Built and independently reconstructed a fixed-second r=0 rank-2 profile CNF using reusable unary totalizers, then compared it with the semantically identical legacy leaf in four paired proof-producing runs.

proof-producing incidence/PB cube decomposition

Fix the unique disjoint second block, classify columns 2..29 by first-block intersection weights 2, 3, and 5, impose counts 24, 1, and 3 with reusable totalizers, and measure fail-closed LRAT search against the legacy leaf.

Hypothesis: For the identical fixed-second rank-2 leaf, reusable unary totalizers either yield a terminal result or improve geometric-mean conflict throughput by at least 1.25x without increasing LRAT bytes per conflict.

Test: Four alternating seed-0/seed-1 CaDiCaL runs at five seconds each, followed by independent raw-log parsing and replay of every trace with lrat-check and CakeLPR.

The exact formula and independent checker are reusable research infrastructure, which is net-new validated progress at the artifact level. The solver observations are explicitly nonterminal and fail the predeclared scale-up gate, so they support redirecting the route but no profile exclusion or covering-number claim.

• The reusable formula exactly encodes the same normalized profile leaf as the legacy formula and has 33654 variables and 164858 clauses.
• The legacy formula has 33302 variables and 186668 clauses, so the reusable formula removes exactly 21810 clauses while adding 352 variables.
• Under the frozen four-run protocol, all statuses were UNKNOWN, the geometric-mean conflict-rate ratio was 1.0117982588743892, and the proof-bytes-per-conflict ratio was 1.0222869657785219.
• Fresh lrat-check and CakeLPR builds explicitly rejected all four incomplete LRAT traces.
• No cover or UNSAT leaf was obtained; the maintained exact range remains 30 to 31.

• sha256sum -c artifacts/epoch46-20260810/SHA256SUMS passed for every listed source and artifact.
• Independent formula receipt SHA-256 1fab1575bfa9cac4a3566dfe054a0e609b9d274b216f85cfbdd19e0ab3fa56c0 records exact reconstruction and 12375 semantic/control cases.
• Run receipt SHA-256 dd8e98131142b7856e32da9666aa7509e2ff94161109220114ebee69c87be398 records four UNKNOWN runs.
• Independent run receipt SHA-256 31ba91349eb2c6c411f57c9966cb5274c7113ccfe20b497c3085a6b60bbc303e reparses all logs, reconstructs both metrics, and records eight explicit incomplete-proof rejections across two replayers.

• .proof-experiments/20260810-071905-8f154d: deterministic formula build, 0.171 seconds, return code 0.
• .proof-experiments/20260810-071917-d83703: independent byte and semantic audit, 0.774 seconds, PASS.
• .proof-experiments/20260810-071938-973e78: four paired five-second CaDiCaL runs, all UNKNOWN.
• .proof-experiments/20260810-072017-9b0393: independent log audit and dual replay, PASS with route gate false.

check_fixed_second_reusable_profile_v1.py independently reconstructed every CNF byte and local semantic boundary. check_fixed_second_reusable_calibration_v1.py independently reparsed logs, checked any SAT witness path, compiled fresh lrat-check and CakeLPR binaries, and required explicit rejection of every incomplete trace.

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

• Shared cardinality networks -> predicted that reusable profile counters would reduce proof cost -> clauses fell 11.68% but proof bytes/conflict worsened 2.23%, falsifying the operational prediction.
• Certified cube-and-conquer -> predicted that fail-closed replay should precede scale-up -> both independent replayers rejected all nonterminal prefixes, correctly preventing false UNSAT promotion.
• Second-block stabilizer normalization -> predicted that fixing the unique r=0 owner preserves the profile leaf -> independent reconstruction confirmed the fixed blocks and exact residual counts.

