (csSup_le hne hub) + obtain ⟨c, hc, hhc⟩ := hex + exact superlevelBar_clearance_downward g M hhc hc + · intro hh + obtain ⟨c, hhc, hc⟩ := superlevelBar_clearance_improve g hg M hh + exact hhc.trans_le (le_csSup (superlevelBar_clearance_bddAbove g M) hc) + +/-- Proves: the death value is at most the initial height when clearance is +nonempty. Method: the supremum is at most this upper bound. +Does not establish: the bound on the empty default branch. -/ +theorem superlevelBar_death_le_start_of_nonempty {d : ℕ} + (g : superlevelBar_E d → ℝ) (M : superlevelBar_E d) + (hne : (superlevelBar_clearanceSet g M).Nonempty) : + superlevelBar_deathHeight g M ≤ g M := by + rw [superlevelBar_death_eq_sSup_of_nonempty g M hne] + exact csSup_le hne (fun h hh => (superlevelBar_mem_lt_start g M hh).le) + +/-- Proves: the death upper bound for continuous nonessential inputs. +Method: establish nonempty clearance and apply its guarded supremum bound. +Does not establish: a strict inequality or an attained optimal path. -/ +theorem superlevelBar_death_le_start {d : ℕ} (g : superlevelBar_E d → ℝ) + (hg : Continuous g) (M : superlevelBar_E d) + (hM : ¬ superlevelBar_isEssential g M) : superlevelBar_deathHeight g M ≤ g M := by + exact superlevelBar_death_le_start_of_nonempty g M + (superlevelBar_clearance_nonempty g hg M hM) + +/-- Proves: the clearance set is the open lower interval below death on the +continuous nonessential locus. Method: extensionality and the strict cut. +Does not establish: a closed-threshold path at the excluded endpoint. -/ +theorem superlevelBar_clearance_eq_Iio {d : ℕ} (g : superlevelBar_E d → ℝ) + (hg : Continuous g) (M : superlevelBar_E d) + (hM : ¬ superlevelBar_isEssential g M) : + superlevelBar_clearanceSet g M = Set.Iio (superlevelBar_deathHeight g M) := by + ext h + exact (superlevelBar_strict_cut g hg M hM h).symm + +/-- Proves: death is not a strict clearance for continuous nonessential inputs. +Method: the cut would otherwise imply that death is strictly below itself. +Does not establish: failure or success of closed-threshold attainment. -/ +theorem superlevelBar_death_not_mem_clearance {d : ℕ} + (g : superlevelBar_E d → ℝ) (hg : Continuous g) (M : superlevelBar_E d) + (hM : ¬ superlevelBar_isEssential g M) : + superlevelBar_deathHeight g M ∉ superlevelBar_clearanceSet g M := by + intro hh + have hlt := (superlevelBar_strict_cut g hg M hM (superlevelBar_deathHeight g M)).2 hh + exact (lt_irrefl _) hlt + +/-- Proves: uniqueness of a real number with this strict cut on the valid locus. +Method: evaluate either cut at the competing endpoint. +Does not establish: equality of arbitrary total interface records off that locus. -/ +theorem superlevelBar_cut_unique {d : ℕ} (g : superlevelBar_E d → ℝ) + (hg : Continuous g) (M : superlevelBar_E d) + (hM : ¬ superlevelBar_isEssential g M) (a : ℝ) + (ha : ∀ h : ℝ, h < a ↔ h ∈ superlevelBar_clearanceSet g M) : + a = superlevelBar_deathHeight g M := by + apply le_antisymm + · by_contra hn + have hlt : superlevelBar_deathHeight g M < a := lt_of_not_ge hn + exact superlevelBar_death_not_mem_clearance g hg M hM ((ha _).1 hlt) + · by_contra hn + have hlt : a < superlevelBar_deathHeight g M := lt_of_not_ge hn + have hc := (superlevelBar_strict_cut g hg M hM a).1 hlt + exact (lt_irrefl a) ((ha a).2 hc) + +/-- Proves: any constant height selects the zero default at every point. +Method: every point is globally maximal, hence essential in these data. +Does not establish: any finite ordinary bar for that constant height. -/ +theorem superlevelBar_death_const (d : ℕ) (b : ℝ) (M : superlevelBar_E d) : + superlevelBar_deathHeight (fun _ : superlevelBar_E d => b) M = 0 := by + apply superlevelBar_death_eq_zero_of_essential + intro y + exact le_rfl + +/-- Proves: the essential default forbids an unconditional death/start bound. +Method: constant height minus one has empty clearance and death zero. +Does not establish: any finite ordinary bar on a constant field. -/ +theorem superlevelBar_control_negative_essential (d : ℕ) (M : superlevelBar_E d) : + superlevelBar_deathHeight (fun _ : superlevelBar_E d => (-1 : ℝ)) M = 0 ∧ + ¬ superlevelBar_deathHeight (fun _ : superlevelBar_E d => (-1 : ℝ)) M ≤ (-1 : ℝ) := by + rw [superlevelBar_death_const d (-1) M] + norm_num + +end TorusFiltration diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/SuperlevelBarField.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/SuperlevelBarField.lean new file mode 100644 index 000000000..f0300a6b4 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/SuperlevelBarField.lean @@ -0,0 +1,205 @@ +import Mathlib +import SuperlevelBar +import Specifications.Field + +/-! +# C263 adapter: an inhabitant of the Field superlevel-bar interface + +Proves: the C263 strict-clearance data (`superlevelBar_deathHeight`, +`superlevelBar_isEssential`) together with the elder clause's own right-hand side +inhabit `UniversalLaw.Spec.Field.SuperlevelBarInterface d L` for every `d` and `L`; +on smooth `L`-periodic heights every inhabitant agrees with this one on essentiality, +on the death height at nonessential points and, on the Morse distinct-value locus, on +the elder relation; and the interface leaves the death height at essential points free +(two inhabitants differ there). + +Method: `IsSmooth` at `n = 0` gives continuity, so `superlevelBar_strict_cut` is the +maximin clause and `superlevelBar_cut_unique` forces agreement of any other death +height; `essential_iff` is `superlevelBar_essential_iff`; `ElderPartner` is defined as +the right-hand side of `elderPartner_iff`, so that clause holds by `Iff.rfl`. + +Does not establish: an identification of `ElderPartner` with a persistence-module elder +rule, the source's elder event or its Borel measurability (the clause holds by +definition, not as a theorem about persistence); any use of periodicity or of `L` in the +construction (the uniqueness theorems take the periodicity hypothesis only to apply an +arbitrary inhabitant's clauses, and are false without it); a torus quotient; Morse +genericity; any meaning of the value `0` that the C263 default +gives at essential points; anything about the separate +`UniversalLaw.Spec.Lifetime.SuperlevelBarInterface` (not inhabited here) or about any +Lifetime statement; scientific acceptance. The statement packet's own text "no +inhabitant is constructed" is about that packet and stays true of it; nothing in it +changes. No claim-level label moves; alignment stays PENDING_INDEPENDENT_REVIEW. + +Authored by the Anthropic/Claude lane (owner of this companion's gate), delegated AI +work for Dylan Roy, under the adapter route of Math- #428 comments 6112932068 and +6124097060; organizational-independence credit 0; no human review. It consumes the +C263 definitions, whose FORMAL HOLD extends to this module. The interface is +`specifications/lean_statements_20261010/Specifications/Field.lean`, blob +cc40e9d19446b8a150401c574d3cfd56177f195f, compiled in the same replay. +Target pin: Lean 4.34.1 / Mathlib d13f23b723b8a846827a245b89c10fc7d3f11612. +Elaboration, kernel audit and executed controls are recorded in separate source-bound +receipts; this text makes no execution claim. +-/ + +noncomputable section + +open UniversalLaw.Spec.Field + +namespace TorusFiltration + +/-- Proves: a smooth height in the Field sense is continuous. +Method: the `n = 0` instance of `IsSmooth` and `ContDiff.continuous`. +Does not establish: any regularity beyond continuity. -/ +theorem superlevelBarField_continuous {d : ℕ} {g : E d → ℝ} (hg : IsSmooth d g) : + Continuous g := + (hg 0).continuous + +/-- Proves: the C263 data inhabit the Field interface for every dimension and period. +Method: death height and essentiality are the C263 definitions, the maximin clause is +`superlevelBar_strict_cut`, and the elder relation is the right-hand side of its clause. +Does not establish: a persistence-module elder rule; `elderPartner_iff` holds by +definition, and periodicity and `L` are not used. -/ +def superlevelBarField_interface (d : ℕ) (L : ℝ) : SuperlevelBarInterface d L where + deathHeight := superlevelBar_deathHeight + IsEssential := superlevelBar_isEssential + ElderPartner := fun g M S => + IsNondegMax d g M ∧ IsIndexSaddle d g S ∧ ¬ superlevelBar_isEssential g M ∧ + superlevelBar_deathHeight g M = g S + essential_iff := fun g _ _ M => superlevelBar_essential_iff g M + deathHeight_maximin := fun g hs _ M hM h => + superlevelBar_strict_cut g (superlevelBarField_continuous hs) M hM h + elderPartner_iff := fun _ _ _ _ _ _ => Iff.rfl + +/-- Proves: the Field superlevel-bar interface is inhabited for every `d` and `L`. +Method: the record above. Does not establish: that any inhabitant is the source's +superlevel H0 object. -/ +theorem superlevelBarField_nonempty (d : ℕ) (L : ℝ) : + Nonempty (SuperlevelBarInterface d L) := + ⟨superlevelBarField_interface d L⟩ + +/-- Proves: the death height of the record is the C263 death height. +Method: `rfl`. Does not establish: a finite death for an essential point. -/ +theorem superlevelBarField_deathHeight_eq {d : ℕ} {L : ℝ} (g : E d → ℝ) (M : E d) : + (superlevelBarField_interface d L).deathHeight g M = superlevelBar_deathHeight g M := + rfl + +/-- Proves: essentiality in the record is global maximality. +Method: `Iff.rfl`. Does not establish: agreement with a persistence essential class. -/ +theorem superlevelBarField_isEssential_iff {d : ℕ} {L : ℝ} (g : E d → ℝ) (M : E d) : + (superlevelBarField_interface d L).IsEssential g M ↔ ∀ y, g y ≤ g M := + Iff.rfl + +/-- Proves: the elder relation of the record is, for every height, the right-hand side +of the interface clause. Method: `Iff.rfl`, since it is defined that way. +Does not establish: that this relation is ordinary elder death in any persistence sense. -/ +theorem superlevelBarField_elderPartner_iff {d : ℕ} {L : ℝ} (g : E d → ℝ) (M S : E d) : + (superlevelBarField_interface d L).ElderPartner g M S ↔ + (IsNondegMax d g M ∧ IsIndexSaddle d g S ∧ ¬ superlevelBar_isEssential g M ∧ + superlevelBar_deathHeight g M = g S) := + Iff.rfl + +/-- Proves: on smooth periodic heights every inhabitant's essentiality is global maximality. +Method: its own `essential_iff` clause. Does not establish: anything off that locus. -/ +theorem superlevelBarField_essential_unique {d : ℕ} {L : ℝ} (I : SuperlevelBarInterface d L) + (g : E d → ℝ) (hs : IsSmooth d g) (hp : IsLPeriodic d L g) (M : E d) : + I.IsEssential g M ↔ superlevelBar_isEssential g M := + I.essential_iff g hs hp M + +/-- Proves: on smooth periodic heights every inhabitant's death height at a nonessential +point is the C263 death height. Method: its maximin clause is a strict cut for the +clearance set, and `superlevelBar_cut_unique` leaves one such number. +Does not establish: agreement at essential points (see the control below). -/ +theorem superlevelBarField_deathHeight_unique {d : ℕ} {L : ℝ} (I : SuperlevelBarInterface d L) + (g : E d → ℝ) (hs : IsSmooth d g) (hp : IsLPeriodic d L g) (M : E d) + (hM : ¬ I.IsEssential g M) : + I.deathHeight g M = superlevelBar_deathHeight g M := by + have hM' : ¬ superlevelBar_isEssential g M := + fun h => hM ((superlevelBarField_essential_unique I g hs hp M).2 h) + exact superlevelBar_cut_unique g (superlevelBarField_continuous hs) M hM' _ + (fun h => I.deathHeight_maximin g hs hp M hM h) + +/-- Proves: on the Morse distinct-value locus of smooth periodic heights every inhabitant's +elder relation is the record's. Method: both are their clause's right-hand side, whose +essentiality and death-height terms agree by the two uniqueness theorems above. +Does not establish: agreement off the Morse distinct-value locus, where the interface +constrains no elder relation. -/ +theorem superlevelBarField_elderPartner_unique {d : ℕ} {L : ℝ} (I : SuperlevelBarInterface d L) + (g : E d → ℝ) (hs : IsSmooth d g) (hp : IsLPeriodic d L g) (hm : IsMorseDistinct d L g) + (M S : E d) : + I.ElderPartner g M S ↔ (superlevelBarField_interface d L).ElderPartner g M S := by + rw [I.elderPartner_iff g hs hp hm M S] + constructor + · rintro ⟨hmax, hsad, hne, hdeath⟩ + exact ⟨hmax, hsad, fun h => hne ((superlevelBarField_essential_unique I g hs hp M).2 h), + (superlevelBarField_deathHeight_unique I g hs hp M hne).symm.trans hdeath⟩ + · rintro ⟨hmax, hsad, hne, hdeath⟩ + have hne' : ¬ I.IsEssential g M := + fun h => hne ((superlevelBarField_essential_unique I g hs hp M).1 h) + exact ⟨hmax, hsad, hne', (superlevelBarField_deathHeight_unique I g hs hp M hne').trans hdeath⟩ + +/-- Proves: a second record, equal to the first except that its death height is the +constant `c` at essential points, also inhabits the interface. Method: off the essential +points it is the C263 death height, so the same strict cut applies; its elder relation is +again its clause's right-hand side. Does not establish: any preferred value at essential +points. -/ +def superlevelBarField_shift (d : ℕ) (L c : ℝ) : SuperlevelBarInterface d L where + deathHeight := fun g M => by + classical + exact if superlevelBar_isEssential g M then c else superlevelBar_deathHeight g M + IsEssential := superlevelBar_isEssential + ElderPartner := fun g M S => + IsNondegMax d g M ∧ IsIndexSaddle d g S ∧ ¬ superlevelBar_isEssential g M ∧ + (by classical + exact if superlevelBar_isEssential g M then c else superlevelBar_deathHeight g M) = g S + essential_iff := fun g _ _ M => superlevelBar_essential_iff g M + deathHeight_maximin := fun g hs _ M hM h => by + classical + rw [ite_eq_right hM] + exact superlevelBar_strict_cut g (superlevelBarField_continuous hs) M hM h + elderPartner_iff := fun _ _ _ _ _ _ => Iff.rfl + +/-- Proves: at a constant height the shifted record's death height is `c`. +Method: every point of a constant height is essential. Does not establish: a finite bar +for a constant height. -/ +theorem superlevelBarField_shift_deathHeight_const (d : ℕ) (L c b : ℝ) (M : E d) : + (superlevelBarField_shift d L c).deathHeight (fun _ => b) M = c := by + classical + show (if superlevelBar_isEssential (fun _ : E d => b) M then c + else superlevelBar_deathHeight (fun _ : E d => b) M) = c + have hM : superlevelBar_isEssential (fun _ : E d => b) M := fun _ => le_rfl + rw [ite_eq_left hM] + +/-- Proves: constant heights are smooth. Method: `contDiff_const`. +Does not establish: anything about nonconstant heights. -/ +theorem superlevelBarField_const_smooth (d : ℕ) (b : ℝ) : IsSmooth d (fun _ : E d => b) := + fun _ => contDiff_const + +/-- Proves: constant heights are `L`-periodic. Method: both sides are the constant. +Does not establish: anything about the lattice. -/ +theorem superlevelBarField_const_periodic (d : ℕ) (L b : ℝ) : + IsLPeriodic d L (fun _ : E d => b) := + fun _ _ => rfl + +/-- Proves: the nonessential hypothesis of `superlevelBarField_deathHeight_unique` cannot +be dropped: at the smooth periodic constant height `0` every point is essential for both +records, the record's death height there is `0`, its shift by `1` has death height `1`, +and the two differ. Method: `superlevelBar_death_const` and the shifted value. +Does not establish: anything about nonessential points, where all inhabitants agree. -/ +theorem superlevelBarField_control_essential_free (d : ℕ) (L : ℝ) (M : E d) : + IsSmooth d (fun _ : E d => (0 : ℝ)) ∧ IsLPeriodic d L (fun _ : E d => (0 : ℝ)) ∧ + (superlevelBarField_interface d L).IsEssential (fun _ => (0 : ℝ)) M ∧ + (superlevelBarField_shift d L 1).IsEssential (fun _ => (0 : ℝ)) M ∧ + (superlevelBarField_interface d L).deathHeight (fun _ => (0 : ℝ)) M = 0 ∧ + (superlevelBarField_shift d L 1).deathHeight (fun _ => (0 : ℝ)) M = 1 ∧ + (superlevelBarField_interface d L).deathHeight (fun _ => (0 : ℝ)) M ≠ + (superlevelBarField_shift d L 1).deathHeight (fun _ => (0 : ℝ)) M := by + have h0 : (superlevelBarField_interface d L).deathHeight (fun _ => (0 : ℝ)) M = 0 := by + rw [superlevelBarField_deathHeight_eq, superlevelBar_death_const] + have h1 : (superlevelBarField_shift d L 1).deathHeight (fun _ => (0 : ℝ)) M = 1 := + superlevelBarField_shift_deathHeight_const d L 1 0 M + refine ⟨superlevelBarField_const_smooth d 0, superlevelBarField_const_periodic d L 0, + fun _ => le_rfl, fun _ => le_rfl, h0, h1, ?_⟩ + rw [h0, h1] + norm_num + +end TorusFiltration diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/TorusTopology.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/TorusTopology.lean new file mode 100644 index 000000000..ed678ea32 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/companions/torus_filtration_20261011/TorusTopology.lean @@ -0,0 +1,247 @@ +import Mathlib + +/-! +# The actual period-lattice quotient and specified-start path compatibility + +C254 proposed source. Compilation and kernel audit: NOT_RUN. +The ambient space and lattice-vector formula below are exactly those in the +source-bound Field.lean; this module does not import or alter that owner's file. +All quotient, compactness, covering and descent properties are constructed. +The final lift endpoint is existential. No critical-level attainment is claimed. +Compile fixes at the pin (Lean v4.34.1, Mathlib d13f23b7), no statement altered: `Basis` → `Module.Basis`, `Basis.unitsSMul_apply` → `Module.Basis.unitsSMul_apply`, lattice discreteness passed via `SetLike.isDiscrete_iff_discreteTopology.mpr` instead of `DiscreteTopology.isDiscrete`, quotient path-connectedness via `inferInstanceAs (PathConnectedSpace (Quotient _))` instead of `infer_instance`. +Three unused simp arguments (`PiLp.single_apply` twice, `hij`) removed from `torus_basis_sum_eq_latticeVec` so the +file elaborates with warnings as errors; namespace `TorusFiltration` in this companion (handoff: `ResearchFormalCoreR1`). +-/ + +noncomputable section + +open Topology + +namespace TorusFiltration +abbrev torus_E (d : ℕ) := EuclideanSpace ℝ (Fin d) + +/-- The exact `Field.latticeVec` formula, retained without any coordinate rescaling. -/ +def torus_latticeVec (d : ℕ) (L : ℝ) (n : Fin d → ℤ) : torus_E d := + ∑ i, EuclideanSpace.single i (L * (n i : ℝ)) + +theorem torus_field_latticeVec_formula (d : ℕ) (L : ℝ) (n : Fin d → ℤ) : + torus_latticeVec d L n = ∑ i, EuclideanSpace.single i (L * (n i : ℝ)) := rfl + +def torus_periodBasis (d : ℕ) (L : ℝ) (hL : 0 < L) : + Module.Basis (Fin d) ℝ (torus_E d) := + (EuclideanSpace.basisFun (Fin d) ℝ).toBasis.unitsSMul + (fun _ => Units.mk0 L hL.ne') + +def torus_periodLattice (d : ℕ) (L : ℝ) (hL : 0 < L) : Submodule ℤ (torus_E d) := + Submodule.span ℤ (Set.range (torus_periodBasis d L hL)) + +abbrev torus_quotient (d : ℕ) (L : ℝ) (hL : 0 < L) := + torus_E d ⧸ (torus_periodLattice d L hL).toAddSubgroup + +def torus_proj (d : ℕ) (L : ℝ) (hL : 0 < L) : torus_E d → torus_quotient d L hL := + QuotientAddGroup.mk + +/-- Concrete adapter from integer basis coefficients to Field's exact vector formula. -/ +theorem torus_basis_sum_eq_latticeVec (d : ℕ) (L : ℝ) (hL : 0 < L) + (n : Fin d → ℤ) : + ∑ i, n i • torus_periodBasis d L hL i = torus_latticeVec d L n := by + classical + unfold torus_latticeVec + apply Finset.sum_congr rfl + intro i hi + ext j + by_cases hij : i = j + · subst j + simp [torus_periodBasis, Module.Basis.unitsSMul_apply, OrthonormalBasis.coe_toBasis, + EuclideanSpace.basisFun_apply, Units.smul_def, PiLp.smul_apply, + zsmul_eq_mul, mul_comm] + · simp [torus_periodBasis, Module.Basis.unitsSMul_apply, OrthonormalBasis.coe_toBasis, + EuclideanSpace.basisFun_apply, Units.smul_def, PiLp.smul_apply, + Ne.symm hij] + +theorem torus_lattice_mem_iff (d : ℕ) (L : ℝ) (hL : 0 < L) (x : torus_E d) : + x ∈ torus_periodLattice d L hL ↔ ∃ n : Fin d → ℤ, torus_latticeVec d L n = x := by + classical + change x ∈ Submodule.span ℤ (Set.range (torus_periodBasis d L hL)) ↔ _ + rw [Submodule.mem_span_range_iff_exists_fun] + simp only [torus_basis_sum_eq_latticeVec] + +instance torus_lattice_discrete (d : ℕ) (L : ℝ) (hL : 0 < L) : + DiscreteTopology (torus_periodLattice d L hL) := by + unfold torus_periodLattice + infer_instance + +instance torus_lattice_addSubgroup_discrete (d : ℕ) (L : ℝ) (hL : 0 < L) : + DiscreteTopology (torus_periodLattice d L hL).toAddSubgroup := by + unfold torus_periodLattice + infer_instance + +instance torus_lattice_isZLattice (d : ℕ) (L : ℝ) (hL : 0 < L) : + IsZLattice ℝ (torus_periodLattice d L hL) := by + unfold torus_periodLattice + infer_instance + +theorem torus_proj_surjective (d : ℕ) (L : ℝ) (hL : 0 < L) : + Function.Surjective (torus_proj d L hL) := QuotientAddGroup.mk_surjective + +theorem torus_proj_continuous (d : ℕ) (L : ℝ) (hL : 0 < L) : + Continuous (torus_proj d L hL) := continuous_quotient_mk' + +theorem torus_proj_eq_iff (d : ℕ) (L : ℝ) (hL : 0 < L) (x y : torus_E d) : + torus_proj d L hL x = torus_proj d L hL y ↔ + ∃ n : Fin d → ℤ, y = x + torus_latticeVec d L n := by + change (QuotientAddGroup.mk x : torus_quotient d L hL) = QuotientAddGroup.mk y ↔ _ + rw [QuotientAddGroup.eq] + change -x + y ∈ torus_periodLattice d L hL ↔ _ + rw [torus_lattice_mem_iff] + constructor + · rintro ⟨n, hn⟩ + exact ⟨n, by rw [hn]; abel⟩ + · rintro ⟨n, rfl⟩ + exact ⟨n, by abel⟩ + +/-- Duplicate representatives have exactly the same quotient image. -/ +theorem torus_control_lattice_translate (d : ℕ) (L : ℝ) (hL : 0 < L) + (x : torus_E d) (n : Fin d → ℤ) : + torus_proj d L hL (x + torus_latticeVec d L n) = torus_proj d L hL x := + ((torus_proj_eq_iff d L hL x _).2 ⟨n, rfl⟩).symm + +theorem torus_proj_isCoveringMap (d : ℕ) (L : ℝ) (hL : 0 < L) : + IsCoveringMap (torus_proj d L hL) := + ((torus_periodLattice d L hL).toAddSubgroup.isAddQuotientCoveringMap_of_comm + (SetLike.isDiscrete_iff_discreteTopology.mpr + (torus_lattice_addSubgroup_discrete d L hL))).isCoveringMap + +theorem torus_compact (d : ℕ) (L : ℝ) (hL : 0 < L) : + IsCompact (Set.univ : Set (torus_quotient d L hL)) := by + have hc := IsZLattice.isCompact_range_of_periodic (torus_periodLattice d L hL) + (torus_proj d L hL) (torus_proj_continuous d L hL) + (fun z w hw => QuotientAddGroup.mk_add_of_mem z hw) + rwa [Set.range_eq_univ.mpr (torus_proj_surjective d L hL)] at hc + +theorem torus_pathConnectedSpace (d : ℕ) (L : ℝ) (hL : 0 < L) : + PathConnectedSpace (torus_quotient d L hL) := by + exact inferInstanceAs <| PathConnectedSpace (Quotient _) + +/-- Exact Field periodicity predicate with the literal lattice-vector adapter above. -/ +def torus_isPeriodic (d : ℕ) (L : ℝ) (g : torus_E d → ℝ) : Prop := + ∀ (x : torus_E d) (n : Fin d → ℤ), g (x + torus_latticeVec d L n) = g x + +theorem torus_field_periodic_formula (d : ℕ) (L : ℝ) (g : torus_E d → ℝ) : + torus_isPeriodic d L g ↔ + ∀ (x : EuclideanSpace ℝ (Fin d)) (n : Fin d → ℤ), + g (x + ∑ i, EuclideanSpace.single i (L * (n i : ℝ))) = g x := Iff.rfl + +theorem torus_periodic_respects (d : ℕ) (L : ℝ) (hL : 0 < L) + (g : torus_E d → ℝ) (hg : torus_isPeriodic d L g) (x y : torus_E d) + (hxy : QuotientAddGroup.leftRel (torus_periodLattice d L hL).toAddSubgroup x y) : + g x = g y := by + have hq : torus_proj d L hL x = torus_proj d L hL y := Quotient.sound hxy + obtain ⟨n, rfl⟩ := (torus_proj_eq_iff d L hL x y).1 hq + exact (hg x n).symm + +def torus_descend (d : ℕ) (L : ℝ) (hL : 0 < L) + (g : torus_E d → ℝ) (hg : torus_isPeriodic d L g) : torus_quotient d L hL → ℝ := + Quotient.lift g (torus_periodic_respects d L hL g hg) + +theorem torus_descend_proj (d : ℕ) (L : ℝ) (hL : 0 < L) + (g : torus_E d → ℝ) (hg : torus_isPeriodic d L g) (x : torus_E d) : + torus_descend d L hL g hg (torus_proj d L hL x) = g x := rfl + +theorem torus_descend_continuous (d : ℕ) (L : ℝ) (hL : 0 < L) + (g : torus_E d → ℝ) (hg : torus_isPeriodic d L g) (hgc : Continuous g) : + Continuous (torus_descend d L hL g hg) := + hgc.quotient_lift (torus_periodic_respects d L hL g hg) + +/-- Lift from a prescribed representative; the endpoint is determined existentially. -/ +theorem torus_path_lift_from (d : ℕ) (L : ℝ) (hL : 0 < L) + {M : torus_E d} {y : torus_quotient d L hL} + (γ : Path (torus_proj d L hL M) y) : + ∃ z : torus_E d, ∃ Γ : Path M z, + torus_proj d L hL z = y ∧ + ∀ t : unitInterval, torus_proj d L hL (Γ t) = γ t := by + let cov := torus_proj_isCoveringMap d L hL + let Γ := cov.liftPath γ M γ.source + refine ⟨Γ 1, ⟨Γ, cov.liftPath_zero γ M γ.source, rfl⟩, ?_, ?_⟩ + · exact (congrFun (cov.liftPath_lifts γ M γ.source) 1).trans γ.target + · intro t + exact congrFun (cov.liftPath_lifts γ M γ.source) t + +/-- Control: a constant projected path cannot be lifted from M to a different lift. +This refutes replacing the existential endpoint by an arbitrary chosen representative. -/ +theorem torus_control_constant_path_endpoint (d : ℕ) (L : ℝ) (hL : 0 < L) + {M z : torus_E d} (Γ : Path M z) + (hΓ : ∀ t : unitInterval, torus_proj d L hL (Γ t) = torus_proj d L hL M) : z = M := by + have heq : (Γ : unitInterval → torus_E d) = fun _ => M := + (torus_proj_isCoveringMap d L hL).eq_of_comp_eq Γ.continuous continuous_const + (funext hΓ) 0 Γ.source + exact Γ.target.symm.trans (congrFun heq 1) + +/-- Combining the duplicate-image and uniqueness controls gives a direct rejection +of any different preselected endpoint for a constant quotient path. -/ +theorem torus_control_no_arbitrary_endpoint (d : ℕ) (L : ℝ) (hL : 0 < L) + (M : torus_E d) (n : Fin d → ℤ) (hn : torus_latticeVec d L n ≠ 0) : + torus_proj d L hL (M + torus_latticeVec d L n) = torus_proj d L hL M ∧ + ¬ ∃ Γ : Path M (M + torus_latticeVec d L n), + ∀ t : unitInterval, torus_proj d L hL (Γ t) = torus_proj d L hL M := by + refine ⟨torus_control_lattice_translate d L hL M n, ?_⟩ + rintro ⟨Γ, hΓ⟩ + have heq := torus_control_constant_path_endpoint d L hL Γ hΓ + apply hn + exact add_left_cancel (show M + torus_latticeVec d L n = M + 0 by simpa using heq) + +/-- Higher-endpoint compatibility for any scalar path predicate, including strict +and closed thresholds. No conclusion here asserts critical-level attainment. -/ +theorem torus_higher_path_iff (d : ℕ) (L : ℝ) (hL : 0 < L) + (g : torus_E d → ℝ) (hg : torus_isPeriodic d L g) (M : torus_E d) (P : ℝ → Prop) : + (∃ y : torus_E d, g M < g y ∧ ∃ γ : Path M y, ∀ t : unitInterval, P (g (γ t))) ↔ + (∃ y : torus_quotient d L hL, + torus_descend d L hL g hg (torus_proj d L hL M) < torus_descend d L hL g hg y ∧ + ∃ γ : Path (torus_proj d L hL M) y, + ∀ t : unitInterval, P (torus_descend d L hL g hg (γ t))) := by + constructor + · rintro ⟨y, hy, γ, hγ⟩ + exact ⟨torus_proj d L hL y, hy, γ.map (torus_proj_continuous d L hL), hγ⟩ + · rintro ⟨y, hy, γ, hγ⟩ + obtain ⟨z, Γ, hz, hΓ⟩ := torus_path_lift_from d L hL γ + refine ⟨z, ?_, Γ, ?_⟩ + · have hz' := congrArg (torus_descend d L hL g hg) hz + change g z = torus_descend d L hL g hg y at hz' + change g M < torus_descend d L hL g hg y at hy + rwa [← hz'] at hy + · intro t + have ht := hγ t + rw [← hΓ t] at ht + exact ht + +theorem torus_strict_higher_path_iff (d : ℕ) (L : ℝ) (hL : 0 < L) + (g : torus_E d → ℝ) (hg : torus_isPeriodic d L g) (M : torus_E d) (h : ℝ) : + (∃ y : torus_E d, g M < g y ∧ ∃ γ : Path M y, ∀ t : unitInterval, h < g (γ t)) ↔ + (∃ y : torus_quotient d L hL, + torus_descend d L hL g hg (torus_proj d L hL M) < torus_descend d L hL g hg y ∧ + ∃ γ : Path (torus_proj d L hL M) y, + ∀ t : unitInterval, h < torus_descend d L hL g hg (γ t)) := + torus_higher_path_iff d L hL g hg M (fun a => h < a) + +theorem torus_closed_higher_path_iff (d : ℕ) (L : ℝ) (hL : 0 < L) + (g : torus_E d → ℝ) (hg : torus_isPeriodic d L g) (M : torus_E d) (h : ℝ) : + (∃ y : torus_E d, g M < g y ∧ ∃ γ : Path M y, ∀ t : unitInterval, h ≤ g (γ t)) ↔ + (∃ y : torus_quotient d L hL, + torus_descend d L hL g hg (torus_proj d L hL M) < torus_descend d L hL g hg y ∧ + ∃ γ : Path (torus_proj d L hL M) y, + ∀ t : unitInterval, h ≤ torus_descend d L hL g hg (γ t)) := + torus_higher_path_iff d L hL g hg M (fun a => h ≤ a) + +theorem torus_global_max_iff (d : ℕ) (L : ℝ) (hL : 0 < L) + (g : torus_E d → ℝ) (hg : torus_isPeriodic d L g) (M : torus_E d) : + (∀ y : torus_E d, g y ≤ g M) ↔ + ∀ y : torus_quotient d L hL, + torus_descend d L hL g hg y ≤ torus_descend d L hL g hg (torus_proj d L hL M) := by + constructor + · intro h y + obtain ⟨z, rfl⟩ := torus_proj_surjective d L hL y + exact h z + · intro h y + exact h (torus_proj d L hL y) + +end TorusFiltration \ No newline at end of file diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/ElderBridge.