Complete dual-implementation enumeration of all degree-preserving q=2 two-point exchanges between two blocks of the hash-bound seed-5 U=5 state.

degree-preserving constructive local search

Swap two exclusive points in each direction between two blocks, preserving all point degrees, then fully rescore and canonicalize resulting block families.

Hypothesis: At least one legal q=2 two-block exchange from the hash-bound U=5 state has uncovered-triple deficit at most 4.

Test: Enumerate and fully rescore every legal q=2 exchange, canonicalize resulting families, and independently reconstruct the complete scored neighborhood with integer masks.

Two materially different exhaustive implementations agreed on the complete scored neighborhood and survived relabeling, ordering, corruption, and replay controls. This certifies the local exclusion but cannot be extrapolated beyond the fixed state and move definition.

• The frozen source has no directly improving legal degree-preserving q=2 two-block exchange.
• Its q=2 neighborhood contains exactly 11626 unique nonidentity block families.
• The unique-neighbor deficit histogram has minimum 5 with multiplicity one and contains no cover.
• No global conclusion about C(15,6,3) follows.

• python3 scripts/audit_degree_u5_q2_v1.py artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --receipt artifacts/epoch14-20260809/degree_u5_q2_receipt.json
• python3 checkers/check_degree_u5_q2_v1.py artifacts/epoch13-20260809/degree-local-search-run-v1/result.json artifacts/epoch14-20260809/degree_u5_q2_receipt.json --receipt artifacts/epoch14-20260809/degree_u5_q2_checker_receipt_final.json
• Deterministic /tmp replay reproduced both receipt files byte-for-byte.
• Manifest verification recomputed every listed SHA-256 and checked the histogram and count identities.

• .proof-experiments/20260809-005059-8d47da: primary full-set enumeration completed in 3.904 seconds and found minimum deficit 5.
• .proof-experiments/20260809-005258-4d616d: independent bit-mask reconstruction and relabeling control completed in 17.946 seconds and matched exactly.

checkers/check_degree_u5_q2_v1.py uses integer block masks and direct scans of all 455 triples, unlike the primary frozenset/triple-union implementation; all controls passed.

not_requested

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

Original model outcome

progress

Public classification

progress

• Exact-degree block trades -> use balanced exchanges to remain inside the forced degree-12 slice -> complete q=2 enumeration found no direct improvement.
• Canonical state deduplication from graph search -> distinguish syntactic moves from resulting families -> 15146 legal operations collapsed to 11626 unique nonidentity states.

• The q=2 transformation A,B -> A-X+Y,B-Y+X preserves every point degree.
  Each point in X loses one incidence in A and gains one in B; each point in Y changes oppositely; all other incidences are unchanged. · Every defined q=2 exchange. · proved
• The source state's complete q=2 neighborhood has 11626 unique nonidentity families.
  Matching exhaustive receipts and canonical scored-neighborhood SHA-256 30b2929f16d362f1bdca6c86df583ee3a2e5c97969ba155a4d230b2f38184828. · Source SHA-256 c6667e129e912f0c68d7f72a479266b16042bb1872a489973514ed93c014b10f. · computed
• No unique q=2 neighbor of the source has deficit below 5.
  Both implementations found histogram minimum 5 with multiplicity one. · All 11626 unique nonidentity q=2 neighbors. · computed

• Directly improve the frozen seed-5 U=5 state using one legal degree-preserving q=2 two-block exchange.
  All 15146 legal operations and all 11626 unique nonidentity resulting families. · The exact minimum neighbor deficit is 5, equal to the source deficit. · artifacts/epoch14-20260809/degree_u5_q2_receipt.json and artifacts/epoch14-20260809/degree_u5_q2_checker_receipt_final.json · Change the starting family, allow a path through equal or worse states, or use three or more blocks.

• Two-step q<=2 plateau escape through the unique q=2 tie state.
  It is deterministic, costs seconds, and directly tests whether the certified local optimum can be escaped without accepting a worse deficit. · Run complete q=1 and q=2 audits from best_neighbor_blocks in the primary q=2 receipt. · high · open
• Degree-preserving three-block trades.
  They cross the exhausted direct two-block neighborhoods while retaining the forced degree slice. · Define a canonical balanced three-block move and measure its exact or sampled branching factor before search. · normal · open
• Exact SAT/CP replacement shells around the U=5 state.
  A near-cover with only five missed triples may give small proof-oriented radius shells. · Compile and independently audit exact replacement radii 2 through 6, comparing preprocessed dimensions before solving. · normal · open

Objective: Determine whether the unique q=2 U=5 tie state exposes a two-step route to deficit at most 4.

First action: Implement scripts/audit_degree_u5_q2_v2.py to accept --start-receipt artifacts/epoch14-20260809/degree_u5_q2_receipt.json --start-key best_neighbor_blocks, then exhaust q=1 and q=2 from that state.

Stop condition: Stop and independently validate on U<=4; redirect to canonical plateau-component enumeration or exact repair if neither neighborhood improves.

• Audit q=1 and q=2 neighborhoods of the unique U=5 tie family listed as best_neighbor_blocks in the primary receipt.
• If no two-step improvement exists, enumerate the canonical U<=5 q<=2 plateau component under a predeclared cap or test degree-preserving three-block trades.
• Compile exact SAT or CP replacement shells around the original U=5 state as the proof-oriented alternative.
• Stop immediately on U=0 and validate the 30-block list with independent Python and C cover checkers.

• 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 acted as principal investigator. A GPT-5.6 Terra experiment-verification delegate supplied advisory route triage and the predicted legal-operation count; Sol independently audited and implemented the result. Deterministic tools were Python 3.12.3, itertools/frozensets, integer bit masks, SHA-256, the computational-researcher experiment wrapper, Git read-only inspection, and web source/search retrieval. No SAT solver, CAS, or proof assistant was used; model agreement was not validation.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1087.9s

Review state

not a result claim

Attempt ID

covering-c1563-20260809-010012-4e3ba3

No human review recorded.