Replaced the overlapping six second-block representatives with a disjoint frontier owned by the minimum intersection with a fixed first block, then exhaustively and independently checked its coverage, symmetry normalization, static bans, and representation counts.

proof-producing incidence/PB cube decomposition

Assign each 29-block remainder to its unique minimum intersection r with the fixed block; map one minimum block to a canonical representative under S_6 x S_9 and ban all lower-intersection blocks.

Hypothesis: The minimum intersection with fixed block 012345 defines six exhaustive, mutually exclusive branches, and the associated lower-intersection bans strictly reduce the normalized branch-capacity upper bound relative to the old overlapping scheme.

Test: Enumerate all 5004 non-fixed blocks and all 278256 six-part intersection profiles of a 29-block remainder; require exactly one owner, explicit stabilizer normalization, exact weighted coverage, and rejection of gap and overlap controls in two implementations.

Minimum intersection is unique for every nonempty 29-block remainder. The fixed-block stabilizer preserves that value and maps any attaining block to its canonical representative. Exhaustive tuple and bitmask implementations agree on every finite count, while omission and overlap controls fail closed.

• The six minimum-intersection branches are exhaustive and mutually exclusive before normalization.
• Their intersection-class sizes are exactly (84,756,1890,1680,540,54), with allowed suffix sizes (5004,4920,4164,2274,594,54).
• The normalized branch-capacity upper bound is 3.686941086759067 times smaller than the previous overlapping bound.
• No C(15,6,3) bound changed; the maintained range remains 30 <= C(15,6,3) <= 31.

• Primary PASS receipt SHA-256 2769ca4f8a27407bccbf6afe3589af06419305d197e855379f26764a9f47b9fa
• Independent PASS receipt SHA-256 0d6c7b7ed63b641f2a3996174557bfbfbe3222b55e79b1ae6e0695599f33ba7d
• sha256sum -c artifacts/epoch34-20260809/second-block-min-frontier-v1/SHA256SUMS passed for all listed artifacts
• Primary run: 5004 block owners, 5004 stabilizer maps, 278256 profiles, return code 0
• Independent run: 5004 bitmasks, 5004 stabilizer images, 278256 bar-position profiles, return code 0

• .proof-experiments/20260809-221324-54f52f: primary frontier audit PASS in 1.830 seconds
• .proof-experiments/20260809-221336-7221a6: independent bitmask/profile audit PASS in 1.077 seconds

checkers/check_second_block_frontier_v1.py uses 15-bit masks and bar-position profile enumeration rather than the primary tuple and recursive-composition representation; it reproduced all exact counts and rejected four control types.

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

• C(12,6,4) least-orbit case index -> predict that the least fixed-block intersection gives a disjoint C(15,6,3) frontier -> exhaustive profile and block audits passed.

• Every simple 29-block remainder after fixing F has exactly one minimum-intersection branch r in {0,1,2,3,4,5}.
  Elementary minimum argument plus exhaustive audit of all 278256 intersection profiles · Simple selections of 29 distinct non-F blocks · proved
• Every block in intersection class r maps to the recorded representative under a permutation stabilizing F.
  Explicit permutation construction checked independently for all 5004 non-F blocks · All 6-subsets of a 15-set other than F · computed
• The six exact pre-normalization owner counts partition choose(5004,29).
  Weighted profile enumeration and independent closed-form differences choose(N_r,29)-choose(N_{r+1},29) · All simple 29-block remainders · computed
• The new normalized capacity upper bound improves the old overlapping bound by exactly the recorded rational ratio, approximately 3.686941086759067.
  Both receipts independently recompute the binomial expressions · Raw normalized branch-capacity accounting, not runtime or isomorphism classes · proved

• Treat the old six representative branches as a disjoint case split.
  Intersection-profile ownership of 29-block remainders · Any profile using multiple intersection classes belongs to multiple contains-class branches; 278250 of 278256 profiles overlap. · Overlap witness (0,0,0,0,1,28) and exhaustive profile census · None; use the minimum-intersection owner instead.
• Omit the r=5 branch.
  Six-class minimum-intersection frontier · Profile (0,0,0,0,0,29) has no remaining owner. · Independent missing-branch control · A proved impossibility of the entire r=5 family with a replayed certificate.

• Ban-augmented incidence leaf
  It is the cheapest test of whether the structural reduction translates into smaller proof-producing CNFs. · Generate one r=5 or r=4 genuine leaf with intersection lower bounds, reconstruct its clauses independently, emit LRAT, and replay with CakeLPR. · high · open
• Within-branch canonical minimum-block ownership
  Multiple minimum blocks can still yield repeated normalized representations inside one r branch. · Test a lexicographically canonical attaining-block rule on profile-level and small labelled samples before attempting full canonical augmentation. · normal · open
• Fixed-C5 constructive tail lookup
  This remains a materially different witness route if proof-size calibration fails. · Run the saved matched one-orbit-tail pilot with direct verification of any witness. · normal · open

Objective: Measure whether lower-intersection bans produce a semantically correct, independently replayable genuine proof leaf of tractable size.

First action: Implement a ban-augmented incidence generator and independent clause reconstructor, then run one 60-second, 512-MiB LRAT calibration through the experiment harness.

Stop condition: Redirect on CNF semantic mismatch, UNKNOWN, proof-format failure, CakeLPR rejection, or projected aggregate proof volume beyond the declared certificate budget.

• Add exact intersection-at-least-r constraints to residual columns in each incidence branch.
• Write a separate CNF checker that reconstructs the ban clauses from the mathematical specification.
• Generate one genuine bounded leaf, request LRAT output, and replay it with hash-pinned CakeLPR.
• Project total proof bytes only after the genuine leaf passes; redirect if the projection exceeds the certificate budget.
• Directly verify any SAT witness against all 455 triples.

• 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/html/2607.23766v1 ↗

GPT-5.6 Sol served as principal investigator. GPT-5.6 Terra delegates supplied advisory leads, but their memo files were truncated and hash-inconsistent, so no delegate assertion was counted as evidence. Deterministic work used Python 3.12.3, exact integer arithmetic, SHA-256, and the computational-researcher experiment harness. Web search checked the maintained source, current range, and nearby literature. No SAT solver, CAS, proof assistant, or external lab job was used this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

690.7s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260809-221759-2ae2be

No human review recorded.