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/ElderBridge.lean new file mode 100644 index 000000000..a59ee9daa --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/ElderBridge.lean @@ -0,0 +1,83 @@ +import ElderClearance +import ElderInterval + +/-! +Geometric input join for the actual ordinary-H0 interval construction. +Root01a11cbf, local author candidate, 17 October 2026. +No analytic fold, Morse uniqueness or torus theorem is assumed proved here. +-/ + +noncomputable section +open CategoryTheory Limits + +namespace TorusFiltration +universe u + +/-- Every active closed level includes continuously into the strict level +above d. Pulling back the proved strict separation gives closed-level separation. -/ +theorem elderTrap_closedTrace_isClopen {X : Type u} [TopologicalSpace X] + (g : X → ℝ) (D : Set X) (d t : ℝ) + (htrace : IsClopen (elderTrap_strictTrace g D d)) (ht : t < -d) : + IsClopen {x : superlevel_set g t | x.val ∈ D} := by + have hdt : d < -t := by linarith + let i : superlevel_set g t → {x : X // d < g x} := + fun x => ⟨x.val, lt_of_lt_of_le hdt x.property⟩ + have hi : Continuous i := continuous_subtype_val.subtype_mk _ + exact htrace.preimage hi + +/-- Actual geometric boundary, ceiling and closed path data inhabit the interval +trap. There is no H0 equation, interval split, clearance or death among inputs. -/ +theorem elderTrap_of_boundary {X : Type u} [TopologicalSpace X] + (g : X → ℝ) (D : Set X) (m d : ℝ) (M P : X) + (hD : IsClosed D) (hboundary : ∀ x ∈ frontier D, g x ≤ d) + (hdm : d < m) (hMheight : g M = m) (hM : M ∈ D) + (hheight : ∀ x ∈ D, g x ≤ m) (hP : m < g P) + (γ : Path M P) (hγ : ∀ t : unitInterval, d ≤ g (γ t)) : + ElderTrap g M P m d D where + death_lt_birth := hdm + maximum_height := hMheight + older_height := hP + maximum_inside := hM + older_outside := by + intro hPD + exact (not_lt_of_ge (hheight P hPD)) hP + height_ceiling := hheight + active_clopen := fun t ht => elderTrap_closedTrace_isClopen g D d t + (elderTrap_strictTrace_isClopen g D d hD hboundary) ht.2 + closed_path := by + refine ⟨γ, ?_⟩ + intro t + change -(-d) ≤ g (γ t) + simpa only [neg_neg] using hγ t + +/-- The constructed singular-H0 direct summand, instantiated from geometric +inputs, rather than a premise asserting that a persistence summand exists. -/ +def elderBoundaryDirectSummand (k : Type u) [Field k] + {X : Type u} [TopologicalSpace X] + (g : X → ℝ) (D : Set X) (m d : ℝ) (M P : X) + (hD : IsClosed D) (hboundary : ∀ x ∈ frontier D, g x ≤ d) + (hdm : d < m) (hMheight : g M = m) (hM : M ∈ D) + (hheight : ∀ x ∈ D, g x ≤ m) (hP : m < g P) + (γ : Path M P) (hγ : ∀ t : unitInterval, d ≤ g (γ t)) : + superlevel_h0Filtration g (ModuleCat.of k k) ≅ + elderInterval k m d ⊞ kernel (elderRetraction (k := k) + (elderTrap_of_boundary g D m d M P hD hboundary hdm hMheight hM hheight hP γ hγ)) := + elderDirectSummand (k := k) + (elderTrap_of_boundary g D m d M P hD hboundary hdm hMheight hM hheight hP γ hγ) + +/-- On Euclidean heights the same geometric inputs also give the literal C263 +death. This is a proved join, not identification of an abstract death field. -/ +theorem elderBoundary_death_eq {n : ℕ} + (g : superlevelBar_E n → ℝ) (hg : Continuous g) + (D : Set (superlevelBar_E n)) (m d : ℝ) (M P : superlevelBar_E n) + (hD : IsClosed D) (hboundary : ∀ x ∈ frontier D, g x ≤ d) + (hMheight : g M = m) (hM : M ∈ D) + (hheight : ∀ x ∈ D, g x ≤ m) (hP : m < g P) + (γ : Path M P) (hγ : ∀ t : unitInterval, d ≤ g (γ t)) : + superlevelBar_deathHeight g M = d := by + apply elderTrap_death_eq g hg D d + (elderTrap_strictTrace_isClopen g D d hD hboundary) M hM + (fun x hx => (hheight x hx).trans_eq hMheight.symm) P + (hMheight.trans_lt hP) γ hγ + +end TorusFiltration diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/ElderClearance.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/ElderClearance.lean new file mode 100644 index 000000000..bc6136ee5 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/ElderClearance.lean @@ -0,0 +1,129 @@ +import SuperlevelBar + +/-! +# From a geometric exit trap to the literal strict-clearance death + +Author: OpenAI/Codex ROOT01a11cbf. Local candidate, 17 October 2026. +This proves a topological interface used by the reviewed asymmetric fold trap. +It does not assume the desired elder relation, clearance cut or death value. +The hypotheses are an actual closed domain with boundary below the threshold, +a bound on all heights inside, and an actual higher-endpoint path at the closed +threshold. The analytic fold construction of these hypotheses, the torus lift, +Morse saddle uniqueness and the H0 interval split are separate obligations. +Compilation, kernel checking and alignment are recorded separately. +-/ + +noncomputable section + +namespace TorusFiltration + +/-- The trace of a geometric domain in the strict superlevel above d. -/ +def elderTrap_strictTrace {X : Type*} (g : X → ℝ) (D : Set X) (d : ℝ) : + Set {x : X // d < g x} := {x | x.val ∈ D} + +/-- A boundary height bound makes the trace clopen in the strict superlevel. +This derives separation from a geometric domain; it does not assume a path cut. -/ +theorem elderTrap_strictTrace_isClopen {X : Type*} [TopologicalSpace X] + (g : X → ℝ) (D : Set X) (d : ℝ) (hD : IsClosed D) + (hboundary : ∀ x ∈ frontier D, g x ≤ d) : + IsClopen (elderTrap_strictTrace g D d) := by + have heq : elderTrap_strictTrace g D d = + (Subtype.val : {x : X // d < g x} → X) ⁻¹' interior D := by + ext x + constructor + · intro hx + change x.val ∈ D at hx + change x.val ∈ interior D + apply (mem_interior_iff_notMem_frontier hx).2 + intro hfrontier + exact (not_lt_of_ge (hboundary x.val hfrontier)) x.property + · intro hx + change x.val ∈ interior D at hx + change x.val ∈ D + exact interior_subset hx + refine ⟨hD.preimage continuous_subtype_val, ?_⟩ + rw [heq] + exact isOpen_interior.preimage continuous_subtype_val + +/-- A path strictly above the exit threshold cannot leave the geometric trap. -/ +theorem elderTrap_path_stays {X : Type*} [TopologicalSpace X] + (g : X → ℝ) (D : Set X) (d : ℝ) + (htrace : IsClopen (elderTrap_strictTrace g D d)) + {M y : X} (hM : M ∈ D) (γ : Path M y) + (hγ : ∀ t : unitInterval, d < g (γ t)) : y ∈ D := by + let δ : unitInterval → {x : X // d < g x} := fun t => ⟨γ t, hγ t⟩ + have hδ : Continuous δ := γ.continuous.subtype_mk _ + have hpre := htrace.preimage hδ + have hzero : (0 : unitInterval) ∈ δ ⁻¹' elderTrap_strictTrace g D d := by + change γ 0 ∈ D + simpa only [γ.source] using hM + have hall := hpre.eq_univ ⟨0, hzero⟩ + have hone : (1 : unitInterval) ∈ δ ⁻¹' elderTrap_strictTrace g D d := by + rw [hall] + exact Set.mem_univ _ + change γ 1 ∈ D at hone + simpa only [γ.target] using hone + +/-- Every higher-endpoint path must meet or fall below the exit threshold. +The conclusion quantifies over all ambient paths, including remote detours. -/ +theorem elderTrap_higher_path_meets_threshold {X : Type*} [TopologicalSpace X] + (g : X → ℝ) (D : Set X) (d : ℝ) + (htrace : IsClopen (elderTrap_strictTrace g D d)) + (M : X) (hM : M ∈ D) (hheight : ∀ x ∈ D, g x ≤ g M) + {y : X} (hy : g M < g y) (γ : Path M y) : + ∃ t : unitInterval, g (γ t) ≤ d := by + by_contra hnone + push Not at hnone + have hyD := elderTrap_path_stays g D d htrace hM γ hnone + exact (not_lt_of_ge (hheight y hyD)) hy + +/-- The literal C263 clearance is exactly the strict interval below d. +The closed path is supplied by the geometric construction, rather than inferred +from a supremum over paths or from a compactness assertion about path space. -/ +theorem elderTrap_clearance_eq_Iio {n : ℕ} + (g : superlevelBar_E n → ℝ) (D : Set (superlevelBar_E n)) (d : ℝ) + (htrace : IsClopen (elderTrap_strictTrace g D d)) + (M : superlevelBar_E n) (hM : M ∈ D) + (hheight : ∀ x ∈ D, g x ≤ g M) + (P : superlevelBar_E n) (hP : g M < g P) (γ : Path M P) + (hγ : ∀ t : unitInterval, d ≤ g (γ t)) : + superlevelBar_clearanceSet g M = Set.Iio d := by + ext h + constructor + · rintro ⟨y, hy, η, hη⟩ + obtain ⟨t, ht⟩ := elderTrap_higher_path_meets_threshold g D d htrace M hM hheight hy η + exact lt_of_lt_of_le (hη t) ht + · intro hh + exact ⟨P, hP, γ, fun t => lt_of_lt_of_le hh (hγ t)⟩ + +/-- With continuity, the new geometric cut identifies the existing C263 death. +Nonessentiality is derived from the actual higher point P. -/ +theorem elderTrap_death_eq {n : ℕ} + (g : superlevelBar_E n → ℝ) (hg : Continuous g) + (D : Set (superlevelBar_E n)) (d : ℝ) + (htrace : IsClopen (elderTrap_strictTrace g D d)) + (M : superlevelBar_E n) (hM : M ∈ D) + (hheight : ∀ x ∈ D, g x ≤ g M) + (P : superlevelBar_E n) (hP : g M < g P) (γ : Path M P) + (hγ : ∀ t : unitInterval, d ≤ g (γ t)) : + superlevelBar_deathHeight g M = d := by + have hne : ¬ superlevelBar_isEssential g M := fun hess => + (not_lt_of_ge (hess P)) hP + apply (superlevelBar_cut_unique g hg M hne d ?_).symm + intro h + rw [elderTrap_clearance_eq_Iio g D d htrace M hM hheight P hP γ hγ] + rfl + +/-- The displayed closed path actually attains d. This does not replace strict +clearance by a closed inequality or infer an optimal path from a supremum. -/ +theorem elderTrap_closed_path_attains {X : Type*} [TopologicalSpace X] + (g : X → ℝ) (D : Set X) (d : ℝ) + (htrace : IsClopen (elderTrap_strictTrace g D d)) + (M : X) (hM : M ∈ D) (hheight : ∀ x ∈ D, g x ≤ g M) + {P : X} (hP : g M < g P) (γ : Path M P) + (hγ : ∀ t : unitInterval, d ≤ g (γ t)) : + ∃ t : unitInterval, g (γ t) = d := by + obtain ⟨t, ht⟩ := elderTrap_higher_path_meets_threshold g D d htrace M hM hheight hP γ + exact ⟨t, le_antisymm ht (hγ t)⟩ + +end TorusFiltration diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/ElderInterval.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/ElderInterval.lean new file mode 100644 index 000000000..108f880b3 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/ElderInterval.lean @@ -0,0 +1,392 @@ +import OrdinaryH0 + +/-!+# One actual ordinary-H0 interval summand from an asymmetric trap + +Author implementation. The geometric premises remain visible; this file does not +prove the analytic ridge estimates, Morse labeling, or an exhaustive barcode. +The coefficient object is the field itself, and H0 is the existing singular +homology functor, compared with components by the proved C264 isomorphism. +-/ + +noncomputable section + +open CategoryTheory Limits AlgebraicTopology +open scoped Classical + +namespace TorusFiltration + +universe u + +/-- Increasing parameter `t=-height`: closed birth and open death. -/ +def elderActive (m d t : ℝ) : Prop := -m ≤ t ∧ t < -d + +variable (k : Type u) [Field k] + +/-- The fiber is k on the interval and the zero submodule elsewhere. -/ +def elderFiber (m d t : ℝ) : Submodule k k where + carrier := {a | ¬ elderActive m d t → a = 0} + zero_mem' := by simp + add_mem' := by intro a b ha hb h; simp [ha h, hb h] + smul_mem' := by intro a b hb h; simp [hb h] + +theorem elderFiber_eq_zero {m d t : ℝ} (ht : ¬ elderActive m d t) + (x : elderFiber k m d t) : x = 0 := by + apply Subtype.ext + exact x.property ht + +def elderFiberMap (m d : ℝ) {s t : ℝ} (hst : s ≤ t) : + elderFiber k m d s →ₗ[k] elderFiber k m d t where + toFun x := ⟨if t < -d then x.val else 0, by + intro ht + by_cases htd : t < -d + · have hsa : ¬ elderActive m d s := by + intro hs + exact ht ⟨hs.1.trans hst, htd⟩ + simp [htd, x.property hsa] + · simp [htd]⟩ + map_add' := by intro x y; apply Subtype.ext; dsimp; split_ifs <;> simp + map_smul' := by intro a x; apply Subtype.ext; dsimp; split_ifs <;> simp + +@[simp] theorem elderFiberMap_val (m d : ℝ) {s t : ℝ} (hst : s ≤ t) + (x : elderFiber k m d s) : + (elderFiberMap k m d hst x).val = if t < -d then x.val else 0 := rfl + +/-- The actual interval persistence functor; no maps are assumed. -/ +def elderInterval (m d : ℝ) : ℝ ⥤ ModuleCat k where + obj t := ModuleCat.of k (elderFiber k m d t) + map f := ModuleCat.ofHom (elderFiberMap k m d (leOfHom f)) + map_id t := by + apply ModuleCat.hom_ext + apply LinearMap.ext + intro x + apply Subtype.ext + change (if t < -d then x.val else 0) = x.val + by_cases ht : t < -d + · simp [ht] + · have hn : ¬ elderActive m d t := fun h => ht h.2 + simp [ht, x.property hn] + map_comp {r s t} f g := by + apply ModuleCat.hom_ext + apply LinearMap.ext + intro x + apply Subtype.ext + change (if t < -d then x.val else 0) = + if t < -d then (if s < -d then x.val else 0) else 0 + by_cases ht : t < -d + · have hs : s < -d := (leOfHom g).trans_lt ht + simp [ht, hs] + · simp [ht] + +/-- A canonical lift of a coefficient to any active interval fiber. -/ +def elderFiberLift {m d t : ℝ} (ht : elderActive m d t) : + k →ₗ[k] elderFiber k m d t where + toFun a := ⟨a, fun hn => (hn ht).elim⟩ + map_add' := by intros; rfl + map_smul' := by intros; rfl + +@[simp] theorem elderFiberLift_val {m d t : ℝ} (ht : elderActive m d t) (a : k) : + (elderFiberLift k ht a).val = a := rfl + +/-- Endpoint control: birth is included. -/ +theorem elder_control_birth {m d : ℝ} (hdm : d < m) : elderActive m d (-m) := by + constructor <;> linarith + +/-- Endpoint control: death is excluded, despite closed superlevels. -/ +theorem elder_control_death (m d : ℝ) : ¬ elderActive m d (-d) := by + intro h + exact (lt_irrefl _) h.2 + +variable {X : Type u} [TopologicalSpace X] (g : X → ℝ) (M P : X) (m d : ℝ) + +/-- The topological consequences of the asymmetric ridge trap. No H0 maps or +splitting conclusions occur among the premises. The path at d is actual closed +superlevel connectivity, while active cuts are clopen. -/ +structure ElderTrap (D : Set X) : Prop where + death_lt_birth : d < m + maximum_height : g M = m + older_height : m < g P + maximum_inside : M ∈ D + older_outside : P ∉ D + height_ceiling : ∀ x ∈ D, g x ≤ m + active_clopen : ∀ t, elderActive m d t → + IsClopen {x : superlevel_set g t | x.val ∈ D} + closed_path : JoinedIn (superlevel_set g (-d)) M P + +variable {g M P m d} {D : Set X} + +theorem ElderTrap.maximum_mem (T : ElderTrap g M P m d D) {t : ℝ} + (ht : elderActive m d t) : M ∈ superlevel_set g t := by + change -t ≤ g M + erw [T.maximum_height] + linarith [ht.1] + +theorem ElderTrap.older_mem (T : ElderTrap g M P m d D) {t : ℝ} + (ht : elderActive m d t) : P ∈ superlevel_set g t := by + change -t ≤ g P + linarith [ht.1, T.older_height] + +/-- Clopen cuts are constant on actual path components, with no local path +connectivity assumption on the ambient space. -/ +theorem elder_clopen_path_iff {Y : Type u} [TopologicalSpace Y] {U : Set Y} + (hU : IsClopen U) {x y : Y} (p : Path x y) : x ∈ U ↔ y ∈ U := by + have hxy : y ∈ connectedComponent x := + pathComponent_subset_component x (show y ∈ pathComponent x from ⟨p⟩) + have hyx : x ∈ connectedComponent y := + pathComponent_subset_component y (show x ∈ pathComponent y from ⟨p.symm⟩) + exact ⟨fun hx => hU.connectedComponent_subset hx hxy, + fun hy => hU.connectedComponent_subset hy hyx⟩ + +/-- The D indicator is genuinely descended to the path-component quotient. -/ +def ElderTrap.componentInside (T : ElderTrap g M P m d D) {t : ℝ} + (ht : elderActive m d t) : ZerothHomotopy (superlevel_set g t) → Prop := + ZerothHomotopy.lift (fun x => x.val ∈ D) (fun _ _ p => + propext (elder_clopen_path_iff (T.active_clopen t ht) p)) + +@[simp] theorem ElderTrap.componentInside_mk (T : ElderTrap g M P m d D) + {t : ℝ} (ht : elderActive m d t) (x : superlevel_set g t) : + T.componentInside ht (ZerothHomotopy.mk x) ↔ x.val ∈ D := Iff.rfl + +variable {k} + +/-- The actual singular-H0 point-class difference on an active level. -/ +def elderDifference (T : ElderTrap g M P m d D) {t : ℝ} + (ht : elderActive m d t) : + ModuleCat.of k k ⟶ (superlevel_h0Filtration g (ModuleCat.of k k)).obj t := + ordinaryH0_pointClass (TopCat.of (superlevel_set g t)) (ModuleCat.of k k) + ⟨M, T.maximum_mem ht⟩ - + ordinaryH0_pointClass (TopCat.of (superlevel_set g t)) (ModuleCat.of k k) + ⟨P, T.older_mem ht⟩ + +theorem elderDifference_map (T : ElderTrap g M P m d D) {s t : ℝ} + (hs : elderActive m d s) (ht : elderActive m d t) (hst : s ≤ t) : + elderDifference (k := k) T hs ≫ + (superlevel_h0Filtration g (ModuleCat.of k k)).map (homOfLE hst) = + elderDifference T ht := by + exact ordinaryH0_difference_map (ModuleCat.of k k) + (TopCat.ofHom (superlevel_inclusion g hst)) + ⟨M, T.maximum_mem hs⟩ ⟨P, T.older_mem hs⟩ + +theorem elderDifference_map_dead (T : ElderTrap g M P m d D) {s t : ℝ} + (hs : elderActive m d s) (ht : -d ≤ t) (hst : s ≤ t) : + elderDifference (k := k) T hs ≫ + (superlevel_h0Filtration g (ModuleCat.of k k)).map (homOfLE hst) = 0 := by + unfold elderDifference + erw [ordinaryH0_difference_map] + apply ordinaryH0_sameComponent_difference_zero + apply (superlevel_component_eq_iff_joinedIn g t _ _).2 + exact T.closed_path.mono (superlevel_mono g ht) + +/-- The component indicator transported by the existing actual H0 comparison. -/ +def elderProjection (T : ElderTrap g M P m d D) {t : ℝ} + (ht : elderActive m d t) : + (superlevel_h0Filtration g (ModuleCat.of k k)).obj t ⟶ ModuleCat.of k k := by + classical + exact (ordinaryH0_filtrationIso g (ModuleCat.of k k)).hom.app t ≫ + Sigma.desc (fun q => if T.componentInside ht q then 𝟙 _ else 0) + +theorem elderProjection_point (T : ElderTrap g M P m d D) {t : ℝ} + (ht : elderActive m d t) (x : superlevel_set g t) : + ordinaryH0_pointClass (TopCat.of (superlevel_set g t)) (ModuleCat.of k k) x ≫ + elderProjection (k := k) T ht = + if x.val ∈ D then 𝟙 _ else 0 := by + classical + unfold elderProjection + erw [← Category.assoc] + erw [ordinaryH0_pointClass_iso] + erw [Sigma.ι_comp_desc] + rfl + +theorem elderDifference_projection (T : ElderTrap g M P m d D) {t : ℝ} + (ht : elderActive m d t) : + elderDifference (k := k) T ht ≫ elderProjection T ht = 𝟙 _ := by + classical + unfold elderDifference + erw [Preadditive.sub_comp, + elderProjection_point, elderProjection_point] + simp [T.maximum_inside, T.older_outside] + +def elderSectionApp (T : ElderTrap g M P m d D) (t : ℝ) : + (elderInterval k m d).obj t ⟶ + (superlevel_h0Filtration g (ModuleCat.of k k)).obj t := by + classical + exact if ht : elderActive m d t then + ModuleCat.ofHom (elderFiber k m d t).subtype ≫ elderDifference T ht else 0 + +def elderRetractionApp (T : ElderTrap g M P m d D) (t : ℝ) : + (superlevel_h0Filtration g (ModuleCat.of k k)).obj t ⟶ + (elderInterval k m d).obj t := by + classical + exact if ht : elderActive m d t then + elderProjection T ht ≫ ModuleCat.ofHom (elderFiberLift k ht) else 0 + +theorem elderSectionApp_naturality (T : ElderTrap g M P m d D) {s t : ℝ} + (hst : s ≤ t) : + (elderInterval k m d).map (homOfLE hst) ≫ elderSectionApp T t = + elderSectionApp T s ≫ + (superlevel_h0Filtration g (ModuleCat.of k k)).map (homOfLE hst) := by + classical + by_cases hs : elderActive m d s + · by_cases ht : elderActive m d t + · simp only [elderSectionApp, dite_eq_left hs, dite_eq_left ht] + erw [Category.assoc, elderDifference_map T hs ht hst, ← Category.assoc] + congr 1 + ext x + change (if t < -d then x.val else 0) = x.val + simp [ht.2] + · have htd : -d ≤ t := by + by_contra h + exact ht ⟨hs.1.trans hst, lt_of_not_ge h⟩ + simp only [elderSectionApp, dite_eq_left hs, dite_eq_right ht, comp_zero] + erw [Category.assoc, elderDifference_map_dead T hs htd hst, comp_zero] + · apply ModuleCat.hom_ext + apply LinearMap.ext + intro x + have hx : x = 0 := elderFiber_eq_zero k hs x + simp [hx] + +/-- Section from the interval to actual singular H0. -/ +def elderSection (T : ElderTrap g M P m d D) : + elderInterval k m d ⟶ superlevel_h0Filtration g (ModuleCat.of k k) where + app := elderSectionApp T + naturality := by + intro s t f + exact elderSectionApp_naturality T (leOfHom f) + +theorem elderProjection_naturality_active (T : ElderTrap g M P m d D) {s t : ℝ} + (hs : elderActive m d s) (ht : elderActive m d t) (hst : s ≤ t) : + (superlevel_h0Filtration g (ModuleCat.of k k)).map (homOfLE hst) ≫ + elderProjection (k := k) T ht = elderProjection T hs := by + classical + apply ordinaryH0_hom_ext (TopCat.of (superlevel_set g s)) (ModuleCat.of k k) + intro x + erw [← Category.assoc] + erw [ordinaryH0_pointClass_map] + erw [elderProjection_point, elderProjection_point] + rfl + +/-- The birth-crossing square follows from the geometric height ceiling: +there is no source point in D above m. It is not an assumed naturality field. -/ +theorem elderProjection_prebirth (T : ElderTrap g M P m d D) {s t : ℝ} + (hs : s < -m) (ht : elderActive m d t) (hst : s ≤ t) : + (superlevel_h0Filtration g (ModuleCat.of k k)).map (homOfLE hst) ≫ + elderProjection (k := k) T ht = 0 := by + classical + apply ordinaryH0_hom_ext (TopCat.of (superlevel_set g s)) (ModuleCat.of k k) + intro x + erw [← Category.assoc] + erw [ordinaryH0_pointClass_map] + erw [elderProjection_point] + change (if x.val ∈ D then 𝟙 _ else 0) = _ + have hx : x.val ∉ D := by + intro hx + have hceil := T.height_ceiling x.val hx + have hlevel : -s ≤ g x.val := x.property + linarith + simp only [ite_eq_right hx] + erw [comp_zero] + +theorem elderRetractionApp_naturality (T : ElderTrap g M P m d D) {s t : ℝ} + (hst : s ≤ t) : + (superlevel_h0Filtration g (ModuleCat.of k k)).map (homOfLE hst) ≫ + elderRetractionApp T t = + elderRetractionApp T s ≫ (elderInterval k m d).map (homOfLE hst) := by + classical + by_cases ht : elderActive m d t + · by_cases hs : elderActive m d s + · simp only [elderRetractionApp, dite_eq_left hs, dite_eq_left ht] + apply ModuleCat.hom_ext + apply LinearMap.ext + intro x + apply Subtype.ext + change elderProjection T ht + ((superlevel_h0Filtration g (ModuleCat.of k k)).map (homOfLE hst) x) = + if t < -d then elderProjection T hs x else 0 + rw [ite_eq_left ht.2] + have he := congrArg + (fun f : (superlevel_h0Filtration g (ModuleCat.of k k)).obj s ⟶ + ModuleCat.of k k => f x) + (elderProjection_naturality_active (k := k) T hs ht hst) + exact he + · have hsm : s < -m := by + by_contra h + exact hs ⟨le_of_not_gt h, hst.trans_lt ht.2⟩ + simp only [elderRetractionApp, dite_eq_right hs, dite_eq_left ht, zero_comp] + erw [← Category.assoc, elderProjection_prebirth T hsm ht hst, zero_comp] + · simp only [elderRetractionApp, dite_eq_right ht, comp_zero] + by_cases hs : elderActive m d s + · simp only [dite_eq_left hs] + have htd : ¬ t < -d := by + intro h + exact ht ⟨hs.1.trans hst, h⟩ + apply ModuleCat.hom_ext + apply LinearMap.ext + intro x + apply Subtype.ext + change 0 = if t < -d then _ else 0 + simp [htd] + · simp [hs] + +/-- Retraction from actual singular H0 to the interval, using component descent. -/ +def elderRetraction (T : ElderTrap g M P m d D) : + superlevel_h0Filtration g (ModuleCat.of k k) ⟶ elderInterval k m d where + app := elderRetractionApp T + naturality := by + intro s t f + exact elderRetractionApp_naturality T (leOfHom f) + +/-- The constructed section and retraction split the actual interval functor. -/ +@[reassoc (attr := simp)] +theorem elderSection_retraction (T : ElderTrap g M P m d D) : + elderSection (k := k) T ≫ elderRetraction T = 𝟙 (elderInterval k m d) := by + classical + apply NatTrans.ext + funext t + change elderSectionApp (k := k) T t ≫ elderRetractionApp (k := k) T t = 𝟙 _ + by_cases ht : elderActive m d t + · simp only [elderSectionApp, elderRetractionApp, dite_eq_left ht] + erw [Category.assoc, ← Category.assoc (elderDifference T ht), + elderDifference_projection, Category.id_comp] + ext x + rfl + · ext x + have hx : x = 0 := elderFiber_eq_zero k ht x + simp [hx] + +def elderKernelProjection (T : ElderTrap g M P m d D) : + superlevel_h0Filtration g (ModuleCat.of k k) ⟶ kernel (elderRetraction (k := k) T) := + kernel.lift (elderRetraction T) + (𝟙 _ - elderRetraction T ≫ elderSection T) (by + erw [Preadditive.sub_comp, Category.id_comp, Category.assoc, + elderSection_retraction, Category.comp_id, sub_self]) + +/-- An actual natural direct-sum isomorphism, not a postulated decomposition. +Its maps are v ↦ (qv,v-jqv) and (c,w) ↦ jc+w. -/ +def elderDirectSummand (T : ElderTrap g M P m d D) : + superlevel_h0Filtration g (ModuleCat.of k k) ≅ + elderInterval k m d ⊞ kernel (elderRetraction (k := k) T) where + hom := biprod.lift (elderRetraction T) (elderKernelProjection T) + inv := biprod.desc (elderSection T) (kernel.ι (elderRetraction T)) + hom_inv_id := by + erw [biprod.lift_desc] + simp only [elderKernelProjection, kernel.lift_ι] + abel + inv_hom_id := by + apply biprod.hom_ext' <;> apply biprod.hom_ext + · simp [Category.assoc, elderSection_retraction] + · apply (cancel_mono (kernel.ι (elderRetraction T))).1 + simp [Category.assoc, elderKernelProjection, Preadditive.comp_sub] + · simp + · apply (cancel_mono (kernel.ι (elderRetraction T))).1 + simp [elderKernelProjection, Category.assoc, Preadditive.comp_sub] + +/-- Ordinary individual point classes remain nonzero; the dying section uses +their difference. This is valid independently of characteristic. -/ +theorem elder_control_ordinary_point_survives (t : ℝ) (x : superlevel_set g t) : + ordinaryH0_pointClass (TopCat.of (superlevel_set g t)) (ModuleCat.of k k) x ≠ 0 := by + apply ordinaryH0_pointClass_ne_zero + intro h + have hl := congrArg ModuleCat.Hom.hom h + change (LinearMap.id : k →ₗ[k] k) = 0 at hl + exact one_ne_zero (LinearMap.congr_fun hl 1) + +end TorusFiltration diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/provenance/INTERFACE_BOUNDARIES.md b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/provenance/INTERFACE_BOUNDARIES.md new file mode 100644 index 000000000..c1f0ae9cb --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/asymmetric_elder_bridge_20261017/provenance/INTERFACE_BOUNDARIES.md @@ -0,0 +1,43 @@ +# Source-bound local elder interfaces + +This is a dependency map for this local candidate, not a global project claim or +scientific-status register. Actual root pickup: Main419/6240826507. Exact source +baseline: Math `dc7aca2f9adaf6d6d986b0e34b7b39c8fc3cf417`; consumed companion +bytes are identical to the earlier pinned `319f9c45` cut. All new source is a +candidate for `frontiers/asymmetric_elder_bridge_20261017/`, with no remote +branch, PR, hosted event or adoption in this stage. + +| Interface | Actual construction or input | Formal boundary | +|---|---|---| +| Smooth fold chart gives a closed asymmetric domain D, boundary height at most d, inside height at most m, and an actual higher-endpoint path at closed height d | Exact conventional asymmetric proof SHA `6d07631c068019bf9f081ad0b6f50325679e3ab554f221265e002d5c047f17ac`; actual separate integrated review SHA `16678abc7de50986b0fc6d54303449bf1b4c5ba8b10f6a3ccc7f9d03a97a4e9f` | Analytic construction is not implemented in Lean here. These geometric facts remain explicit hypotheses, not a death/bar/splitting certificate. | +| Boundary bounds imply strict-superlevel separation | `elderTrap_strictTrace_isClopen` | Derives clopenness from closed D and the actual frontier height bound. No elder premise. | +| Every higher-endpoint path meets height at most d | `elderTrap_path_stays`, `elderTrap_higher_path_meets_threshold` | Proves global path exclusion, including remote detours, from that separation and inside height ceiling. | +| Literal strict-clearance cut and real death | `elderTrap_clearance_eq_Iio`, `elderTrap_death_eq` | Uses unchanged C263 definitions and cut uniqueness. Continuity is explicit for the real death theorem. Nonessentiality follows from the actual higher endpoint. | +| Closed path attains d | `elderTrap_closed_path_attains` | This is separate from strict clearance at d, which is excluded. No path-space compactness is assumed. | +| Actual singular H0 versus path-component coproduct | Unchanged C264 `ordinaryH0_naturalIso`, filtration isomorphism and point-class formulas | Actual Mathlib singular homology, not an assumed H0 comparison field. Exact current source is consumed; its legacy alignment/HOLD is not lifted here. | +| Interval `[-m,-d)` and its maps | `elderInterval`, `elderFiberMap` | Constructs the interval functor, including endpoint conventions, rather than assuming an interval module. | +| Natural section and retraction | `elderSection`, `elderRetraction` | Constructs the point-class difference and component indicator. The prebirth square follows from the geometric ceiling; the death square follows from the actual closed path. | +| One actual interval direct summand | `elderSection_retraction`, `elderDirectSummand` | Constructs the natural kernel-complement isomorphism. No exhaustive or simultaneous barcode decomposition is concluded. | +| Strict-level separation implies every active closed-level separation | `elderTrap_closedTrace_isClopen` | Pulls back along the actual continuous inclusion of a closed level above d into the strict d-level. | +| Geometric domain/path data give the interval trap and actual split | `elderTrap_of_boundary`, `elderBoundaryDirectSummand` | The constructor derives the older point's outside membership and active clopenness. No H0 equality, natural map, death or splitting conclusion occurs in its hypotheses. | +| The