Exhaustive degree-preserving replacement of three blocks around the fixed hash-bound source U=5 anchor, preceded by sound removed-union and missed-triple assignment filters.

exact radius-three replacement enumeration

Remove each triple of anchor blocks, use point-degree multiplicities to restrict three replacement blocks, derive the third block from two canonical masks, and score all residual triples exactly.

Hypothesis: At least one exact degree-preserving replacement of three blocks in the fixed source U=5 anchor yields a 30-block cover.

Test: Exhaustively enumerate every surviving deletion/replacement case and accept only a directly rescored zero-deficit family; independently reconstruct the result through recursive point-to-slot assignments.

Every possible cover in the scoped shell must pass the two necessary filters. The primary then enumerates every canonical degree-feasible replacement, while the checker independently reconstructs the same universe through recursive slot assignments. Their exact agreement and successful controls support the local exclusion, but its fixed-anchor quantifier prevents promotion to a global covering-number result.

• For the fixed source anchor, exactly 405 of 4,060 deletion triples pass the removed-union gate and exactly 33 pass the missed-triple assignment-capacity gate.
• Those 33 cases contain 296,331 canonical degree-feasible replacement triples, of which 294,624 do not collide with the fixed 27 blocks.
• The minimum deficit among all 294,624 admissible operations is five; none improves the source and none is a 30-block cover.
• The fixed source anchor has no improving degree-preserving replacement involving at most three blocks.

• python3 scripts/enumerate_degree_u5_radius3_v1.py --source-result artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --out artifacts/epoch17-20260809/degree_u5_radius3_primary_receipt.json
• python3 checkers/check_degree_u5_radius3_v1.py --source-result artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --primary artifacts/epoch17-20260809/degree_u5_radius3_primary_receipt.json --out artifacts/epoch17-20260809/degree_u5_radius3_checker_receipt.json
• python3 scripts/control_degree_u5_radius3_v1.py --source-result artifacts/epoch13-20260809/degree-local-search-run-v1/result.json --primary artifacts/epoch17-20260809/degree_u5_radius3_primary_receipt.json --checker artifacts/epoch17-20260809/degree_u5_radius3_checker_receipt.json --out artifacts/epoch17-20260809/degree_u5_radius3_controls_receipt.json
• Primary receipt sha256=26a10c955b48cbd9e0d0a4bf260e65e60875f9ffe34f82cde72b676e3042d767
• Checker receipt sha256=e29ef05e82103c0dbd86b865bbae8d7142929bbaee3297fb1af940e297d35f9f
• Controls receipt sha256=f095b944e87c9eac69a078ef0075bdff6f44b07a81e783186d481432db478fe8

• .proof-experiments/20260809-025258-a35fb5: primary completed in 6.964 seconds; 4,060 -> 405 -> 33, 294,624 admissible operations, minimum deficit 5, zero covers.
• .proof-experiments/20260809-025446-fbf637: tightened independent checker completed in 40.559 seconds and matched all cases.
• .proof-experiments/20260809-025534-14eacf: byte-identical replay and two corruption controls passed in 45.696 seconds.

checkers/check_degree_u5_radius3_v1.py uses frozenset coverage and recursive point-to-slot assignments, rather than deriving a third mask from a candidate pair; it matched all 33 cases and the 294,624-operation signature.

not_requested

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

Original model outcome

progress

Public classification

progress

• C(12,6,4) certificate-first exhaustive proof -> require a separately encoded checker for a negative shell -> the recursive checker matched and failed closed on mutations.
• Terra necessary-capacity prefilter -> predict 4,060 -> 405 -> 33 deletion triples -> independently reconstructed exactly before use.

• The source-anchor necessary filters leave exactly 33 deletion triples.
  Matching primary and independent checker receipts · The fixed source family with hash 876ac1f5f8c35285acff196c9e48747ebc98c99a881c9e6948fbaf1883aaf386 · computed
• No degree-preserving replacement involving at most three blocks improves the fixed source anchor.
  Complete enumeration of 294,624 admissible operations, independently reconstructed with minimum deficit five · One fixed labelled degree-12, deficit-5 anchor · computed

• Obtain a 30-cover or lower-deficit family by an exact degree-preserving replacement of at most three blocks from the source anchor.
  Every such replacement with the other source blocks fixed · The complete admissible operation histogram has minimum deficit five and no zero entry. · artifacts/epoch17-20260809/degree_u5_radius3_primary_receipt.json and degree_u5_radius3_checker_receipt.json · A demonstrated source-byte mismatch, checker defect, or a move changing at least four blocks or passing through another state
• Treat the Terra pair-multiplicity-four edge-star suggestion as current evidence for a global exclusion.
  The proposed canonical edge-star certificate route · No independently complete canonical edge-star catalogue or proof-bearing leaf set exists yet. · Only an advisory delegate memo was available and was not used as mathematical evidence. · Produce a hash-bound complete canonical catalogue with independently checked orbit coverage.

• Exact radius-three replacement shell of the tie U=5 anchor
  It is the cheapest remaining same-family test and can reuse both exact implementations. · Add receipt/key-based anchor loading, reconstruct the 4,060 deletion prefilter, and stop if the survivor set is unexpectedly large or a cover appears. · high · open
• Fresh nonisomorphic degree-12 U<=5 constructive anchor
  The present exclusion is local and another near-cover may have different repair geometry. · Run a bounded canonical-family search that stops on a U<=5 family not isomorphic to either existing anchor. · normal · open
• Certified canonical root-link catalogue
  Unlike local repair, a complete catalogue could support a global UNSAT decomposition. · Enumerate at most 10,000 canonical links or run for 30 minutes, with an independent completeness-frontier checker. · normal · open

Objective: Determine whether the tie U=5 anchor has a degree-preserving radius-three replacement yielding a cover or improvement.

First action: Implement scripts/enumerate_degree_u5_radius3_v2.py with --anchor-receipt artifacts/epoch14-20260809/degree_u5_q2_receipt.json --anchor-key best_neighbor_blocks, then run its 4,060-case necessary prefilter.

Stop condition: Stop and validate immediately on deficit zero; redirect if the independent prefilter disagrees, or hold local radius growth after a fully checked negative tie shell.

• Generalize the enumerator and checker to read the tie anchor from artifacts/epoch14-20260809/degree_u5_q2_receipt.json.
• Run the tie-anchor necessary prefilter before allocating exact enumeration.
• If the tie shell is negative, hold further local radius expansion and compare a new nonisomorphic U<=5 anchor search against the canonical root-link catalogue.

• 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 principal designed, implemented, audited, and interpreted the epoch. GPT-5.6 Terra delegates supplied advisory prior-art and experiment-reduction memos; their agreement was not treated as validation, and the relied-on filter was independently reconstructed in the main workspace with provenance recorded. Python 3.12.3 performed exact enumeration, independent checking, hashing, replay, and mutation tests. Web search checked current source status and prior art. No SAT solver, CAS, proof assistant, external publication action, or lab compute was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1283.2s

Review state

not a result claim

Attempt ID

covering-c1563-20260809-030142-21ee97

No human review recorded.