Derived and independently checked an internal pair-excess coordinate q for a fixed block, compiled its 13 disjoint incidence-CNF cubes, calibrated the two extreme cubes with proof-producing SAT, and combined q with the minimum-intersection frontier to classify necessary arithmetic profiles.

pair-excess and minimum-intersection proof-cube decomposition

The forced 4-regular pair-excess multigraph makes the internal excess q of a fixed six-point block an integer from 0 through 12. Exact counters over existing pair-support variables encode these disjoint cubes, while intersection-count equations eliminate impossible (r,q) cells before SAT.

Hypothesis: The fixed-block internal pair excess yields a sound disjoint 13-cube partition, at least one extreme cube q=0 or q=12 terminates within five seconds and 4 MiB of LRAT, and combining q with the minimum-intersection equations eliminates arithmetic cells before SAT.

Test: Build all q=0..12 cubes from the immutable complete incidence CNF, independently reconstruct every counter, run q=0 and q=12 for five seconds with dual fail-closed replay, then independently enumerate every fixed-block intersection-count profile.

The structural claims follow exact incidence identities and two independent finite reconstructions. Solver timeouts and rejected incomplete proofs support no SAT or UNSAT conclusion, so the correct outcome is progress rather than candidate.

• For every hypothetical 30-block cover and fixed selected block F, q=sum_{{x,y} subset F}(lambda_xy-4) is exactly one integer from 0 through 12, giving a disjoint exhaustive 13-cube partition.
• The 13 retained CNFs are the immutable complete fixed-first-block incidence body plus exact counters targeting 60+q over its 450 internal-pair support variables.
• Exactly 249 necessary fixed-block intersection-count profiles survive the degree, minimum-intersection, and pair-excess equations.
• The nine cells (r=0,q=0), (r=0,q=1), (r=0,q=2), (r=1,q=0), and (r=2,q=8..12) are arithmetically empty.
• The q=0 and q=12 five-second runs are UNKNOWN; neither incomplete LRAT proves an exclusion.

• python3 scripts/build_pair_excess_cubes_v1.py --out-dir artifacts/epoch38-20260810/pair-excess-cubes-v1
• python3 scripts/run_pair_excess_extremes_v1.py --artifact-dir artifacts/epoch38-20260810/pair-excess-cubes-v1 --seconds 5
• python3 checkers/check_pair_excess_cubes_v1.py --artifact-dir artifacts/epoch38-20260810/pair-excess-cubes-v1 --out artifacts/epoch38-20260810/pair-excess-cubes-v1/independent-check-v2.json
• python3 scripts/audit_pair_excess_profile_frontier_v1.py --out artifacts/epoch38-20260810/pair-excess-profile-frontier.json
• python3 checkers/check_pair_excess_profile_frontier_v1.py --input artifacts/epoch38-20260810/pair-excess-profile-frontier.json --out artifacts/epoch38-20260810/pair-excess-profile-frontier-independent-v2.json
• sha256sum -c artifacts/epoch38-20260810/SHA256SUMS

• .proof-experiments/20260810-010250-a8b1fa: built 13 q cubes in 1.827 seconds
• .proof-experiments/20260810-010451-62a02c: retained memory-corrected extreme run; q=0 and q=12 were UNKNOWN
• .proof-experiments/20260810-010850-436736: clean independent reconstruction and dual replay passed
• .proof-experiments/20260810-010651-337ac5: seven fail-closed mutations rejected
• .proof-experiments/20260810-011523-337c92: recursive enumeration found 249 profiles and nine empty cells
• .proof-experiments/20260810-011553-b73768: independent bar-position enumeration reproduced the frontier
• .proof-experiments/20260810-010306-7cbd05: invalidated CakeLPR attempt; default 8 GiB runtime exceeded the 1 GiB limit
• .proof-experiments/20260810-011524-3e8688: failed control exposed incorrect expected intermediate count 530; observed correct value 5004

checkers/check_pair_excess_cubes_v1.py reconstructs all pair-support conjunctions and counters and invokes two LRAT implementations; checkers/check_pair_excess_profile_frontier_v1.py independently uses bar-position enumeration rather than the primary recursive composition generator.

