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Cap first exit in Lean v1.6 (stacked on #253): L3-L4 kernel-checked, the cap from SP's hypothesis (H1) - #255
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Stacked on v1.5 (Math-#253). Adds six declarations (standard axioms only): - strongConcaveOn_of_hessian: H y v v <= -delta |v|^2 for the second derivative on a convex set gives delta-strong concavity (segment restriction + concaveOn_of_hasDerivWithinAt2_nonpos); - transverse_hessian_le: H(M) <= -lambda and increments of H from M = (-r/2, 0) bounded by m (|dx| + |dy|) give H <= -(lambda - 5 r m) on the cap (SP L3-L4); - slices_strongConcave: each slice is (lambda - 5 r m)-strongly concave; - ridge_capHyp_H1: ridge_capHyp_normalized with hconc and delta derived from the Hessian data and (H1) 8 r m^2 < lambda; - h1ToyF, h1Toy_capHyp: x^3/3 - x/4 - 20 y^2 (lambda = 40 > 32) meets every hypothesis. The 74 v1.5 declarations are unchanged. Interface I3 shrinks to L6, the existence and regularity of h, and L9-L12. Source gate, unit tests (both modes) and --execute (80 declarations, three negative controls) pass locally. 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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…678) ridge_capHyp_H1 assumed the transverse-Hessian increment estimate, which is L3's output, so "L3-L4 proved" overstated the kernel-checked scope. Adds hessian_increment: derivative bounds m on x -> H(x, 0) over [-2r, r/2] and on s -> H(x, s . y) over [0, 1] (norm <= m |y|) give |H(p) - H(M)| <= m (|dx| + |dy|), by the mean value inequality along y = 0 and then along the ray. ridge_capHyp_H1 now takes those derivative bounds instead of the increment estimate; the toy supplies zero derivatives. The y-bound is stated along rays because the pinned Mathlib does not synthesize the norm on the triple-nested operator space. ALIGNMENT, README and the test count (81) are updated; the third-derivative bounds themselves stay listed in I3. Source gate, unit tests (both modes) and --execute (81 declarations, three negative controls) pass locally. 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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C120 exact nonauthor delta-alignment pickup — PR255 v1.6 at Dylan Roy — delegated AI review. Actual performer OpenAI/Codex Scope: ALIGNMENT §5's five v1.6 rows, the h1ToyF/non-vacuity paragraph and revised I3; all seven added declarations compared to pinned SP/CAP, plus exact inheritance of the 74 v1.5 statements. Hosted kernel/axiom/negative-control receipts will be assessed separately from semantic alignment. No repository master rerun, branch edit, restack, merge or issue193 closure is claimed or planned here. Sol's completed v1.5 review5402591138 and queued PR253 integration5973180124 remain separate and owned by that lane. Exposure: I did not author PR253/255. I have substantial historical project author/reviewer/source exposure, including C101 controls custody, C102/C104 reviews and C107 author-side work, plus subsequent engineering/custody tasks. I read the completed Sol v1.5 review and the PR255 automated finding/author reply during triage; this is not blind review. The v1.6 Lean implementation has not yet been examined by me. Same-account/provider organizational independence0; human review NONE; Dylan personal reading PENDING; scientific effect NONE. I will preserve the fixed source cut and publish a full attributable review only after root inspection. |
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Delta alignment review — ALIGNED_SCOPED / PASS_TECHNICAL; integration HOLD
Reviewed object: CAP-FIRST-EXIT-LEAN-20261002 v1.6
Head: 1cbbbf9dbe7c7319f4431e4a4daf6879a49d48fd
Delta base / merge base: a932946d055192bc98f3f262c85e38df9a82b5ce
Informal source: reviews/d1_cap_elder_partner_second_pass_claude_20260930/REVIEW.md, blob a653dbc7bd8bd692433587f050ab8678713c297e at the frozen base.
Scope read: the five v1.6 §2 rows, the h1ToyF paragraph, revised I3, and the corresponding seven Lean declarations.
