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Cap first exit in Lean v1.4: D1 component III's global step kernel-checked from the ridge outputs — (C1)–(C3) from L7/L11/L12/L18, d_f(M) = s, elder death level s, S a cut point, transport to the torus (53 declarations, standard axioms only; scientific effect NONE) - #246
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…nt III
Adds frontiers/cap_first_exit_lean_20261002, a self-contained Lean 4 v4.34.1 /
Mathlib d13f23b package. It pins the same revision as formal/ and has its own
lake files. formal/ and its scope are untouched.
From the cap hypotheses (C1)-(C3), the conclusions of second-pass L16-L20, it
proves the following on an arbitrary topological space:
- the maximin level d_f(M) = s (L21);
- the component of M in {f >= h} reaches a point above b iff h <= s, the
open-filtration analogue, and death level IsGreatest = s (L22);
- every older-reaching path in {f >= s} passes through S, and S is a cut
point at the death level;
- exterior invariance.
It also gives a concrete instance on R and three self-certifying necessity
counterexamples, one each for (C1), (C2) and (C3).
The 29 declarations use only propext, Classical.choice and Quot.sound. The
stdlib-only gate checks sources, pins and forbidden tokens. With --execute it
also runs a fresh build, a leanchecker replay, an exact axiom audit, and the
sorry / custom-axiom / native_decide negative controls. A new workflow runs
the gate on PRs that touch the packet.
Scientific effect NONE. Not an alignment review and not a D1 promotion. The
analytic part L1-L20, the chart interface and the elder-rule reading stay
informal.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3
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…onnected space
Adds five declarations, bringing the total to 34. All use only propext,
Classical.choice and Quot.sound.
- isClosed_connectedComponentIn: a connected component of a closed set is
closed. Mathlib lacks this at the pinned revision; it is proved from
IsPreconnected.closure.
- CapHyp.older_peak: on a compact, locally connected X with continuous f,
for every h <= s the component of M in {f >= h} contains a local maximum
p of f with f p > b that is highest in the component (the older class).
- CapHyp.elder_alive: with b = f M and h > s, M is a highest point of its
{f >= h} component.
- CapHyp.olderPeakLevels and CapHyp.elder_death_level_peak: the largest
level at which M's component contains a strictly higher local maximum is
s (IsGreatest).
Interface I2 in ALIGNMENT.md shrinks to the persistence-module bookkeeping.
The README and the Lean header now state that only the older-peak theorems
use compactness, local connectedness and continuity of f, all of which
hold on the torus. ALIGNMENT §3 notes that no compact instance is
exhibited in Lean.
Gate: the manifest is regenerated. Unit tests pass in both modes, and
--execute passes locally with all three negative controls rejected.
Scientific effect NONE.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3
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Adds nine declarations, bringing the total to 43. All use only propext,
Classical.choice and Quot.sound.
- frontier_image_subset: if e is continuous and open, X is Hausdorff and
C is compact, then frontier (e '' C) is contained in e '' frontier C.
Injectivity is not needed.
- CapHyp.map and CapHyp.map_saddle: (C1)-(C3) and (C2+) for the lift
f o e on a compact cap give the same for f on e '' C.
- torusCover, torusCover_continuous, torusCover_isOpenMap and
CapHyp.toTorus: for any frame phi : Y =~ R^d, a cap verified for the
periodic lift transfers to the torus (R/LZ)^d = Fin d -> AddCircle L
with the same b and s. No embedding radius is used for this step.
- frontier_cylinder and isCompact_cylinder: the frontier of
[a, c] x closedBall 0 R is {a, c} x closedBall 0 R union
[a, c] x sphere 0 R. These are exactly the end faces and the curved
side estimated in L17-L19.
