Generate fresh noncyclic degree-12 starts by integral bipartite b-matching, apply bounded deficit-guided exchanges, then exhaust q=1 and q=2 two-block repair shells around the best new anchor.

fresh canonical constructive starts

Seed-shuffled min-cost flow generates simple (12,6)-biregular incidence matrices; nauty rejects known or repeated point-isomorphism classes; exact-degree local search minimizes uncovered triples.

Hypothesis: Among 12 fixed noncyclic biregular starts, targeted exact-degree search reaches a deficit-at-most-five family not point-isomorphic to either certified U=5 anchor.

Test: Run 100000 deterministic proposals from each of 12 canonically distinct starts, rescore all stored families independently, and require a nonanchor U<=5 family; stop immediately on U=0.

The U=2 family is directly checkable and materially improves the campaign's constructive state. Independent coverage and graph-isomorphism implementations validated it. The local shell exclusions are exhaustive only for their declared move classes, so they support progress but neither a candidate optimum nor a new lower bound.

• Seed 6 produced 30 distinct 6-subsets with every point in exactly 12 blocks and exactly two uncovered triples: (4,13,15) and (5,11,12).
• The seed-6 U=2 family is not point-isomorphic to either certified U=5 anchor.
• The complete q=1 shell has 6276 raw exchanges, 6252 distinct nonidentity families, minimum deficit 2 attained three times, and zero covers.
• The complete q=2 shell has 15090 raw exchanges, 11539 distinct nonidentity families, minimum deficit 2 attained twice, and zero covers.

• python3 scripts/generate_fresh_nonisomorphic_starts_v1.py --source-result artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --tie-receipt artifacts/epoch14-20260809/degree_u5_q2_receipt.json --seeds 12 --proposals 100000 --output artifacts/epoch24-20260809/fresh_nonisomorphic_starts_result.json
• python3 checkers/check_fresh_nonisomorphic_starts_v1.py --result artifacts/epoch24-20260809/fresh_nonisomorphic_starts_result.json --source-result artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --tie-receipt artifacts/epoch14-20260809/degree_u5_q2_receipt.json --receipt artifacts/epoch24-20260809/fresh_nonisomorphic_starts_checker_receipt.json
• python3 scripts/audit_degree_u2_q1_v1.py --result artifacts/epoch24-20260809/fresh_nonisomorphic_starts_result.json --seed 6 --receipt artifacts/epoch24-20260809/degree_u2_q1_primary_receipt.json
• python3 checkers/check_degree_u2_q1_v1.py --result artifacts/epoch24-20260809/fresh_nonisomorphic_starts_result.json --primary artifacts/epoch24-20260809/degree_u2_q1_primary_receipt.json --receipt artifacts/epoch24-20260809/degree_u2_q1_checker_receipt.json
• python3 scripts/audit_degree_u2_q2_v1.py --result artifacts/epoch24-20260809/fresh_nonisomorphic_starts_result.json --seed 6 --receipt artifacts/epoch24-20260809/degree_u2_q2_primary_receipt.json
• python3 checkers/check_degree_u2_q2_v1.py --result artifacts/epoch24-20260809/fresh_nonisomorphic_starts_result.json --primary artifacts/epoch24-20260809/degree_u2_q2_primary_receipt.json --receipt artifacts/epoch24-20260809/degree_u2_q2_checker_receipt.json
• Hash manifest PASS; manifest SHA-256 4325fd46efbf2c3bdc602a102825f0b8a9046d8cea64a6007f9e3e1244a0c44e

• .proof-experiments/20260809-074829-745d79: 12 fresh starts, 1200000 requested proposals, best U=2 in 53.195 seconds
• .proof-experiments/20260809-074937-94a081: independent 24-family, 66-pair, and anchor-isomorphism audit PASS
• .proof-experiments/20260809-075303-26e79a: q=1 primary shell, 6252 families, no cover
• .proof-experiments/20260809-075319-c9716e: failed diagnostic checker exposed incompatible minimum tie ordering; no mathematical count changed
• .proof-experiments/20260809-075411-444e8d: corrected q=1 checker PASS
• .proof-experiments/20260809-075535-537b02: q=2 primary shell, 11539 families, no cover
• .proof-experiments/20260809-075551-758be1: q=2 checker PASS

Fresh families were checked using frozenset triple unions and NetworkX VF2 rather than cached bit masks and nauty. The q=1 and q=2 shells were independently reconstructed with integer masks and direct scans of all 455 triples.

