Matched fixed-seed simulated-annealing pilot comparing exact-degree-preserving coupled two-block swaps against equally sized uncoupled two-block controls, followed by exhaustive radius-one auditing of the best pinned state.

degree-preserving constructive local search

Target an uncovered triple, repair it in one block, and couple the second one-point edit so every point degree remains twelve; maintain triple deficits incrementally.

Hypothesis: Under identical targeted two-block one-point-edit search and 100000-proposal budgets, exact degree preservation yields a strictly lower overall best and median uncovered-triple count and wins at least 8 of 12 fixed-seed pairs.

Test: Run seeds 0 through 11 from identical regular starts, compare the best uncovered-triple count in each arm, and apply the frozen overall-best, median, and at-least-8-paired-wins gates.

Every reported best state and metric was reconstructed independently, and the fixed-seed comparison decisively favors the pinned mechanism within the checked protocol. However, a heuristic sample and a local-neighborhood certificate neither exhibit a cover nor exclude all 30-block families, so the result is internal validated progress rather than a candidate exact optimum.

• For seeds 0 through 11 and 100000 proposals per arm, the pinned best deficits were [13,12,13,10,11,5,7,10,13,11,14,10] and the controls were [15,13,17,19,16,12,13,20,14,21,16,16].
• The pinned arm won all twelve fixed-seed pairs, with median deficit 11 versus 16 and best deficit 5 versus 12.
• The seed-5 pinned state has all point degrees twelve and misses exactly {1,7,13}, {3,6,13}, {6,8,10}, {7,10,14}, and {8,13,14}.
• Among all 6276 legal distinct q=1 coupled swaps from that state, the minimum neighbor deficit is five and there are no strictly improving neighbors.
• No conclusion about the global existence of a 30-block cover follows.

• python3 scripts/degree_preserving_local_search_v1.py --proposals 100000 --seeds 12 --burnin 5000 --output artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --checkpoint artifacts/epoch13-20260809/degree-local-search-run-v1/checkpoint.json --progress artifacts/epoch13-20260809/degree-local-search-run-v1/progress.json
• python3 checkers/check_degree_preserving_local_search_v1.py artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --receipt artifacts/epoch13-20260809/degree_local_search_checker_receipt.json
• python3 scripts/audit_degree_u5_radius1_v1.py artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --receipt artifacts/epoch13-20260809/degree_u5_radius1_receipt.json
• python3 checkers/check_degree_u5_radius1_v1.py artifacts/epoch13-20260809/degree-local-search-run-v1/result.json artifacts/epoch13-20260809/degree_u5_radius1_receipt.json --receipt artifacts/epoch13-20260809/degree_u5_radius1_checker_receipt.json
• Three corruption controls altered a score, duplicated a block, and altered the summary; all were rejected.

• .proof-experiments/20260809-001720-1f7848: 24 search cells completed in 102.004 seconds; pinned passed all three gates and reached U=5.
• .proof-experiments/20260809-001924-efaa7c: independent checker reconstructed all 24 best states and comparative metrics.
• .proof-experiments/20260809-002218-99618c: set-based audit checked 6276 legal radius-one swaps and found neighbor minimum 5.
• .proof-experiments/20260809-002231-56dc3b: independent bit-mask audit reproduced the exact radius-one counts.

checkers/check_degree_preserving_local_search_v1.py uses frozenset blocks and itertools triple unions instead of the search's cached bit-mask incidence table. checkers/check_degree_u5_radius1_v1.py independently reconstructs the local neighborhood 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

• Slack-zero exact pinnability from the C(12,6,4) certificate campaign -> enforce point degrees as equalities through coupled edits -> pinned search won all twelve matched pairs.
• Classical simulated-annealing covering construction -> target uncovered triples while retaining uphill moves -> reached U=5, but exhaustive radius-one auditing exposed a plateau requiring larger moves.

• The coupled edit A-a+z, B-z+a preserves every point degree.
  Each affected point loses and gains one incidence across the two blocks; all other incidences are unchanged. · Every legal two-block coupled q=1 proposal. · proved
• The pinned arm won 12/12 fixed-seed pairs with median 11 versus 16 and best 5 versus 12.
  Hash-bound 24-cell result and independent checker receipt. · Seeds 0 through 11, 100000 proposals per arm, the exact v1 protocol. · computed
• The seed-5 pinned state has degree vector (12,...,12) and exactly five uncovered triples.
  Independent frozenset/itertools reconstruction of all 455 triples and fifteen point degrees. · The block family stored in result SHA-256 c6667e129e912f0c68d7f72a479266b16042bb1872a489973514ed93c014b10f. · computed
• The U=5 state has no improving legal distinct q=1 coupled swap.
  Independent set and bit-mask enumerations agree on 6276 moves, neighbor minimum 5, and one minimum neighbor. · The complete radius-one coupled neighborhood of the hash-bound seed-5 state. · computed

• Directly improve the seed-5 U=5 state with one legal coupled one-point two-block swap.
  All 6276 legal distinct q=1 swaps from that exact state. · Both exhaustive encodings found neighbor minimum five and zero strictly improving neighbors. · artifacts/epoch13-20260809/degree_u5_radius1_receipt.json and degree_u5_radius1_checker_receipt.json · Change the starting state, permit q>=2, use three or more blocks, or allow a path through equal/worse states.

• Coupled q=2 and three-block degree-preserving moves
  They preserve the empirically useful degree slice while crossing the certified q=1 plateau. · Implement incremental q=2 exchanges and a bounded matched pilot from fresh and U=5 starts. · high · open
• Exact SAT or CP repair around the U=5 state
  A hash-bound near-cover can define finite Hamming-distance cubes with direct witness certificates. · Encode radius-2 through radius-6 block replacements and compare preprocessing dimensions before solver scale-up. · normal · open
• Materially revised incidence proof route
  It remains the path to a certified negative result, but prior cold-branch gates failed and the canonical excess frontier is incomplete. · Proceed only after an audited proof-prefix reuse mechanism, stronger encoding, or complete canonical frontier is supplied. · low · open

Objective: Determine whether larger exact-degree-preserving exchanges can escape the U=5 plateau and produce a 30-block cover.

First action: Implement scripts/degree_preserving_local_search_v2.py with coupled q=2 exchanges and degree-preserving three-block trades, retaining the v1 checker interface.

Stop condition: Stop on a dual-validated U=0 witness; redirect to exact repair or incidence SAT if the matched pilot neither beats U=5 nor materially improves the deficit distribution.

• Implement coupled q=2 exchanges and degree-preserving three-block trades with incremental triple updates.
• Run a short matched fixed-seed pilot from both fresh regular starts and the hash-bound U=5 state.
• Stop immediately on U=0 and validate the block list with the existing independent Python and C cover checkers.
• If larger moves do not beat U=5, compile an exact SAT or constraint-programming repair model around the U=5 state.
• Reopen global incidence UNSAT work only after a material encoding, canonical frontier, or proof-prefix reuse improvement.

• 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 principal investigator. A GPT-5.6 Terra experiment-design delegate supplied the advisory 12-pair/100000-proposal/three-gate template; Sol independently designed the precise moves, implemented the programs, ran the efficiency pass, and interpreted the evidence. A challenger Terra memo was not used as evidence. Delegate agreement was not validation. Deterministic tools were Python 3.12.3, SHA-256, and the computational-researcher experiment wrapper. No CAS, SAT/SMT solver, proof assistant, or proof-log checker was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1681.4s

Review state

not a result claim

Attempt ID

covering-c1563-20260809-002810-944f57

No human review recorded.