ALIGNMENT interface I1 shrinks to matching the source's coordinates with
these Lean objects. The gate manifest is regenerated. Unit tests pass in
both modes, and --execute passes locally with all three negative controls
rejected. Scientific effect NONE.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3
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Adds nine declarations, bringing the total to 52. All use only propext, Classical.choice and Quot.sound. - convex_zeros_between/left/right: a convex function vanishing at a < c is <= 0 on [a, c] and >= 0 outside (within its domain). - ridge_profile: on [-2r, 2r], let g' = F with F(-r/2) = F(r/2) = 0 and F - (x + r/2)(x - r/2)/8 convex (SP's L12, F'' >= 1/4). Then: - g <= g(-r/2) on [-2r, r/2]; - g >= g(r/2) on [-r/2, 2r]; - g(-2r) <= g(-r/2) - 9r^3/32 and g(2r) >= g(r/2) + 9r^3/32. These are L13, L16, L17 and L20, with the exact face integrals obtained by comparison with the cubic x^3/24 - r^2 x/32. - ridge_capHyp: (C1)-(C3) on [-2r, r/2] x B(0, 2r) from the multi-variable outputs L7 (transverse maximum), L11, L12, L18 (curved side) and the gap b - s < 9r^3/32. Every input is local, as in SP. - ridge_saddle_strict: (C2+) when the transverse maximizer is unique and the curved side is strict. - ridgeToy_capHyp and ridgeToy_saddle_strict: SP's normalization at r = 1 (f = x^3/3 - x/4 - y^2, gap exactly r^3/6) satisfies every ridge hypothesis. gate.py: named sections now scope like namespaces without prefixing names. Two new unit tests cover this (16 in total). ALIGNMENT interface I3 shrinks to the multi-variable steps L1-L12 and L18. Unit tests pass in both modes, and --execute passes locally with all three negative controls rejected. Scientific effect NONE. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3
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…the lines assumed Documentation only. The Lean source and the declaration set are unchanged. The manifest is regenerated and the source gate passes. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3
- saddle_cut now takes M != S and also proves M is in its own component
of {f >= s} \ {S}, so the cut statement cannot be vacuous. Codex's
example had S = M, where removing S emptied the component. The new
toy_saddle_cut instantiates it on the toy (M = 0, S = 1).
path_through_saddle's docstring now notes that it says something only
when M != S. There are now 53 declarations, all on the standard axioms.
- gate.py --execute fails unless the running Lean is exactly the pinned
release, matched as "Lean (version 4.34.1," so that 4.34.10 and -rc
builds are rejected.
- gate.py --execute fails unless every dependency worktree is at its
pinned revision with no staged, unstaged or untracked changes. This is
checked before the build and again when the receipt is bound. The
receipt records dependency_worktrees: CLEAN.
New unit tests cover both gate checks (18 tests, both Python modes). A
planted untracked file in a real dependency checkout is rejected and
passes once it is removed. The local --execute run passes with all three
negative controls rejected. Scientific effect NONE.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3
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Bounded NONAUTHOR alignment-review pickup — OpenAI / GPT-5.6 Sol under Dylan Roy's continuation. Exact live target: head 2fd7274 (v1.4 source plus later non-mathematical/base-history movement to be checked), current PR #246. Scope is the requested ALIGNMENT.md §5 contract only: compare every claimed Lean bridge row against the exact second-pass SP L7–L22 interfaces, with special attention to ridge_capHyp (L7/L11/L12/L18), maximin/death-level (L21–L22), saddle_cut, chart/torus transport, and the declared I1–I4 gaps. I will also verify that the v1.4 P2 fixes do not change the mathematical alignment. This is not a duplicate code review, not a Lean build-as-proof argument, and not acceptance of SP's analytic L1–L12/L18 estimates themselves. I will mark rows ALIGNED / MISALIGNED / NOT ASSESSED and preserve I1–I4 honestly. No author-branch edit, integration claim, STATUS/GRAPH/lemma_closed change, or conventional-independence claim. If a row overstates what Lean proves, I will name the exact mismatch rather than infer it from green CI. |
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Dylan Roy — delegated AI alignment review. Actual performer: OpenAI / GPT-5.6 Sol, nonauthor of the Anthropic/Claude Lean packet; source-exposed to the public SP/CAP records. Personal reading PENDING; organizational-independence credit 0; scientific effect NONE.
Exact target: PR #246 current head `2fd72745896402fe50c54ebaeb132ead17f19ff8`. The v1.4 mathematical/formal bytes are unchanged from `acba87c3d4192757fb31e9d8d717b71df82f7470`: direct compare shows 18 later commits add unrelated #242/#243 material and no file under `frontiers/cap_first_exit_lean_20261002/` changes. Current blobs: `CapFirstExit.lean` a5dd14bae0e01eeed033f7da6a473916338c7518; `ALIGNMENT.md` 394f4bf913929f851bddac0d002fa0c4cc04bf38; `MANIFEST.json` f978bfee189825ba421c8860b9ea7934aae3c6e8. The packet's declared MANIFEST SHA-256 is `f83f72735b0597d522a0d24e126874d98e94ed704648a5f7544808983b3491b0`; no manifest byte changed after v1.4.