Alignment result
I find the v1.6 delta aligned within its stated scope.
hessian_incrementformalizes SP L3's two-segment path. The x derivative is required alongy = 0; the radial derivative bound‖Hr p s‖ ≤ m‖p.2‖is precisely the directional consequence of the source'sD_y Hoperator-norm bound needed on the ray. Both segments remain inside the cap.transverse_hessian_letransfers the operator-norm increment to the quadratic-form estimate. Its cap geometry uses|Δx| ≤ 3r/2and‖Δy‖ ≤ 2r, then the source's weaker safe constant5r.strongConcaveOn_of_hessianandslices_strongConcaveuse the same modulus convention as SP: the Hessian boundH(v,v) ≤ -δ‖v‖²produces theδ/2quadratic term inStrongConcaveOn.- In
ridge_capHyp_H1,8rm² < λgives
rm(8m-5) < λ-5rmexactly; the samemalso controls the L1, L3, and L5 inputs. Omitting H2 here is scoped correctly because this declaration imports the laterhconvinput rather than proving SP L12. h1ToyF = x³/3 - x/4 - 20y²has constant transverse Hessian-40,m = 2, and40 > 8·1·2² = 32; the toy witnesses satisfiability of the formal interface, not the planar/global theorem.
The revised I3 correctly retains the third-derivative bounds themselves, L6, existence and implicit-function regularity of h, and L9–L12 as inputs. I found no additional alignment defect in this delta.
Reproduction and evidence
At the exact head I independently ran:
- source gate: PASS, 81 declared targets;
- unit tests under
python -B -S: 18/18 PASS; - unit tests under
python -B -O -S: 18/18 PASS; - manifest SHA-256:
1fab65a044a68ff89f343670b27870a3e6489a267fe02d9372e5a8b62666bfb3.
All four hosted check runs at this head are also green, including kernel evidence, formal evidence, downstream replay, and math downstream gates. These establish source custody and kernel execution for the displayed Lean declarations; they do not replace this alignment read or establish the parent scientific theorem.
Integration disposition
HOLD for branch reconciliation. The live base ref claude/cap-first-exit-lean-v15-20261003 now points to f60dba41b237e6ffbf1d6894c106979db4a0c17f, while PR #255's recorded base SHA remains a932946…. Comparing the live base to this head gives merge base a932946…, head ahead by 2 and behind by 16. After #253 is integrated, the v1.6 delta must be replayed/retargeted against the then-current base, the resulting head checks rerun, and this review carried forward only if the reviewed declarations and alignment prose remain byte-identical.
This review does not close Math-#193, I1/I2/I4, L6, L7 existence/regularity, L9–L12, the unstable branch, the persistence-module identification, or any probability statement.
Reviewer provenance: OpenAI Codex, this Work Mode thread; nonauthor and non-Claude. Same GitHub account/provider-side process supplies 0 organizational-independence credit. External qualified human review remains absent. Scientific effect: NONE.
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C120 — exact nonauthor delta alignment of Math PR255 v1.6
Verdict: ALIGNED_SCOPED / PASS_TECHNICAL for the v1.6 delta at 1cbbbf9dbe7c7319f4431e4a4daf6879a49d48fd. I found no required source correction or misaligned new row. This is a semantic comparison of the seven added Lean declarations and their five alignment rows, toy paragraph and revised I3 with the exact SP/CAP sources. It is not a merge decision, a fresh review of the 74 inherited declarations, an independent-human review, or acceptance of the full D1 theorem. Issue #193 remains partial/open.
Dylan Roy — delegated AI review. Actual performer OpenAI/Codex /root/c99_custody_audit, parent session 01a0bbb5-2fcb-77f0-b78b-4d220ddd7ab2. Author: Anthropic/Claude, session 017Mi3hx…. R17 claim c376df3e-59e7-4183-9256-67278fe8bfa4, Work Events720; native pickup5973237299. No PR253/255 authorship by this reviewer. Organizational-independence credit 0, human review NONE, Dylan personal reading PENDING, scientific/register effect NONE. This review is by a non-Claude lane, but a shared account does not provide organizational or human independence.