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

• 4-regular excess multigraph -> predict a bounded internal-edge coordinate q -> obtained an exact 13-cube partition, but extreme SAT leaves remained UNKNOWN
• minimum-intersection profile frontier -> predict q is computable from intersection counts -> obtained the exact r=2 map 0,1,2,3,3,4,4,5,6,7
• integer-profile presolve -> predict empty composite cubes before SAT -> eliminated nine of 39 naive (r,q) cells

• The internal pair excess q of any fixed selected block belongs to {0,...,12}.
  Pair lower bound lambda>=4, exact point degree 12, and 24=2q+cut(F); independently checked CNF realization · Every hypothetical 30-block C(15,6,3) cover · proved
• The fixed-block search is covered disjointly by the 13 q values.
  build-manifest.json and independent-check-v2.json · Complete fixed-first-block incidence model · computed
• Exactly 249 necessary intersection-count profiles occupy 30 live (r,q) cells.
  Recursive and independent bar-position enumeration of all 278256 weak profiles · Every hypothetical simple 30-cover after fixing a block and applying the minimum-intersection lemma · computed
• Nine named (r,q) cells are empty.
  pair-excess-profile-frontier.json and independent-v2 receipt · Necessary fixed-block intersection-count arithmetic · proved
• The q=0 and q=12 five-second runs produced no terminal evidence.
  extreme-run-receipt.json; both incomplete proofs explicitly rejected twice · CaDiCaL 1.7.3 seed 0, five-second solver limit, retained formulas · computed

• Scale q alone because an extreme cube is immediately proof-tractable.
  q=0 and q=12, seed 0, five-second and 4 MiB protocol · Both runs returned UNKNOWN with only about 300 conflicts. · extreme-run-receipt.json and independent-check-v2.json · A material conjunction with another invariant, proof-prefix reuse, or terminal evidence from a bounded cell
• The nine empty (r,q) cells contain a 30-cover.
  Fixed-block necessary intersection-count profiles · No weak count vector satisfies all exact incidence, minimum-intersection, and q equations. · Two exhaustive enumerations · A defect in the degree-12 lemma, pair lower bound, minimum-intersection lemma, or either enumeration
• Treat q buckets as symmetry classes.
  All 13 q values · Equal internal excess does not imply isomorphic residual hypergraphs or equivalent SAT instances. · Construction uses q only as a disjoint invariant key and makes no orbit claim. · An independently checked colored-hypergraph isomorphism classification within each q bucket
• Accept CakeLPR from exit code zero.
  Current proof replay pipeline · Rejected incomplete proofs return exit code zero; acceptance requires the explicit verification marker. · Dual replay logs and exit-code mutation control · A pinned CakeLPR version with documented and independently tested failure exit semantics

• Capacity-ranked 249-profile frontier
  Exact arithmetic has already removed 278007 weak profiles; raw block-set capacities can select one genuinely smallest live proof cell without arbitrary cutoffs. · Compute product-of-binomial capacities for all 249 profiles under the fixed representative and verify the ranking independently. · high · open
• Fixed-point-free C5 one-orbit tail lookup
  A constructive witness remains cheaper to validate and the lookup directly targets 888391 measured late failures. · Index unused block orbits by residual cycle profile and exact coverage-mask containment at one remaining slot. · high · open
• Hash-bound composite profile cube union
  A global UNSAT result needs a complete union over all 249 profiles or an equivalent coarser partition. · Specify ownership, source hashes, and independent union coverage before producing more proofs. · normal · open

Objective: Identify the smallest honest composite proof cell among the 249 necessary profiles and decide whether it beats the constructive C5 route.

First action: Run a deterministic exact-capacity census over pair-excess-profile-frontier.json, with a separate binomial-product checker and no solver call.

Stop condition: Redirect to the C5 tail route if the smallest profile still projects beyond the declared CNF/proof budget or if capacity does not predict materially stronger preprocessing.

• Compute exact raw block-set capacities for all 249 profiles and identify the smallest live r=0 or r=1 cell.
• Design a hash-bound union manifest before compiling any composite profile cube.
• Run one short preprocessing/proof-prefix discriminator on the smallest cell and redirect if it remains nonterminal or exceeds the certificate projection.
• In parallel campaign planning, retain the fixed-point-free C5 one-orbit tail lookup as the constructive alternative.

• 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://github.com/marijnheule/drat-trim/tree/2e3b2dc0ecf938addbd779d42877b6ed69d9a985 ↗
• https://github.com/tanyongkiam/cake_lpr ↗

GPT-5.6 Sol principal designed, executed, audited, and interpreted the epoch. GPT-5.6 Terra delegates supplied advisory stubs; their agreement was not validation, the old overlapping-representative interpretation was rejected, and all used claims were independently rederived. Deterministic tools: CPython 3.12.3, CaDiCaL 1.7.3, GCC, lrat-check.c, CakeLPR with recorded 512 MiB heap and 128 MiB stack, exact integer enumeration, SHA-256, and Proof Factory run_experiment.py. No CAS, proof assistant, cloud lab, external proof service, or human validator was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1824.1s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260810-012106-e26df2

No human review recorded.