same Euclidean geometric inputs give C263 death d | `elderBoundary_death_eq` | A proved adapter to literal C263, not identification with an independently assumed abstract death. | +| Compact Morse labeling, unique quotient saddle, torus lift, all-bar counting and stochastic convergence | Explicit conventional inputs and separate research packets | Not newly formalized or established by this local interface candidate. Periodic lifts are not asserted unique. | + +The coefficient field is nonzero by its actual `Field` instance. The ordinary +individual point class is proved nonzero; the interval section is the difference +`[M]-[P]`, including in characteristic two. No desired elder relation, death +value, H0 natural map or split is supplied as a premise in the geometric adapter. + +Executed elaboration, controls, axiom prints, negative cases and kernel replay +are reported only by their actual receipts. The author handoff freezes some +earlier statuses as PENDING; later results are additive records, never rewrites +of that seal. The named-target axiom census and the kernel replay of all new +module constants are different scopes. Standard `leanchecker Target` replays +Target's own new declarations against its imported environment. It is the same +Lean kernel, not an external verifier and not `--fresh` replay of all Mathlib. + +New source authors are Root01a11cbf and the separate interval implementation +agent. The substantive source/result reviewer is a separate source-exposed +OpenAI agent, same provider, organizational-independence credit zero. That +review does not manufacture human/provider independence or institutional +statement alignment. Scientific effect is NONE; full research goal ACTIVE/UNMET. diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/DEPENDENCIES.json b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/DEPENDENCIES.json new file mode 100644 index 000000000..fa6f75412 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/DEPENDENCIES.json @@ -0,0 +1,502 @@ +{ + "schema": "accepted-theory-source-dependencies-v1", + "state": "SOURCE_COMPOSITION_CANDIDATE_NOT_ADOPTED", + "edges": [ + { + "from": "higher_configuration", + "to": "asymmetric_elder", + "meaning": "Exact candidate compact limit input; asymmetric geometry supplies the separate elder selection" + }, + { + "from": "asymmetric_elder", + "to": "reverse_full_projected_ppp", + "meaning": "Actual first elder identification after reverse birth-side localization" + }, 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b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/README.md @@ -0,0 +1,74 @@ +# Exact accepted Gaussian short-bar theory sources — source-adoption candidate + +This directory is a source-author composition proposal carrying full unchanged +proof/review bytes. It is not a new proof, an independent review, a scientific +status register or an integration authorization. The source map gives exact +identities and preserved review scope. Previously accepted conventional +mathematics and historical formal execution remain distinct from current-source +review, hosted checks, institutional alignment and scientific acceptance. + +The four full conventional proofs and their full actual integrated/focused +reviews are organized as follows: + +- [Higher configurations](higher_configuration/PROOF.md), with its + [actual integrated review](higher_configuration/REVIEW.md): every fixed + factorial order and compact regular-mark candidate Laplace/vague limit for + one declared Gaussian cutoff family, one fixed B. This alone is not an + elder-bar or singular-mark theorem. +- [Asymmetric global elder](asymmetric_elder/PROOF.md), with its + [actual integrated review](asymmetric_elder/REVIEW.md): deterministic + first-elder attainment/one ordinary-H0 interval summand and regular-sector + coupling. The older full enclosing fold-height barrier remains false. +- [Reverse localization](reverse_full_projected_ppp/PROOF.md), with its + [actual integrated review](reverse_full_projected_ppp/REVIEW.md): full + lifetime/spatial projected finite-bar Poisson limit for that same ideal + Gaussian family, including the zero boundary through complement bounds. + This is not all-mark higher factorial convergence or general universality. +- [Smooth manifold Gaussian admission](smooth_gaussian_manifold_admission/PROOF.md), + with its [actual integrated review](smooth_gaussian_manifold_admission/REVIEW.md): + expected finite ordinary-H0 short-bar intensity for the explicit positive, + smooth-dense, all-norm-summable independent Gaussian series on each fixed + compact connected boundaryless smooth Riemannian manifold, fixed d>=1. + The density is asymptotic to C ell^(-1/3) as ell decreases to zero; the + cumulative expected count per volume is asymptotic to (3/2)C t^(2/3) as + t decreases to zero. No exact finite-scale law or general process limit + follows from this admission result. + +The distinct [formal geometric source slice](../asymmetric_elder_bridge_20261017/) +contains unchanged Lean geometric-clearance/interval/bridge sources and actual +historical source/result review records. It consumes the exact existing Math +C263/C264 companion sources identified by SOURCE_MAP.json. Its geometric +premises remain explicit; it does not formalize the analytic chart, torus, +canonical Morse, stochastic or all-bar joins. Same-provider reviews earn zero +organizational independence and do not discharge institutional alignment. + +DEPENDENCIES.json gives the exact directed source relationships, existing +Math companions, current three-file Lean/Lake configuration reuse and all nine +immutable package pins. No active Lake project, runner, workflow, formal manifest +or gate is added by this candidate. In particular, the inherited default +ResearchFormalCoreR1 target does not build the staged elder modules. A separately +reviewed new path/import wrapper and explicit target/receipt registration are +needed before any new execution claim. Historical absolute-path wrappers and +all original failures/raw streams/own oleans remain in their exact recovery +packet; no binary, external olean or dependency/cache tree is copied here. + +Original proofs and review text are never edited to repair paths or historical +PENDING labels. SOURCE_MAP.json supplies the relocation crosswalk. Original +sealed metadata containing private filesystem locators stays unchanged in +retained custody, represented publicly by exact digest/role references. The +relative root-sketch link has an explicitly labeled routing wrapper; it is +not a reconstructed original sketch or a replacement proof. Original seals +are historical evidence with original path bases, not claims of complete +physical input membership in this minimal source directory. + +Already existing Main sources are referenced at immutable commits and blobs, +including the open Main429 dependency packet. An open PR object is not a +landed default source or an ownership transfer. The existing higher-homology +source is [Main #331](https://github.com/d6g8k5htny-coder/main/pull/331), with the +[exact existing proof](https://github.com/d6g8k5htny-coder/main/blob/89657c7c48289378949dc59e7fbe6dc1a55d9c62/experiments/universality/higher_homology_fold/PROOF.md) +and [review record](https://github.com/d6g8k5htny-coder/main/blob/89657c7c48289378949dc59e7fbe6dc1a55d9c62/experiments/universality/higher_homology_fold/REVIEW.md) +at its frozen head; reuse/link it for any later adapter +rather than duplicate that mathematics. Its documentary D1 and source-chain +composition remain separate. 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their own dated scope. No project status transition.", + "scientific_effect": "NONE", + "source_adoption": false +} diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/asymmetric_elder/PROOF.md b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/asymmetric_elder/PROOF.md new file mode 100644 index 000000000..66d9c4c7f --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/asymmetric_elder/PROOF.md @@ -0,0 +1,301 @@ +# An asymmetric fold trap, its attained elder death, and an actual H0 interval summand + +16 October 2026 UTC. **New local author proposal, pending separate nonauthor review.** Authors: OpenAI/Codex Root (the sketch and interval-splitting steering) and `/root/core_theorem_audit_20261015` (reconstruction, attack, quantitative join and this report). Actual same-provider organizational-independence credit **0**. No helper was spawned for this task. No production/native/remote/runtime/Lean edit, source adoption, ownership transfer or scientific-status transition. The frozen higher-configuration proposal, its evidence, and the accepted source-publication packet are excluded and unchanged. + +The asymmetric trap works under the explicit hypotheses below. It directly proves that the strict-clearance death is the actual saddle height, with a path attaining that level, and excludes every earlier remote elder connection. It also gives an explicit interval **direct summand** of actual ordinary singular H0 by a section/retraction argument, without assuming a barcode certificate or a general interval-decomposition theorem as an input. This is conventional mathematics, not a completed Lean implementation. Identifying the unique globally labeled elder barcode and all bars still uses the compact Morse/distinct-value setting and the existing finite-event result. The full-boundary height barrier remains false. + +## 1. Source and evidence binding + +The new root sketch is [ASYMMETRIC_ELDER_TRAP_ROOT_SKETCH.md](ASYMMETRIC_ELDER_TRAP_ROOT_SKETCH.md). Its exact hash is bound in the new freeze manifest. It is an author sketch, not an independent review or accepted premise. The accepted analytic inputs are the exact frozen `QUANTITATIVE_FOLD_ISOLATION_PROPOSAL.md` and `UNIFORM_LOCAL_NORM_DIAGONAL_PROPOSAL.md`, with their separate ACCEPT_SCOPED reviews: + +| Frozen input | SHA-256 | +|---|---| +| isolation author report | `827c82953845eb12005542ff6a4e1c99886b68d1ea9f1aeae16f981622ac6a30` | +| isolation nonauthor review | `f06c61e3f3e60c795a1a14e824640f1689d4143dba4f6698cada7a41929eec90` | +| uniform local-norm author report | `8e1eefca9888d43824a9fd9caca34c44d2de4bf8e30c668b474fa2097491c76b` | +| uniform local-norm nonauthor review | `ff552dfacadede53405530a219302c1163dca8ba97d518f02d407970d3de3c1c` | +| finite critical-count author report | `621cee96854aa440498cecf5452a8d1e23f90a4458779c3286c18eba106fe138` | +| finite critical-count nonauthor review | `16152427f7013822ac1bba81142429800334e6bf0f354a6439b23a1997bc4524` | + +The pinned mathematical source is Main `85b928b326ff0586715a56ae91f51d3fbd89557a`, not a claim about the current live head: + +* [finite_gaussian_contact/PROOF.md](https://github.com/d6g8k5htny-coder/main/blob/85b928b326ff0586715a56ae91f51d3fbd89557a/experiments/universality/finite_gaussian_contact/PROOF.md), 25367 B, SHA-256 `7fea2656e11d986e1a5d6bfb0a05c8421bd93262d28051b8cfef05320595f474`: ideal real Gaussian coefficient law, literal endpoint distance r and cubic mark k. +* [finite_h0_lifetime/PROOF.md](https://github.com/d6g8k5htny-coder/main/blob/85b928b326ff0586715a56ae91f51d3fbd89557a/experiments/universality/finite_h0_lifetime/PROOF.md), 27064 B, SHA-256 `5ea91111fa5f3d78d7c4188ebe84b3bc2c452b05dfdfc62c74b03123cb2696d1`: §3 intrinsic Borel elder threshold from open superlevels, §4 actual finite Morse component/elder-basis induction and selector (7), §5 Borel selected area formula. That conventional proof is consumed with its stated local-Morse/critical-free-flow imports; its mere presence is not a Lean theorem or a new scientific approval. + +The formal source cut is Math- `319f9c45f66dbcce4a71b9c2f3e0fb62a666bfec`, as retained and audited in `GLOBAL_ELDER_INTERFACE_AUDIT.md`. The relevant exact declarations are: + +| Module | SHA-256 | What can be reused | +|---|---|---| +| [SuperlevelBar.lean](https://github.com/d6g8k5htny-coder/Math-/blob/319f9c45f66dbcce4a71b9c2f3e0fb62a666bfec/companions/torus_filtration_20261011/SuperlevelBar.lean) | `346eea948e18d5904422904006827930e27d3141ccbc19ee8ac2c24bcf65a4a9` | `superlevelBar_clearanceSet`, `deathHeight`, `strict_cut`, `cut_unique`; strict paths to a higher endpoint, with a guarded real supremum | +| [OrdinaryH0.lean](https://github.com/d6g8k5htny-coder/Math-/blob/319f9c45f66dbcce4a71b9c2f3e0fb62a666bfec/companions/torus_filtration_20261011/OrdinaryH0.lean) | `3d6b0d6b53413c484b1d010bccf0d53ae7f75becb2d747868f22dbd673fe0a88` | `ordinaryH0_naturalIso`, `ordinaryH0_filtrationIso`, `ordinaryH0_filtration_naturality`, `ordinaryH0_filtration_difference_zero_iff_joinedIn` | +| [ClosedSuperlevel.lean](https://github.com/d6g8k5htny-coder/Math-/blob/319f9c45f66dbcce4a71b9c2f3e0fb62a666bfec/companions/torus_filtration_20261011/ClosedSuperlevel.lean) | `90d1481b75d9c6d14e462c3fbf0bf7c7cdd0efb9eaad68d6732400658753b1c2` | actual filtration `superlevel_set g t={x|-t<=g x}` and singular H0 inclusions | +| [SuperlevelBarField.lean](https://github.com/d6g8k5htny-coder/Math-/blob/319f9c45f66dbcce4a71b9c2f3e0fb62a666bfec/companions/torus_filtration_20261011/SuperlevelBarField.lean) | `8a2470431c3fda5e98859aa9372cbb8866ed5c5c722cea5dfc28cb577840bbad` | Field adapter; its `elderPartner_iff` is a definition of the relation, not an independent H0 interpretation | + +The exact definitions and relevant proofs were read again. No compiler, kernel audit or runtime was run. The finite analytic argument below is C4; the finite Fourier source is C-infinity and therefore also meets the Field interface's stronger `IsSmooth`. The existing quotient smooth-atlas/Hessian/inertia alignment and interval-object implementation are not silently supplied by that observation. + +## 2. Deterministic hypotheses: a chart, not the desired elder conclusion + +Let F be C4 on an open neighborhood of a closed embedded chart square `Q_R=[-R,R]^2` in the interior of a boundaryless surface. The chart is a Hessian eigenframe about the anchored nondegenerate maximum M=(0,0). Assume exactly the accepted ridge conclusions, with fixed a,tau,R>0: + +```text +F_yy <= -a/2, +F_y(x,h(x))=0 uniquely, h in C3, |h(x)|<=R/32, +h(0)=h'(0)=0, +f(x)=F(x,h(x)), f'''(x)>=tau/2, +F(x,y)<=f(x)-(a/4)(y-h(x))^2, +F(x,+/-R)<=f(x)-aR^2/8. (1) +``` + +Let the candidate saddle be `S=(s,h(s))`, a second ambient critical point, with + +```text +00, +ell0 and the hypotheses (1)-(2) are derived from endpoint stationarity and the good derivative event. + +The compact set D will be the image of `[-R,s]x[-R,R]` in the surface. Its image must have exactly that chart boundary and interior. A sufficiently small torus chart has this property; an arbitrary self-overlapping image would not. + +## 3. Strong convexity supplies the asymmetric boundary + +Put `g=f'`. Then `g(0)=g(s)=0`, `g'(0)=F_xx(0)<0`, and `g''>=tau/2`. Strong convexity gives `g<0` on (0,s), and positive g on [-R,0) and (s,R]. For example, + +```text +0=g(s)>=s g'(0)+tau s^2/4, g'(0)<=-tau s/4, +0=g(0)>=-s g'(s)+tau s^2/4, g'(s)>= tau s/4. +``` + +For x<=0 and x>=s respectively, Taylor's integral inequality gives + +```text +g(x)>=g'(0)x+tau x^2/4>=tau x^2/4, +g(x)>=g'(s)(x-s)+tau(x-s)^2/4>=tau(x-s)^2/4. +``` + +Thus f increases to its unique maximum m at 0 on [-R,s], decreases to d at s, and + +```text +m-f(-R) >= tau R^3/12 > ell, +f(R)-d >= tau(R-s)^3/12 >= tau R^3/96 > ell. (3) +``` + +In particular `f(-R)m`, where `P=(R,h(R))` lies in the chart. From (1)-(3) the boundary of D satisfies + +```text +left face: F(-R,y)<=f(-R)=d}={S}`. The full enclosing boundary of `Q_R` is not below m: its positive ridge point P is deliberately **above** m. This is the feature that provides an older side, not a repair of the false full-boundary claim. + +## 4. Closed-superlevel separation and the exact attained cut + +Write `X_alpha={F>=alpha}` in the full ambient surface. For every alpha>d, + +`X_alpha intersect D = X_alpha intersect interior(D)`. + +It is both closed and open relative to X_alpha, because D is closed and its boundary misses X_alpha. No connected subset or path of X_alpha can meet both this set and its complement. This is a global separation statement, even if the exterior has arbitrary topology or remote critical points. + +For `dm it is empty. Also F<=m throughout D. Consequently, for `dm. Every path in X_d from M to any higher point must pass through S: its endpoint is outside D, so its first departure from interior(D) meets boundary(D), whose only point in X_d is S. This first-exit assertion uses the closed embedded D and continuity of the path, not compactness of a family of paths. + +For the actual C263 strict-clearance definition + +```text +C_M={h: exists y with F(y)>m and a path M->y + whose every value is strictly >h}, +``` + +the ridge path proves every h=d a strict path would cross boundary(D) at a point with height <=d<=h, which is impossible; h>=m is also ruled out by its starting point. Thus + +```text +C_M=(-infinity,d), C_M nonempty, +superlevelBar_deathHeight(F,M)=d0`, `F_yy(S)<0` and `det Hess F(S)=F_yy(S)f''(s)<0`. Thus S is a nondegenerate index-one saddle. Its two upper ridge sectors are in different global path components just above d. The left sector joins M on the decreasing ridge branch, while the right sector joins P on the increasing branch; an above-d path between them would contradict the clopen separation by D. At d the ridge joins them through S. This rules out the loop-saddle alternative. Local index alone would not have done so. + +On a compact C4 Morse surface with distinct critical values, choose a regular `h0 in (m,F(P))`. The component of the compact regular superlevel `{F>=h0}` containing P is path connected and compact. F attains its maximum at an interior critical point O with `F(O)>=F(P)>m`; Morse nondegeneracy makes this point a local maximum. A path P->O stays above h0, so the closed ridge path at d extends to an older maximum. No maximizing path compactness theorem is needed. Equivalently one may use the conventional higher-point/higher-birth argument in finite-H0 §4. + +Equation (5) shows that the component labeled by M cannot have joined any older component above d, and joins one at d. It is therefore its first **elder** merger. In this particular isolated strip it also cannot receive an outside component above d; the statement is not promoted to a claim that every maximum's first merger of any kind is its elder death. + +Distinct critical values imply that S is the unique critical point at height d **on the surface/torus quotient**. Its periodic Euclidean lifts are not unique points. This discharges the unique attained-saddle implication for candidates satisfying this trap; it does not prove the corresponding result for every nonessential maximum. Without the distinct-value premise (5) and the gate S remain true, but a remote critical point of the same height need not be absent. In particular a definitional adapter that labels every saddle at death height cannot be given an unguarded uniqueness interpretation in that case. + +For the exact conventional finite-H0 selector (7), the compact Morse/distinct locus, nondegenerate endpoint types, (5), and d H_beta` when beta<=alpha. Let I be k on `dH_alpha` by + +```text +j_alpha(1)=[M]-[P] when dI_alpha` to be zero when I_alpha=0. At an active level use the actual H0/component isomorphism: give the unique component `X_alpha intersect D` coefficient 1, and every outside component coefficient 0. Equivalently q sends a point class to 1 if its point is in D and to 0 otherwise. This is well-defined on path components because D cuts X_alpha into clopen pieces. + +Here are **all** potentially nontrivial naturality cases for q: + +1. If both heights are active, membership in the fixed D is unchanged under inclusion, and the clopen separation prevents an outside component from mapping into its inside component. Thus the component-indicator maps commute. +2. If the source height exceeds m and the target is active, the source superlevel has no point in D, since F<=m there. Every source point class therefore maps to an outside point class and is annihilated by the target retraction. This is the birth-crossing square; it must not be omitted. +3. If the target height is <=d, its interval space and retraction are zero. Both sides of the square are zero. The section square across this same death is already supplied by the closed path. +4. If both interval spaces are zero, or the source is before birth and the target after death, the square is zero. Descending height rules out a dead source mapping back into an active target. + +At active levels M is inside D and P is outside, so `q_alpha j_alpha(1)=1-0=1`. At inactive levels both sides are the identity of the zero vector space. Hence + +```text +q j = id_I, +H0({F>=alpha};k) as a persistence module + is isomorphic to I direct_sum ker(q). (7) +``` + +Explicitly `v -> (qv, v-jqv)` and `(c,w)->jc+w` are inverse natural maps. This proves an **actual interval direct summand**, not only a vanishing difference or a postulated abstract module. It uses the geometry (4)-(5), actual point-class transport, and the actual ordinary-H0/component natural isomorphism. The existing C264 `ordinaryH0_filtrationIso` supplies exactly that last isomorphism, and `ordinaryH0_filtration_difference_zero_iff_joinedIn` supplies the death/nondeath test for (6). C264 does not already implement (6)-(7). + +For a globally unique barcode label and multiplicity-one birth at m, compact Morse/distinct values and the conventional local birth/regular-band argument in finite-H0 §4 apply: only M is critical at m, its component is newly born, and a saddle at d merges precisely the two distinguished components. The finite-event elder basis identifies the interval with that label. An exhaustive decomposition of all intervals, all nonessential maxima and the essential interval is not proved by this one split. + +The precise remaining formal work is smaller than inventing a global bar certificate, but real: express the embedded D/ridge geometry in the smooth quotient; prove the clopen/fiber paths and (5) for the existing C263 data; align Hessian inertia and quotient points; define the ordinary field-valued interval module; construct the component-indicator natural map through `ordinaryH0_filtrationIso`; prove its birth/death squares and transport the split. The Field adapter's `elderPartner_iff` is still definitional, and the separate Lifetime interface is not automatically inhabited or aligned. No Lean statement or proof file was added here. + +## 7. Uniform quantitative join from actual endpoint marks + +Keep the source pair convention `M=z-r u/2`, `S=z+r u/2`; r is the **endpoint distance**, `ell=k r^3`. Suppose both endpoints are critical, M is a nondegenerate maximum, S is an index-one saddle, `k>=k_->0`, and `F_vv(M)<=-a` for the pair-normal v, with fixed a>0. Let B,C bound the multilinear operator norms of D3F,D4F throughout the local region. The accepted endpoint calculation gives + +```text +12k=integral_0^r [6s(r-s)/r^3] F_uuu(M+s u) ds, +|F_uuu(M)-12k|<=Cr/2, +||Hess F(M)u||<=Br/2. (8) +``` + +If rB<=a, the hard eigenvalue is <=-a, the soft eigenvalue has magnitude <=Br/2, and the soft unit eigenvector e can be oriented by e.u>0. With hard unit vector e_h, + +```text +|u.e_h|<=Br/(2a)<=1/2, e.u>=sqrt(3)/2, +F_eee(M)>=12k-r(C/2+3B^2/a). (9) +``` + +These are derived from gradient stationarity; the pair frame is not assumed to diagonalize the Hessian. The factor 12 would be wrong if r were silently treated as a half-distance. + +Now consume exactly the uniform local-norm construction. Assume normalized variance and uniform physical M12, with `L>=Lmin>0`. Set + +```text +b=min(1,Lmin/16), p3=1/15, p4=2/15, +T=(1+M12)b^-10 *430299201784277045424464881586844996. +``` + +For its deterministic n-by-n cover, `n=ceil(L/b)`, write `S_g=||chi_b F(g+.)||_H6`. The accepted estimate is + +`P(S_g^2>v)<=sqrt(2) exp[-v/(4T)]`, `n^2<=b^-2(1+L)^2`. + +For Q>=4 define the unconditional global event + +```text +G_Q={S_g^2<=Q/4 for every deterministic covering center g}. +P(G_Q^c)<=sqrt(2)b^-2(1+L)^2 exp[-Q/(16T)]. (10) +``` + +At the covering center of any selected maximum M, `W=1+S_g` obeys `W^2<=Q`, since `1+sqrt(Q)/2<=sqrt(Q)`. The whole rotated radius-b chart about M lies inside the cutoff-one region: `b/2+sqrt(2)b<2b`. Its support is embedded since 6b0. (11) +``` + +Suppose all lifetimes under consideration obey `ell<=Tgap t` and put `r_t=(Tgap t/k_-)^(1/3)`. The following explicit eventual conditions suffice: + +```text +r_t <= kappa/(2Q), +Tgap t < a kappa^2/(8Q^2), +Tgap t < tau kappa^3/(96Q^3). (12) +``` + +They imply every conversion premise, rather than assuming the desired cubic. Indeed + +```text +rB <= kappa p3/(2sqrt(Q)) <=a/(64sqrt(Q)) <=a, +r(C/2+3B^2/a)<=r QD0<=kappa D0/2<=3tau/64=12k-3tau/64>=2tau-3tau/64>=tau. (13) +``` + +Use the accepted chart in this e,e_h eigenframe with + +```text +R=min(b,a/(32B),tau/[32(C+B^2/a)]) >= kappa/Q. (14) +``` + +The actual S lies inside the half chart because `|S-M|=r<=R/2`. Its x coordinate is `s=r(e.u)` in `[sqrt(3)r/2,r]`, so `00, + +```text +Q_p(t)=16T[4p log K+2log(1+L)+u0 log(1/t)] (17) +``` + +(eventually >=4) gives + +```text +expected factorial discrepancy + <=sqrt(2)b^-2 16^(2p) product ||f_i||_infty t^u0. (18) +``` + +The p-dependent bound is chosen before using the count cap. It does not multiply a prechosen pair error by uncontrolled powers. The good event may be different in separate fixed-p estimates; the two processes and the source law are unchanged. For the Laplace comparison alone the K term in Q is unnecessary. + +For the declared normalized Gaussian truncations `L~t^-1/3`, `Lambda^2~B log(1/t)`, `K=ceil(L Lambda/(2pi))`, the accepted grid comparison supplies normalized variance and a uniform M12. Equation (17) is O_p(log(1/t)). Thus `r_t Q_p->0`, `t Q_p^2->0` and `t Q_p^3->0`, proving all of (12) eventually. More generally the sufficient growth condition is `t^(1/3)[log K+log(1+L)+log(1/t)] ->0`, with the fixed mark bounds retained. Very fast K growth or moving singular lower marks can defeat this particular bound. + +Consequently, **if** the separate frozen higher-configuration candidate proof survives its nonauthor review, (16) and (18) transfer its compact regular-mark Laplace and fixed factorial conclusions to this global-elder selected sector. This conditional join is not a review or promotion of that proposal, nor a source-adoption decision. The present finite quantitative coupling is the new claim submitted for review here. + +Remote microscopic elder pairs can have physical radius of order one and `k=ell/r^3` tending to zero, and soft transverse marks can approach zero. Such bars escape every compact subset of this regular mark space. No tightness at k=0 or A=0, unmarked lifetime projection, all-mark coefficient equality, universal exponent, higher-dimensional/Hk theorem, essential-class limit or full persistence/universality theorem follows from (15). Literal lifetime density `ell^(-1/3)` and cumulative mass `t^(2/3)` remain distinct; neither is newly established by this coupling alone. + +## 9. Attacks, controls and formal boundary + +**The earlier false barrier is explicitly preserved.