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome

progress

Public classification

progress

• Degree-constrained graph generation -> seed-shuffled bipartite b-matching should escape cyclic start components -> produced 12 new start classes and a U=2 family.
• Canonical graph generation -> colored incidence canonicalization should prevent false component novelty -> nauty and VF2 agreed that the U=2 family is outside both U=5 anchor classes.
• Local-search plateau auditing -> exact small exchange shells should reveal whether a near-cover is trivially repairable -> q=1 and q=2 both had minimum U=2 and no cover.

• The fixed seed-6 family has 30 distinct blocks, point degrees all 12, and exactly two uncovered triples.
  Fresh result and independent checker receipt · One explicitly stored labelled family · computed
• All 12 generated starts are pairwise nonisomorphic and outside both stored U=5 anchor classes.
  nauty primary keys and independent VF2 checks of 66 start pairs plus anchor comparisons · The fixed 12-seed protocol · computed
• The complete q=1 shell contains 6252 distinct nonidentity families, minimum deficit 2, and no cover.
  Primary and integer-mask checker receipts agree · Fixed labelled seed-6 U=2 anchor · computed
• The complete q=2 shell contains 11539 distinct nonidentity families, minimum deficit 2, and no cover.
  Primary and integer-mask checker receipts agree · Fixed labelled seed-6 U=2 anchor · computed

• Repair the seed-6 U=2 anchor by exchanging one exclusive point between two blocks.
  All q=1 two-block exchanges of the fixed labelled family · The complete shell has minimum deficit 2 and zero covers. · artifacts/epoch24-20260809/degree_u2_q1_checker_receipt.json · A defect in the checker or a materially different anchor or move class
• Repair the seed-6 U=2 anchor by exchanging two exclusive points between two blocks.
  All q=2 two-block exchanges of the fixed labelled family · The complete shell has minimum deficit 2 and zero covers. · artifacts/epoch24-20260809/degree_u2_q2_checker_receipt.json · A defect in the checker or a materially different anchor or move class

• Exact q=3 two-block shell around the U=2 anchor
  It is the next complete local class and should remain computationally small. · Enumerate all exclusive three-subset exchanges with a separate integer-mask reconstruction. · high · open
• Exact three-block replacement model
  The two missing triples are disjoint and may require coordinated repairs beyond any two-block move. · Delete three blocks under a sound missing-triple capacity prefilter and solve the exact degree/coverage replacement constraints. · high · open
• Continue stochastic search from distinct U=2 q<=2 plateau neighbors
  The new component is materially closer than the prior U=5 starts, but cross-q deduplication is required first. · Construct the exact union of U=2 q=1/q=2 minima, quotient by point isomorphism, and run matched short continuations. · normal · open

Objective: Determine whether the U=2 anchor can be repaired by the next-smallest exact move class.

First action: Implement and run `python3 scripts/audit_degree_u2_q3_v1.py --result artifacts/epoch24-20260809/fresh_nonisomorphic_starts_result.json --seed 6 --receipt artifacts/epoch25-20260809/degree_u2_q3_primary_receipt.json`, followed by a separate integer-mask checker.

Stop condition: Stop on a dual-validated U=0 cover; otherwise close only the checked q=3 shell and redirect to a three-block exact replacement model.

• Implement scripts/audit_degree_u2_q3_v1.py and a materially independent checker.
• Cross-deduplicate the U=2 neighbors from q=1 and q=2 before launching continuation searches.
• If q=3 has no cover, compile an exact three-block replacement model centered on the U=2 anchor.
• Retain incidence/PB search as the proof-oriented alternative; do not infer a global exclusion from local shells.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=6&t=3&v=15 ↗
• https://arxiv.org/abs/math/9502238 ↗
• https://arxiv.org/abs/2607.23766 ↗

GPT-5.6 Sol was the principal investigator. GPT-5.6 Terra delegates supplied advisory prior-art and verification constraints, promoted with provenance but not treated as evidence. Deterministic Python 3.12.3, NetworkX 3.3, Debian nauty labelg, the computational-researcher experiment wrapper, and custom frozenset/integer-mask programs produced and checked the artifacts. Web fetch and curl were attempted but returned no usable source content because network resolution was unavailable. No SAT solver, CAS, proof assistant, external publication, or system modification was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1432.6s

Review state

not a result claim

Attempt ID

covering-c1563-20260809-080339-458005

No human review recorded.