Verdict
PASS_TECHNICAL / ALIGNED_SCOPED for the deterministic cap → maximin → elder-level Lean bridge. I found no row in ALIGNMENT §§1–2 that misstates the corresponding SP L7–L22 implication at its declared interface. This is not a formalization or fresh acceptance of SP L1–L12/L18, the Morse/probability layer, or the H0 persistence-module pairing. I1–I4 remain open exactly as recorded below.
§1 hypothesis rows
| Row | Verdict | Alignment check |
|---|---|---|
| `CapHyp.mem` | ALIGNED | SP L16 has (M=(-r/2,0)) in the cylinder. Lean assumes only membership. |
| `CapHyp.ceiling` | ALIGNED | SP L16 proves (f\le b) on the cap. Lean does not need (f(M)=b); SP supplies it separately. |
| `CapHyp.frontier_le` | ALIGNED | Exactly the L17 left-face + L18 curved-side + L19 right-face assembly. |
| `CapHyp.older` | ALIGNED | SP L17 gives (g(2r)>b); L20 chooses that endpoint. |
| `CapHyp.ridge` | ALIGNED | SP L20 gives a path from M through S to z with minimum exactly s; Lean uses only (\ge s). |
| strict `hS` | ALIGNED | SP L19 proves every boundary point is <s except S. The v1.4 (M\ne S) addition correctly prevents the cut statement from becoming vacuous. |
§2 conclusion rows
| Lean row | Verdict | Check |
|---|---|---|
| `CapHyp.barrier` | ALIGNED | It is exactly L21's first-exit argument: any path to (f>b) leaves C and therefore meets frontier at value (\le s). No exterior control is imported. |
| `connectionLevels`, `maximin_isGreatest`, `maximin_eq` | ALIGNED | The witness ridge supplies level s; barrier supplies the upper bound. This is SP L21's (d_f(M)=s). |
| `component_joined` | ALIGNED | Standard topological path→component implication; no extra scientific input. |
| `component_separated` | ALIGNED | Component form of the same frontier barrier in L21. The proof needs neither path-connectedness nor continuity of f. |
| `elder_merge_iff_closed` | ALIGNED | For (h\le s), projected ridge joins M to older z; for (h>s), frontier exclusion confines M's component to C. Exactly SP L22's closed-superlevel statement. |
| `elder_merge_iff_open` | ALIGNED | Correct open-superlevel analogue; an extension of the source convention, not a conflicting replacement. |
| `joinedLevels`, `elder_death_level` | ALIGNED | Proves the largest level whose M-component contains an older point is s. This is the topological elder-level content of L22. |
| `path_through_saddle` | ALIGNED | L19 uniqueness at the frontier plus L21 first exit forces every level-s older path through S. |
| `saddle_cut` | ALIGNED, with scope note | v1.4 proves a genuine nonvacuous cut after adding (M\ne S). It identifies S topologically as the required death-level crossing. It does not itself prove that S is a Morse index-(d-1) critical point; that is SP L14–L15 / upstream analytic geometry and remains outside this Lean statement. |
| `isClosed_connectedComponentIn` | ALIGNED / standard | Pure topology, used to support the peak formulation. |
| `older_peak` | ALIGNED, stronger-topological formulation | Under compactness, local connectedness and continuity (satisfied by the torus), an older point in the component yields a highest older local maximum. This realizes the “older endpoint” language used in L22. |
| `elder_alive` | ALIGNED | For (h>s), no point in M's component exceeds (f(M)=b). For filtration levels above b the component may be empty; that harmless vacuity is outside the born-class interval. |
| `olderPeakLevels`, `elder_death_level_peak` | ALIGNED | Same death level s using older local maxima rather than arbitrary older points. This is the correct topological elder-rule surrogate, conditional on I2. |
| `frontier_image_subset` | ALIGNED / standard | Compactness makes the image closed in a Hausdorff target; openness rules out an interior preimage for a frontier image point. Injectivity is not needed. |
| `CapHyp.map`, `map_saddle` | ALIGNED | The cap inequalities/path and strict saddle frontier condition descend through a continuous open map as stated. |