** For `F=-mu x^2/2+tau x^3/6-a y^2/2`, the accepted chart radius can be fixed independently of tiny mu. Then `S=(2mu/tau,0)`, `ell=2mu^3/(3tau^2)`, and `F(R,0)>F(0)` for `mu=tau/2`. Strong convexity gives a strictly negative g between its two roots, and the displayed Taylor inequalities imply `g'(0)<=-tau s/4` and `g'(s)>=tau s/4`. On the negative branch, `g(x)>=g'(0)x+tau x^2/4>=tau x^2/4`; on the outgoing branch, `g(x)>=tau(x-s)^2/4`. Integrating yields + +`m-f(-R)>=tau R^3/12`, and `f(R)-d>=tau(R-s)^3/12>=tau R^3/96`. + +Thus f is unimodal on [-R,s], is at most m there, and the strict gap hypotheses place the negative face below d and the positive ridge point P strictly above m. The vertical estimate then places both horizontal faces of D strictly below d. On the right face, the quadratic loss from its unique ridge maximum makes equality at d possible only at S. Hence + +`boundary(D) intersect {F>=d}={S}`. + +Both the endpoint half-chart bound and the two strict gap bounds are used. The argument does not derive a full enclosing height barrier for the larger square. In fact its outgoing boundary point P is above m. The accepted cubic counterexample remains negative evidence against that false barrier and is the local model explaining why the asymmetric domain works. My nonconstant polynomial controls retain this distinction: their outgoing ridge heights are positive while all strip exit faces have the stated saddle-level gate. A second family with the second root at R shows why the mere endpoint types cannot replace the radius/gap premises. + +The analytic argument uses the chart already accepted at its exact radius and operator-norm hypotheses. It does not reuse a pair-coordinate cubic as an eigenframe cubic without the error calculation in §7, or omit the mixed `B^2/a` contribution. Those earlier falsifiers remain attached. + +## The trap proves a global first elder death, not just a local index + +For every alpha>d, the boundary misses the full ambient superlevel `X_alpha={F>=alpha}`. Consequently `X_alpha intersect D=X_alpha intersect interior(D)` is clopen in X_alpha. No remote path or connected set can cross it. This is a global exclusion derived from an actual closed embedded domain, not a guessed gradient basin or a probabilistic assertion that remote paths are unlikely. + +For `dm. No point in D has height above m. Every connection to any strictly higher point must therefore leave D and meet its boundary. + +At d the explicit ridge to P has minimum exactly d at S. Every older-endpoint path at this closed level must pass through that same gate. For the literal strict-clearance set, this path witnesses every h=d is blocked by the forced boundary crossing; h>=m is also excluded at the starting endpoint. Therefore the set is exactly `(-infinity,d)`, nonempty, with death height dm. The path in this component extends the ridge to an older maximum. M has no older connection above d and does at d, so this is its first elder merger. The argument neither equates first elder merger with the first merger of an arbitrary maximum nor asserts a compactness theorem for the family of paths. + +Distinct critical values make S the unique quotient critical point at d. They do not make its periodic Euclidean lifts unique. Without distinct values the gate and cut remain valid, while a remote critical point with the same height may exist. This is a necessary qualification for any adapter that defines a partner solely by endpoint type and death height. + +## The actual ordinary-H0 interval split and its canonical-label boundary + +The section is the point-class difference `[M]-[P]` on `d (qv,v-jqv)` and `(c,w) -> jc+w` are inverse natural maps, with `ker(q)` a submodule. This proves an actual interval direct summand on `(d,m]`, equivalently `[-m,-d)` for increasing parameter t=-alpha. It is stronger than a vector merely becoming zero. It does not construct a simultaneous splitting from all independently chosen traps or prove an exhaustive decomposition of the whole persistence module. + +Canonical elder labeling uses the separate explicit compact Morse/distinct finite-event premise. In that setting only M is critical at birth m, so its birth event has multiplicity one; the gate S is the unique merging critical event at d; and the conventional finite component/elder-basis induction identifies the bar born at M and killed at S. A particular section can depend on the chosen outside point P without making the endpoint label noncanonical. Without those global finite-event premises, the local result remains the one direct summand and attained cut; no canonical all-bar conclusion is inferred. + +The exact conventional source read for this join is `finite_h0_lifetime/PROOF.md`, 27064 B / `5ea91111fa5f3d78d7c4188ebe84b3bc2c452b05dfdfc62c74b03123cb2696d1`, especially §§3–4 and selector (7). It states the intrinsic Borel open-superlevel threshold, finite Morse events, elder basis and once-counting. We consume its stated local-Morse and critical-free-flow imports; we did not repeat its first source review or a primary-book audit. The trap supplies the needed local-to-global conclusion rather than inserting the desired elder relation as a certificate assumption. + +## Exact C263/C264 reuse does not pretend to implement the new theorem + +The actual frozen source was inspected, not only the prose interface audit. The topology reviewer read the full relevant definitions and proofs; I additionally read the literal C263 clearance/default and cut-uniqueness declarations, C264 filtration isomorphism/naturality/difference theorem, and the conventional selector passages. The principal exact imports are: + +| Source at Math-319 | SHA-256 | Actual reused content | +|---|---|---| +| `SuperlevelBar.lean` | `346eea948e18d5904422904006827930e27d3141ccbc19ee8ac2c24bcf65a4a9` | Euclidean strict paths to a higher endpoint, nonempty guarded real supremum, strict cut and cut uniqueness | +| `OrdinaryH0.lean` | `3d6b0d6b53413c484b1d010bccf0d53ae7f75becb2d747868f22dbd673fe0a88` | Actual singular-H0 point classes and transport; proved natural component comparison; filtration isomorphism, its naturality, and mapped difference zero iff target `JoinedIn` with nonzero coefficients | +| `ClosedSuperlevel.lean` | `90d1481b75d9c6d14e462c3fbf0bf7c7cdd0efb9eaad68d6732400658753b1c2` | Actual closed sets `{x|-t<=g x}`, inclusions and ordinary singular-H0 filtration | +| `TorusTopology.lean` | `ddb145b3f791c181d4f2a51e54e711937297bd6476ba55bcd5252b7ddb57cfe1` | Quotient descent and actual higher-path transfer, with strict and closed predicates preserved separately | +| `SuperlevelBarField.lean` | `8a2470431c3fda5e98859aa9372cbb8866ed5c5c722cea5dfc28cb577840bbad` | Field adapter and definitional elder relation; no independent H0 elder interpretation | + +C263 itself is a Euclidean definition. For the finite torus application, the periodic Euclidean lift and the actual strict/closed quotient-transfer theorems provide the semantic join. An arbitrary surface is not silently identified with Euclidean space. C264's comparison is a genuine proved natural isomorphism, rather than a compatibility field assumed in a record. It supplies the component-indicator bridge used here. Its difference theorem does not itself prove the gate, attained death, interval object or split. + +The new interval module, indicator natural map, its birth/death squares, quotient chart geometry, Hessian/inertia alignment and transported splitting remain formal implementation work. The separate Lifetime interface is not thereby inhabited. No Lean file, compilation, kernel audit, source HOLD or alignment status was changed by this conventional proof/review. + +## The uniform norm event supplies every mark-to-chart premise + +The source radius r is the endpoint distance, with gap `ell=k r^3`. The endpoint stationarity formula has normalized kernel `6s(r-s)/r^3`, integral one and first moment r/2. Consequently the endpoint third derivative differs from 12k by at most Cr/2, and vector Taylor integration bounds `||Hess F(M)u||<=Br/2`. These are endpoint statements, without silently replacing M by the midpoint or r by a half-distance. + +The pair-normal bound `F_vv(M)<=-a` forces a hard eigenvalue at most -a by the Rayleigh quotient. At a nondegenerate maximum, the other eigenvalue has magnitude at most Br/2. If rB<=a the eigenvalues are separated, and the soft direction e can be chosen with `e.u>0`. The hard projection satisfies `|u.e_h|<=Br/(2a)<=1/2`, so `e.u>=sqrt(3)/2`. A telescoping multilinear comparison costs at most `3B||e-u||`, bounded conservatively by `3B^2r/a`. This proves the eigenframe cubic bound (9). Positivity of k alone is not used as a substitute for this error control. + +For each deterministic covering center, the accepted H6 tail has `P(S_g^2>v)<=sqrt(2)exp[-v/(4T)]`. The event threshold is exactly Q/4. A finite union bound gives (10). On G_Q, with Q>=4, + +`W=1+S_g<=1+sqrt(Q)/2<=sqrt(Q)`, hence `W^2<=Q`. + +The deterministic grid, rather than a field-selected maximum, is where the unconditioned Gaussian estimate is applied. Its cutoff-one plateau contains the entire rotated radius-b chart around every possible M, because the grid offset plus rotated-square extent is less than 2b. The support is embedded since 6b=12k_-=2tau`, the resulting eigenframe cubic is at least tau. The slack is more than sufficient; no upper k mark is required for this implication. + +The radius bound has the right power of Q. Each of `b`, `a/(32B)` and `tau/[32(C+B^2/a)]` is at least its corresponding fixed minimum constant divided by Q, using W<=sqrt(Q), Q>=1 and `C+B^2/a<=QD1`. Therefore `R>=kappa/Q`. The radius premise puts the actual S in the half chart. Its soft coordinate is `s=r(e.u)` between `sqrt(3)r/2` and r, so it is positive and at most R/2. Its hard coordinate is the unique ridge root by stationarity. The accepted ridge hypotheses, rather than a guessed direction alignment, now apply to this actual candidate. + +Finally, `ell<=Tgap t` and the strict remaining conditions in (12) give `ell0. Retuning the proof event with p changes neither process nor source. A prechosen first-order event could leave growing K powers and would not prove every factorial discrepancy. For Laplace convergence alone one may omit that K term, as claimed. + +In the declared Gaussian truncation family, the accepted moment bound is uniform and `Q_p=O_p(log(1/t))`. Thus `r_tQ_p->0`, `tQ_p^2->0` and `tQ_p^3->0`; with fixed positive kappa,a,tau and lifetime cap, every strict condition in (12) holds eventually. The broader sufficient logarithmic growth condition in §8 also has the right strength: `t^(1/3)Q_p->0` implies the required cubic condition and the weaker quadratic one. Fixed lower marks and fixed Tgap are material; moving them requires new inequalities. + +The accepted higher-configuration theorem uses this same finite Gaussian family, same regular mark space, same ordered candidate identity and same source pins. Its intensity has coefficient 144 and its convergence is compact/vague. Equations (16) and (18) therefore transfer both its compact Laplace conclusion and every fixed ordered factorial conclusion to Y_t. The Poisson intensity on this regular sector is unchanged. This mathematical join uses the exact accepted input; it does not amend that frozen theorem or claim that a source has been adopted by a project implementation. + +This proves only a sector of actual finite elder bars with fixed regular endpoint marks. A small lifetime can accompany an order-one endpoint distance and hence k approaching zero; transverse curvature may approach zero; births and other marks may escape every compact set. The present trap supplies no uniform complement estimates for those bars and no unmarked lifetime projection or all-mark coefficient equality. A full global microscopic-bar theorem would need those reverse/escape estimates as additional work. + +## Final disposition and retained boundaries + +| Claim | Disposition | +|---|---| +| Strong-convexity strip boundary and outgoing older point | ACCEPT_SCOPED with the actual half-chart, strict gap and embedded-domain premises | +| Strict clearance `(-infinity,d)`, closed attainment at d, no remote earlier elder connection | ACCEPT_SCOPED | +| Nonloop merger, older maximum and unique quotient critical event | ACCEPT_SCOPED on the explicit compact Morse/distinct locus | +| Actual ordinary-H0 interval summand `[-m,-d)` | ACCEPT_SCOPED over a nonzero field, with all section/retraction naturality squares | +| Canonical multiplicity-one elder label for this trapped pair | ACCEPT_SCOPED using the explicit conventional finite-event source premise | +| Global good-event derivative bounds, eigenframe conversion and `R>=kappa/Q` | ACCEPT_SCOPED with all errors and radius/gap premises retained | +| Regular candidate/elder equality and fixed-p factorial discrepancy | ACCEPT_SCOPED; exact source subset, once-counting and count-cover cancellation | +| Transfer of accepted compact regular-mark Poisson/factorial conclusions | ACCEPT_SCOPED for the one declared ideal Gaussian family | +| Simultaneous/exhaustive decomposition constructed from all local splits | NOT ESTABLISHED by this argument | +| All-bar reverse regularity, singular/unbounded mark tightness, full unmarked bar law | NOT ESTABLISHED | +| New Lean implementation, quotient/inertia alignment, source adoption or scientific promotion | NOT PERFORMED | + +The full enclosing height barrier remains FALSE and is preserved as negative evidence. An ordinary point class is not claimed to die. No global elder predicate is factorized by a local Gaussian density lemma. The previous source HOLD/alignment obligations, essential-class and higher-homology limits, and the distinction between lifetime density `ell^(-1/3)` and cumulative mass `t^(2/3)` remain intact. The broader persistence/universality goal remains unmet. + +All outputs of this review are new files in this review directory. The final machine verdict and manifest bind the actual evidence, the topology review and the accepted candidate-process input. No author source, prior sealed report, production/native/remote record, Lean artifact, ownership/status record or scientific register was edited. Scientific effect NONE; organizational-independence credit 0. diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/smooth_gaussian_manifold_admission/PROOF.md b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/smooth_gaussian_manifold_admission/PROOF.md new file mode 100644 index 000000000..d61fd11a8 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/frontiers/gaussian_short_bar_theory_20261017/smooth_gaussian_manifold_admission/PROOF.md @@ -0,0 +1,739 @@ +# Actual fold admission for smooth Gaussian fields with dense smooth support + +Author proposal, 17 October 2026. Mathematical authors: OpenAI/Codex +smooth_full_support_universality_20261017 and root (mechanism coauthor). +Status: AUTHOR PROPOSAL, not an independently reviewed or adopted theorem. +This is a new admission theorem for an explicit class of actual infinite +Gaussian laws. It consumes, rather than republishes as new, the existing +conditional structural theorem ST and its deterministic elder-transfer lemma. +The exact consumed files and their full read extents are in INPUT_MANIFEST.json. +All output is local. There is no Lean, numerical field realization, new +scientific acceptance, point-process assertion, or homology-degree extension. + +## 1. The law, theorem and distinction from the previous conditional result + +Fix a connected compact boundaryless smooth manifold X, a smooth Riemannian +metric g, and a fixed dimension d >= 1. Write V = vol_g(X). Choose deterministic +real functions phi_j in C-infinity(X), j >= 1, such that their real linear span +is dense in the usual C-infinity topology. Choose strictly positive a_j with + + A_q = sum_j a_j ||phi_j||_Cq < infinity, for every integer q >= 0. (1) + +The norms can be the increasing norms sum_{h<=q} sup_X |nabla^h f|_g; +coordinate norms from a finite atlas are equivalent. Let Z_j be independent +standard real Gaussians and define the actual law by + + F = sum_j a_j Z_j phi_j. (2) + +Neither stationarity, constant variance, isotropy, parity, a flat metric, nor +a distinguished global frame is required. The independent Gaussian law and +the full dense support are substantive hypotheses. The covariance is the +actual smooth kernel K(p,q) = sum_j a_j^2 phi_j(p) phi_j(q). + +**Admission theorem (subject to the explicit classical imports in Section 12).** +Every law (2) satisfying (1) and the dense-span condition supplies all five +interfaces COUNT, CONTACT, SELECT, ENVELOPE and FAR of ST. In particular the +expected finite ordinary superlevel-H0 elder-bar measure, per volume, has the +canonical finite nonnegative Borel density representative nu defined below, +with + + nu(ell) ~ C_F,g ell^(-1/3), + E N_bar((0,t]) / V ~ (3/2) C_F,g t^(2/3), t -> 0+, (3) + +and 0 < C_F,g < infinity. The essential class is excluded. The coefficient +is the exact local covariance functional (25), or equivalently (27). +The same leading coefficient holds for the ordered maximum/index-(d-1) +candidate-pair density. No selected-density continuity is asserted. + +All constants in the proof belong to this one fixed law, metric and dimension. +The class genuinely allows changes in covariance, geometry and dimension; +it asserts admission separately for each member. It does not give constants +uniform over all such changes, degenerating spectra, varying metrics, or +unbounded dimensions. It does not give a small-lifetime numerical window. +For d=1 no transverse block is present; the argument uses the one-dimensional +version of ST's elder lemma and remains valid. + +ST alone did not prove full jet rank, conditional remote genericity, or field +admission on arbitrary manifolds. The new content here is proving those facts +for (2) from the dense smooth span, together with same-law infinite Gaussian +regression and finite-block incidence at literal conditioned targets. The +previous exact periodized-Gaussian adapter IG is a model for some arguments, +not a premise asserting this broader admission. + +## 2. Nonemptiness, smoothness, moments and full smooth support + +The hypotheses are nonempty on every such X. A direct dense system can be +built as follows. Choose finitely many coordinate cubes U_i covering X, +coordinates z_i with image (-pi,pi)^d, a smooth partition of unity rho_i with +compact support in U_i, and smooth cutoffs chi_i supported in U_i and equal +to one on supp rho_i. Include the globally extended smooth functions + + chi_i cos(n.z_i), chi_i sin(n.z_i), n in Z^d, + +and optionally the constant 1, in a countable list. For any smooth f, +rho_i f in these coordinates extends by zero to a smooth periodic function. +Repeated integration by parts makes its Fourier coefficients decrease faster +than every polynomial. Its Fourier partial sums converge in every Cq norm. +Multiplying those sums by chi_i and summing i converges to sum_i rho_i f = f +in C-infinity. This proves dense real span. The only manifold construction +used is the named smooth finite partition/cutoff fact, not an unproved +spectral completeness claim. Fourier coefficient decay proves the required +smooth convergence directly. No Laplace eigenfunction claim is needed. + +For any such list one explicit choice is + + a_j = 2^(-j) / (1 + max_{0<=q<=j} ||phi_j||_Cq). (4) + +For a fixed q, the finitely many j=q term of +(1) is at most 2^(-j). Thus (4) meets every summability requirement and +has a_j>0. One may vary the positive weights and the dense list. + +Let B_q = sum_j a_j |Z_j| ||phi_j||_Cq. Its expectation is finite, so B_q +is finite almost surely. Intersect over integer q. Uniform absolute +convergence of all derivative series proves that (2) is a smooth field, +and ||F||_Cq <= B_q. For p>=1, Minkowski gives + + ||B_q||_Lp <= A_q ||Z_1||_Lp < infinity. (5) + +Lower positive moments follow from the p=1 bound or larger moments. +Thus every Cq norm has all finite positive moments. Differentiating the +covariance termwise is justified: the required products of summable +nonnegative derivative bounds are summable. This also proves that K is +smooth on X times X. The smooth random element is Borel as an almost sure +C-infinity limit; define it arbitrarily on the null complement. + +For completeness, its topological support is all of C-infinity(X). A basic +neighborhood of h contains a Cq ball of radius epsilon. Choose a finite +linear combination h_N of the phi_j with ||h-h_N||_Cq < epsilon/3. Increase +N, adding zero coefficients, until E||sum_{j>N} a_j Z_j phi_j||_Cq is less +than epsilon/6. By Markov the tail has Cq norm < epsilon/3 with positive +probability. Independently, the first N Gaussian coefficients fall in a +small neighborhood of the coefficients of h_N/a_j with positive probability, +giving prefix error < epsilon/3. Their intersection is positive. This +support proof uses no assertion that the infinite standard coefficient +sequence belongs to unweighted l2; it does not. + +## 3. Dense smooth span proves all needed finite minors + +We first prove a finite-dimensional lemma. Let L_1,...,L_m be continuous +linear functionals on C-infinity(X) that are linearly independent. Then the +matrix with columns (a_j L_i phi_j)_i has row rank m, and some finite set +of columns already has row rank m. Indeed, if its column span were proper, +a nonzero c in R^m would annihilate all columns. Since a_j>0 and the span +of phi_j is dense, sum_i c_i L_i would vanish on all smooth functions, +contradicting independence. A spanning subset of R^m has a finite basis. +Consequently + + Cov(LF) = sum_j (a_j L phi_j)(a_j L phi_j)^T > 0. (6) + +For the local rows used here independence is proved, not postulated. +At finitely many distinct sites take disjoint coordinate neighborhoods +and cutoffs equal to one near each site. A polynomial of degree at most +s in a neighborhood, multiplied by its cutoff, realizes any prescribed +s-jet at that site while killing all jets at the other sites. Coordinate +jets and covariant scalar derivatives up to a finite order differ by an +invertible triangular transformation. In a normal chart, the particular +third axial derivative along a geodesic and parallel frames is its third +Taylor coefficient when the gradient is zero; independence in any case +also follows from the triangular jet transformation. Thus independent +components of finitely many jets at distinct sites are independent +continuous distributions on C-infinity(X). + +Fix a unit u at x and an orthonormal frame (u,v_1,...,v_{d-1}). Precisely +the following maps therefore have full row rank for the actual field and +for a suitable finite coefficient block: + +1. grad F(p), and (grad F(p),grad F(q)) for p!=q. +2. O_pq = (F(p),grad F(p),F(q),grad F(q)), 2d+2 rows. +3. U_0 = (F,F_u,F_uu,F_uuu,(F_vi,F_uvi)_i) at x, 2d+2 rows. +4. U_0 enlarged by the (d-1)d/2 independent symmetric entries of + A = Hess F restricted to u-perp. No axial-column entry is duplicated. +5. U_0 together with (F(p),grad F(p)), p!=x. +6. U_0 together with both first jets at p,q, where x,p,q are distinct. + +Items 3-4 are independent subsets of the full third jet at x; items 5-6 +join disjoint-site jets. This is also a proof of the needed conditional +remote ranks. If the block matrix (U_0,J) is surjective, then J restricted +to ker U_0 is surjective: lift (0,y) for any y. Therefore the free +coefficient directions after the exact contact pin still realize every +listed remote first jet, even though the pinned contact coordinates +have zero conditional variance. The conditional covariance of A is +positive on all its independent symmetric entries by item 4. + +There is no claim of a uniform finite block over colliding site tuples. +On any compact separated tuple domain, continuity and a finite cover +let us take the union of finitely many witnessing column sets. That +one finite block has the required rank throughout that domain. Full +covariance eigenvalues have positive floors on such compact domains. +For U_0 and the enlarged contact map, compactness of the orthonormal +frame bundle gives a uniform floor. Frames may be handled in finitely +many local trivializations; a global perpendicular frame is unnecessary. + +For fixed x,u the remote domains can be exhausted by dist(x,p)>=1/n, +and dist(x,p),dist(x,q),dist(p,q)>=1/n. A finite block exists for each +nonempty compact domain. Countably many blocks and covers are enough +for the conditional arguments below. A single finite block working +arbitrarily close to the contact is neither assumed nor needed. + +## 4. Exact contact observations and same-law Gaussian regression + +Choose r_0 below a uniform short-geodesic scale, to be decreased later. +For 00; + v_0 = (b,0,0,12k,(0,0)_i). (8) + +The transform from O_MS has absolute determinant 12 r^(-(d+3)). +The contact rows in Section 3 are exactly the r=0 extension. Integral +finite-difference identities for a one-dimensional restriction and its +parallel transverse derivatives show + + |U_r f| <= C ||f||_C3, + |(U_r-U_0)f| <= C r ||f||_C4. (9) + +Constants are uniform over the compact frame bundle. The fourth row +is a bounded weighted average of the third axial derivative (its +constant-third-derivative value is that derivative); the other divided +rows are averages of derivatives of order at most two. These integral +identities avoid divergent separate bounds on the summands of (7). + +Set Sigma_r=Cov(U_r F) and c_r(z)=Cov(F(z),U_r F). Equations (5),(9) +and the covariance series give uniform Sigma_r -> Sigma_0 and +c_r -> c_0 in global Cq for each fixed q. For the latter, apply the +row difference to the smooth covariance in its observation variable, +then differentiate its field variable; (1) supplies every mixed bound. +The contact rank and compactness yield, after shrinking r_0, + + lambda I <= Sigma_r <= Lambda I, 0<=r<=r_0. (10) + +On one underlying copy of the actual infinite field define + + F_r(z) = F(z)+c_r(z) Sigma_r^(-1)(v_r-U_r F). (11) + +The observation of this field is exactly v_r. The residual +R_r=F-c_r Sigma_r^-1 U_rF is Gaussian and uncorrelated with U_rF, +hence independent. Independence for a countable dense evaluation set +and continuity proves independence of the entire smooth field. +Thus (11) is the genuine Gaussian conditional law Q_r at every target, +not a pin event with positive probability. It is jointly Borel in +parameters using local frame charts and the explicit formula. + +Its residual has full support in exactly the smooth kernel of U_r. +Indeed H_r:f -> f-c_r Sigma_r^-1 U_r f is a continuous linear map on +C-infinity, U_r H_r=0, and H_r f=f whenever U_r f=0. Since F has full +smooth support, every relatively open nonempty subset of ker U_r has +positive H_r F probability: take a point in that subset and its nonempty +open preimage under H_r. The support is contained in the closed kernel, +so it is precisely ker U_r. Translating by c_r Sigma_r^-1 v gives full +support on the affine fiber U_r f=v. This support observation alone +is not used to apply finite incidence; the original finite independent +coefficient blocks and their residual Gaussian coordinates are constructed +separately in Sections 6-7. + +At fixed marks and frame, F_r converges to F_0 globally in C4, almost +surely and in every finite Lp norm of C4. The same holds along convergent +parameter sequences. Uniform deterministic bounds on c_r Sigma_r^-1, +combined with (9), give for every q + + ||F_r||_Cq <= ||F||_Cq + C_q ||F||_C3 + C_q(|b|+k). (12) + +The constants are uniform in small r,x,u and all b,k; the random part +has every finite moment. This proves the mean bound for growing targets +as well as compact-mark coupling. It does not require any boundedness +of the infinite raw coefficient sequence or finite-dimensional norm +equivalence for that sequence. The conditional mean is exactly +c_r Sigma_r^-1 v_r; its Cq norm is at most C_q(|b|+k). + +## 5. CONTACT, positivity and all-mark ENVELOPE + +Under Q_r the exact endpoint gradients vanish. The physical Hessians +in the parallel frames have form + + H_i = [ r alpha_i r beta_i^T ; r beta_i A_i ], i=M,S. (13) + +The axial-column average along the geodesic is zero because the endpoint +gradient components agree. The fundamental theorem of calculus bounds +alpha_i,beta_i by C||F_r||_C3. Global C4 convergence and (8) give +alpha_M -> -6k, alpha_S -> 6k, bounded beta_i, and A_i -> A_0. +The exact polynomial identity is + + det H_i / r = alpha_i det A_i - r beta_i^T adj(A_i) beta_i. (14) + +For d=1 the second term is absent and det A=1. The correct inertia +congruence rescales by diag(r^(-1/2),I). Therefore with + + W_r = |det H_M det H_S| 1{H_M<0} 1{index H_S=d-1}, + V_r = W_r/r^2, + +one has almost surely + + V_r -> V_0 = 36k^2 (det A_0)^2 1{A_0<0}, + V_r <= C(1+||F_r||_C3)^(2d). (15) + +If A_0 is nonsingular and not negative definite the maximum mark fails; +if A_0 is singular the limiting determinant vanishes. Hence no cone-boundary +null assertion is needed for this weighted limit. Section 3 in fact proves +that A_0 has full conditional Gaussian density on Sym(u-perp), at every +literal target. Its mean may depend on b,k. For k>0 the negative-definite +open cone has positive probability and the determinant is nonzero there. +Thus z_0=E_Q0 V_0 is finite and strictly positive. For d=1 it is 36k^2. +Dominated convergence using (12) proves E_Qr V_r -> z_0. Density +convergence pi_r(v_r)->pi_0(v_0) follows from (8),(10). +This discharges CONTACT. + +For every b in R,k>0, (12) implies the uniform moment bound +E_Qr V_r <= C(1+|b|+k)^(2d). Moreover the first and fourth entries of +v_r are b-kr^3/2 and 12k. On the bounded r interval this linear map +of (b,k) has a uniformly bounded inverse on its image, so +|v_r|^2 >= c(b^2+k^2). Equation (10) gives the all-mark bound + + 12 pi_r(v_r) E_Qr V_r + <= H(b,k) = C(1+|b|+k)^(2d) exp[-c(b^2+k^2)]. (16) + +The integral of k^(-2/3)H over SX times R times (0,infinity) is finite: +SX has finite volume, k^-2/3 is integrable at zero, and Gaussian tails +beat the polynomial. This supplies ENVELOPE, with one common r_0. +It uses neither inverse curvature moments nor an inverse normalizer. + +## 6. COUNT: actual all-Borel Rice and unconditional genericity + +Here we prove the finite-incidence reduction for this infinite law. +For a compact site or separated pair domain choose a finite block B of +original independent coefficients whose gradients (or first jets) have +the ranks proved in Section 3. Split F=F_B+T. The complement T is an +independent smooth Gaussian tail. The block keeps its original weights; +there is no truncation approximation or variance renormalization. + +Fix a smooth tail realization. In a point chart choose d invertible +coefficient columns in the gradient matrix and split block coordinates +as (a,eta). On a neighborhood with the same minor, the exact equation +for a critical point solves + + a(p,eta)=-B(p)^(-1)[grad T(p)+D(p)eta]. (17) + +The map (p,eta)->(a(p,eta),eta) is smooth between equal-dimensional +spaces. At a zero its derivative is singular exactly when the physical +Hessian is singular; moving-frame derivative terms vanish when the +gradient is zero. Equal-dimensional smooth Sard makes its critical +values Lebesgue-null in coefficient space. The finite Gaussian block +has a density. A finite/countable minor and chart cover, followed by +integration over T, proves that F is Morse almost surely before using +Rice. Compactness then makes its critical set finite. + +For the pair-gradient equation use 2d invertible block columns. Its +incidence projection has absolute Jacobian +|det H_p det H_q|/|det B(p,q)| in orthonormal volume coordinates. +Coordinate versions have the matching Riemannian volume factors. +The weighted equal-dimensional area formula applied to this map and +the finite Gaussian density yields, for every nonnegative Borel mark +w of (p,q,F), the root-counting identity on that minor chart. +Partition overlapping chart/minor neighborhoods into disjoint Borel +pieces. Exhaust unbounded coefficient domains by relatively compact +ones, so only local Lipschitzness of the smooth projection is needed. +Integrate eta and T and sum by Tonelli. Exhaust the off-diagonal pair +space by separated compact domains. The result is the extended identity + + E sum_{p!