Exhaust the genuinely new q=3 degree-preserving two-block exchanges from both certified U=5 plateau anchors, using complement equivalence to remove operations already covered by q<=2 receipts.

degree-preserving local exchange enumeration

Complement-reduce q=3 exchanges to disjoint block pairs, canonicalize labelled families, and rescore every surviving family against all 455 triples.

Hypothesis: At least one genuinely new q=3 two-block exchange from either certified U=5 anchor has deficit at most 4.

Test: Enumerate every disjoint-pair q=3 exchange from both anchors, fully rescore each unique family, and independently reconstruct the full q=3 space while checking complement equivalence operation by operation.

The primary and materially independent implementations agree on all counts, histograms, representatives, and scored-family digests. The checker also validates the reduction over the unreduced space and binds the prior component receipt, so the negative claim is complete for the stated local scope.

• For an unordered pair of six-blocks with exclusive-set size d, every q-exchange has the same result as its complementary d-q exchange.
• The two anchors have exactly four disjoint block pairs total and exactly 788 unique new q=3 neighbor families.
• The minimum deficit among those 788 families is 13, attained twice; none has deficit at most 4 and none is a cover.
• Neither anchor admits a one-step q<=3 improvement.
• Neither anchor admits a degree-preserving two-block replacement yielding a 30-cover.

• python3 scripts/audit_degree_u5_q3_v1.py --source-result artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --tie-receipt artifacts/epoch14-20260809/degree_u5_q2_receipt.json --component-receipt artifacts/epoch15-20260809/degree_u5_plateau_component_receipt.json --receipt artifacts/epoch16-20260809/degree_u5_q3_receipt.json
• python3 checkers/check_degree_u5_q3_v1.py --source-result artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --tie-receipt artifacts/epoch14-20260809/degree_u5_q2_receipt.json --component-receipt artifacts/epoch15-20260809/degree_u5_plateau_component_receipt.json --primary-receipt artifacts/epoch16-20260809/degree_u5_q3_receipt.json --receipt artifacts/epoch16-20260809/degree_u5_q3_checker_receipt.json
• Primary receipt SHA-256 1504a9fc5ef12bfce59d6468aff6b4b60bf2a5d816d788e4f58975f3a42c090c
• Checker receipt SHA-256 020cd21ef335cdb5df4e16c9066409b3858d0b8ab04f64f63e735aaa901af327
• Combined scored-family digest 84fc3b9a3538ac55286ea14a3a2d55342a39b46af0b84e5c88a7522816b70807

• .proof-experiments/20260809-021215-20743a: primary PASS, raw_full=20475, raw_reduced=1600, union=788, best_u=13
• .proof-experiments/20260809-021322-94551c: checker PASS, complement equivalence and controls PASS, union=788, best_u=13
• .proof-experiments/20260809-021131-5fa58e: failed-closed control exposed a tied-representative ordering mismatch; no score, histogram, or digest disagreement

checkers/check_degree_u5_q3_v1.py uses integer masks and direct containment scans, reconstructs the full q=3 operation space, and does not reuse the primary frozenset scoring implementation.

not_requested

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

Original model outcome

no_progress

Public classification

no_progress

• Complement duality for bipartition exchanges -> predict that non-disjoint q=3 moves duplicate q<=2 -> operation-wise audit confirmed this and removed 92.186% of raw operations.

• q and d-q two-block exchanges are equivalent as unordered results.
  Algebraic proof plus full operation-wise checker audit. · All unordered two-block exchanges. · proved
• The source anchor's genuinely new q=3 shell has 394 families and minimum deficit 13 with multiplicity two.
  Matching primary and checker receipts. · All disjoint-pair q=3 exchanges from the source anchor. · computed
• The tie anchor's genuinely new q=3 shell has 394 families and minimum deficit 17 with multiplicity nine.
  Matching primary and checker receipts. · All disjoint-pair q=3 exchanges from the tie anchor. · computed
• No one-step q<=3 move improves either U=5 anchor.
  Hash-bound composition of epoch-15 q<=2 component and epoch-16 q3 checker receipts. · The two fixed labelled anchors. · computed
• Degree-preserving replacement of two blocks is exhausted by q<=3 exchanges.
  Multiplicity argument in q3_complement_reduction_lemma.md. · Two-block replacement with all other blocks fixed. · proved

• Escape either plateau anchor to deficit at most 4 using one q=3 two-block exchange.
  All 20,475 raw q=3 operations and all 788 genuinely new labelled families. · The new-shell minimum is 13; non-disjoint operations reproduce already checked q<=2 families. · artifacts/epoch16-20260809/degree_u5_q3_checker_receipt.json · Change the anchor, permit a path through worse states, or replace at least three blocks.
• Find a 30-cover in the exact degree-preserving radius-two replacement shell of either anchor.
  All replacements of exactly two blocks with the other 28 fixed and all point degrees preserved. · Every such replacement is a q<=3 exchange, and the composed exact shells contain no cover. · artifacts/epoch16-20260809/q3_complement_reduction_lemma.md and degree_u5_q3_checker_receipt.json · Use another anchor, replace at least three blocks, or relax degree preservation.

• Exact radius-three replacement SAT/CP around both U=5 anchors
  It is the smallest unexamined degree-preserving repair shell and can return either a direct cover or proof-bearing local exclusions. · Compile both radius-three models, independently audit semantics, and compare preprocessed dimensions. · high · open
• Certified canonical root-link catalogue plus extension instances
  It is materially different from local repair and targets a global symmetry-reduced exhaustive frontier. · Run a 10,000-orbit or 30-minute canonical-link frontier pilot with independent reconstruction and stabilizer checks. · normal · open
• Fresh degree-12 constructive anchors
  The local exclusion applies to only two states; another near-cover could have a different repair shell. · Run a bounded fixed-seed search that stops only on a new U<=5 family not isomorphic to the existing anchors. · normal · open

Objective: Determine whether either anchor has an exact degree-preserving radius-three replacement yielding a cover.

First action: Implement scripts/build_degree_u5_radius3_cnf_v1.py using the two receipt-bound anchors, fixed 27-block complements, exact point degrees, distinct replacement blocks, and all 455 coverage clauses.

Stop condition: Stop and directly validate on SAT; record only proof-replayed UNSAT; redirect if semantic audit fails or preprocessing gives no material reduction.

• Implement exact radius-three replacement models around both anchors, holding the other 27 blocks fixed.
• Independently audit degree preservation, distinct-block semantics, and triple-coverage equivalence before solving.
• Compare preprocessed dimensions and run only short proof-producing pilots.
• Redirect to a certified canonical root-link catalogue if radius-three preprocessing or proof throughput is poor.

• 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/2607.23766 ↗

OpenAI GPT-5.6 Sol principal designed, audited, implemented, and interpreted the epoch. GPT-5.6 Terra delegates supplied advisory prior-art and experiment memos; their agreement was not treated as validation, and no delegate artifact supports the mathematical claim. Deterministic Python 3.12.3 programs performed exact enumeration and independent checking. No SAT solver, CAS, proof assistant, external publication action, or untracked long-running compute was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1320.0s

Review state

not a result claim

Attempt ID

covering-c1563-20260809-021920-52db42

No human review recorded.