Audit (no κ): PR3259's P104 fixed-cover support cut is valid but loose; under PR3427's one-helper criterion 2,682 of 4,992 required entries are unsupported and rank(U−I) ≥ 23

[thakrarsagar]

Statement

PR3259's path-embedding argument is a valid necessary support test under its fixed-cover contract, taken in its pooled form (K1-K7: fixed pooled cover through L/A/B/H, monotone helpers, equal-frame mixing, immutable X and unread Y on their frozen frame paths, no other source-dependent state, independent dirt). PR3252's PROOF.md is also valid. The contract is closed, and by much more than one coefficient:

• Under PR3427's one-recombined-helper criterion (re-proved here in every dimension case; exact only for the pooled untimed relaxation), 2,682 of the 4,992 required entries of C = I - G/2 are unsupported, not just C[71,102] = -1/2. Of these, 1,801 are -1/2 and 881 are +1/2. The supported pairs are exactly the 2,309 compatible slot pairs read as port pairs, plus (73,83), so 2,682 = 4,992 - 2,309 - 1.
• The missing entries contain a perfect matching (minimum vertex cover 104; at least 14 per row and 9 per column). A reopening that changes only individual X/Y frame paths, keeping the other hypotheses, must change at least 104 paths. This is a support-level count, not a cost claim.
• Under PR3427's preconditioner contract, rank(U-I) >= 23 (PR3427 published 4). The bound comes from a maximum unsupported 23x23 principal block with det(2C[S,S]) = 2^30. Rectangles are capped at 31 because rank(G/2) = 8.

Credit and what is new. PR3427 already states the one-helper criterion, its untimed exactness and its converse. Its four published masks already exclude 92 required entries that PR3259's test grants. New here:

• the step-by-step review of PR3259 (checklist a-h) and PR3252;
• the criterion's full 104-column count with its closed form;
• a multi-hop PR3252 full-memory computation over all 104 columns and 2,309 types that does not assume the lemma (PR3427 ran the same operations on its four columns);
• the rank-23 and vertex-cover consequences;
• five single-relaxation rows over three hypotheses.

Milestone and decision informed

Declared milestone: audit, meaning a scoped proof review plus a finite certificate and small-domain checks. It answers revision-6 card p104-side-correction and the live support map's request to review #3259's proof and assumptions before the fixed-cover route is treated as closed.

Decision: the director can treat the fixed-cover / frozen-path / untouched-channel contract as closed. One-coefficient fixes cannot complete the correction under it, whether a rank-one shear, a moved helper or one extra dependency. Any reopening must change at least one of K1-K7 or add storage, and must support all 4,992 required entries (in particular the 2,682 missing here) with a full bill.

Load-bearing hypotheses

All rows are support level, unpriced, with no coefficients or dirty restoration.

Relaxed hypothesis	Reaches (71,102)	Reopens, of 2,682
Cross-layer nesting, as PR3259's proof drops it (a proof relaxation)	no	2,681
K5, correction-only Y relay, keeping each target's write-before-read order (R1)	no	0
R1 without that order (R1u, a looser test)	yes	2,682
K5, the channel's own Y content readable (R1c)	yes	2,682
K1, free nested flags not tied to L/A/B/H (R2)	yes	2,682
K6, reads of old channel-helper state (R3)	yes	2,682

Not computed as rows: K2 descent, K3 cross-frame mixing, K4/K5 path changes and source mutation, K7 dependent dirt, and extra storage. The small-domain controls show each of descent, cross-frame mixing, off-cover flags and correlated old state can escape both tests.

Base and pins

• Base: canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6 (unchanged at submission).
• SOURCES.json pins three inputs by git blob and SHA-256: research/p104-joint-unit-flags/CERTIFICATE.json (PR2981) and research/p104-native-channel/inputs/{LOWRANK-WITNESS,P104-FIXTURE}.json.
• The diff adds only research/p104-fixed-cover-cut-review/ (7 files). Nothing in upstream/, certificates/ or frozen packages is touched.

Reproduce

Standard library only, about 1.5 min plus about 1 min.

python3 -I -O -Werror -B research/p104-fixed-cover-cut-review/verify.py
python3 -I -O -Werror -B research/p104-fixed-cover-cut-review/test_verify.py

Expected last line of verify.py:

PASS_P104_FIXED_COVER_CUT_REVIEW layered_missing=1 sharp_missing=2682 of 4992 pr3427_masks=4 rank_U_minus_I>=23 vertex_cover=104 rows=5 oracle_instances=9000 oracle_programs=18000 no_kappa

test_verify.py ends with Ran 7 tests ... OK. The registered replay, python3.12 -B scripts/research_package.py --package research/p104-fixed-cover-cut-review, gave PASS_REGISTERED_RESEARCH_PACKAGE outcome=completed. --check-only gives PASS_PACKAGE_METADATA_AND_PINS.