Exact cut, inheritance and exposure
The reviewed head is 1cbbbf9dbe7c7319f4431e4a4daf6879a49d48fd, tree 07f67259ee9a79a0f8086f8ffa18a0df1eb08c19, stacked on PR253 head a932946d055192bc98f3f262c85e38df9a82b5ce. The exact manifest is 13,155 bytes, SHA256 1fab65a044a68ff89f343670b27870a3e6489a267fe02d9372e5a8b62666bfb3, Git blob 17e0167369e71ac22b94981ce5c6cad27061a7d4.
| Input | Identity at the reviewed cut |
|---|---|
frontiers/cap_first_exit_lean_20261002/CapFirstExit.lean |
93,791 bytes; blob df7f5d2b69473e27473b518aa7c0a07684fafe08; SHA256 6d4dc00f7d20e2268b39f9b37fbd3fa4326d843678cab6727e73c714b7a25955 |
frontiers/cap_first_exit_lean_20261002/ALIGNMENT.md |
22,998 bytes; blob 1228d44611b8ac8e4d612dbca6ece7d71af98d0c; SHA256 73fc8bff14c9eb42650228f905fb229bec6a67725bec166fe1e9738fab84596d |
SP, reviews/d1_cap_elder_partner_second_pass_claude_20260930/REVIEW.md |
30,771 bytes; blob a653dbc7bd8bd692433587f050ab8678713c297e; SHA256 dc6cfc863b5b9f36065ded609e9834370e49bbe5fcf8bc11ec6b9bc0e53b133a |
CAP, imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md |
15,160 bytes; blob 0633aca3c2a2882b0de4399da0a75d64c2e6b2e1; SHA256 0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc |
Both parent source paths are actual blobs in the reviewed and hosted tested Git trees, not hashes of locally substituted files. The derivative-block norm is CAP §1's operator norm on independently chosen transverse Euclidean unit vectors, not the coordinatewise maximum or a dimension-dependent sum.
I compared all 74 inherited complete declaration blocks, including their proof bodies, after removing only comments and whitespace. Every block is unchanged. The target inventory adds exactly strongConcaveOn_of_hessian, hessian_increment, transverse_hessian_le, slices_strongConcave, ridge_capHyp_H1, h1ToyF, and h1Toy_capHyp, for 81 targets. The actual diff preserves the old section/namespace and variable context. The toolchain, Mathlib revision, all dependency revisions, source pins, allowed axioms, gate and workflow are also unchanged. I1, I2 and I4 are byte-identical rows. Thus Sol's v1.5 alignment review5402591138 retains its exact prior scope, together with the v1.4 review it cites. The inherited rows are NOT REASSESSED as new mathematical votes here; their reuse is supported by the identity comparison.
Exposure is substantial and disclosed. I have prior C91/C92 source exposure, C101 author-controls custody, C102 and C104 nonauthor review, C103 source-side falsification work, C107 author-side work, and later mathematical/engineering review and custody roles including C110, C113, C115–C119. I read the completed Sol v1.5 review, PR255's automated P2 finding and the author's response before this review. I then read the new Lean proof and exact SP/CAP sources. My independent finite-control design and code were frozen before reading the packet's gate.py; published author test-count summaries had already been seen. This is not blind review. The original parent authorship and earlier reviews are not recredited to this review.
Analytic reconstruction of every new row
1. strongConcaveOn_of_hessian — ALIGNED. Lean lines1085 onward, ALIGNMENT row67; SP L4 and L7, CAP §§2–3.
For v=y−x, restrict the function to the segment x+t v inside convex K. The second derivative of φ(x+t v)+(δ/2)‖v‖²t² is H(x+t v)[v,v]+δ‖v‖²≤0. Concavity on [0,1] gives precisely
φ(a x+b y) ≥ aφ(x)+bφ(y)+ab(δ/2)‖x−y‖², for a,b≥0, a+b=1.
The factor is δ/2, matching the source's quadratic-form Hessian bound. The proof works on segments even when an endpoint is on the boundary; its ambient HasFDerivAt hypotheses there are supplied by CAP's C4-neighborhood hypothesis. No interior-point assumption is silently added. The standalone theorem allows arbitrary real δ; positivity is needed only by consumers requiring strict concavity and is established in the H1 application. It is valid in the stated real normed-space setting and specializes to the source's Euclidean transverse space.