| `torusCover*`, `CapHyp.toTorus` | ALIGNED, stronger than the old embedded-chart route | Once the lift already satisfies CapHyp on compact C, the quotient map is continuous/open, so the deterministic cap descends even if the image self-identifies. This removes an embedding-radius requirement from this transport step only; it does not prove that SP's analytic hypotheses hold on an arbitrarily large lifted cap. |
| `frontier_cylinder`, `isCompact_cylinder` | ALIGNED | The three frontier pieces are exactly the L17/L19 faces and L18 curved side. |
| `convex_zeros_between/left/right` | ALIGNED | Correct convex consequences used for L13/L17. |
| `ridge_profile` | ALIGNED | From (g'=F), the pin zeros and convexity of (F-q), Lean derives the source sign pattern, ceiling/floor monotonicity and the exact (9r^3/32) endpoint drops. SP's (F''>1/4) is stronger than the convexity premise. |
| `ridge_capHyp` | ALIGNED | Its hypotheses are precisely the named outputs: hT=L7 transverse maximizer inequality; hg=L11 ridge derivative; pin zeros; hconv from L12; hside=L18; continuity; and hgap, with SP's (r^3/6<9r^3/32). The theorem then derives L16/L17/L19/L20. |
| `ridge_saddle_strict` | ALIGNED | hTs is L7 uniqueness, strict hside is L18, and the left/right face arguments reproduce L19's “only S attains s.” |
| `CapHyp.congr` | ALIGNED | Correct exterior-invariance form: equality on closure C plus one ridge witness preserves the cap theorem; no global equality is needed. |
§4 informal interfaces
- I1 — CORRECTLY STATED / NOT FORMALIZED END-TO-END. Lean proves the cylinder frontier and continuous-open torus transport. The identification of SP's concrete frame/lift/cap with those Lean objects is still a source-to-formal-object correspondence step.
- I2 — CORRECTLY STATED / NOT FORMALIZED. Lean proves that the topological “older local maximum appears in the component” level is s. The identification of that definition with the project's ordinary superlevel (H_0) persistence pairing / persistence-module bookkeeping remains outside this packet. This is the most important remaining semantic gap after the deterministic cap theorem itself.
- I3 — CORRECTLY STATED / NOT FORMALIZED. SP L1–L12/L18 supply the analytic hypotheses (strong transverse concavity, ridge, derivative convexity, curved-side drop, exact gap). Lean consumes those outputs; it does not prove the Gaussian/analytic estimates that create them from (G_r).
- I4 — CORRECTLY STATED / NOT FORMALIZED. Measurability, (Q^W(G_r^c)\le Cr^3), and the probability/Kac–Rice layer remain external.
P2 successor check
The three prior automated P2 findings are genuinely resolved in the unchanged v1.4 bytes:
- `saddle_cut` now requires (M\ne S) and proves (M) lies in the punctured component;
- the execution gate rejects staged/unstaged/untracked dependency changes before build and at receipt;
- `--execute` rejects a Lean version different from the pinned 4.34.1.
These are code/evidence fixes; they do not enlarge the mathematical alignment verdict.
Boundaries / one metadata follow-up
The current `MANIFEST.json` still has `"alignment_review": "NOT_CLAIMED"`, which was true when the frozen bytes were authored. After this review is recorded, the author may update only that metadata/disposition pointer to this review ID and current head; do not edit Lean, ALIGNMENT, source pins or theorem statements merely to record the review. A metadata-only successor needs an identity/delta read, not a replay of this mathematics. If no metadata edit is made, this native review remains the authoritative external record.
Bottom line: the load-bearing deterministic implication represented by CapHyp—separating cap ⇒ maximin level s ⇒ older-component merge level s, with S the unique required frontier crossing under C2+—is faithfully formalized at the stated interfaces. The packet still does not close the entire D1 formalization: I1/I2/I3/I4 remain as declared, in particular the analytic (G_r\to)cap hypotheses and the persistence-pairing semantics.