• The reusable fixed-second formula is byte-reproducible, represents profile (1,0,24,1,0,3), and has 33654 variables and 164858 clauses.
  Formula SHA-256 815e39973774f1d0eb2aff960700d001d191485c0fb98fd1275c5f6a931bdb99 and independent audit SHA-256 1fab1575bfa9cac4a3566dfe054a0e609b9d274b216f85cfbdd19e0ab3fa56c0. · One normalized fixed-second rank-2 profile leaf. · computed
• The reusable encoding removes 21810 clauses and adds 352 variables relative to the semantically matched legacy formula.
  Independent DIMACS parsing of hashes 815e39973774f1d0eb2aff960700d001d191485c0fb98fd1275c5f6a931bdb99 and f4affe090af0f00387ca006de89b27fe005082c48008249dc5a226e0b6177b40. · Raw DIMACS dimensions for the two matched encodings. · computed
• Both proof-oriented continuation gates fail under the frozen two-seed five-second protocol.
  Independent receipt SHA-256 31ba91349eb2c6c411f57c9966cb5274c7113ccfe20b497c3085a6b60bbc303e gives ratios 1.0117982588743892 and 1.0222869657785219. · Four specified CaDiCaL 1.7.3 runs only. · computed

• Scale the unchanged fixed-second reusable-totalizer formula by merely increasing the timeout.
  The exact formula SHA-256 815e39973774f1d0eb2aff960700d001d191485c0fb98fd1275c5f6a931bdb99 with CaDiCaL 1.7.3 and the current two-seed protocol. · All runs were nonterminal and both predeclared proof-search efficiency gates failed. · Run and independent-check receipts dd8e98131142b7856e32da9666aa7509e2ff94161109220114ebee69c87be398 and 31ba91349eb2c6c411f57c9966cb5274c7113ccfe20b497c3085a6b60bbc303e. · A material counter architecture, verified shared proof prefix, different proof-producing solver with a bounded advantage, or terminal SAT/UNSAT evidence.
• Treat any of the four bounded LRAT files as an UNSAT certificate.
  The four epoch-46 proof prefixes. · CaDiCaL returned UNKNOWN and both lrat-check and CakeLPR explicitly rejected every prefix. · independent-run-check.json proof_replay entries. · A complete proof ending in a replay-accepted empty-clause derivation.

• Nine forced multiplicity-four pair-star preprocessing cubes.
  The structural pair normalization changes the decomposition boundary and can be rejected cheaply before proof search. · Independently enumerate the nine templates, compile all semantic formulas, and compare post-preprocess dimensions against the fixed-first control with a 2x geometric-mean gate. · high · open
• Proof-producing complete 249-profile ownership frontier.
  It remains the most direct negative route, but needs certificate-sized leaves rather than nonterminal prefixes. · After a material decomposition passes preprocessing, solve the smallest owned leaf and dual-replay any terminal proof. · normal · open
• Materially changed constructive witness search.
  A direct 30-cover remains symmetric in value to exclusion, while current deterministic C5 ordering is held. · Define a new witness-bearing heuristic or sound bulk prune before reopening any capped constructive family. · low · open

Objective: Decide whether forced multiplicity-four pair stars create a genuinely smaller proof-producing decomposition.

First action: Promote and independently reimplement the nine-template enumerator, prove the pair-multiplicity-four normalization, and generate a hash-bound template manifest before formula construction.

Stop condition: Redirect on template coverage mismatch, semantic CNF mismatch, or less than 2x geometric-mean reductions in both post-preprocess variables and clauses; continue only on a direct witness or verified proof-producing shrinkage.

• Independently reconstruct the forced multiplicity-four pair lemma and all nine canonical four-block pair-star templates.
• Compile all nine semantic formulas only through pinned preprocessing and independently reconstruct fixed blocks, candidate pools, degree RHSs, and residual triple masks.
• Continue the pair-star route only if every semantic audit passes and geometric-mean post-preprocess variables and clauses are both at least 2x smaller than the fixed-first control.
• Do not scale the current reusable profile formula without a material counter change, proof-prefix reuse mechanism, or terminal evidence.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://arxiv.org/abs/math/9502238 ↗
• https://github.com/tanyongkiam/cake_lpr ↗

GPT-5.6 Sol principal designed, implemented, executed, and audited the epoch. GPT-5.6 Terra delegates supplied advisory challenger/prior-art and verification memos; their agreement was not evidence and no delegate artifact supports a claim. Deterministic tools: CPython 3.12.3, CaDiCaL 1.7.3, GCC, lrat-check, hash-pinned CakeLPR/CakeML assembly, SHA-256, and the Proof Factory experiment harness. Web search checked current status and exact-parameter literature. No CAS, proof assistant, cloud lab, external validator, or human reviewer was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1393.0s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260810-073031-745fc9

No human review recorded.