=q, grad F(p)=grad F(q)=0} w(p,q,F) + = integral_{p!=q} p_(grad F(p),grad F(q))(0) + E[|det H_p det H_q| w | both gradients=0] + dvol(p)dvol(q). (18) + +The determinant appears once. Neither finiteness nor a conditional +Morse assertion is a premise of this all-Borel identity. + +We specify why the kernels in (18) agree with (11) even at zero, not +merely almost every target. In the finite-tail calculation, linear +coefficient change of variables defines the fiber integral at every +gradient target: the selected coefficient minor is invertible and its +Gaussian density is everywhere positive. The tail receives the Bayes +density equal to the block observation density at target-minus-tail, +divided by the full observation density. Given that tail, the block +has its exact affine Gaussian conditional law. Their integrated field +law at each target is the Gaussian regression law: its characteristic +function for any finite evaluation vector is the ordinary finite joint +Gaussian conditional characteristic function, obtained by completing +the finite observation square. This calculation is valid at every +target. Countably many evaluations identify Borel laws on smooth +functions: derivatives are measurable limits of value difference quotients +at a countable set of chart points, and each Cq seminorm is a supremum +of derivative magnitudes over countably many chart points. Rational finite +linear combinations from Section 2 give a countable dense set and hence +a countable norm-ball basis. Thus the smooth Borel sigma-field is generated +by countably many value evaluations, and the law is the same whole-field +kernel. Arbitrary Borel +marks were present in the area formula from the start; continuity of +such marks on a null fiber is never presumed. + +On each separated compact domain, first-jet covariance floors and +conditional Hessian moments make the unmarked integral finite. The +conditional two-height Gaussian density given both gradients is +nondegenerate by Section 3 item 2. In (18) the tie mark F(p)=F(q) +therefore has zero integral after height disintegration. Countable +annulus exhaustion proves distinct critical values almost surely. +The analogous point identity and compact gradient covariance floor +also give finite expected total critical-point count. + +For clarity, the everywhere Borel actual elder selector can be defined +intrinsically. Fix a countable dense set and a countable basis of small +connected geodesic balls with compact closures. Connection in {f>c} +is equivalent to a finite chain of such balls with all minima of f +on their closures > c. This is a countable Borel test. For a point p, +take the supremum over rational cf(p), with empty supremum -infinity. +Call the result D_f(p). On the open Morse/distinct-critical-values locus +set sigma(p,q,f)=1 exactly when p is a maximum, q has index d-1, +D_f(p)=f(q)0 + + nu(ell)=V^(-1) integral_{p!=q} integral_R + p_O(b,0,b-ell,0) E_Q[W_pq sigma(p,q,F)] db dvol(p)dvol(q). (19) + +It is initially nonnegative Borel; its finiteness and local integrability +follow from the near and far bounds below. This supplies COUNT for the +same versions used in CONTACT, not independently chosen fibers. + +## 7. Fixed-target conditional remote genericity + +Fix a literal x,u,b,k with k>0 and condition on U_0 F=v_0. This law has +a forced degenerate critical point at x and is not globally Morse. +We prove its precise off-contact genericity independently of the +unconditional result. For each compact domain away from x, or pair +domain separated from x and the diagonal, take a finite block B with +the joint contact/remote ranks of Section 3. The block contact covariance +Sigma_B is positive. For F=F_B+T, under the canonical contact law the +tail distribution is exactly + + dmu_tail,v(t)/dmu_tail(t) + = p_(U_0 F_B)(v_0-U_0 t)/p_(U_0 F)(v_0). (20) + +The denominator is positive and the density integrates to one by +Gaussian convolution. This is an actual fixed-target Bayes identity. +The conditional tail remains smooth since it is absolutely continuous +with respect to the original smooth tail law. + +Given T=t, let L_B be the contact matrix in the standard finite block +coordinates. The conditional block coefficients are + + xi_B = L_B^T (L_B L_B^T)^(-1)(v_0-U_0 t) + P_B z, + P_B = I-L_B^T (L_B L_B^T)^(-1)L_B. (21) + +Choose an orthonormal basis of ker L_B. The free coordinates of P_B z +in that basis are independent standard Gaussians, for every literal +target and every smooth tail. The deterministic mean in (21) need +not vanish. Joint rank implies that these free directions realize +arbitrary remote first jets; this is exactly the restricted-kernel +linear algebra proved in Section 3. + +For the one-point residual gradient, repeat (17) in the free coordinates. +Smooth Sard proves that all critical points in the compact domain are +Morse, for almost every conditional block vector. Integrate (20) and +exhaust X minus {x} by countably many compact domains. Thus every +other critical point is Morse under this fixed contact law. + +All residual covariance floors needed for the following Rice estimates +hold only on the particular separated compact domain. Additional finite +Gaussian regression there and (12) bound the conditional Hessian moments, +so the local unmarked expectations are finite. No floor as a site approaches +x or another site is used. + +The same residual one-point area formula has a nondegenerate conditional +height given residual gradient, since the residual first-jet map is +surjective. The mark F(p)=b has zero expectation on each domain, so +no other critical value equals b. The residual two-point area formula +has a nondegenerate two-height law given both gradients; the tie mark +F(p)=F(q) has zero expectation on each separated domain. Exhaustion +proves pairwise distinct off-contact critical values. Covariance floors +are only used on the separated domains, where they are available. +Different domains may use different finite blocks; each conclusion +is a property of the same canonical whole-field Q_0, so their countable +intersection is still probability one. No conclusion about all literal +targets simultaneously has been drawn. + +The conditional A_0 is invertible almost surely, by its full Gaussian +density (or is empty in d=1). Its transverse implicit ridge exists near +x. The reduced function has first and second derivatives zero at x +and third derivative 12k>0. Its reduced derivative is positive at every +sufficiently small nonzero axial coordinate, since its Taylor expansion +is 6k s^2 + o(s^2). Thus x is the only critical point in a neighborhood. +This isolation does not require A_0<0, only invertibility. The remaining +critical set is contained in a compact set away from x and all its +points are isolated Morse points. It is therefore finite: an infinite +subset of a compact set would have an accumulation point which is also +critical and hence would fail isolation. Its critical values have +finite positive gaps from b and one another. + +This proves exactly the contact-global genericity premise of ST's +deterministic elder theorem, for each fixed specified target. It is +not an invocation of unconditional Morse genericity, and it does not +assert Morse behavior at the forced contact itself. + +## 8. Actual SELECT through the deterministic elder argument + +Use the coupling (11) at fixed parameters. On V_0>0, A_0<0, k>0, +and Section 7 holds almost surely. Thus F_r -> F_0 globally in C4, +with exact endpoint value/gradient pins, a single local cubic contact, +and finitely many other Morse critical branches with distinct heights +separated from b. These are the actual hypotheses of ST Section 4. + +To identify its topological content, in a small fixed cylinder a unique +transverse ridge h_r(s) exists, and the reduced derivative has positive +second derivative. The exact critical pins give its only two zeros +at s=+-r/2. The left zero is the maximum, the right zero is the +index-(d-1) point. A left cylinder cut at the right zero has left and +side boundaries strictly below the death level, while its right face +is at most the death level by transverse concavity. The newborn +component is enclosed at every intermediate level. The positive ridge +beyond the right zero reaches a fixed point with height greater than b, +so that branch is older. The closed death-level path through the right +zero reaches that older branch. No earlier remote detour can escape +the cylinder. Persisting other critical branches and the compact +gradient floor make F_r globally Morse with distinct critical values +eventually. Consequently sigma(M,S,F_r)=1 eventually. This is the +actual deterministic elder transfer, rather than candidate identification. +For d=1 the side boundary is absent and the same left interval/right +cut/older ridge argument applies. The argument does not impose a false +height barrier on the entire boundary of a symmetric fold box. + +On V_0=0, V_r(1-sigma)->0 regardless of the selector, by (15). +On V_0>0 it vanishes eventually. The moment domination (12),(15) +then gives + + E_Qr[V_r(1-sigma)] -> 0, + E_Qr[V_r sigma] -> z_0>0. (22) + +This supplies SELECT. It also gives the normalized weighted selector +probability tending to one at each fixed mark, but no all-mark inverse +normalizer, rate, or common samplewise physical chart is required. +If desired, compact-mark uniformity follows by convergent-parameter +couplings and a subsequence argument applied to the single limiting +parameter's probability-one genericity event. This extra uniformity +is qualitative and is not needed for (3). + +## 9. FAR and finite lifetime representatives + +On dist(p,q)>=r_0, the value/gradient covariance O_pq has a positive +floor and finite upper bound, by rank, compactness and continuity. +The actual same-field Gaussian regression at target (b,0,b-ell,0) +has conditional C2 moments bounded by a polynomial in 1+|b| when +ell belongs to a fixed compact interval. In particular for 00, k=ell/r^3 cannot approach zero as r<=r_0; Gaussian decay at +large k dominates every power as r->0. Thus that near integral is finite +for every ell>0 and locally bounded on every compact positive-lifetime +interval. This proves the stated finite Borel representative and +local integrability in (19). + +## 10. Actual geometry cancellation and positive local coefficient + +The short-pair midpoint coordinates (x,h) map to +(exp_x(-h/2),exp_x(h/2)). Their natural volume Jacobian J(x,h) is smooth, +positive, has J(x,0)=1, and is even in h. Its derivative at h=0 is +(a,b)->(a-b/2,a+b/2), absolute determinant one. Pair swapping is +h->-h and preserves both measures. Compactness gives uniformly +J(x,ru)=1+O(r^2) and a common bound. Therefore + + dvol(M)dvol(S)=J(x,ru) dvol(x) r^(d-1)dr d sigma_x(u). + +Let B_r=12 pi_r(v_r) E_Qr[V_r sigma]. The radial ledger is + + spatial volume: r^(d-1), height change: r^3, + observation density: r^(-(d+3)), determinant product: r^2. + +Their product is r, in every fixed d>=1. The physical Hessians give +the determinant order; the literal linear observation transform gives +the contact-density order. They are not inferred from the cubic height +gap alone. Curvature changes J at finite r, with leading value one. + +The direct fixed-ell substitution in (19) gives + + ell^(1/3) nu_near(ell) + = (1/(3V)) integral_{SX,R,(0,infinity)} + 1{k>=ell/r_0^3} k^(-2/3) + J(x,(ell/k)^(1/3)u) B_((ell/k)^(1/3)) db dk dvol(x)d sigma. (24) + +Equations (15),(22) give B_r -> 12 pi_0(v_0)z_0 and (16) dominates +the integrand. Dominated convergence and (23) yield + + C_F,g = (4/V) integral_{SX,R,(0,infinity)} + k^(-2/3) pi_0(b,0,0,12k,0,...) z_0(b,k,x,u) + db dk dvol(x)d sigma_x(u). (25) + +This is exactly ST's functional, now with all five premises actually +proved for law (2). Each integrand at k>0 is strictly positive: the +contact Gaussian density is positive and the transverse conditional +Gaussian has positive negative-cone determinant weight. Finite volume +and (16) give finiteness; integrating over any nonempty finite mark +window gives strict positivity. Candidate weight has the same limit, +so its leading coefficient agrees. The cumulative factor 3/2 follows +by integration of ell^-1/3; it does not identify the lifetime density +with the cumulative exponent 2/3. + +## 11. General Gaussian jet invariant, parity boundary and falsifiability + +One may express (25) without a birth integral and without a parity +hypothesis. At a fixed x,u write + + G=grad F(x), V_u=Hess F(x)(u,.), T=nabla_u^3 F(x), + A=Hess F(x)|_(u-perp). + +All components are actual covariant jets in orthonormal frames. Under +the pin G=0,V_u=0, T is the third derivative along the geodesic used +above. It is not assumed independent of A or of those pins. Integrating +b in (25) by the actual Gaussian height disintegration gives + + C_F,g = (144/V) integral_SX integral_0^infinity k^(4/3) + p_(G,V_u,T)(0,0,12k) + E[(det A)^2 1{A<0} | G=0,V_u=0,T=12k] dk dvol d sigma. (26) + +Substitute t=12k. Since 144*12^(-7/3)=12^(-1/3), this is + + C_F,g = (12^(-1/3)/V) integral_SX p_(G,V_u)(0,0) + E[(T_+)^(4/3)(det A)^2 1{A<0} | G=0,V_u=0] dvol d sigma. (27) + +The conditional Gaussian is centered because (2) is centered and the +conditioning target is zero; its full joint T,A covariance is positive +by the jet-rank proof. The expectation is an exact Gaussian fractional/cone +moment of the covariance Schur complement. It remains meaningful without +stationarity or reflection symmetry. In d=1 A is empty, its determinant +and cone indicator are one, and sphere measure on S^0 counts both directions. + +There is also an antipodally invariant form without an oriented positive +part. At the same point replace u by -u and keep the perpendicular frame. +The gradient axial entry changes sign; for the Hessian column its axial +entry F_uu stays the same and its transverse entries F_uvi change sign; +T changes sign; A is unchanged. The zero pin is invariant and the pin +coordinate transformation has absolute determinant one. Thus its density +at zero is unchanged and the conditional T,A law is transported to -T,A. +Changing u to -u preserves sphere area, so integrating T_+^(4/3) gives +one half of the integral with |T|^(4/3). Consequently + + C_F,g = [1/(2*12^(1/3)*V)] integral_SX p_(G,V_u)(0,0) + E[|T|^(4/3)(det A)^2 1{A<0} | G=0,V_u=0] dvol d sigma. (28) + +The universal scalar 1/(2*12^(1/3)) is now separated from a local joint-jet +invariant. This averaging uses both ordered directions in the full sphere; +it is not an additional unordered-pair division by two. Equations (25)-(27) +remain the primary derivation and independently specify the normalization. + +Only if the additional odd/even independence is separately established +can (27) be factored into p_G(0)p_Vu(0), a half-Gaussian T moment and +a transverse determinant-cone moment as in ST equation (19). That +factorization has NOT been asserted for the general nonstationary +manifold law. In particular G may correlate with V_u, and T may +correlate with A, even though every field coefficient is Gaussian. +The coefficient can vary with x,u and with the metric and covariance. + +The formula has a falsifiable amplitude check: F -> aF, a>0, gives +p_(G,V_u)(0,0) -> a^(-2d) times itself, while the moment gives + a^(4/3+2(d-1)); hence C_(aF),g=a^(-2/3)C_F,g. If the metric is +multiplied by lambda^2 with the same scalar field, its volume changes +by lambda^d, G by lambda^-1, V_u by lambda^-2, T by lambda^-3, +and A by lambda^-2. The local density product in (27) scales as +lambda^(3d-4-4(d-1))=lambda^(-d), so C scales as lambda^-d, +as required for a per-volume density; the total expected count is +metric-independent. The exponent remains -1/3 because the metric's +short-pair radial order is still d-1. + +The hypotheses are not covariance-only conditions for arbitrary laws. +The already preserved amplitude-mixture counterexample F=RH, with +R>0 bounded, E R^2=1 and P(R<=s) of order s^alpha, alpha<2/3, +keeps H's covariance, smooth positive moments and Morse pairings, +but its expected short-bar count is bounded below by c t^alpha. +It violates the claimed t^(2/3) conclusion. This theorem admits the +specified Gaussian law, not every law sharing its covariance. +Likewise dropping dense support may force contact rank collapse or +zero transverse negative-cone weight. Approaching such degeneracy +need not preserve any common constants or usable asymptotic scale. + +## 12. Imports, evidence boundaries and status ledger + +The result is a conventional proof under these explicit imports: + +* Smooth finite coordinate refinements, bump cutoffs and partitions of + unity on compact smooth manifolds; inverse/implicit-function theorem; + existence of a uniform short-geodesic scale and smooth parallel frames. +* Equal-dimensional smooth Sard: critical values of a smooth map between + equal-dimensional finite-dimensional manifolds are Lebesgue-null. +* Weighted equal-dimensional Euclidean area formula for a locally Lipschitz + map and every nonnegative Borel integrand, with multiplicity; applied + only on compact subdomains of smooth finite-dimensional incidence maps. +* Classical Morse local form, critical-free compact level-band flow and + isolated handle attachment for an actual smooth Morse function. ST's + Borel elder construction and its vector-ridge deterministic elder lemma + supply the precise H0 use; no separate stochastic selection import exists. +* Elementary finite Gaussian regression, finite joint Gaussian uniqueness, + Tonelli, Minkowski, dominated convergence and ordinary change of variables. + +The finite-jet bump interpolation, actual finite coefficient minors, +residual-rank implication, fixed-target Bayes identity, infinite-field +regression, norm bounds and all five stochastic interfaces were proved +here. No general infinite-dimensional transversality theorem, conditional +Morse theorem, or all-Borel Kac-Rice theorem is imported without proof. +The only global H0 topology imported is the stated deterministic Morse +level/handle package, as already explicit in ST. + +Bounded current source checks: the [UT differential-topology notes](https://web.ma.utexas.edu/users/abhishek/files/notes/Differential%20Topology.pdf) +state smooth Sard at Theorem 1.5.1 (page 14 by one-based PDF indexing). +The [author's area/coarea preprint abstract](https://arxiv.org/abs/1305.2968) identifies the general weighted-volume +area/coarea setting. The [Stanford differential-geometry handout index](https://math.stanford.edu/~conrad/diffgeomPage/handouts.html) lists bump, +local-form and Morse handouts. These bounded checks are source pointers; +there was no new full classical-book read or proof of the imported +area formula. The required theorem statements above remain explicit +classical premises. An unsuccessful attempted secondary link resolution +in the browsing session is recorded in READ_EXTENT.json and supplies no +additional evidence. + +The exact ST source is 43927 bytes, SHA256 +3d952a906553d578876456539a8f0dcd64aedd45099e8394cbe9db99866ec531; +its source-bound review records the fresh incorporated-file PASS after +the preserved earlier half-moment correction. The IG source is 34629 +bytes, SHA256 c8bd489ca3fbb3760f7b2d53e7385131ac5336c51ba983e00e4d540fe2f00aaa; +its actual finite-tail/conditional genericity route was read fully. +Their full actual reviews were also read. Exact copies in inputs/ +are frozen; no original file was changed. This author is source-exposed, +and root supplied the dense-support mechanism. A distinct nonauthor +mathematical review of these exact new bytes is still required. + +Result scope: actual expected finite H0 lifetime intensity for every +fixed member of the explicitly constructed Gaussian class on every +fixed compact connected boundaryless smooth Riemannian manifold of +dimension d>=1. No arbitrary smooth Gaussian law lacking the hypotheses +is silently admitted. No higher-singularity taxonomy theorem, Hk result, +all-mark factorial moment or Poisson limit, blind 50-model validation, +coefficient enclosure, sampler/grid correctness, Lean proof, quantitative +window, manuscript acceptance, external uptake, or scientific promotion +follows. The larger research goal remains active. diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/specifications/lean_statements_20261010/Specifications/Field.lean b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/specifications/lean_statements_20261010/Specifications/Field.lean new file mode 100644 index 000000000..cc40e9d19 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/Math-/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/frontiers/compact_morse_elder_adapter_20261018/inputs/main/experiments/universality/finite_h0_lifetime/PROOF.md b/frontiers/compact_morse_elder_adapter_20261018/inputs/main/experiments/universality/finite_h0_lifetime/PROOF.md new file mode 100644 index 000000000..7a3518edd --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/main/experiments/universality/finite_h0_lifetime/PROOF.md @@ -0,0 +1,526 @@ +# Finite Gaussian H0 elder-bar lifetime intensity: conventional analytic source + +Prepared on 8 October 2026 by OpenAI/Codex `/root/universality_review`, at +`/root`'s request. Incorporated by the source and integration owner OpenAI/Codex +`/root`, task `01a11cbf-068d-7102-861e-75814e715c98`, under the +[five-path claim](https://github.com/d6g8k5htny-coder/main/issues/307#issuecomment-6070045125). +This source incorporates the frozen external preparation +`finite-selected-bar-intensity-preparation.md`, 25,439 UTF-8 bytes / 501 lines, +SHA256 `5f529e5ed2632c90819746848146e3a24acda7e888cd3c9342df76edcbcbb7a3`. +Its mathematical text in Sections 2–8 is unchanged. Source locations, +elder-source identity and review chronology are updated here. Actual new-byte +reviews and integration evidence are separately bound in the +[review record](REVIEW.md). + +The original author authored the candidate-lifetime and weighted-window +preparations and reviewed the contact, fold and contact-genericity arguments. +The elder-transfer author implemented the catalogue machinery and contributed +to the earlier mathematical routes. Root fully read the original saved source. +`/root/benchmark_formal_audit` and `/root/final_contract_review` each freshly +read and hashed all 501 lines of that original preparation and returned full +PASS, without a mathematical must-fix finding. Those verdicts bind the frozen +preparation; they are not rebound to this successor. This is source and +proof-route exposure within the same OpenAI/Codex team; organizational- +independence credit is zero. No human, blind or provider-distinct acceptance, +formal Lean proof, sampler execution, seed consumption or scientific promotion +is claimed. The source supplies a Borel selector, exact once-counting and the +selected density composition using the bound elder source below. + +## 1. Exact consumed sources and read scopes + +The original author fully reread C, CG and the frozen ET preparation for the +original composition. Current consumed repository identities are: + +* **C**, incorporated candidate theorem: + [candidate proof](../finite_candidate_lifetime/PROOF.md), + 29,190 UTF-8 bytes, 577 lines, SHA256 + `ed31f562f0ffaaf6d4bd2ceb1ae3d234798e983a55d55c6983b128fce27133f5`. +* **CG**, incorporated contact-conditioned genericity: + [contact-genericity proof](../finite_contact_genericity/PROOF.md), + 23,431 UTF-8 bytes, 473 lines, SHA256 + `233e80421ccf78e9a9015116aaaf2d355a8425107ff109c022ddc98cd1f87fd9`. +* **ET**, incorporated elder-transfer proof: + [elder proof](../finite_elder_transfer/PROOF.md), 23,774 UTF-8 bytes / 458 lines, + SHA256 + `c9e4fb09900888cd2b1bd96b49f85072b51c822b2503fe9f3d8891049bf66201`. + +Root freshly read the complete incorporated ET source and its complete real +predecessor diff, and independently reproduced these current C/CG/ET byte +identities. The mathematical sections of ET are byte-identical to its frozen +preparation; the successor rebinds CG to the incorporated source and makes the +completed original-review chronology explicit. This composition consumes the +stated and proved analytic result at ET's current exact bytes; any source +correction requires reassessing this dependency. New-byte ET review verdicts +are separately recorded in [ET's companion](../finite_elder_transfer/REVIEW.md). + +The original composition instead consumed +`finite-contact-elder-transfer-preparation.md`, 21,138 UTF-8 bytes / 418 lines, +SHA256 `1cac9b525831b7e1f37f0d0561d3231fa73c8713c4c92283dca1d5b51b8a67ce`. +It recorded pending ET review when written; those original bytes remain frozen. +That original ET subsequently received a full exact-file nonauthor +`/root/final_contract_review` PASS and root complete read. Reading a source and +completing its mathematical review do not establish formal or scientific +admission. The original C/CG worktree locations are historical exposure paths; +the relative links above identify the current consumed repository copies. + +The underlying [**FG** contact proof](../finite_gaussian_contact/PROOF.md) was +previously fully read and reviewed: +25,367 UTF-8 bytes, 554 lines, SHA256 +`7fea2656e11d986e1a5d6bfb0a05c8421bd93262d28051b8cfef05320595f474`. +The original author rehashed its conditional-worktree copy; root freshly +reproduced the same current repository identity for this incorporation. +Its named moment, normalizer and envelope inputs were reread through C and ET. +No fresh full classical-paper or book fetch was performed in this composition. + +C explicitly imports equal-dimensional Sard and the weighted area formula. +Those imports, their prior primary-source exposure and their limits remain as +recorded in C; no new proof of Sard or the area formula is claimed here. +ET explicitly records its specified primary Milnor and Edelsbrunner-Letscher- +Zomorodian reads and imported Morse/elder facts. That is ET's author's read, +not a fresh primary-book read by this author. Sections 3-4 below independently +spell out the component argument and elementary H0 once-counting used here. + +The source chain used is finite Gaussian ranks/regression and envelope (FG), +unconditional genericity and all-Borel pair counting (C), contact-conditioned +genericity (CG), deterministic and weighted elder transfer (ET), then the +selector/counting/density composition proved here. No arbitrary finite-jet +independence premise is transferred across this chain. + +## 2. Fixed ideal law, observable, and precise theorem + +Fix an integer K>=2, L>0 and omega=2pi/L. On X=R^2/(L Z^2), with area V=L^2, +use the ideal real Gaussian field + + S_K={-K,...,K}^2, + q_n>0, q_n=q_-n, sum q_n=1, + F(z)=sqrt(q_0) C_0 + +sum_pairs sqrt(2q_n) + [C_n cos(omega n.z)+S_n sin(omega n.z)], (1) + +where the coefficient coordinates are independent standard real Gaussians +and one representative of each nonzero pair {n,-n} is used. The coefficient +vector xi has dimension N=(2K+1)^2. The field is smooth, stationary and +variance one. It need not be rotationally isotropic. + +Use superlevel H0 over any fixed coefficient field, with increasing filtration +parameter t=-c and sets {F>=c}. A finite elder bar born at a local maximum p +and dying at a component-merging index-one saddle q has height lifetime + + ell=F(p)-F(q)>0. + +Its interval in increasing parameter is [-F(p),-F(q)); endpoint conventions +do not alter its lifetime. The one essential class is excluded. For a Borel +I subset of (0,infinity), let N_bar(I) count all such finite bars and put + + mu_bar(I)=E N_bar(I)/V. + +The observable can be assigned value zero on the unconditional null set of +non-Morse or equal-critical-value fields. Section 4 gives a Borel version +which equals the actual finite-bar count almost surely. + +**Composition theorem.