Clean-room clause-exact reconstruction of all six fixed-second incidence CNFs, explicit S6 x S9 case-cover enumeration, differential checking, and DIMACS mutation tests.

proof-producing incidence/PB cube decomposition

Bind every retained CNF clause to the mathematical covering model and explicitly verify that six overlapping second-block representatives cover every normalized candidate.

Hypothesis: Every clause in each of the six retained fixed-second incidence CNFs exactly matches a clean-room reconstruction of the sound covering model, and the six cases exhaust all 5004 second-block choices after fixing the first block.

Test: Stream-compare all 938352 clauses against a generator-independent reconstruction, enumerate explicit stabilizer maps for all 5004 second blocks, then run a differently structured checker and actual CNF mutations.

Every hypothetical 30-cover has exact point degree 12 and pair degrees between 4 and 8, extends to the reconstructed support and totalizer variables, and can be normalized into at least one of the six formulas. Exact clause agreement binds this argument to the retained artifacts. No solver result was generated or promoted.

• Every hypothetical 30-block cover extends to a satisfying assignment of at least one of the six hash-identified CNFs.
• Each of the six CNFs has 33162 variables and exactly 156392 clauses matching the clean-room reconstruction.
• The six second-block orbits have sizes 84, 756, 1890, 1680, 540, and 54 and cover all 5004 non-first blocks.
• The six cases may overlap and do not constitute a disjoint partition of solution isomorphism classes.
• No covering-number bound changed.

• python3 scripts/audit_six_case_compiler_v1.py --out artifacts/epoch65-20260810/six-case-compiler-primary.json
• python3 checkers/check_six_case_compiler_v1.py --receipt artifacts/epoch65-20260810/six-case-compiler-primary.json --out artifacts/epoch65-20260810/six-case-compiler-checker.json
• python3 checkers/test_six_case_compiler_fail_closed_v1.py
• sha256sum -c artifacts/epoch65-20260810/SHA256SUMS

• .proof-experiments/20260810-205657-00083d: primary final audit passed in 3.797 seconds, checking 938352 clauses and 5004 maps.
• .proof-experiments/20260810-205701-7f03a6: independent differential checker passed in 3.957 seconds.
• .proof-experiments/20260810-205732-7fabc2: three actual DIMACS mutations were rejected in 1.176 seconds.

checkers/check_six_case_compiler_v1.py uses differential shared-core/suffix hashing, direct fixed-unit reconstruction, and an independently implemented stabilizer enumeration; checkers/test_six_case_compiler_fail_closed_v1.py separately mutates real CNF files.

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

• Certified covering-number proof architecture from C(12,6,4) -> predict that case coverage must be hash-bound to exact proof inputs before scale-up -> the audit passed and removed an opaque-core verification gap.

• Every point of a hypothetical 30-block cover has degree exactly 12.
  Schonheim link lower bound gives degree at least 12; total incidence is 30*6=15*12. · Any 30-block C(15,6,3) cover. · proved
• Every pair degree lies between 4 and 8.
  Each pair must cover 13 third points four per block, giving at least 4; incident pair-degree sum is 12*5=60, giving at most 8. · Any 30-block C(15,6,3) cover. · proved
• All 938352 retained clauses match the clean-room reconstruction.
  artifacts/epoch65-20260810/six-case-compiler-primary.json and independent differential receipt. · The six exact hash-identified epoch-9 CNFs. · computed
• All 5004 possible second blocks map to the six declared representatives.
  Two independent explicit stabilizer enumerations with matching orbit sizes. · Six-subsets other than 012345. · computed

• Treat the six second-block cases as a disjoint partition of full solutions.
  The six retained root formulas. · A solution with several residual intersection types can satisfy multiple root cases. · The normalization proves coverage only; both receipts explicitly record overlap. · An exactly-once ownership predicate with independently checked union and disjointness.
• Treat comparison with an inherited baseline prefix as sufficient semantic validation.
  The 153905-clause shared core of the six retained formulas. · It does not independently establish what the shared clauses encode. · The new clean-room audit reconstructs the full core and detects injected core corruption. · Only relevant if the formula hashes or compiler change; rerun the clean-room audit then.

• Multi-anchor support-three constructive escape search.
  It retains a direct terminal witness path and combines fresh degree-12 starts with transitions larger than the closed support-two neighborhoods. · Compare support-two and exact support-three transitions under identical deterministic proposal budgets. · high · open
• Proof-producing exactly owned incidence frontier.
  The root formulas are now fully validated, but exhaustive descendants require hash-bound ownership and replayable terminal proofs. · Specify one smallest owned frontier and dual-replay one legitimate UNSAT leaf before scale-up. · normal · open
• Canonical root-link catalogue.
  A complete independently checked catalogue could convert local blockers into a global exclusion. · Measure canonical augmentation growth under a 10000-orbit cap and audit the completeness frontier. · normal · open

Objective: Determine whether multi-anchor exact support-three transitions improve the constructive search over the matched support-two control.

First action: Run the retained fresh-start checker on artifacts/epoch24-20260809/fresh_nonisomorphic_starts_result.json with its epoch-13 source and epoch-14 tie receipt, then freeze equal proposal budgets.

Stop condition: Stop on degree drift, duplicate blocks, validation disagreement, no deficit or canonical-diversity improvement, or projected runtime above two minutes without lab submission.

• Revalidate the epoch-24 fresh nonisomorphic starts and freeze matched proposal counts.
• Implement the exact multi-anchor support-three challenger against a support-two control.
• Directly validate every zero-deficit state with both retained Python and C cover checkers.
• Reopen proof scale-up only with a materially different proof-producing owned frontier or a pinned PB proof emitter and replayer.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?v=15&k=6&t=3 ↗
• https://arxiv.org/abs/2607.23766 ↗

GPT-5.6 Sol acted as principal investigator. Two GPT-5.6 Terra delegates supplied advisory prior-art and verification leads; their claims were not treated as evidence and promoted summaries record provenance. Python 3.12 deterministic programs performed exact CNF reconstruction, enumeration, hashing, and mutation tests. SHA-256 bound the artifacts. Web search checked the maintained status and related primary source. No SAT solver, CAS, proof assistant, cloud lab, or external human validator produced a mathematical result this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

937.8s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260810-210250-948a6d

No human review recorded.