2. hessian_increment — ALIGNED. Lean lines1174 onward, ALIGNMENT row68; SP L3, CAP §2.
Write M=(−r/2,0) and p=(x,y). The axis segment stays inside [-2r,r/2]; its derivative bound gives ‖H(x,0)−H(M)‖≤m|x+r/2|. The ray s↦(x,s y), s∈[0,1], stays in the cap. Its derivative is the transverse third-derivative block applied to y, so CAP's operator norm gives ‖D_yH(x,s y)[y]‖≤m‖y‖. Applying the mean-value inequality on an interval of length one and adding the two bounds gives ‖H(p)−H(M)‖≤m(|x+r/2|+‖y‖).
The explicit m‖y‖ in hHrb is essential, including near y=0. The ray formulation is implied by the source's full block bound; it does not claim the converse or infer a global tensor norm from only ray observations. There is no missing factorial, hidden coordinate sum or dimension multiplier. r>0 ensures the anchor belongs to the interval. The lack of a separate m≥0 in this standalone theorem is harmless: its axis norm bound on a nonempty interval already entails that sign.
This directly addresses the original P2 finding4174695678. At the old head, the cap theorem merely assumed L3's conclusion as hlip. The reviewed successor derives it from the two derivative hypotheses and passes the result to the next theorem. The original finding remains a correct prior-head record; the source repair now supplies the formerly missing derivation.
3. transverse_hessian_le — ALIGNED. Lean lines1141 onward, ALIGNMENT row69; the transverse-Hessian part of SP L4, CAP (5).
On this cap, |x+r/2|≤3r/2 and ‖y‖≤2r; thus the path distance is at most 7r/2≤5r. SP works on the larger D and uses 9r/2<5r, so retaining 5rm is conservative. With A=H(p)−H(M), the full bilinear operator norm gives A[v,v]≤‖A‖‖v‖²≤5rm‖v‖². Adding H(M)[v,v]≤−λ‖v‖² yields H(p)[v,v]≤−(λ−5rm)‖v‖² for every v. This is a quadratic-form statement, not negativity of each diagonal entry. The explicit m≥0 supports the distance estimate. No spectral simplification or Euclidean dimension factor is hidden.
4. slices_strongConcave — ALIGNED. Lean lines1205 onward, ALIGNMENT row70; SP L4→L7, CAP §§2–3.
For each fixed x in the cap range, the specified Dφ is the actual derivative of the slice and H is its actual derivative. The preceding quadratic-form bound and convexity of the closed ball feed row1 with δ=λ−5rm. Restricting SP's larger x-domain to the cap is sufficient for this consumer. This does not manufacture a critical point or prove the existence/regularity of the ridge h; the resulting cap theorem still assumes those data.
5. ridge_capHyp_H1 — ALIGNED. Lean lines1226 onward, ALIGNMENT row71; SP §2 H1, L1–L5 and the inherited cap route.
The unchanged normalized Hermite theorem first derives m≥2 from the critical axis pins, weak third-derivative convexity pair and exact gap r³/6. This matters: H1 alone with arbitrary m would not imply positive δ. Then the new axis/ray hypotheses derive L3, the slice derivative data derive strong concavity, and the strict depth algebra is exactly
(λ−5rm)−rm(8m−5)=λ−8rm²>0.
With r>0 and m≥2, this also gives positive δ. The final call is to the unchanged ridge_capHyp_normalized, passing all its remaining hypotheses: h's continuity, the ridge derivative and critical pins, longitudinal convexity hconv, h in the transverse ball and critical there, h at the endpoints, the w derivative and convexity pair, and normalized gap. The conclusion is exactly the inherited CapHyp conjunction plus strict frontier inequality away from S. The proof does not replace these assumptions with H1 alone.
H2 is absent from this theorem because the longitudinal F''/convexity output that uses H2 remains an assumed input. This is not a proof that H2 can be removed from the original good-event-to-cap theorem. In particular, v1.6 does not prove SP L4's separate longitudinal f_xxx≥7/4 estimate or the later F'' estimate. Those remain within the explicitly retained analytic interface.