** Consuming the analytic elder-transfer result of ET +for the same fixed law (1), mu_bar has a finite nonnegative Borel density +representative nu_bar on ell>0, specified by the marked kernel below, with + + nu_bar(ell) ~ C_loc ell^(-1/3), ell decreasing to zero, (2) + E N_bar((0,t])/V ~ (3/2) C_loc t^(2/3). (3) + +The density coefficient is exactly C's finite positive candidate coefficient: + + C_loc=Gamma(7/6)/(24^(1/3) sqrt(pi)) + *integral_{S^1} p_G(0) p_Vu(0) tau_u^(4/3) + [s_u^2/2] d sigma(u), (4) + +where G=(F_u,F_v), V_u=(F_uu,F_uv), T_u=F_uuu, A_u=F_vv, +tau_u^2=Var(T_u|G=0), and s_u^2=Var(A_u|V_u=0). Every derivative and +covariance is from (1). Angular measure is unnormalized arclength of mass +2pi. The numeral 24 in (4) is the Jacobian/gamma factor, not the side L. +Whole-torus expected counts have coefficient V C_loc; at side24, V=576. + +Equation (2) concerns the particular finite Borel representative constructed +in Section 5. An arbitrary Radon-Nikodym representative can be changed on a +null set and need not retain a pointwise asymptotic. No continuity of nu_bar +is proved, and no O(1) near-term remainder is claimed. Equation (3) is an +intrinsic statement about the expected measure and follows by integration, +not by differentiating a cumulative asymptotic. + +## 3. A Borel elder-death threshold from open superlevels + +Let D be a countable dense subset of X. Fix a countable basis of open metric +balls with centers in D and rational radii below one eighth of the torus +injectivity radius. Their closures are compact, and the balls are path +connected. This basis is fixed independently of F. + +For a real level lambda, define C_lambda(p,z;xi) to mean that p,z belong to +the same path component of the open set {F_xi>lambda}. It has the following +countable description: there is a finite chain of basis balls B_1,...,B_m +such that + + p in B_1, z in B_m, + B_i intersects B_(i+1), + min_{closure B_i} F_xi > lambda for every i. (5) + +Each chain condition is Borel jointly in (lambda,p,z,xi). The minimum on +each fixed compact ball is continuous in xi, because coefficient convergence +implies uniform field convergence. The union over finite chains is countable. +Conversely, a path in {F>lambda} has compact image and can be covered by +finitely many basis balls with closures inside that open set; a chain along +the path gives (5). A chain gives a path because its balls are path connected +with consecutive nonempty intersections. Thus (5) is the exact connectivity +relation, not an approximation by a fixed finite mesh. + +For every p define the extended-real threshold + + d_xi(p)=sup{lambda in Q: + lambdaF_xi(p), + C_lambda(p,z;xi)}, (6) + +with supremum of the empty set equal to -infinity. This is a Borel function +of (p,xi), by countability of the levels and dense points. It never equals ++infinity because qualifying levels are below F_xi(p). + +The dense-point condition is equivalent to connecting to any point of height +strictly greater than F(p): strict inequality supplies an open neighborhood +containing a dense point, and a short extension of the path stays in the open +superlevel. Consequently (6) is intrinsic, despite the chosen basis. In +particular it is equivariant under torus translations. + +If no point has greater height, d=-infinity. If some point has greater height, +every sufficiently low level qualifies, since X is connected and lambda can +be taken below min F. Therefore d is finite in that case. For a nondegenerate +local maximum, a small enclosing boundary has height strictly below F(p), +which makes d0. (12) + +C already proves finiteness of its displayed candidate density at every +positive lifetime, and finite total expected candidate count. Thus (11) is +finite everywhere on ell>0, and mu_bar has finite total mass as well. + +The unconditional null exceptional set G^c need not be null in every fixed +six-pin regression fiber. Formula (9) with its indicator implies only that +the determinant-weighted exceptional contribution vanishes for almost every +location/height tuple, after disintegration. That is sufficient for (11). +We do not turn that almost-everywhere statement into a static all-target +conditional Morse assertion. The separate ET coupling proves the small-r +selection limit needed next. + +The selected factor is global and need not be continuous. C's continuity +proof uses the continuous determinant-weighted type W without S. It does +not imply continuity of the expectation in (10) or of nu_bar. Measurability, +finiteness and domination are the statements proved here. + +## 6. Exact consumption of the elder-transfer result + +In the near band let M=x-r u/2, S=x+r u/2 and use the literal six-pin law Q_r +at heights b,b-k r^3, with k>0. Its divided-difference target is + + v_r=(b-k r^3/2,-k r^2,0,12k,0,0). + +Write V_r=W_r/r^2 and z_r=E_Qr V_r. ET uses the actual coupled coefficients + + xi_r=m_r+P_r Z, + +with m_r,P_r from the six divided-difference rows. They converge to their +contact versions and give global C4 convergence of the same actual fields. +Under the contact law, CG supplies isolated contact, all other critical points +Morse, finite distinct other critical heights different from b. On A_0<0, +ET Sections 2-9 prove that the actual pinned pair is eventually the finite +elder pair and that F_r is globally Morse with distinct values. These are +exactly the conditions in the selector (7). + +ET Section 11 proves, for each fixed b,k>0,x and frame, + + E_Qr[V_r(1-1_Er)]->0, + z_r->z_0=36k^2 E[A_0^2 1{A_0<0}]>0, (13) + +where E_r is that globally good actual-pair event. The selector agrees with +1_Er on the actual pinned fields. ET proves (13) by the polynomial Gaussian +envelope and dominated convergence: V_r converges to +36k^2 A_0^2 1{A_0<0}, and selection is eventually true on its positive +support. It does not assume that every fixed finite-r pin law is globally +generic. ET Section 12 separately proves uniform normalized selection on +compact birth/gap marks, locations and frames, by varying-parameter +subsequences; pointwise convergence alone is not represented as uniform. + +Consequently the exact consumption needed here is + + E_Qr[V_r S(M,S,xi)]->z_0. (14) + +It follows by subtracting the first term in (13) from z_r. We need only the +pointwise form for every fixed mark and frame in the all-mark dominated +convergence argument. The stronger compact uniformity is available from ET, +but no all-mark normalized inverse-z bound is used. + +When z_r>0, this also reads Q_r^W(S=1)->1. If a normalizer vanishes outside +the eventual compact-mark positive band, no ratio is needed: the selected +amplitude below is defined directly by its unnormalized expectation. No +division by k, a transverse curvature, or a random inverse normalizer enters. + +## 7. Selected near-diagonal density and all-mark limit + +Choose C's common near band 0=ell/r_0^3} + B_((ell/k)^(1/3))(b,k,u)/(3k^(2/3)) + db dk d sigma(u). (18) + +This is direct height/lifetime disintegration of the selected count; it is +not a derivative of a cumulative estimate. Values at the band boundary can +be assigned by either convention without changing the integral. +Equivalently, substitute k=ell/r^3 directly in the fixed-ell polar integral +from (11): its radial factor is r^-2 B_r dr. This gives (18) pointwise for +the same representative, not just equality of two density versions almost +everywhere. + +For every fixed b,k>0,u, pi_r(v_r)->pi_0(v_0) by the contact covariance +limit, and (14) gives + + B_r->12 pi_0(v_0) z_0. (19) + +As ell decreases to0 the indicator in (18) tends to1. Equation (16) gives +the integrable majorant H/(3k^(2/3)). The product is Borel, so dominated +convergence over all marks and angles gives + + ell^(1/3) nu_bar,near(ell) + ->4 integral k^(-2/3) pi_0(v_0) z_0 db dk d sigma + =C_loc. (20) + +Each fixed mark belongs to some compact mark window, so ET's compact-mark +scope supplies its pointwise limit wherever needed. No common samplewise +event over all marks or frames is required. The integral over noncompact +marks is controlled by the unnormalized envelope, rather than by a claimed +uniform selection probability on that noncompact set. + +## 8. Far complement, cumulative conclusion, and coefficient + +At torus separation at least r_0, the original six-pin covariance has a +compact-domain positive floor. C's far candidate bound gives + + 0<=nu_bar,far(ell)<=nu_cand,far(ell)<=C, + 00`, by C's near limit and (20). +This is only an `o(ell^-1/3)` complement estimate; no `O(1)` missed-near +bound follows from the dominated-convergence argument. + +Since C_loc>0, for every relative error tolerance there is a small lifetime +band on which nu_bar is bounded above and below by the corresponding +multiples of C_loc ell^-1/3. Integrating from0 to t gives (3), with + + integral_0^t ell^-1/3 d ell=(3/2)t^(2/3). + +The cumulative statement is independent of the representative's null-set +values. No stronger near remainder or continuity is inferred. + +The coefficient in (20) is C's already evaluated contact functional. Its +parity ledger retains all within-block correlations. In particular the +law of A conditioned on F=b,V_u=0 need not be centered; birth disintegration +first reduces it to A|V_u=0, whose negative-halfline second moment is +s_u^2/2. The remaining half-Gaussian integral gives precisely the factor +Gamma(7/6)/(24^(1/3)sqrt(pi)) in (4). Thus selection changes no leading +coefficient or angular/area factor. It supplies no numerical enclosure or +model-independent coefficient. + +## 9. Finite family and remaining boundaries + +C, CG and ET cover every fixed positive symmetric full-square ideal Gaussian +law with integer K>=2 and L>0, with its own constants. In particular the ten +catalogue spectra with ideal Gaussian coefficients at L24 and K2/3 give +twenty fixed cases. The Borel and H0 arguments here are deterministic and +unchanged between them. Finite minima of the input bands and constants, and +ET's separately proved finite-family compact uniformity, are valid. Finite +many individual density limits then yield + + max_j |ell^(1/3) nu_bar,j(ell)-C_loc,j| ->0. + +Each coefficient is positive, so uniform relative leading asymptotics over +this fixed finite family follow as well. There is no uniform claim over +spectra approaching a positivity boundary, unbounded K, variable L or an +unspecified infinite-field law. K1 and non-Gaussian coefficient laws are +outside the consumed route. No finite-word or quantized sampler is identified +with the continuous ideal Gaussian coefficient law. + +This source supplies all finite superlevel H0 bars in the expected +lifetime measure, not just candidate pairs. It does not supply higher +homology, factorial moments, a point-process limit, a quantitative usable +asymptotic window, coefficient digits, blind-field validation, scientific +promotion, publication acceptance or external uptake. The exact contact +itself remains degenerate. No formal owner lane or old source is amended. + +The only new model-specific selection input consumed by the composition is +ET at the stated current exact source hash. Its CG/contact prerequisites, +original pending-review history and subsequently completed original review +remain explicit. The Borel selector, once-counting and selected near/far +dominated-convergence steps are proved here. New-byte exact-source nonauthor +verdicts are separately bound in the companion review record. diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/main/experiments/universality/fold_isolation_diagonal/QUANTITATIVE_FOLD_ISOLATION_PROPOSAL.md b/frontiers/compact_morse_elder_adapter_20261018/inputs/main/experiments/universality/fold_isolation_diagonal/QUANTITATIVE_FOLD_ISOLATION_PROPOSAL.md new file mode 100644 index 000000000..86c9715b9 --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/main/experiments/universality/fold_isolation_diagonal/QUANTITATIVE_FOLD_ISOLATION_PROPOSAL.md @@ -0,0 +1,293 @@ +# Quantitative fold isolation in arbitrary pair directions + +16 October 2026 UTC. Local conventional-mathematics proposal by OpenAI/Codex `/root/core_theorem_audit_20261015`, independently reconstructing and attacking the sketch supplied by OpenAI/Codex `/root`. Organizational-independence credit: **0**. This document has not yet received a nonauthor review. It changes no production source, native record, claim, graph node or scientific status. In particular, it is not a Lean theorem, a factorial Rice bound, or a spatial point-process limit. + +## 1. Concrete advance and source binding + +An explicit Lyapunov–Schmidt chart isolates **at most two ambient critical points**, including at most one maximum, around a maximum with a nonvanishing soft cubic and a uniformly negative hard eigenvalue. Its radius is bounded below by a constant times the inverse square of a finite Gaussian coefficient norm. A literal positive cubic-gap mark can imply the required eigenframe cubic, but only with the derivative errors displayed below. Combining these statements gives an arbitrary-angle **occurrence-probability** estimate `C exp(-c/eta)` for two nearby short pairs on fixed positive transverse/cubic mark windows. It does not bound their factorial counting kernel. + +The source consumed for the pair convention and finite Gaussian law is Main `85b928b326ff0586715a56ae91f51d3fbd89557a`, [experiments/universality/finite_gaussian_contact/PROOF.md](https://github.com/d6g8k5htny-coder/main/blob/85b928b326ff0586715a56ae91f51d3fbd89557a/experiments/universality/finite_gaussian_contact/PROOF.md). Git blob `5f345201309cfa1781e92d6292f542de4bcb2999`, 25,367 bytes, SHA-256 `7fea2656e11d986e1a5d6bfb0a05c8421bd93262d28051b8cfef05320595f474`. Its complete local body was checked against this tree identity. Its lines 41–57 define the actual finite source, 68–76 define the pins and radius, 145–186 give the divided differences, and 253–307 specify physical Hessian weights. The source is pinned, rather than asserted to be the latest moving head. + +The previous frozen [COALESCING_CONTACT_CALCULATION.md](COALESCING_CONTACT_CALCULATION.md), [NONAUTHOR_COALESCENCE_REVIEW.md](NONAUTHOR_COALESCENCE_REVIEW.md), and [AUTHOR_ALIGNMENT_ADDENDUM.md](AUTHOR_ALIGNMENT_ADDENDUM.md) were read. This proposal is a separate arbitrary-angle calculation. Their review verdict does not extend to this document. Their collinear `exp(-c/eta^4)` Rice-kernel estimate is **not** the estimate derived here. Their four-site height-covariance rank obstructions remain in force. + +## 2. Deterministic chart, including domain and constants + +Use Euclidean coordinates in a Hessian eigenframe at an anchored critical point, moved to the origin. Let `F` be `C^4` on an open neighborhood of the closed square `Q_R0=[-R0,R0]^2`, with `R0>0`. Suppose + +```text +grad F(0)=0, F_xy(0)=0, +F_xx(0)<0, F_yy(0)<=-a<0, +|F_xxx(0)|>=tau>0. +``` + +Let `B>=sup_Q_R0 ||D^3 F||op` and `C>=sup_Q_R0 ||D^4 F||op`. These are multilinear operator norms with unit vectors in each slot, so the bounds survive frame changes. In particular, each coordinate partial is bounded by the corresponding constant. Necessarily `B>0`. If only the maximum absolute coordinate partials `M3,M4` in an initial orthonormal frame are available, the conservative substitutions `B=2 sqrt(2) M3`, `C=4 M4` suffice by `||v||_1<=sqrt(2)||v||_2` in each slot. + +Set + +```text +D = C+B^2/a > 0, +R = min(R0, a/(32B), tau/(32D)). (1) +``` + +The following statements hold on the **closed** square `Q_R`. Classifications refer to ambient critical points of `F`, not constrained extrema of the square. + +For each `|x|<=R`, there is a unique `h(x)` in `(-R,R)` with `F_y(x,h(x))=0`. The ridge has `C^3` regularity (locally extending at the two endpoints), and + +```text +h(0)=h'(0)=0, +|h(x)|<=B x^2/a<=R/32, +|h'(x)|<=4BR/a<=1/8. (2) +``` + +For `f(x)=F(x,h(x))`, + +```text +|f'''(x)-F_xxx(0)|<=2CR+16B^2 R/a<=tau/2. (3) +``` + +Thus `f'''` has the fixed sign of `F_xxx(0)` and magnitude at least `tau/2`. There are at most two critical points of `F` in `Q_R`. The anchored maximum is its only local maximum. If a second critical point exists, including on `x=+/-R`, it is nondegenerate and has Morse index one. The faces `y=+/-R` contain no ambient critical points. This is an isolation result containing a possible maximum–saddle pair, not an ordinary quadratic Morse chart about the maximum. + +### Proof of the chart and its vertical barrier + +On the square, integration along coordinate segments gives + +```text +|F_yy(x,y)-F_yy(0)|<=B(|x|+|y|)<=2BR, +F_yy(x,y)<=-a/2. +``` + +Since `F_y(0)=F_xy(0)=0`, the one-dimensional Taylor integral on `y=0` gives `|F_y(x,0)|<=B x^2/2`. Hence + +```text +F_y(x,+R)<=B R^2/2-aR/2<=-31 aR/64, +F_y(x,-R)>=-B R^2/2+aR/2>=+31 aR/64. (4) +``` + +The intermediate value theorem and strict decrease in `y` give the unique root. The implicit function theorem gives its stated regularity; the strict face margins permit local continuation at the endpoints. Subtracting `F_y(x,0)` from `F_y(x,h)=0` bounds `|h|<=B x^2/a`. Also + +```text +h'=-F_xy/F_yy, +|F_xy(x,h)|<=B(|x|+|h|)<=2BR, +|h'|<=4BR/a. +``` + +All derivatives in the next display are evaluated at `(x,h(x))`. Exact differentiation, using the first and second derivatives of `F_y(x,h(x))=0`, yields + +```text +f'=F_x, +f''=F_xx+F_xy h'=F_xx-F_xy^2/F_yy, +f'''=F_xxx+3F_xxy h'+3F_xyy(h')^2+F_yyy(h')^3. (5) +``` + +For example, differentiating `f''=F_xx+F_xy h'` initially produces `F_xy h''`; the second derivative of `F_y=0` and `F_xy=-F_yy h'` convert this into the remaining mixed terms in (5). Omitting that cancellation would be an error. + +The first term differs from `F_xxx(0)` by at most `2CR`. For `p=|h'|<=1/8`, `3p+3p^2+p^3<=4p`. The remaining terms have total absolute value at most `4Bp<=16B^2R/a`, proving (3). Here `2C+16B^2/a<=16(C+B^2/a)` and (1) give the final `tau/2` bound. + +Put `g=f'`. Fixed-sign `g''=f'''` makes `g` strictly convex or strictly concave, so it has at most two zeros. Every critical point of `F` in the square lies on `h`, and its `x` coordinate is a zero of `g`. At the anchored zero, `g'(0)=f''(0)=F_xx(0)<0`. If a second zero exists, strict monotonicity of `g'`, or Rolle's theorem between the two roots, implies its derivative is strictly positive. Thus the second root is simple and `f''>0`. At a critical point, + +```text +det Hess F = F_yy f''. +``` + +The second point has a negative determinant and is an index-one saddle. This also proves that a second degenerate root cannot coexist with the anchored nondegenerate maximum. Without that anchor hypothesis, a double root `g=g'=0` is possible; the two-root bound by itself would not prove nondegeneracy. No critical point outside the square or constrained boundary extremum is covered. + +Finally, integrating `F_yy<=-a/2` twice from the ridge gives the useful **vertical** height barrier + +```text +F(x,y)<=f(x)-(a/4)(y-h(x))^2, +F(x,+/-R)<=f(x)-aR^2/8. (6) +``` + +The last bound follows from `|h|<=R/32` and `(31/32)^2/4>=1/8`. It is relative to `f(x)`, rather than to the height at the anchored maximum. In particular, a distinct maximum at Euclidean distance strictly less than `R` from the origin is excluded, regardless of its own eigenframe or pair direction. + +## 3. Falsification attempts that must remain attached + +**A height drop on the entire boundary relative to `F(0)` is false.** For + +```text +F(x,y)=-mu x^2/2+tau x^3/6-a y^2/2, mu,a,tau>0, +``` + +the origin is a nondegenerate maximum with the required transverse curvature and cubic. But `F(R,0)>F(0)` whenever `mu0`, let + +```text +F(x,y)=-A x^2(x-delta)^2-a y^2/2, A=tau/(12delta). +``` + +There are nondegenerate maxima at `x=0,delta`, both with `|F_xxx|=tau`; but `|F_xxxx|=2tau/delta`. Their distance can vanish as the fourth-derivative bound diverges. They share a local saddle; they are not being asserted to represent two actual finite persistence bars. + +These polynomial examples are deterministic local controls. They do not assert positive-probability occurrences within a particular finite Fourier law. + +## 4. Literal endpoint distance and conversion of marks to the eigenframe + +The pinned source uses `M=z-r u/2`, `S=z+r u/2`, so **`r=|S-M|` is the endpoint distance**, and `ell=F(M)-F(S)=k r^3`. It is not the half-distance. Assume the whole endpoint segment is in the domain on which the operator bounds `B,C` hold, and both endpoints are critical. The source's fourth divided-difference row becomes exactly + +```text +12k = integral_0^r [6s(r-s)/r^3] F_uuu(M+s u) ds. (8) +``` + +The nonnegative kernel has mass one and mean `s=r/2`. Consequently, + +```text +|F_uuu(M)-12k|<=C r/2. (9) +``` + +This is an endpoint estimate. It must not be replaced by a midpoint estimate without moving the evaluation point. + +Let `H=Hess F(M)`. Vector Taylor expansion of `grad F(S)-grad F(M)=0` gives + +```text +H u = -(1/r) integral_0^r (r-s) D^3F(M+s u)[u,u,.] ds, +||H u||<=B r/2. (10) +``` + +Suppose `M` is a nondegenerate maximum and its most negative eigenvalue is at most `-a`. For example the source transverse condition `F_vv(M)<=-a` for `v` perpendicular to the **pair** direction implies this by the Rayleigh quotient; it does not assert that the pair frame diagonalizes `H`. Assume `rB<=a`. The soft eigenvalue has absolute value at most `Br/2<=a/2`, since the smallest singular value of `H` is at most `||Hu||`. Thus the hard and soft eigenspaces are distinct. Let `v_h` be a unit hard eigenvector and `e` a unit soft eigenvector oriented by `e.u>=0`. Projecting (10) onto the hard eigenspace gives + +```text +|u.v_h|<=Br/(2a)<=1/2, +||u-e||<=sqrt(2)|u.v_h|, +|F_eee(M)-F_uuu(M)|<=3B||e-u||<=3B^2 r/a. (11) +``` + +The last constant is deliberately conservative. Combining (9) and (11), + +```text +F_eee(M)>=12k-r(C/2+3B^2/a). (12) +``` + +Therefore `k>=k_->0`, `rB<=a`, and + +```text +r(C/2+3B^2/a)<=6k_- (13) +``` + +imply `F_eee(M)>=tau:=6k_-`. Choose the isolation chart's `x` direction to be this `e` and its `y` direction the hard eigenvector. The original pair direction can have any angle; only the estimate (11) is used. An upper mark bound is not needed for this implication. The derivative conditions are retained explicitly. + +The normalization check requested by the sketch author is exact: + +```text +F=x^3/6-delta^2 x/2-a y^2/2, +M=-delta, S=+delta, r=2delta, +ell=2delta^3/3, k=1/12, F_xxx=1=12k. (14) +``` + +Conversely, positive `k` alone does **not** give a nonzero cubic at the maximum. The polynomial + +```text +F=(k/r)x^4-2kr x^2-a y^2/2 +``` + +has a nondegenerate maximum at `M=0`, a saddle at `S=r`, and gap `k r^3`, but `F_xxx(M)=0` and `F_xxxx=24k/r`. The error `Cr/2=12k` in (9) is exactly saturated. This falsifies a uniform mark-to-endpoint-cubic inference without controlling `rC`. It asserts no global elder relation. + +## 5. Consequence for the actual fixed finite Gaussian source + +Keep exactly the pinned source law: fixed `L>0`, `omega=2pi/L`, square support `S_K={-K,...,K}^2`, fixed positive symmetric weights `q_n` summing to one, and its `N=(2K+1)^2` independent standard real coefficients `Z`. No covariance isotropy or independent-copy replacement is made. For `j=3,4`, put + +```text +beta_j = [sum_(n in S_K) q_n (omega|n|)^(2j)]^(1/2)>0, +W=1+||Z||_2. +``` + +For arbitrary unit vectors in the `j` derivative slots, the squared Euclidean norm of the coefficient row for that directional derivative is at most `beta_j^2`: within each sine/cosine pair the squares add to `2q_n` times the squared product of the frequency projections. Cauchy–Schwarz then gives, uniformly in location and frame, + +```text +sup_X ||D^jF||op <= beta_j ||Z||_2 <= beta_j W. (15) +``` + +Fix an embedded square scale `R0>0` small enough for all translated/rotated charts being used, for example `R0<=L/16`, and use global derivative bounds `B=beta_3 W`, `C=beta_4 W`. For a maximum satisfying the direct transverse/cubic hypotheses with fixed `a,tau>0`, define + +```text +kappa=min(R0, a/(32beta_3), + tau/[32(beta_4+beta_3^2/a)])>0. +``` + +Because `W>=1`, each member of the minimum in (1) is at least the corresponding constant divided by `W^2`. Thus its isolation radius obeys + +```text +R>=kappa/W^2. (16) +``` + +If another distinct maximum lies at Euclidean chart distance `delta`, (16) implies `W^2>=kappa/delta`. In particular, for any measurable occurrence event `E_delta` of two such maxima at positive distance at most `delta`, and `delta<=kappa/4`, + +```text +P(E_delta)<=2^(N/2) exp[-kappa/(16delta)]. (17) +``` + +Indeed `W>=sqrt(kappa/delta)` forces `||Z||>=sqrt(kappa/delta)-1>=sqrt(kappa/delta)/2`, and `E exp(||Z||^2/4)=2^(N/2)`. Markov's inequality yields (17). The same statement applies to outer probability if an occurrence set's measurability has not been separately supplied. It does not require a first/second-jet covariance inverse. + +### Pair marks without silently assuming their errors small + +Here is a precise norm dichotomy that incorporates the errors (13). Fix `a,k_->0`, `00`. Suppose two candidate pairs `(M_i,S_i)`, with distinct maxima `M_1!