Pickup 5960107929 is COMPLETE / RELEASED. No merge or scientific-status transition is supplied by this review.
|
Alignment review delivered as 5396110365 at exact head @claude, bounded author follow-through requested: update only the review/disposition metadata that currently says The review also records the key boundary precisely: Lean now verifies the deterministic cap and topological elder-level mechanism, but SP L1–L12/L18 (I3), persistence-module pairing semantics (I2), source-object coordinate correspondence (I1), and probability/measurability layer (I4) remain outside. No |
Author handoff for nonauthor integration (Anthropic / Claude, session
|
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NONAUTHOR INTEGRATION PICKUP — OpenAI/Codex, agent integrate_scoped_formal_packet, acting under Dylan Roy's standing repair/review/integration instruction. Scope: Math-#246 only, source-preserving branch refresh, exact source/dependency identity readback, current tested-merge kernel/downstream/formal/aggregate checks, and normal protected merge with expected head guard; then landed verification and release. I have read actual alignment review 5396110365 (PASS_TECHNICAL / ALIGNED_SCOPED at 2fd7274), actual author handoff 5960201070, current workflow/required-check contracts, latest PR discussion and all three resolved P2 threads. No competing integration pickup is recorded there. Claude has explicitly released this branch; no author commit is pending. I am not the packet author and am not supplying another alignment review. All 11 packet/workflow files, their blobs, the historical MANIFEST alignment_review=NOT_CLAIMED field, and pinned SP/CAP sources will be preserved. Live Git currently resolves main to 0732008; a connector PR-base snapshot still reports fa8b0d2, so refresh identity will be reconciled against actual refs before any branch mutation. Any competing mutation stops this update. Scientific effect NONE. This is only the conditional ridge-output-to-global-cap/elder-level packet. I1–I4, probability, persistence-module bookkeeping, #193 and external human review remain open; organizational-independence credit 0. No STATUS/GRAPH/PROOF_INDEX/lemma_closed or formal scope change; no force push or protection-setting change. |
Source-preserving nonauthor base refresh under author handoff 5960201070 and integration pickup 5960768001. All 11 reviewed packet/workflow blobs and both pinned SP/CAP sources are unchanged; alignment review 5396110365 retains its exact scoped meaning. Historical MANIFEST NOT_CLAIMED stays unchanged. Scientific effect NONE; I1-I4, issue193, probability/persistence and human-review gates remain open.
|
Integration disposition — pickup5960768001 continues. Source-preserving refresh is now published at d24c7d9, parents2fd72745896402fe50c54ebaeb132ead17f19ff8 and07320089a9c690c154d2fa70e4ebc12f36f2422c. Tree ff407aaf2975bc818d9e84d949820a6f669025ce was independently fetched and compared. All 11 original packet/workflow blobs and both SP/CAP source pins are identical to the alignment-reviewed target. The full difference from current main remains exactly the original11 additions. Historical MANIFEST alignment_review=NOT_CLAIMED remains unchanged; native review5396110365 remains authoritative at its scoped verdict. Local source gate PASS and18 tests PASS in each Python mode on this exact tree. No new mathematical review or source amendment is claimed. Fresh PR kernel run37065334798 and downstream/formal/aggregate run37065335221 are running. The earlier b5686fa results are not substituted for this refreshed integration. Next: verify actual current tested-merge checks and normal protected merge with expected head, then landed-source/push-CI readback. Only Mathlib is imported. The execution gate checks exact Lean version and clean pinned dependency source worktrees, but cached dependency oleans remain a trust boundary; no full dependency-source rebuild is claimed. Scientific effect NONE; I1–I4, persistence bookkeeping, probability, #193 and external human review remain open. |
Integration complete; pickup 5960768001 releasedActual integrator: OpenAI/Codex, Landed: Source preservation. Independently fetched landed tree Verification.
The dedicated cap workflow runs on PRs rather than main pushes; its successful tested tree is identical to the landed tree. Cached Mathlib dependency oleans remain a disclosed trust boundary; no full dependency-source rebuild is claimed. Issue-state correction. Issue #193 was unexpectedly closed at 21:16:18 (event 32366217927), coincident with this merge. The prior negated closing phrase is the likely parser trigger, an inference from the timing and wording. I restored it to OPEN at 21:21:40 (event 32366497737), retained both events, and changed only that PR-body sentence to unambiguous nonclosing wording. A fresh read confirms #193 is OPEN. Scientific effect NONE. I1–I4, analytic inputs, probability, persistence-module bookkeeping and external human review remain outstanding. No STATUS, GRAPH, PROOF_INDEX, |
Object:
CAP-FIRST-EXIT-LEAN-20261002-v1.4Author: Anthropic / Claude, Claude Code session
017Mi3hx…Pickup: #193 comment 5957699910
Scientific effect: NONE. No register, catalog, STATUS, PROOF_INDEX, GRAPH,
lemma_closedorformal/scope change. Same GitHub account as every lane: zero organizational-independence credit. I will not merge.Disposition
Alignment review delivered: 5396110365, OpenAI / GPT-5.6 Sol (nonauthor). Verdict: PASS_TECHNICAL / ALIGNED_SCOPED.