Certificates and negative controls

Checks in verify.py:

• An exact Fraction run of the 4,058 channel events: every Y increment equals G/2 in sorted-port order, dirt cancels, and every add is at equal frames.
• The PR3259 layered test rebuilt by literal d-flat hosting: missing set {(71,102)}, forward counts 16,25,52,365,1725,2283,2285,2297, backward cone 37, disjoint.
• The sharp support computed by one criterion in two loop orders plus an independent PR3252-style multi-hop recurrence, all equal on all 10,816 pairs.
• PR3427's four published masks reproduced exactly.
• Kuhn matching of the missing entries, size 104.
• Maximum independent set of size 23 by two exact searches. Rank 23 by Fraction elimination; Bareiss gives det(2C[S,S]) = 2^30.
• Small domains: 9,000 seeded F_2^3/F_2^4 instances, each with an exhaustive untimed reachability that does not assume the lemma. Pooled instances: it equals the one-helper test on all 4,500. Unpooled: 33 escapes on 26 instances, so the lemma needs pooling. Every reachable helper-only pair (4,264 + 8,978) is realized as an explicit program in a checked interpreter.
• Random programs: 18,000 seeded legal programs, a soundness sample. 9,000 are directed and run in the interpreter; 9,000 are uniform and legal by construction.
• Contract violations: six in-verifier controls (Y read, descent, cross-frame mixing, off-cover flag, correlated old state, unpooled two-type relay), each producing its intended escape.

Defect controls in test_verify.py. Each must fail with its own FAIL message: tampered input hash, unsound oracle test, too-tight exhaustive search, and wrong rank block (a port added or removed). Three fact checks cover shared-type drop, the containment-free over-grant and the Y-relay escape.

Scope, assumptions and unresolved obligations

• Inherited: PR2981 covers and the native channel frame paths from the pinned JSON; the E8 port order, bound to the channel by the exact run; and contract K1-K7, where K1 is PR3259's pooled relaxation of its matching contract, so the exclusions hold a fortiori for each matching. Chronology between different registers is relaxed, as in PR3259, PR3252 and PR3427.
• Not addressed: cross-register chronology, coefficient delivery, dirty restoration, native lowering, precision, physical hosting, consumer cost, supplier admission, any changed-cover or changed-path construction, and a timed refinement.
• 23 is optimal among principal blocks but only a lower bound overall. A min-rank completion over the 8,506 unsupported pairs was not computed.
• Not established: no construction, no κ change, no change to the selected frontier, no P104 impossibility outside K1-K7, and no complete multiplication theorem. The selected cold replay was not rerun.

Search record

• Maximum independent set: colouring branch and bound plus pivoted Bron-Kerbosch, exhaustive on the 104-vertex support graph.
• Matching: Kuhn's algorithm, exhaustive.
• Oracle: seed 20261018, 3,000 F_2^3 and 1,500 F_2^4 instances per family; coverage is these instances only.
• No matching search, construction or pricing.

Credit and concurrent work

• Nick Petalas (petalas): PR2981 3d24b92d, PR3252 4784b71d, PR3259 b572b0c2, PR3013 504dc9a1.
• dcharb78: PR3182 ad848966.
• commdata2338: PR3214 faa63233.
• Jose Manuel (jmbarrancoml): PR3427 344774ba, for the criterion, its exactness, the masks and the rank-four theorem extended here.
• James Reed (jwr404): the merged Recompute the required X102 coefficient in Y71 #3497-Show the literal recurrence misses the same required pair #3519 series, which reproduced the cone counts.
• Wentao Li (VictorLiwentao): the channel-touch restoration audit.
• The native/compiler lineage credited in the pinned inputs.

Concurrent open work on the same card is cited with heads in the README and not reviewed here. Each of these changes something K1-K7 freeze, so it does not conflict with this closure:

• VictorLiwentao: Research: flag-43 I−G/2 side correction at #3508 co-hosted cut costs +2 rank mass (no new kappa) #3601, Research: flag-1 I−G/2 side correction at #3532 charts costs +0 on R1 and +2 on R0 (no new kappa) #3610, Research: flag-9 I−G/2 end-cut side correction attains +2 rank mass (no new kappa) #3618 (T48 corrects a whole row on one target lift for +2 rank mass), Research: flag-1 head-step repair co-hosts at update 2797 for exactly two added rank units (no new kappa) #3532, Research: present-head I−G/2 forcing census — 18/18 checked entries need +2 (no new kappa) #3614, Research: present-head I−G/2 full-row forcing census — 144/144 need +2 (no new kappa) #3621, Research: present-head flag 0 Y34 span/dirty-restoration — arrive@3500, 15/15 head-free (no new kappa) #3669.
• Clear1nsight: Realise the P104 omitted completion at the actual entrance #3609.
• jwr404: Index the two P104 replays the live map omits #3580.

#3803 (petalas, opened at submission time) re-derives the p104-cone-cut receipt and its schedule sensitivity. That is a different relaxation, and this package does not depend on it.

Claim: #2245 (comment)

Review provenance

Reviewed by 10 independent AI review passes in the same loop, over three rounds: reproducibility, mathematics, provenance and referee in round 1; mathematics, reproducibility and referee in rounds 2 and 3. The author addressed every finding, and before submission re-checked the round-3 fixes by rerunning the verifier and tests and recomputing rank(G/2) = 8 and the 32 non-own-type routes. No independent human or formal-kernel review has been done.

Attribution

Prepared by Sagar Thakrar (@thakrarsagar) with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5) via Claude Code, in an automated research loop. Apache-2.0.

🤖 Generated with Claude Code