=M_2`, satisfy + +```text +grad F(M_i)=grad F(S_i)=0, +M_i is a nondegenerate local maximum, +u_i=(S_i-M_i)/r_i, r_i=|S_i-M_i|>0, +F(M_i)-F(S_i)=k_i r_i^3, k_i>=k_-, +F_(v_i v_i)(M_i)<=-a, v_i perpendicular to u_i, +00. (18) +``` + +Then these two pairs force + +```text +W^2>=c_*/eta. (19) +``` + +For otherwise `W^2=kappa/W^2>kappa eta/c_*>=(1+rho)eta>=delta`, placing the other maximum strictly inside the isolation square, a contradiction. Global derivative bounds justify using the same `B,C` along the first pair segment and throughout this chart. + +For an occurrence event `E_eta` of such configurations with center distance at most `eta<=min(eta0,c_*/4)`, (19) gives + +```text +P(E_eta)<=2^(N/2) exp[-c_*/(16eta)]. (20) +``` + +For varying center distances `s<=eta`, apply (19) with `s` and obtain the stronger lower bound `W^2>=c_*/s>=c_*/eta`. Thus this is a bound on the whole close-pair occurrence event, not just one fixed configuration. It is fixed-law and fixed-mark-window; no uniformity in vanishing `a` or `k_-`, unbounded `K`, varying `L`, or varying spectra is claimed. + +## 6. What this does and does not change + +The next finite mathematical task can use (19) as an arbitrary-angle collision obstruction on positive transverse/cubic windows. This removes a collinearity restriction from this **occurrence** bound and avoids demanding a nonexistent uniform twelve-row height density on the K2/K3 coordinate stratum. It does not repair that density, calculate a conditional four-site covariance determinant, or supply conditional determinant/selection-weight estimates. In particular, an exponentially small probability of some collision does not by itself bound the expectation of a possibly large factorial count. A separate weighted counting argument, with its actual conditioning list and finite-support rank control, remains necessary. + +Small transverse curvature, small cubic marks, pairs whose radii are not bounded by a fixed fraction of center separation, shared-maximum incidences, and other collision geometries are outside (18)–(20). The height barrier (6) proves no actual local selection probability. Global H0 max–saddle pairing, interval decomposition, essential-class handling, higher homology, leading complement estimates, same-field expanding-domain control, process convergence, and the literal lifetime density `ell^(-1/3)` rather than merely cumulative `t^(2/3)` remain separate obligations. No theorem closure is inferred from this local normal form. + +## 7. Finite algebra controls and review custody + +[fold_isolation_controls.py](fold_isolation_controls.py) is a standard-library exact-rational polynomial checker. [FOLD_ISOLATION_CONTROLS.json](FOLD_ISOLATION_CONTROLS.json) is its output. It checks the full mixed reduced-third-derivative identity on a prescribed nontrivial stationary ridge, the endpoint-distance factor 12, the positive-mark/zero-endpoint-cubic counterexample, the mixed-cubic sign-loss example with `D^4F=0`, the false full-height-barrier example, the two-maxima derivative-scale example, and the displayed rational chart margins. Every check passed using Python `Fraction`; no floating arithmetic, field draws, sampling, persistence experiment or process test occurred. + +These controls catch finite algebra and normalization mistakes. They do not mechanically verify the analytic proof, geometric selection, measure-theoretic counting, or a Lean statement. The current review lineage is root sketch -> this agent's independent conventional derivation plus falsification controls -> pending separate nonauthor review. All are OpenAI/Codex in the same parent thread; there is no provider independence or human review. The report and its control source/output are to be frozen by SHA-256 for that next review. Any correction after freezing should be an append-only addendum. diff --git a/frontiers/compact_morse_elder_adapter_20261018/inputs/main/experiments/universality/fold_structural_universality/PROOF.md b/frontiers/compact_morse_elder_adapter_20261018/inputs/main/experiments/universality/fold_structural_universality/PROOF.md new file mode 100644 index 000000000..483d779cf --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/inputs/main/experiments/universality/fold_structural_universality/PROOF.md @@ -0,0 +1,804 @@ +# Regular cubic folds and the structural H0 lifetime exponent + +Source proof by OpenAI/Codex universality_review, 8 October 2026; repository +composition and current-source binding by OpenAI/Codex root, task +01a11cbf-068d-7102-861e-75814e715c98, under standing owner authorization and +seven-path claim6070584604. Initial source base is main2c84170712c59d9de580c172815bd30bac5d93cd. +This proof record preserves the corrected preparation's mathematical Sections1-10 +byte for byte and supplies repository source bindings below. It changes no model, +sampler, applicability label, scientific register, compiler or Lean target. + +This is a conditional structural theorem with explicit field-admission interfaces. +The particular [periodized Gaussian adapter](../periodized_gaussian_h0/PROOF.md) +discharges them for its actual law separately. The [review record](REVIEW.md) +binds the original error, its one-line correction, the fresh full corrected-source +review and the separately reviewed incorporated bytes. Same-provider source and +route exposure is substantial; organizational independence is zero. Source +incorporation and technical review do not constitute scientific promotion. + +## 1. Structural claim and falsifiable interface table + +A cubic height gap alone does not determine a lifetime density exponent. +For a regular fold, the actual two-height/two-gradient contact density, +spatial volume and two Hessian determinants must also have their indicated +orders. When those orders combine to give `r dr`, the cubic pushforward gives +the lifetime exponent `-1/3`. This dimensional cancellation is proved here. +Verifying that a particular field has those orders, the needed domination, +and the actual elder selection remains a substantive admission task. + +Let X be a connected compact smooth Riemannian manifold without boundary, +of dimension d>=1 and volume V_X. Samples of F are smooth real functions. +Use superlevel H0, with increasing filtration parameter -c for {F>=c}, +over any fixed coefficient field. Count finite elder bars, excluding the one +essential class. A finite bar is born at a maximum M and dies at a merging +critical point S of Morse index d-1 for F. Its lifetime is + + ell=F(M)-F(S)>0. + +The following five interfaces are sufficient. They are conditions on the +actual field, its actual conditional laws and the actual H0 selector. They +are not substitutes for one another. + +| Interface | Exact consumed contract | A direct refusal or falsifier | +| --- | --- | --- | +| COUNT | F is unconditionally Morse with distinct critical values almost surely; an everywhere Borel elder selector sigma counts each finite H0 bar once. The all-nonnegative-Borel ordered critical-pair formula and two-height disintegration in Section 2 hold for specified density and conditional-law versions. | Atomic height laws, a selector counting candidates instead of bars, a determinant counted twice, or failure of once-counting. | +| CONTACT | For literal rows/targets in Section 3, pi_r(v_r) converges to pi_0(v_0), and E_Qr[W_r/r^2] converges to z_0=36k^2 E_Q0[(det A)^2 1(A<0)] for each fixed (x,u,b,k), k>0. These quantities are finite. | Contact rank collapse, a singular contact density, a different determinant order, or zero negative-cone weight. | +| SELECT | For those same specified fibers, E_Qr[(W_r/r^2)(1-sigma)] tends to zero. No all-target finite-r conditional genericity is inferred from unconditional genericity. | Local extra critical points, nonisolated limit contact, a younger competing branch, or a global collision defeating the elder event. | +| ENVELOPE | A common r_0>0 and nonnegative H(x,u,b,k) satisfy 12 pi_r(v_r) E_Qr[W_r/r^2]<=H for 00 boundary. | +| FAR | The complement integral in the specified kernel representative (4) is a finite, locally integrable Borel lifetime density nu_far, with ell^(1/3) nu_far(ell)->0. A uniform O(1) far candidate bound is sufficient. | A remote small-lifetime contribution of comparable or larger order. | + +All integrals in the table use dvol(x), ordinary sphere area d sigma_x(u), +db and dk. In d=1 sphere area means counting measure on S^0={-1,+1}. +The envelope may depend on x and direction; its integral is over the unit +tangent bundle SX. For a family of models, a common envelope is an additional +uniformity requirement, not a consequence of the individual limits. + +The theorem also requires the resulting contact coefficient C in Section 6 +to be strictly positive. If C=0, the proof gives a zero scaled limit, not a +positive `-1/3` asymptotic. Finite C follows from the stated envelope. + +**Conditional structural theorem.** Under COUNT, CONTACT, SELECT, ENVELOPE +and FAR, the expected finite-bar measure per volume has a particular finite +nonnegative Borel density representative nu_bar on ell>0 satisfying + + nu_bar(ell) ~ C ell^(-1/3), ell down to 0, + E N_bar((0,t])/V_X ~ (3/2) C t^(2/3), t down to 0. (1) + +Its coefficient is the actual contact functional (14) below. Under the +additional centered-Gaussian parity contract it simplifies to (19). +The exponent is independent of d once these interfaces hold. It is not an +unconditional dimension-independent admission result. The coefficient depends +on the field law and local geometry. No continuity of the selected density, +point-process law or quantitative asymptotic window is asserted. + +## 2. Counting and the measurable elder selector + +For distinct p,q write + + O_(p,q)=(F(p),grad F(p),F(q),grad F(q)), + Delta_(p,q)=|det Hess F(p) det Hess F(q)|, + tau_(p,q)=1{Hess F(p)<0} 1{Hess F(q) has index d-1}, + W_(p,q)=Delta_(p,q) tau_(p,q). (2) + +The index indicator is restricted to nonsingular Hessians. A change on a +singular Hessian has no effect because its determinant is zero. Densities +of gradients use orthonormal-coordinate Euclidean volume in the tangent +spaces. Thus rotations of the local frames have absolute determinant one. + +COUNT's all-Borel formula means, initially as an extended nonnegative +integral, that for every nonnegative Borel mark w on the field and pair, + + E sum_(p!=q,grad F(p)=grad F(q)=0) w(p,q,F) + =integral_(p!=q) p_(grad F(p),grad F(q))(0) + E[Delta_(p,q) w | both gradients=0] + dvol(p) dvol(q). (3) + +Applying (3) to `w=tau sigma 1{F(p)-F(q) in I}` gives W sigma. Delta +appears exactly once. Disintegrating the two heights with actual density +p_O and an actual regular conditional field kernel Q^p,q_(b,s) gives + + K_sel(p,q,b,s)=p_O(b,0,s,0) E_Q[W_(p,q) sigma(p,q,F)], + nu_bar(ell)=V_X^(-1) integral_(p!=q) integral_R + K_sel(p,q,b,b-ell) db dvol(p) dvol(q). (4) + +Joint Borel versions of p_O and Q are part of COUNT. They are explicitly +available through Gaussian coefficient projection in Section 8. Tonelli +proves (4) as a density identity; this does not differentiate a cumulative +asymptotic. A conditional kernel can be singular in unused second-jet +coordinates. Full two-site Hessian rank is not required. + +The field law is a Borel law on smooth functions. All contact density and +conditional-law versions below are induced from these same specified +COUNT versions by the invertible observation transform, rather than chosen +independently on exceptional fibers. FAR likewise refers to the complement +of (4) itself. This fixes the representative for the pointwise conclusion. + +Here is a dimension-independent Borel selector construction. Fix a countable +dense subset D of X and a countable basis of relatively small geodesic balls +with compact closures. Choose radii below a common convexity radius, so the +balls are path connected. For an open superlevel {f>lambda}, p and z lie +in the same path component exactly when there is a finite chain of basis +balls from p to z, consecutive balls intersect, and + + min_(closure B_i) f > lambda for each ball in the chain. + +A compact path is covered by such balls; conversely a chain supplies a +path. Each minimum is continuous in f in the uniform norm. The chain +relation is Borel in (f,p,z,lambda), by its countable description. Put + + D_f(p)=sup{lambda in Q: lambdaf(p) connected to p in {f>lambda}}, (5) + +with empty supremum -infinity. This is jointly Borel. The dense-point +condition is equivalent to reaching any point of greater height. Hence +(5) is intrinsic and invariant under any isometry transporting the field. + +On the open set G of Morse functions with distinct critical values, define +sigma to be one exactly when p,q are critical, p is a maximum, q has index +d-1, D_f(p) is finite and + + D_f(p)=f(q)0 let + + M=exp_x(-r u/2), S=exp_x(r u/2). + +Use the frame obtained by parallel transporting u,v_i along this geodesic. +Write h_M=F(M), h_S=F(S), g_uM,g_uS for the axial gradients, and +g_iM,g_iS for the transverse gradient coordinates. Reorder O as + + (h_M,h_S,g_uM,g_uS,(g_iM,g_iS)_(i=1,...,d-1)). + +This is just a permutation of the original observations. Define U_r by + + (h_M+h_S)/2, + (h_S-h_M)/r, + (g_uS-g_uM)/r, + (6/r^2)[g_uM+g_uS-2(h_S-h_M)/r], + ((g_iM+g_iS)/2,(g_iS-g_iM)/r)_(i=1,...,d-1). (6) + +For heights b,b-k r^3 and zero gradients, k>0, the exact target is + + v_r=(b-k r^3/2,-k r^2,0,12k,(0,0)_(i=1,...,d-1)). (7) + +There are 2d+2 rows. The first four transformation rows have matrix + + [ 1/2 1/2 0 0 ] + [ -1/r 1/r 0 0 ] + [ 0 0 -1/r 1/r ] + [ 12/r^3 -12/r^3 6/r^2 6/r^2 ]. + +The two diagonal 2-by-2 blocks have determinants 1/r and -12/r^3. +The last row's height entries do not change this block-triangular +determinant. Each transverse pair has determinant 1/r. Therefore + + |det(O -> U_r)|=12 r^(-(d+3)), + p_O(b,0,b-k r^3,0)=12 r^(-(d+3)) pi_r(v_r). (8) + +This identity is exact on a manifold as well, for the specified parallel +orthonormal gradient coordinates. It is a finite-dimensional linear +observation transform, not a spatial-coordinate Jacobian. + +In the contact limit the rows are, with the order in (6), + + U_0=(F,F_u,F_uu,F_uuu,(F_vi,F_uvi)_(i=1,...,d-1)), + v_0=(b,0,0,12k,(0,0)_(i=1,...,d-1)). (9) + +Directional derivatives are covariant derivatives along the geodesic and +parallel frame. In particular F_uu=Hess F(u,u), F_uvi=Hess F(u,v_i), +and F_uuu is the derivative of Hess F(u,u) along that geodesic. A local +normal-coordinate third derivative agrees with this at the contact, +since grad F=0 there. Finite-r difference rows converge to (9) for a C4 +field. Convergence of random rows by itself does not prove convergence +of their densities or conditional laws; CONTACT demands those facts. + +Different bases of u-perp act orthogonally on the transverse rows. Densities +with respect to their Euclidean volume, the transverse determinant and the +cone A<0 are invariant. One uses local frame charts (or a Borel partition), +not an assumed global section of the perpendicular frame bundle. + +## 4. Determinant order and a sufficient elder-transfer adapter + +Let A_i denote the physical transverse Hessian block at i=M,S in the +parallel frame. Under the exact pins, the physical Hessians have the form + + H_i=[ r alpha_i r beta_i^T ] + [ r beta_i A_i ]. (10) + +Zero endpoint gradients make the geodesic average of every axial Hessian +column entry zero. The fundamental theorem of calculus bounds alpha_i +and beta_i by a constant times a C3 norm. Under an actual C4 contact +coupling they satisfy + + alpha_M -> -6k, alpha_S -> 6k, + beta_M,beta_S bounded, A_M,A_S -> A_0. + +For example the first limit follows from integrating F_uu along the +geodesic, whose average is zero, and whose derivative tends to F_uuu=12k. +No conditional covariance of a pinned coordinate is asserted positive. + +The determinant is a polynomial identity, so no inverse A is needed: + + det H_i/r=alpha_i det A_i-r beta_i^T adj(A_i) beta_i. (11) + +For d=1 use det A=1 and omit the second term. The correct inertia +congruence uses diag(r^(-1/2),I), giving the block matrix with alpha_i, +sqrt(r) beta_i and A_i. If A_0 is negative definite, the left Hessian +is negative definite and the right Hessian has index d-1 for small r. +If A_0 is nonsingular and is not negative definite, the maximum condition +fails for small r. If A_0 is singular, the limit of det H_i/r is zero +and the determinant product kills any type ambiguity. Consequently, + + V_r:=W_r/r^2 -> V_0:=36k^2 (det A_0)^2 1{A_0<0}. (12) + +This pointwise weighted limit needs no boundary-null theorem for A_0. +Uniform integrability still has to be proved to pass to expectations. +The merger type must be the inertia index d-1. A negative determinant +alone is the correct saddle test only in d=2; in d=3 the intended +one-positive/two-negative saddle has a positive determinant. + +Here is a deterministic sufficient condition for SELECT, proved in every +dimension; it is not a verification that an unspecific random law meets it. +Suppose smooth f_j->f_0 in global C4, r_j->0, x_j,u_j and their local +frames converge, b_j->b and k_j->k>0. Suppose the exact pins (7) hold. +At the limiting point suppose the contact rows are (9), A_0<0, and the +critical set of f_0 is finite; every other critical point is Morse, with +distinct critical values different from b. + +In moving normal coordinates write g_j(s,t), t in R^(d-1). At contact +g_0,t=0, g_0,st=0, g_0,tt=A_0<0. On a fixed small cylinder +[-delta,delta] times B_tau, eventually g_j,tt<=-a I. The implicit +transverse ridge h_0(s) has h_0(0)=h_0'(0)=0. Shrink delta so +|h_0(s)|=c>0. The exact critical pins make h_j(+-r_j/2)=0 and +phi_j'(+-r_j/2)=0. Thus phi_j' is strictly convex, has exactly these +two zeros, is negative between them, and is positive outside them. +Its derivatives at the left and right zeros are respectively negative +and positive. The Hessian Schur pivot is phi_j'' and the transverse +block is negative definite, giving the required maximum/index-(d-1) +types and no additional local critical points. + +The limit ridge is strictly increasing near zero except at zero; +phi_0(-delta)b; +its perturbed height exceeds b_j. Its component is older. At d_j the +ridge from M_j through S_j to that point lies in the closed superlevel. +Only S_j is at that critical level. Its upper sectors therefore merge +the newborn component with an older component. The H0 elder rule gives +exactly the pair (M_j,S_j), of lifetime k_j r_j^3. + +For d=1 there are no transverse variables or side boundary. The same +left interval, right cut and older positive ridge prove the deterministic +statement on a circle. This is a separate topological argument; no d=2 +conditional-global genericity theorem is silently used in dimension one. + +For a random-law adapter, suppose the specified Q_r and Q_0 can be +coupled with global C4 convergence, exact pins and an integrable dominating +variable for V_r at each fixed mark. Suppose the preceding contact-global +genericity holds Q_0-almost surely on V_0>0. Then sigma is eventually one +on V_0>0. On V_0=0, V_r(1-sigma)->0 regardless of the selector. +Dominated convergence proves E[V_r(1-sigma)]->0 and E[V_r]->E[V_0]. +Uniform integrability with convergence in probability is an alternative +to one dominating variable. This supplies CONTACT's weight limit and +SELECT. It supplies neither a common physical chart radius nor a +simultaneous samplewise event over all marks and frames. + +With continuous varying-parameter couplings, common moment domination and +contact genericity at every fixed limiting parameter, a compact-mark +sequence argument supplies uniform convergence of the unnormalized weighted +failure E[V_r(1-sigma)] to zero on compact birth/gap windows with k bounded +away from zero. A uniform normalized Q_r^W(sigma=1)->1 additionally requires +a continuous, strictly positive z_0 on that compact window, hence a positive +minimum. Neither uniform statement follows from pointwise convergence alone. +The theorem below needs only pointwise convergence followed by all-mark +domination. + +## 5. The geometric measure factor on a manifold + +Choose r_0 below a common short-geodesic scale. The midpoint map + + Psi(x,h)=(exp_x(-h/2),exp_x(h/2)) + +is a smooth diffeomorphism from a neighborhood of the zero section of TX +onto the corresponding neighborhood of the diagonal in X times X. +The inverse is the unique short geodesic's midpoint and oriented tangent +separation. With the natural base/fiber volume its Jacobian is a positive +smooth function J(x,h). At h=0 its derivative is + + (a,b) -> (a-b/2,a+b/2), + +whose absolute determinant is one. Hence J(x,0)=1. Swapping the pair +is h->-h and preserves both volume measures, so J is even in h. +Smoothness and compactness give J(x,r u)=1+O(r^2) uniformly in x,u, +and a common finite bound for small r. In particular + + dvol(M) dvol(S) + =J(x,r u) dvol(x) r^(d-1) dr d sigma_x(u). (13) + +The tangent-fiber polar factor r^(d-1) is exact. The midpoint metric +Jacobian J is generally not exactly one at finite r. Curvature therefore +enters finite-r formulas, but its leading factor in this contact band is +one. No global Euclidean spatial-coordinate formula is asserted. + +## 6. Proof of the structural theorem and the dimensional cancellation + +Write z_0(b,k,x,u)=E_Q0 V_0 and define the selected amplitude directly by + + B_r(b,k,x,u)=12 pi_r(v_r) E_Qr[V_r sigma]. + +One never needs a global inverse of E V_r. Normalized Q_r^W is useful +only where that expectation is positive; the product here is defined +even when it vanishes. CONTACT and SELECT give +B_r->12 pi_0(v_0) z_0 at every fixed mark and direction. Also 0<=B_r<=H. + +The complete near-diagonal ledger is + + spatial polar volume r^(d-1) dr, + height change s=b-k r^3 r^3 db dk, + original observation density 12 r^(-(d+3)) pi_r(v_r), + selected determinant weight r^2 E_Qr[V_r sigma]. + +Their product is exactly `J r B_r dr db dk dvol(x) d sigma_x(u)`. +The dimensional powers cancel to r, since + + (d-1)+3-(d+3)+2=1. + +The physical cubic gap alone is only one entry in this ledger. +In particular no exponent can be inferred while omitting the transformed +contact-density order or the determinant-product order. + +Push forward ell=k r^3 at fixed k. For the very same representative (4), +its near part satisfies pointwise for ell>0 + + ell^(1/3) nu_near(ell) + =(1/(3 V_X)) integral_(SX,R,(0,infinity)) + 1{k>=ell/r_0^3} k^(-2/3) + J(x,(ell/k)^(1/3)u) + B_((ell/k)^(1/3))(b,k,x,u) + dvol(x) d sigma_x(u) db dk. + +This can also be obtained by the substitution k=ell/r^3 directly in +(4)'s fixed-ell near integral, whose radial factor is J r^-2 B_r dr. +Thus it does not replace (4) merely by an almost-everywhere equal +representative before making a pointwise asymptotic claim. + +For every fixed k>0 the radial point tends to zero. The indicator tends +to one, J tends to one, and the bounded-J version of k^-2/3 H is +integrable. Dominated convergence gives + + C=(4/V_X) integral_(SX,R,(0,infinity)) + k^(-2/3) pi_0(v_0) z_0(b,k,x,u) + dvol(x) d sigma_x(u) db dk. (14) + +FAR supplies the entire remaining finite-bar contribution, because COUNT +already counts all bars once. Its scaled limit is zero. Hence +ell^(1/3) nu_bar(ell)->C. Positivity gives the equivalent asymptotic in +(1); integration uses integral_0^t ell^-1/3 d ell=(3/2)t^(2/3). +The cumulative statement is intrinsic to the expected measure. + +Candidate domination makes this a finite Borel representative, but the +global selector need not be continuous in the conditional field. No +selected-density continuity is inherited from a continuous candidate +type-weight kernel. Arbitrary Radon-Nikodym representatives can be changed +on a null set and need not keep a pointwise asymptotic. Also, equality of +the selected and candidate contact coefficients gives only an +o(ell^-1/3) missed-near contribution; it gives no O(1) near remainder. + +For a stationary flat-torus field, sigma is translation equivariant and +the midpoint integral cancels V_X. Directional dependence remains unless +an actual rotational symmetry is proved. On a manifold the x integral +in (14) remains; local laws can depend on x. The theorem is independent +of local coordinate choices, not independent of the law or metric in +its coefficient. + +## 7. Centered Gaussian parity and the exact coefficient + +The additional parity contract is that the Gaussian contact jets are +centered, their full U_0 covariance is positive definite, and the odd +block is independent of the even block: + + odd: G=grad F (dimension d), T=F_uuu (one scalar); + even: F, V=Hess F(u,.) (dimension d), A=Hess F|_(u-perp). (15) + +Here V=(F_uu,F_uv_1,...,F_uv_(d-1)). It is the full axial column, +not the single entry F_uu. In Euclidean stationary real Gaussian fields, +covariance C(h)=C(-h) makes derivatives of odd total order vanish at +zero, proving this odd/even independence. Rotational isotropy is not +needed. On a general manifold an appropriate local parity symmetry or +the stated block-independence condition must actually be verified; +the word stationarity alone is not used to prove (15). + +Let + + tau_(x,u)^2=Var(T|G=0)>0, + D_(x,u)=E[(det A)^2 1{A<0} | V=0]. (16) + +All covariances are unconditional covariances of actual jets; (16) is +their actual Gaussian regression. CONTACT's limit can be written + + pi_0(v_0)=p_(F,V)(b,0) p_G(0) phi_tau(12k), + z_0=36k^2 E[(det A)^2 1{A<0} | F=b,V=0]. + +The law of A conditioned on F=b,V=0 need not be centered. Integrating +birth first, with the true disintegration identity, gives + + integral_R p_(F,V)(b,0) + E[(det A)^2 1{A<0} | F=b,V=0] db + =p_V(0) D_(x,u). (17) + +This removes the F condition and retains every correlation between V and +A. Replacing V by F_uu, or centering the b-conditioned A, is incorrect. + +The remaining scalar factor in (14) is + + 144 integral_0^infinity k^(4/3) phi_tau(12k) dk + =12^(-1/3) integral_0^infinity t^(4/3) phi_tau(t) dt + =Gamma(7/6) tau^(4/3)/(24^(1/3) sqrt(pi)). (18) + +Indeed the last half-Gaussian moment is +2^(-1/3) Gamma(7/6) tau^(4/3)/sqrt(pi); multiplication by +12^(-1/3) gives the stated 24^(-1/3). Thus the Gaussian coefficient is + + C=Gamma(7/6)/(24^(1/3) sqrt(pi) V_X) + *integral_SX p_G(0) p_V(0) tau_(x,u)^(4/3) + D_(x,u) + dvol(x) d sigma_x(u). (19) + +For a stationary flat torus, cancel the x volume. Ordinary sphere area +is used, not probability-normalized angular measure. The roles are +ordered maximum/merger, so there is no additional angular factor 1/2. + +In d=2, A is a centered scalar under V=0, giving +D=Var(A|V=0)/2. In higher dimensions there is no such scalar half-variance +simplification: indefinite matrices also contribute to E(det A)^2, but +not to D. In d=1 A is empty, det A=1 and the negative-definite condition +is vacuous, so D=1. This algebra does not verify the separate circle +COUNT or contact-global SELECT adapter for a particular law. + +Full U_0 covariance rank ensures p_G,p_V and tau exist and are positive. +It does not ensure D>0. For the centered conditional Gaussian A, D>0 +exactly when its linear support meets the open negative-definite cone. +Full conditional covariance on Sym(u-perp) is a sufficient condition, +not a necessary one. For example a scalar Gaussian times the identity +has rank-one matrix support and positive D. Conversely, a trace-zero +two-by-two transverse support has no negative-definite matrix, so D=0 +even if the separate U_0 rows have positive covariance. Such Gaussian +jet data can be realized locally by independent polynomial coefficients; +this is an algebraic counterexample to the inference from contact-row +rank alone, not an admission claim for a globally stationary model. + +As unit checks, multiplying the field by a positive constant a scales +C by a^(-2/3). In (19), p_G p_V contributes a^(-2d), tau^(4/3) +contributes a^(4/3), and D contributes a^(2d-2). A spatial length +rescaling contributes length^(-d). Thus C has units +field^(-2/3) length^(-d), and C ell^(-1/3) has units +field^(-1) length^(-d), as a lifetime density per volume should. + +## 8. A finite Gaussian sufficient adapter and its genuine boundary + +Consider a fixed finite smooth linear Gaussian field F_xi with xi in R^N +standard Gaussian. This subsection proves conditional implications from +specified ranks; it does not prove the ranks or contact-global genericity +for every dimension or manifold. + +Suppose the literal U_r matrix L_r extends continuously at r=0 and +L_0 has row rank 2d+2 for every x,u. On compact SX, its covariance has +a positive floor, and the same is true for all sufficiently small r. +The conditional coefficients at target v_r have the canonical realization + + xi_r=m_r+P_r Z, + m_r=L_r^T (L_r L_r^T)^(-1) v_r, + P_r=I-L_r^T (L_r L_r^T)^(-1) L_r, (20) + +where Z is a standard Gaussian vector. P_r is an orthogonal projector, +not a full-rank residual covariance requirement. This coupling is continuous +in actual coefficients. It gives global C4 convergence at each fixed mark, +and on compact marks ||F_r||Cq<=C(1+||Z||) for each fixed q. Thus it +supplies the conditional moment and uniform-integrability part of CONTACT. +The Gaussian densities also converge at the actual targets, supplying +its density part. SELECT still needs the contact-global genericity premise +on positive limiting weight, supplied to the deterministic result in +Section 4. Row rank alone does not prove that nonlocal premise. + +For all marks, the covariance floor gives conditional mean norm at most +C(|b|+k) and centered covariance at most I. Finite-dimensional norm +equivalence gives ||F||Cq<=C_q||xi||. The exact zero-gradient endpoint +averages and (11) give + + |det