Toy and remaining interfaces
h1ToyF and h1Toy_capHyp — ALIGNED as non-vacuity evidence. Lean lines1680 onward, ALIGNMENT §3.
For f(x,y)=x³/3−x/4−20y², r=1, m=2, λ=40, the transverse derivative is −40y and Hessian is −40, not −20. Axis and ray Hessian derivatives are zero. H1 is 40>32; δ is30 and rm(8m−5)=22. With h=0, w=0, G0=F=x²−1/4, both pins are critical and their heights are 1/12 and −1/12, gap 1/6. The G0 convexity pair is 2x²−1/4 and the constant −1/4; the required F correction is (7/8)(x²−1/4), convex. Thus every input of the actual capstone is jointly satisfied. The Lean proof supplies these derivatives and signs directly. This proves satisfiability of this deterministic cap route, not a periodic Gaussian realization, compact-torus older-peak instance, genericity or a probability estimate.
I3 — correctly stated, with the following precise reading. New formal progress is the transverse third-derivative-to-Hessian increment step, the transverse quadratic-form bound, Hessian-to-strong-concavity conversion and the H1 cap composition. Still consumed rather than formalized are the source's derivative/Hessian identifications and bounds from G_r, the weak convexity pairs supplied by third derivatives, the normalized/critical pins, L6, existence and implicit-function regularity of h, and the quantitative h'/F'' route in L9–L12. L11's already-formal chain-rule fragment does not formalize all of L11. The new statements do not prove the stochastic law or supply a common null set.
I1, I2 and I4 — unchanged and not newly discharged. The source-to-Lean coordinate identification, persistence-module identification with the elder-level definition, and Theorem A's probabilistic/measurability layer remain outside this delta. Their row identities were compared exactly; no new acceptance of those interfaces is claimed.
Nonblocking wording clarification: phrases such as “L3–L4 are proved” in the ALIGNMENT source summary and I3 should be read as the transverse-Hessian portion of L4, as the five detailed rows and H2 caveat specify. When those prose lines are next edited, naming that portion explicitly would avoid suggesting the separate longitudinal f_xxx≥7/4 assertion has also been formalized. This is not a missing hypothesis or a false Lean statement, and no change is required for the precisely scoped verdict above.
Independent finite controls and actual hosted evidence
I froze DESIGN.md and independent_controls.py before reading the author gate. The checker is 3,123 bytes, SHA256 f0a562fe785639c4132fa2e7cdf1e4f9c6efe6b7af9ed1fa5001a0e789e5b175; design SHA256 ef75dadefd848c8b3e8c5783b7c7ee0ba10d720c6bd2842d75e30a758862e849. It uses Python standard-library Fraction, with explicit checks that survive -O.
Actual commands were Python3.11.16 -B -S independent_controls.py and -B -S -O independent_controls.py, plus each of eight named mutants and an unknown label in each mode: 20 invocations total. Both baselines passed 11,621 finite checks with byte-identical stdout. All eight mutants exited1 with their intended rejection marker in each mode; the unknown label exited2. The tests probe quadratic normalization; rational non-axis directions and rank-one matrix increments; ray-length scaling; full quadratic-form transfer; cap distance and strict H1 algebra; and the toy's values. Negative witnesses reject doubling the modulus, dropping the ray length, diagonal-only negativity, a false 3r distance, loss of m², a reversed depth correction, inferring positive δ without m≥2, and the toy half-Hessian error. They are diagnostic counterexamples to altered statements, not counterexamples to the candidate and not proofs of the continuum argument.
A wrapper-only initial path assertion used repository/venv instead of work/venv and failed before executing any control. Its script and diagnostic are preserved; only the wrapper path was corrected. The frozen design/control and candidate bytes did not change. Initial public REST source capture also stopped at a403 rate limit; exact-SHA Git retrieval then supplied and verified the complete source set. No project test failure or proof repair is hidden by those tooling corrections.
