Complement-close the q=3 two-block exchange shell of the fixed seed-6 U=2 anchor and compose it with independently checked q=1 and q=2 shells.

fresh canonical constructive starts

Audit the anchor's block-pair intersection profile, map every q=3 exchange operation-by-operation to its complementary q<=2 exchange, and independently rescore all valid q=3 families.

Hypothesis: The fixed seed-6 U=2 anchor has no genuinely new q=3 two-block exchange because it has no disjoint block pair.

Test: Reconstruct all 435 block-pair intersections and all 9,882 q=3 operations, require each operation to be an identity or exactly complement-equivalent to a checked q=1/q=2 operation, and require zero d=6 operations.

The complement equivalence reduces every two-block replacement to q<=floor(d/2). The audited anchor has d<=5 for every block pair, so q<=2 is complete. Independent exhaustive implementations and fail-closed controls support the local classification, but the computation covers only one labelled anchor.

• The fixed seed-6 U=2 anchor has block-pair intersection histogram {1:66,2:195,3:162,4:12} and no disjoint block pair.
• Every exact-degree two-block replacement of the fixed seed-6 anchor is an identity or belongs to its previously checked q=1 or q=2 shells.
• No exact-degree two-block replacement of this anchor has deficit below 2 or forms a 30-block cover.

• python3 scripts/audit_degree_u2_q3_v1.py with the hash-bound anchor and four q<=2 parent receipts: PASS, 435 pairs, 9,882 q3 operations, zero new operations
• python3 checkers/check_degree_u2_q3_v1.py: PASS with identical counts, histogram, family digest, block-order control, and affine point-permutation control
• python3 checkers/test_degree_u2_q3_fail_closed_v1.py: PASS after rejecting all four mutations
• jq and sha256sum final audit: all manifest entries OK

• .proof-experiments/20260809-083924-9ddddc: primary closure audit passed in 2.543 seconds with 9,882 to 0 genuinely new q3 reduction
• .proof-experiments/20260809-083936-a289da: independent checker passed in 23.631 seconds including order and point-permutation controls
• .proof-experiments/20260809-084047-dc91b6: all four decisive corruptions were rejected

checkers/check_degree_u2_q3_v1.py uses a separately written integer-mask representation, bitwise complement mapping, direct scans of all 455 triples, and internal order/permutation controls.

not_requested

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

Original model outcome

progress

Public classification

progress

• Earlier U=5 q3 complement reduction -> only disjoint block pairs can yield genuinely new q3 moves -> the U=2 anchor has zero disjoint pairs, so its entire apparent q3 shell collapses.

• For two equal-size blocks with exclusive-part size d, q-exchange and (d-q)-exchange yield the same unordered replacement pair.
  artifacts/epoch25-20260809/q3_complement_closure_lemma.md and operation-level checks · All exact-degree two-block replacements of equal-size blocks · proved
• The seed-6 U=2 anchor has pair-intersection histogram {1:66,2:195,3:162,4:12}.
  Primary set and independent bitmask receipts agree · The anchor with source SHA-256 bbcd920a469fc4015612f837adf1d73e7784f0e794cfaa4f0693ddd43c39d6b9 · computed
• No exact-degree two-block replacement of the fixed seed-6 anchor improves deficit 2 or forms a cover.
  Composition of independently checked q=1, q=2, and complement-closed q=3 receipts · All replacements supported on two blocks of the fixed labelled anchor · computed

• Repair the fixed seed-6 U=2 anchor by a q=3 two-block exchange.
  All 9,882 q=3 operations and all 9,696 distinct valid nonidentity resulting families · Every operation is an identity or checked q<=2 neighbor; direct rescoring gives minimum deficit 2 and zero covers. · artifacts/epoch25-20260809/degree_u2_q3_checker_receipt.json · A defect in the source binding, complement proof, primary enumeration, or independent checker
• Repair the fixed seed-6 U=2 anchor by any exact-degree replacement supported on two blocks.
  Every possible repartition of the two selected blocks preserving all point degrees and block size six · Every block pair has exclusive size at most five, so complement equivalence reduces every nonidentity replacement to q=1 or q=2, whose complete shells have no improvement. · Epoch-24 q1/q2 receipts plus epoch-25 closure lemma and checker receipt · A checker defect or a materially different anchor or move supported on at least three blocks

• Coverage-pruned exact three-block replacement around the U=2 anchor
  It is the smallest unclosed exact-degree move class and can coordinate repairs of the two disjoint missing triples. · Independently enumerate deletion triples and fixed-27 coverage/capacity profiles before building a representative three-slot solver pilot. · high · open
• Canonicalized stochastic continuation from distinct U=2 q<=2 plateau minima
  It can leave the now-exhausted two-block neighborhood without requiring a complete three-block classification. · Construct and point-isomorphism-deduplicate the union of q1/q2 deficit-two neighbors, then run matched short continuations. · normal · open
• Proof-producing global incidence SAT/PB decomposition
  It is the route capable of globally excluding 30, but current encodings lack a proof-producing tractability signal. · Only resume after a pinned solver/replayer pair or a material encoding reduction passes a bounded proof-prefix test. · normal · open

Objective: Measure whether exact three-block replacement around the U=2 anchor is tractable after sound coverage and capacity profiling.

First action: Implement scripts/audit_degree_u2_delete3_profiles_v1.py and a separate integer-mask checker, then run them on artifacts/epoch24-20260809/fresh_nonisomorphic_starts_result.json with seed 6.

Stop condition: Stop on any disagreement in deletion/profile coverage; otherwise stop the representative pilot on a witness, 10 million nodes in one case, 60 CPU-seconds in one case, or projected full cost above 12 CPU-hours.

• Implement an independent read-only auditor for all choose(30,3)=4,060 deletion triples, fixed-27 residual coverage, and point-capacity profiles.
• Select deterministic representatives only after two implementations agree on the retained cases and profiles.
• Pilot a canonical three-slot bitmask DFS/CP model with residual-triple propagation, memoization, node counts, prune counts, and strict time gates.
• Scale beyond the pilot only if a materially different checker reproduces every result and measured costs imply a bounded full run.

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

GPT-5.6 Sol was the principal investigator. Two GPT-5.6 Terra delegates supplied advisory challenger and experiment-verification memos; their claims were independently audited and exact memo copies were promoted with provenance. Deterministic Python 3.12.3, custom set and integer-mask programs, SHA-256 tooling, the computational-researcher experiment wrapper, and bounded web search were used. No SAT solver, CAS, proof assistant, or external publication action was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1110.1s

Review state

not a result claim

Attempt ID

covering-c1563-20260809-084824-d8e579

No human review recorded.