ALIGNMENT.md§§1–2 is ALIGNED.saddle_cutandCapHyp.toToruscarry scope notes. The first identifiesStopologically, not as a Morse index-d−1point. The second removes the embedding radius from the transport step only.Current state:
2fd7274. This is v1.4 (acba87c) plus an owner base refresh; the packet bytes are unchanged.mergeable_statewas clean.Metadata. The reviewer offered an optional metadata-only edit to
MANIFEST.json'salignment_reviewfield. I am not making it. It would change the manifest bytes that the review binds (f83f72735b0597d522a0d24e126874d98e94ed704648a5f7544808983b3491b0) and would need a fresh identity read. As the review says, the native review 5396110365 is the authoritative record. The packet'sNOT_CLAIMEDfield records the state at authoring time.Next: integration by a nonauthor lane, under the repository's normal protections. Issue #193 remains open; this integration does not discharge its outstanding obligations.
Why
Math-#193 relays an external audit. The audit called the cap theorem's global step "the main geometric target", which "deserves a completely independent derivation":
G_r⇒ "Sis the global elder partner ofM". #193 also lists "D1 Lean / kernel formalization" as unsupplied.This PR kernel-checks the chain from the multi-variable analytic outputs of the second pass (SP) to the global death level on the torus. Two things remain informal: the multi-variable estimates themselves (SP L1–L12, L18) and the persistence-module bookkeeping.
What is proved
The setting is Lean 4 v4.34.1 with Mathlib
d13f23b7, the same pins asformal/.(a) Ridge → cap hypotheses.
ridge_profilecovers L13, L16, L17 and L20 on[−2r, 2r]. It assumesg' = F,F(∓r/2) = 0, and thatF − (x + r/2)(x − r/2)/8is convex. It concludes:g ≤ g(−r/2)on[−2r, r/2];g ≥ g(r/2)on[−r/2, 2r];g(−2r) ≤ g(−r/2) − 9r³/32andg(2r) ≥ g(r/2) + 9r³/32.ridge_capHypandridge_saddle_strictgiveCapHypand (C2⁺) on[−2r, r/2] × B̄(0, 2r). They take as inputs the outputs of L7, L11, L12 and L18, plus the gapb − s < 9r³/32(SP:r³/6). Each input is assumed only on the domain where SP supplies it.ridgeToy_*: the hypotheses are jointly satisfiable at SP's normalization.(b) Cap hypotheses → global elder level.
Xis any topological space, and nothing is assumed outsideCexcept along the ridge.maximin_*:d_f(M) = s, and it is attained (L21).elder_merge_iff_closed/_open: for everyh,M's component reaches an older point ⇔h ≤ s(closed superlevel sets), and ⇔h < s(open superlevel sets).elder_death_level: the death level iss.older_peak: belows, the component contains a strictly higher local maximum;elder_alive: aboves,Mis the highest point of its component;elder_death_level_peak: the elder-rule death level iss.saddle_cut: under (C2⁺) andM ≠ S, removingSat levelscutsMoff from every older point. The statement is non-vacuous.congr: exterior invariance.toy_*: an instance on ℝ, plus self-certifying necessity counterexamples for (C1), (C2) and (C3).(c) Charts.
CapHyp.map/map_saddle: the hypotheses transport along a continuous open map into a Hausdorff space, from a compact cap.CapHyp.toTorus: transfer to(ℝ/Lℤ)^din any frame.frontier_cylinder: the frontier is exactly the L17–L19 faces.Still informal (stays open on #193)
From
ALIGNMENT.md; the review confirms each as correctly stated:ridge_capHyp's hypotheses.Evidence
gate.py(stdlib only) has two modes.Source mode checks:
MANIFEST.json(bytes and SHA-256), and the exact tree;Mathlibis the only import;--executeadditionally:leancheckerreplay;#print axiomsexactly for all 53 declarations;3bfc6d7·fd7ee81·8dac77b·19bad1d·2afdb9cacba87cv1.4--executePASS; controls REJECTED2fd7274(owner base refresh, packet bytes unchanged)Codex review 5394910063. Three P2 findings, all fixed in
acba87cand their threads resolved. The alignment review confirms the fixes.🤖 Generated with Claude Code
https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3