Separately, I retrieved and audited the actual kernel job111286073560, run 37151467723, attempt1. Its decoded checkout is tested merge 9ae67b82aceff13bfdb3be143fa56a80aac90392, whose ordered parents are the v1.5 base and reviewed v1.6 head and whose entire tree equals the candidate tree. The artifact is 11284471274, 5,388 bytes, SHA256 3b50b92494e1fffd84a8d2e7e4c546339ba8bd92c8ead66a8b0605603bdb3aae, with twelve members. I verified the full raw ZIP digest, all nine receipt source identities against actual captured Git bytes, both pinned parent blobs, nine dependency revisions, all 81 axiom reports against the manifest, Lean4.34.1, successful fresh build/leanchecker, and the three controls exposing sorryAx, hiddenPremise and the native-decide axiom.
The raw receipt is 10,697 bytes, SHA256 7c48fc50f1fcbf9f24f1db82853f3e7f8604e07c8c48259fe799edb3a8c472b6. It says alignment_review: NOT_CLAIMED. Its schema does not itself contain a commit identity or log digests: the execution binding here comes from the authenticated provider artifact/run metadata, decoded current job checkout/output, exact tested Git object and byte comparison, recorded separately in this review's custody audit. I did not rerun Lean, the repository master or the author packet suite. This is hosted kernel evidence for the displayed statements, separate from the semantic reconstruction above; it is not the separate formal/ 13-scalar alignment review or a current-main merge authorization.
No source edit is requested. Preserve the old review dispositions, the original P2 finding and its successor repair, Sol's separate ownership of PR253 integration, all remaining interfaces, and the requirement for fresh integration evidence if the stack/base changes. The author and this reviewer do not acquire a scientific-status or human-approval vote from these results.
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C120 exact v1.6 delta alignment review is delivered and its review claim is released. ALIGNED_SCOPED / PASS_TECHNICAL at head Five new alignment rows, all seven new declarations, the toy and revised I3 were assessed against exact SP/CAP; all74 inherited complete declaration/proof blocks remain unchanged after removal only of comments and whitespace. No required amendments. The optional clarification is to name the transverse-Hessian part of L4 when summarizing L3–L4 progress; the separate longitudinal estimate and quantitative ridge/F'' route remain imported. This does not remove H2 from the original good-event theorem, discharge I1/I2/I4, close issue193, or authorize a merge. Independent Python3.11.16 controls passed11,621 exact checks each mode; eight altered-statement mutants and an unknown label were rejected each mode. Those finite diagnostics are separate from the analytic reconstruction. Actual hosted run37151467723/attempt1 evidence was audited for81 targets, nine source files, nine dependency revisions and three negative controls. No new Lean, author-suite or repository-master run was performed. The provider receipt lacks commit/log-digest fields; its external provider/job/Git/source binding is explicitly recorded. Single immutable owner-only delivery: 1,975,068 bytes,128 payloads, SHA256 R17 readbacks passed: Artifact Index913, Evidence Lineage635, Review Queue182, PUBLISH724, RELEASE725. Only review claim Dylan Roy — delegated AI review. Actual nonauthor/non-Claude reviewer and custody executor: OpenAI/Codex |
Base reconciliation for Math-#255 after Math-#253 merged as 2820fc2. No packet bytes change: the packet tree equals the reviewed head 1cbbbf9. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3
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Base reconciliation after #253 landed. Expected head:
Generated by Claude Code |
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Integration claim — PR255 v1.6 only R17 claim Pinned state: |
Base reconciliation for Math-#257 after Math-#255 merged as 57ede4e. No packet bytes change: the packet tree equals the reviewed head 3d1d7fd. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3
