Complete exact coverage census of the degree-balanced four-delete/three-add shell around Iorio's maintained 55-block cover

degree-pinned local-shell exhaustive search

Enumerate every canonical three-block degree completion for every feasible four-block deletion header and score the resulting 54-block family against all 455 triples.

Hypothesis: At least one of the 43928064 exact degree-balanced four-delete/three-add candidates covers all 455 triples.

Test: For each retained 51-block mask, accept a canonical addition triple A<B<C exactly when it realizes the header demand, avoids all 55 source blocks, and covers every missing triple.

Every candidate appears exactly once in each materially different enumeration. Both report zero hits and per-header agreement, while the preserved defect-ten family supplies the matching upper witness for the local minimum.

• Exactly zero of the 43928064 exact degree-balanced four-delete/three-add candidates is a C(15,5,3) cover.
• The exact minimum defect in that shell is ten uncovered triples.
• Deleting source blocks 2,3,15,40 and adding {1,2,7,11,12}, {1,3,7,8,14}, and {1,5,11,13,14} yields a degree-18 54-block family covering exactly 445 triples.

• Producer: 33100 headers, 43928064 candidates, zero hits, minimum defect 10 in 11.069 seconds.
• Independent checker: all rows matched, all candidates reconstructed, zero hits, minimum defect 10, four mutations rejected in 3.326 seconds.
• Direct audit: 54 distinct blocks, degrees 18^15, 445 covered triples, and ten explicit misses.
• Source SHA-256 e8188ed441e9548b6e7bb1f8d6d70d0c70007d9ad758157033209f9811f6f5d2; manifest SHA-256 e45fa5d6d41c4ee623482782efdca449dc51bd6d85ee0f1b064f1f3fce781fe5.

• .proof-experiments/20260812-103013-1955d4: producer scored 43928064 candidates, found zero hits, and minimum defect ten.
• .proof-experiments/20260812-103035-ea35f2: independent checker reconstructed all candidates and matched every row.
• .proof-experiments/20260812-103113-ab6ee9: direct set audit verified the defect-ten endpoint.

check_degree18_4for3_coverage_v1.cpp uses least-first-block enumeration and residual complementary-pair recovery, unlike the producer's recursive three-bin assignment. audit_degree18_4for3_best_v1.py independently uses Python sets and combinations.

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

• Exact-pinnability local search -> the next shell might contain a cover or reduce defect -> all 43928064 candidates retained minimum defect ten, so radius expansion is demoted.
• Certified exhaustive-search discipline -> use different producer/checker encodings and a compact endpoint certificate -> all three audits agreed.
• Packed set-cover kernels -> eight 64-bit words should remove repeated 455-triple scans -> producer throughput was about 3.97 million candidates per second.

• The exact four-delete/three-add shell has no covering family.
  Zero hits in producer-result.json and independent-check.json after exact reconstruction of 43928064 candidates. · All 33100 feasible headers, with three distinct additions outside all 55 source blocks and final degree 18. · computed
• The minimum defect in the exact shell equals ten.
  Two exhaustive scans give minimum ten; best-defect10-certificate.json directly realizes ten. · The same exact local shell. · computed

• Obtain a cover in the exact degree-balanced four-delete/three-add shell.
  43928064 families across 33100 headers. · Complete independent enumerations found zero hits and minimum defect ten. · producer-ledger.jsonl, producer-result.json, and independent-check.json · A demonstrated source-boundary or enumeration discrepancy; otherwise only a family outside this shell is relevant.
• Continue expanding local radius solely because the prior shell failed.
  Strategy selection after the exact three-delete/two-add and four-delete/three-add results. · Both shells have minimum defect ten despite a more than 2897-fold candidate-count increase. · Accepted prior 15159-candidate census and this 43928064-candidate census. · A bounded preflight showing a feasible next shell plus a new pruning invariant or measured defect-improvement signal.

• Literal block-to-internal-pair-excess coupled type-4 pilot
  It adds a genuine block-identity/pair-excess correlation absent from the stale fixed-pair relaxation and has a cheap matched falsifier. · Generate baseline and coupled proof-producing formulas, reconstruct semantics, prepare dual replay, and run each to 25000 conflicts. · high · open
• Global pair-normalized proof pipeline
  It retains global scope and would settle nonexistence if every branch received independently replayed proofs. · Before live solving, validate a pinned proof-producing solver plus two replay paths on SAT and UNSAT controls. · normal · open

Objective: Determine whether literal block-to-internal-pair-excess coupling gives a measurable proof-search gain on the canonical type-4 branch.

First action: Write the matched baseline/coupled semantic protocol and pin the proof-producing solver and dual replay commands before generating formulas.

Stop condition: Redirect on zero semantic delta, failed checker/replay control, UNKNOWN without at least 20 percent improvement, or evidence that another route has higher marginal value.

• Close and do not repeat the exact four-delete/three-add shell.
• Predeclare the literal block-to-internal-pair-excess coupling on the canonical type-4 branch.
• Run a matched 25000-conflict proof-producing pilot and retain it only after semantic reconstruction, dual replay setup, and at least 20 percent propagation/decision improvement.

• https://www.coveringrepository.com/systems.aspx?k=5&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=5&t=3&v=15 ↗
• https://central.bac-lac.gc.ca/.item?app=Library&id=nq24295&oclc_number=46548328&op=pdf ↗

GPT-5.6 Sol principal investigator; two GPT-5.6 Terra delegates supplied advisory reconnaissance promoted with provenance but not treated as evidence. Deterministic C++20 producer and checker compiled with g++ 13.3.0; Python 3.12.3 direct witness checker and experiment recorder; web reader for source and prior-art checks. No CAS, SAT solver, proof assistant, cloud lab, installation, external publication, Git commit, or remote write.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1181.8s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260812-104024-a25485

No human review recorded.