H_i|/r<=C(1+||F||C3)^d, + W_r/r^2<=C(1+||F||C3)^(2d). (21) + +The polynomial degree is 2d, not the dimension-two degree four. +For d>=2 the two terms in (11) have degrees d and d in Hessian/third +derivative bounds after allowing r<=1; d=1 is the axial bound alone. +The target (7) is uniformly coercive in (b,k): its first and fourth +coordinates are b-k r^3/2 and 12k, so |v_r|^2>=c(b^2+k^2). +Uniform upper and lower covariance bounds then yield + + 12 pi_r(v_r) E_Qr[V_r] + <=C(1+|b|+k)^(2d) exp[-c(b^2+k^2)]. (22) + +This is an unnormalized all-mark envelope. Its product with k^-2/3 +is integrable for every finite d. No inverse random curvature or +global inverse normalizer is used. It supplies ENVELOPE from the stated +finite Gaussian contact rank and smooth finite basis. + +For COUNT and FAR, suppose in addition the original value/gradient +observations at every distinct pair have rank 2d+2. On each compact +annulus away from the diagonal their covariance has a positive floor. +The point-gradient incidence gives unconditional Morse genericity without +a Morse-to-Rice cycle: choose d invertible coefficient columns locally, +write the zero-gradient coefficients as a smooth function of the point +and remaining coefficients, and project that N-dimensional incidence +chart to R^N. Its derivative is singular precisely when the physical +Hessian is singular. Equal-dimensional smooth Sard makes the critical +values null. A countable minor/chart cover proves almost sure Morse. + +Similarly, the pair-gradient incidence uses 2d invertible coefficient +columns. In local orthonormal/coordinate charts, the determinant of its +coefficient projection contains |det H_p det H_q| divided by the chosen +coefficient minor. The equal-dimensional weighted area formula, followed +by coefficient change of variables and a countable disjoint partition +of minor charts, gives (3) for all nonnegative Borel marks, initially +possibly infinite. Start with that extended identity; no finiteness or +Morse conclusion is assumed to justify it. On compact annuli, the +covariance floor and finite conditional determinant moments make the +unmarked integral finite. No arbitrary-jet rank is used. + +The conditional two-height covariance is positive by the original +2d+2 row rank. The tie indicator F(p)=F(q) has zero contribution after +height disintegration on each annulus. Countable annulus exhaustion +and (3) give distinct critical values almost surely. Point Morse +genericity already gave a finite critical set on compact X. This supplies +the unconditional genericity and the Borel H0 count portion of COUNT. + +At separation>=r_0, the same original covariance floor and conditional +determinant moment bound give the candidate density estimate + + integral_R p_O(b,0,b-ell,0) E_Q[W] db <=C, + 00; such an interval +exists because H's expected finite-bar measure is nonzero. For sufficiently +small t, + + E N_F((0,t]) + >=E N_H([a,b]) P(R<=t/b) + =c t^alpha, c>0. (24) + +Since alpha<2/3, the ratio to t^(2/3) diverges. A finite positive coefficient +in (1) is impossible. This is a lower-bound falsifier, not a proof of an +alternative alpha density law. The small-amplitude boundary is not +controlled by covariance or positive moments. Indeed E R^(-2/3) is +infinite, matching the amplitude-scaling diagnostic C_(aH)=a^-2/3 C_H. +Some of the structural regularity/domination interfaces must therefore +fail for this mixture. No covariance-only Gaussian coefficient formula +can be transferred to this non-Gaussian law. + +For a non-Gaussian law which actually meets the five interfaces, (14) is +still the coefficient. It involves the actual contact density and actual +conditional transverse weight, with all-mark integrability. Odd/even +uncorrelatedness is not independence outside the Gaussian setting, and +does not justify the factorization (17)-(19). + +## 10. Conditional order taxonomy; no unproved higher-singularity class law + +There is a useful general scaling test, which is an algebraic pushforward +lemma under its stated analytic domination, not a higher-order field theorem. +Suppose a particular admitted local family has lifetime ell=k r^m, m>0, +radial volume r^gamma dr, height Jacobian r^m, original observation density +order r^-a, and determinant/selected weight order r^w. Suppose its remaining +joint amplitude B_r converges to B_0 and has an integrable majorant for the +substitution below. The joint radial power is + + s=gamma+m-a+w, + r^s B_r dr db dk d(other parameters). + +The pushforward has exponent + + e=(s+1)/m-1=(gamma+1-a+w)/m, + +and coefficient + + (1/m) integral k^(-(s+1)/m) B_0 db dk d(other parameters). (25) + +The appropriate k-weighted majorant and negligible complement have to be +proved for that family. In ordinary spatial dimension d, gamma=d-1, +so e=(d-a+w)/m. The regular cubic contract has m=3, a=d+3 and w=2, +giving s=1 and e=-1/3. The cubic order by itself supplies only m. + +The following changes are therefore falsifiable failures of admission, +or possible different regimes only after new proofs: + +* If a contact density acquires an extra radial singularity or vanishing + factor, a changes. Covariance rank is a direct diagnostic but not a + substitute for the actual density asymptotic. +* If the transverse block has a forced extra null direction, the physical + determinant order can change, or the maximum cone can have zero weight. + One cannot keep (12) with a positive coefficient without checking it. +* If the axial contact has higher order, m and usually the observation and + determinant orders change together. Merely substituting a new m while + keeping the cubic a,w is not a derived higher-order universality law. + The strict convexity of the reduced derivative used for elder transfer + can fail; additional local critical points may occur. +* If geometry or support supplies a different radial order gamma, the + dimension cancellation must be recomputed. A smooth manifold's leading + metric Jacobian is regular, so ordinary curvature alone does not change + gamma=d-1 in this theorem. +* If selected probability has a nontrivial limiting loss, its actual + selected contact weight replaces z_0; a vanishing selection order can + change w. Candidate counting cannot decide that input. +* If a far contribution or noncompact-mark tail dominates, the contact + calculation does not determine the total small-lifetime asymptotic. + Equation (24) is an explicit such boundary failure. + +For s>-1 the local cumulative power predicted by (25) is (s+1)/m. +For s<=-1 a strictly positive limiting coefficient would be nonintegrable +at zero, incompatible with a finite expected short-bar count. This is a +consistency test, not an assurance that a proposed singularity is realized. +Even within the regular fold class, varying a spectrum changes the local +jet coefficient; constants need not be uniform near spectral degeneracy. +The proven toy-control lesson that radial/contact weight matters is exactly +what (25) retains, instead of identifying a cubic gap with a complete law. + +## 11. Exact repository bindings, source custody and residual scope + +The current operational sources at the initial base are: + +| Input | Repository source | UTF-8 bytes / lines | SHA256 | +| --- | --- | --- | --- | +| FG | [Finite contact](../finite_gaussian_contact/PROOF.md) | 25367 / 554 | 7fea2656e11d986e1a5d6bfb0a05c8421bd93262d28051b8cfef05320595f474 | +| C | [Candidate intensity](../finite_candidate_lifetime/PROOF.md) | 29190 / 577 | ed31f562f0ffaaf6d4bd2ceb1ae3d234798e983a55d55c6983b128fce27133f5 | +| CG | [Contact-conditioned genericity](../finite_contact_genericity/PROOF.md) | 23431 / 473 | 233e80421ccf78e9a9015116aaaf2d355a8425107ff109c022ddc98cd1f87fd9 | +| ET | [Actual elder transfer](../finite_elder_transfer/PROOF.md) | 23774 / 458 | c9e4fb09900888cd2b1bd96b49f85072b51c822b2503fe9f3d8891049bf66201 | +| S | [Actual finite H0 intensity](../finite_h0_lifetime/PROOF.md) | 27064 / 526 | 5ea91111fa5f3d78d7c4188ebe84b3bc2c452b05dfdfc62c74b03123cb2696d1 | + +FG,C,CG retain the author's original consumed byte identities. ET and S are +separately reviewed incorporated successors of the original preparations: +ET21138B/418lines SHA1cac9b525831b7e1f37f0d0561d3231fa73c8713c4c92283dca1d5b51b8a67ce; +S25439B/501lines SHA5f529e5ed2632c90819746848146e3a24acda7e888cd3c9342df76edcbcbb7a3. +Their mathematical ETSections2-13 and SSections2-8 are byte-identical, with +common-block SHAeef0cc2fba152da76d83c6f738ca7f5fd9d531c2e919fe9d74b261dd1fbc3abe +and SHA64b1ced2e0027c74ca63dcd60666e7381c388330b3a83d3428181b47df83e288 respectively. +The operational rebind records root's complete current-source reads and exact +block comparisons. It does not relabel the original author's prior read as a +fresh whole-file read of later wrapper bytes. + +Original structural preparation43760B/813lines SHA79e8cb54386551d335a901451f9ab9f72f311563e385934add006e4002b525a4 +remains frozen with its intermediate half-Gaussian moment error and full +final_contract_review AMEND. Corrected successor43761B/813lines SHAb077e2465174cadf78f2df120a80992534f99b838e18b5ba15fed250f4c28cca +changes exactly2^(1/6) to2^(-1/3); its final coefficient equations18-19 were already +correct. The complete corrected delta received final_contract_review PASS, and +benchmark_formal_audit separately read the entire corrected file and returned PASS. +The present source has its own identity and review in REVIEW.md; those verdicts +are not silently rebound to it. Original preparations remain in the coordinator's +ordinary workspace custody, rather than becoming unverified repository links. + +The author had prior full reads of all five original inputs, fully reread S for +this derivation and reread ET's deterministic/weighted sections and FG's row +contract. C/CG prior unchanged full reads were hash-bound. No fresh full C/CG or +classical-primary fetch is attributed to that derivation. The source-exposed +corrected-file reviewer freshly hashed all five and additionally reread +FG205-339,C250-362,CG97-120,ET20-333,S89-252. C and ET record named classical +Sard, area-formula, Morse and elder imports and prior bounded primary statements; +this composition claims no new full-book/paper/raw-PDF custody. + +The sourced finite Gaussian chain supplies actual two-dimensional ideal-law +adapters at each fixed positive full-square spectrum,K>=2,L>0. The twenty fixed +catalogue Gaussian cases retain individual coefficients and qualitative +finite-family uniformity; they are not independent empirical confirmations or +sampler certificates. General dimensions, manifolds and non-Gaussian fields +still require the actual five interfaces and positive cone coefficient. The +separate infinite periodized Gaussian proof is one specific completed analytic +adapter, not an automatic admission rule for arbitrary laws. + +Higher homology, spatial expanding-domain processes, numerical windows, certified +computational coupling, blind-field validation, publication uptake and critical +Lean discharge remain separate. No source-status or governance record is promoted. diff --git a/frontiers/compact_morse_elder_adapter_20261018/reviews/NONAUTHOR_REVIEW.md b/frontiers/compact_morse_elder_adapter_20261018/reviews/NONAUTHOR_REVIEW.md new file mode 100644 index 000000000..955357eee --- /dev/null +++ b/frontiers/compact_morse_elder_adapter_20261018/reviews/NONAUTHOR_REVIEW.md @@ -0,0 +1,124 @@ +# C292 nonauthor mathematical review + +Decision: **ACCEPT_SCOPED_CONVENTIONAL**. + +Reviewer: OpenAI/Codex `/root/c292_geometry_review`, a fresh nonauthor reviewer delegated for Dylan Roy by the C292 owner. Review date: 18 October 2026 UTC. Provider and organization are the same as the author and parent; organizational-independence credit is **0**. This is an AI mathematical/source review, not human review, provider-distinct review, Lean elaboration, kernel verification, or scientific acceptance. + +There are no must-fix mathematical or interface findings against the frozen artifact identified below. The accepted conclusion is exactly the deterministic construction of the closed trap and a higher-endpoint closed path at the literal C263 death, followed by conventional consumption of the existing single ordinary-H0 interval constructor. The exclusions in the proof remain operative. + +## Exact artifact and exposure + +The complete `PROOF.md` was read after the parent declared it ready for review. Its 268 lines contain **25,662 bytes** and have SHA-256: + +```text +39f92734da35aa4c3ad97ea9d8e194cea4ee3a0b23054a8a412f331294d74762 +``` + +The accompanying `SOURCE_IDENTITIES.json` has **13,452 bytes** and SHA-256: + +```text +8b0fc8892a1936ad6a37c49b45e5fcf987154068fb7107e07b0b48d55c9f5d44 +``` + +Both identities were computed from the actual local bytes. The relevant copied source bodies also matched their manifest byte lengths and SHA-256 values. The C292 `inputs/C291/inputs/Lifetime.lean` copy was additionally compared byte for byte with `continuation291/publication-package/inputs/Lifetime.lean`; they are equal, with 51,383 bytes and SHA-256 `90f4892608e953ecc696bb32ea159d1c178ceb8a1c6edfd11acfceb14af462c6`. + +The task instruction disclosed the intended adapter and exact sources before review. Before opening the new proof, I independently reconstructed the necessary compact-Morse argument from those source interfaces and sent the parent interim checkpoints about the fixed older endpoint and critical-level path-connectivity obligations. I did not edit the author's proof, supply replacement author text, or communicate proposed repairs to its author. I then read the complete frozen proof and compared each substantive step with that reconstruction. This is fresh nonauthor review of the artifact, with task/design exposure and same-provider limitations explicitly disclosed; it is not an independence claim based merely on a different agent name. + +Read scope: + +- Complete `SuperlevelBar.lean`, `TorusTopology.lean`, `OrdinaryH0.lean`, `ElderClearance.lean`, `ElderBridge.lean`, and `ElderInterval.lean` at the supplied Math- cut `c8ec07034608741585111603f1b506393c340908` through the frozen local copies. +- Complete C291 `SourceLifetimeGeometry.lean` and `SourceLifetimeField.lean`; relevant literal smoothness, periodicity, derivative, inertia, nondegenerate-maximum, Morse-distinct, and persistence-interface predicates in the frozen Lifetime and Field files. +- Contextual finite-H0 §§3–4, structural-source manifold convention and §2 Morse/elder discussion, and asymmetric-source §6 ordinary-H0 splitting discussion. I did not reopen the full stochastic proofs, integrated historical reviews, or all claims in those source families. + +The repository cuts and Git blob associations are those recorded by the parent source manifest. I independently verified the local byte identities, not a fresh remote raw-file fetch or Git commit ancestry. Hashes identify what was reviewed; they do not establish the mathematics. No numerical experiment, test suite, Lean command, compiler, kernel audit, source edit, or remote post was performed in this review. + +## Reconstructed theorem and premise audit + +The theorem fixes `n >= 1`, `L > 0`, a literal smooth periodic Euclidean function `g`, the literal `IsMorseDistinct` hypothesis, a literal nondegenerate maximum `M`, and nonessentiality in the actual C263 sense. On the genuine period quotient `X`, with descended `F`, birth `m = g(M)`, and the existing number `delta = superlevelBar_deathHeight g M`, it defines `C` as the path component of the projected maximum in `{F > delta}` and `D = closure C`. + +It derives `delta < m`, closedness and maximum membership of `D`, `F <= m` on `D`, `F <= delta` on its frontier, and an actual path in `{F >= delta}` to one point `Q` with `F(Q) > m`. It additionally derives that `Q` can be a nondegenerate local maximum and that the path attains `delta`. + +I checked that no desired closed path, trap, death equality, elder pairing, saddle label, finite-bar count, H0 equation, or interval split occurs among the new theorem's inputs. The nondegeneracy conjunct of `IsMorseDistinct` is genuinely used. The distinct-critical-height conjunct is retained but is not used to smuggle in a unique merger or a quotient-distance identity. The proof says this explicitly. + +The declared classical imports are adequate and remain explicit: smooth quotient charts for the discrete translation lattice; finite-dimensional smooth calculus and the inverse function theorem; the local Morse lemma; elementary compact and locally path-connected topology. The proof does not pretend these imports have been formalized in the current companion files. No handle-attachment theorem, critical-free flow, global elder basis, or all-bars theorem is needed to close its stated argument. + +## Substantive step checks + +### 1. Literal inertia really supplies a Morse Hessian + +The source predicate gives complementary negative and positive subspaces, not an orthogonal eigenspace decomposition. The proof correctly addresses this distinction. For a symmetric Hessian `B`, a radical vector `z = v + w` satisfies + +```text +B(v,v) = -B(w,v) = -B(v,w) = B(w,w). +``` + +The left expression is nonpositive and is zero only if `v = 0`; the right expression is nonnegative and is zero only if `w = 0`. Thus both vanish and the radical is zero. Finite dimension makes the associated Hessian map invertible. Smoothness supplies the symmetry that this argument requires. At the specified maximum the negative subspace has full dimension and is the whole tangent space, giving negative definiteness. This step does not assume a stronger Morse predicate in place of the literal one. + +Translation quotient charts preserve the function and its coordinate derivatives without a scale change. The critical set of the descended function is closed, and the inverse function theorem makes each of its points isolated. Compact metrizability then forces finiteness. The proof includes closedness; an arbitrary discrete nonclosed subset of a compact space would not suffice. + +### 2. Closed superlevels are locally path connected, including critical levels + +For `F > a`, a small open manifold neighborhood lies in `{F >= a}`. At a regular point of `F = a`, the submersion form gives a half-space. At a nondegenerate critical point of `F = a`, the local Morse form gives the homogeneous quadratic cone `||u||^2 - ||v||^2 >= 0`. + +Intersections of this cone with centered open coordinate balls are relatively open neighborhoods of the vertex and are star shaped about it: multiplication by any scalar in `[0,1]` preserves the quadratic inequality and the ball. Their radii can be made arbitrarily small. At nonvertex boundary points the gradient of the quadratic form is nonzero, giving the regular case. Positive- and negative-definite cases give an open neighborhood or a singleton as appropriate. Thus local path connectivity is proved at all relevant points, for `n = 1` as well as higher dimensions. + +Consequently each connected component of a closed superlevel is path connected. The proof does not make the false assertion that arbitrary compact connected sets are path connected. + +### 3. The literal death lies strictly below birth + +C263 continuity and nonessentiality give a nonempty bounded clearance set and its exact strict cut. A small Euclidean ball at `M` has all heights at most `m`, with all punctured-ball heights strictly below `m`. Its nonempty compact boundary sphere has a maximum `c0 < m` because `n >= 1`. + +Every endpoint of height greater than `m` lies outside that ball. Every continuous path from `M` to such an endpoint crosses its boundary, so any strict clearance is strictly below `c0`. The real supremum therefore obeys `delta <= c0 < m`. This is a uniform barrier argument, not the invalid inference that a supremum is strictly below `m` merely because each member is. + +This step is conducted in the actual Euclidean space of C263. It requires no arbitrary choice of a torus lift endpoint and is unaffected by winding paths. The exact existing projection/lifting equivalence then transfers the strict cut to the quotient. + +### 4. A fixed genuinely older endpoint survives the limiting levels + +For every `a < delta`, the open-superlevel path component `U_a` of the maximum contains a point of height greater than `m`. Local path connectivity makes `U_a` open and relatively closed in `{F > a}`, so `closure(U_a) intersect {F > a} = U_a`. + +The maximum of `F` on this compact closure has height greater than `m > a`, hence belongs to the open set `U_a`. It is therefore a local maximum of `F` itself, not merely a constrained maximum. Its derivative vanishes; its Hessian is nonpositive by the second-derivative test and nonsingular by the proved Morse property, hence negative definite. It belongs to the finite set of older local maxima. + +A finite pigeonhole argument on the levels `delta - 1/(j+1)` yields one fixed older maximum `Q` along an infinite, hence cofinal, increasing subsequence. This resolves the endpoint problem: the strict inequality `F(Q) > m` is never replaced by a limiting inequality `F(Q) >= m`. Neither the global maximum nor a preselected older point is incorrectly required to be the endpoint at this death. + +### 5. Nested compact components give the needed closed path + +At each subsequence level, the connected component `K_l` of the maximum in the closed superlevel is compact and connected and contains the same `Q`. Raising the level gives a connected subset of the previous superlevel containing the same maximum, so component maximality proves nesting. Their compact connected intersection contains both endpoints and lies in `{F >= delta}`. + +The proof uses this intersection only to put the endpoints in the same connected component of `{F >= delta}`. Section 4's local path connectivity of that ambient closed superlevel then gives a path in its component. It does not assert that the intersection itself is path connected and does not take a convergent subsequence in an unjustifiably compact path space. + +The path's attained minimum is at least `delta`. If it were greater than `delta`, it would be a strict higher-endpoint path at `delta`, contradicting the exact C263 cut after quotient transport. Thus the minimum equals `delta`; strict exclusion and closed attainment are used consistently. + +### 6. The constructed closure has exactly the required trap bounds + +The strict-death component `C` is open in the quotient and relatively closed in `{F > delta}`. Thus `D intersect {F > delta} = C` and `C` lies in the interior of `D`. A point of `C` higher than `m` would itself give a forbidden strict higher-endpoint path at `delta`, so `F <= m` on `C` and, by continuity, on its closure `D`. + +For a frontier point of the closed set `D`, height greater than `delta` would put it in `C` and then in the interior of `D`, a contradiction. This proves the frontier bound. The older endpoint is outside `D` because of the height ceiling. Nothing requires `D` to be a ball, smooth domain, or bounded Euclidean lift. Its active trace may have several components; the consumer only requires a clopen trace and a height ceiling. + +### 7. The ordinary-H0 consumer matches without hidden premises + +I compared the argument directly with `elderTrap_of_boundary` and `elderBoundaryDirectSummand` in the frozen `ElderBridge.lean`. The geometric data supply every argument: closed `D`, frontier height bound, strict death drop, maximum height and membership, height ceiling, a strictly higher point, and an actual closed-level path. The constructor is generic in a topological space, so applying it to the quotient is legitimate. No conversion of a quotient path to an arbitrarily prescribed Euclidean endpoint is made. + +`ElderInterval.lean` defines the actual interval on `-m <= t < -delta`. Its section is the difference of the two point classes and its retraction is the indicator of membership in the fixed `D`, descended to path components and transported by `ordinaryH0_filtrationIso`. Active clopen separation makes that indicator well-defined. The height ceiling gives the prebirth naturality square; the closed path gives the active-to-dead section square. These are actual requirements, and both are supplied. + +The composite retraction after section equals the identity because the maximum is inside `D` and the older point is outside. The resulting kernel complement gives the displayed natural direct-sum isomorphism. Ordinary point classes remain nonzero under augmentation; it is their difference that vanishes. Characteristic two causes no defect. This is one interval summand, without a claim of a simultaneous or exhaustive decomposition. + +## Counterexample and boundary checks + +1. **Dropping Hessian symmetry:** the complementary-sign predicate alone is insufficient for a nonsymmetric bilinear form. For example the matrix `[[1,1],[-1,-1]]` has a positive first coordinate line and negative second coordinate line but is singular. The proof explicitly obtains symmetry from smooth scalar calculus before its radical argument. +2. **Replacing Morse geometry by compact connectedness:** a compact connected space such as the closed topologist's sine curve need not be path connected. The proof supplies the quadratic-cone local argument and uses it on the ambient closed superlevel, avoiding this failure. +3. **Allowing moving higher endpoints to converge without a fixed gap:** strict inequalities `F(y_j) > m` can be lost under limits. The finite older-maxima and cofinal-subsequence step fixes one endpoint before taking compact intersections. +4. **Bounding each clearance only by birth:** values all less than `m` can have supremum `m`. The compact sphere supplies the genuinely smaller common bound `c0`. +5. **Using the total essential default as a death:** the C263 constant function of height `-1` has default death `0`, so an unconditional death/start bound is false. The present nonessentiality assumption invokes the nonempty supremum branch and excludes this case. +6. **Confusing strict and closed levels:** the proof excludes a strict higher path at `delta` while deriving one with heights at least `delta`. Its interval includes birth and excludes death, in agreement with the actual formal convention. +7. **Demanding arbitrary lifts:** the topology source's constant-path control refutes an arbitrary preselected lift endpoint. The proof uses the higher-endpoint equivalence with its existential endpoint and states its actual closed path on the quotient. +8. **Treating the dying class as an ordinary point class:** ordinary augmentation sends each point class to `1`. The proof and consumer use the point-class difference, so no reduced-H0 substitution is hidden. +9. **Promoting one split to all bars or saddle labels:** the constructed regions and endpoints can depend on the maximum. No compatibility, once-counting, unique merger saddle, or full elder relation has been derived here. The proof excludes each of those promotions. +10. **Promoting deterministic source applicability to an almost-sure result:** C291 supplies smoothness and periodicity of its literal sample, but the reviewed theorem still assumes the sample's Morse predicate. No measurable family of choices or probability-one Morse statement is produced. + +These are mathematical controls and interface checks, not executed experiments or counts of passing tests. + +## Findings and permitted use + +No mathematical amendment is required for the frozen proof. The explicit imports, their scope, and the exclusion of newly formalized or stochastic conclusions are sufficient for conventional acceptance of this adapter. The proof's historical author label remains “pending separate nonauthor review”; this separate report records the review decision without altering or pretending to rehash a different author artifact. + +The accepted composition may be used as a conventional mathematical source for the future formal adapter from literal smooth-periodic Morse inputs to the existing geometric interval constructor. It does not certify the new adapter in Lean, re-run historical source builds, audit all imported kernels/axioms, establish almost-sure Morse genericity of the literal Gaussian source, derive all-bars once-counting or saddle pairing, resolve collision or remote-complement estimates, confer provider-distinct/human approval, or change scientific status. Scientific effect: **NONE**.