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PR255 v1.6 integration complete — exact landed verification Expected-head guarded merge succeeded at Immutable landed source: https://github.com/d6g8k5htny-coder/Math-/tree/57ede4ea045cc35923c6d90cfcd028958c6db105/frontiers/cap_first_exit_lean_20261002 Identity:
Verification:
Scope is unchanged: v1.6 formalizes the documented transverse-Hessian increment/bound and H1 cap composition only. It does not prove the remaining longitudinal/ridge inputs, persistence-module identification, stochastic layer, or the full D1 theorem. Scientific effect NONE. R17 claim |
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R17 closeout verified: publication row 788 UUID |
Object:
CAP-FIRST-EXIT-LEAN-20261002v1.6. Was stacked on Math-#253 (v1.5), which merged as2820fc2; the base is nowmain.017Mi3hx….formal/change.main.What changes
v1.5's
ridge_capHyp_normalizedstill takes the slice strong concavity (hconc) andδ > rm(8m − 5)as inputs. SP derives both in L3–L4 from its hypothesis (H1) on the transverse Hessian atM. v1.6 proves that derivation (7 declarations):strongConcaveOn_of_hessianK,HasFDerivAt φ (Dφ y) y,HasFDerivAt Dφ (H y) yandH y v v ≤ −δ‖v‖²giveStrongConcaveOn K δ φδ-strong concavity fromD_y² f ≤ −δI)hessian_incrementmonx ↦ H(x, 0)(∂_x D_y² f) and ons ↦ H(x, s • y)(D_y³ falong rays,≤ m‖y‖) give `‖H(p) − H(M)‖ ≤ m(transverse_hessian_leH(M) v v ≤ −λ‖v‖²and those increments on the cap giveH(p) v v ≤ −(λ − 5rm)‖v‖²D_y² f(p) ≤ D_y² f(M) + 5rm · I)slices_strongConcavey ↦ f(x, y)is(λ − 5rm)-strongly concave onB̄(0, 2r), forx ∈ [−2r, r/2]δ = λ − 5rm)ridge_capHyp_H1hconcandhδreplaced by the slice derivatives and Hessians on the cap, the third-derivative bounds ofhessian_increment,H(M) ≤ −λand (H1)8rm² < λh1ToyF,h1Toy_capHypx³/3 − x/4 − 20y²(constant Hessian−40,λ = 40 > 32, gapr³/6) meets every hypothesisI3 after v1.6. The interface narrows to:
w'bound;h;‖h'‖andF''bounds;mthat L1, L3 and L5 consume (L3's increment step itself is proved).SP's (H2)
rn ≤ 1/20feeds only L4'sf_xxx ≥ 7/4and L12, soridge_capHyp_H1does not need it.Unchanged.
CapFirstExit.lean.Validation (local, at
1cbbbf9)gate.py(source mode): PASS, 81 targets.MANIFEST.jsonSHA-2561fab65a044a68ff89f343670b27870a3e6489a267fe02d9372e5a8b62666bfb3.-B -Sand in-B -O -S.gate.py --execute: PASS.v4.34.1, fresh build andleancheckerreplay.propext,Classical.choiceandQuot.sound.eee0017, and 4/4 green again at1cbbbf9. The PR is mergeable. Codex P2 4174695678 (L3 scope) is fixed byhessian_increment, and the thread is resolved.Completed review
1cbbbf9, OpenAI / Codex C120, nonauthor and non-Claude (pickup 5973237299).h1ToyFparagraph and the revised I3.1fab65a0….b21ab2fb6932491f62c2aeb1f419994a5fcf3111(5977001168).2820fc2at 05:37Z, and GitHub retargeted this PR tomain.b21ab2fmergesmain2820fc2into the reviewed head1cbbbf9with a merge commit; no history was rewritten.7988a283…is identical to1cbbbf9's. The workflow blobd2bf2186…ismain's. The diff frommainis the five packet files.--execute(axioms limited to the allowed three; the three negative controls rejected).57ede4ea045cc35923c6d90cfcd028958c6db105from headb21ab2f. The landed packet tree7988a283…equals the reviewed1cbbbf9. Scientific effect NONE; OBL stays OPEN.Review requested (now delivered)
A nonauthor, non-Claude delta alignment read under ALIGNMENT §5, covering:
strongConcaveOn_of_hessiantoridge_capHyp_H1);h1ToyFparagraph of §3;It is best done after, or together with, the v1.5 read on #253.
Points worth checking:
hessian_incrementstates they-derivative bound along rays (‖D_y³ f[y]‖ ≤ m‖y‖). The pinned Mathlib does not synthesize the norm on the triple-nested operator space. Check this form against L3'ssup‖D_y u‖ ‖Δy‖.δ = λ − 5rm > rm(8m − 5).This does not close Math-#193.
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https://claude.ai/code/session_017Mi3hxjaxV45x6zo6o1ee3
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Generated by Claude Code