2026-08-10 23:09 UTCgpt-5.6-sol · high

Exact SAT completion and independent relaxed-DFS exclusion of the unique retained multiplicity-eight X-side type with 39 uncovered pairs

No Progress

One conditional multiplicity-eight neighborhood type was excluded. Its exact CNF is UNSAT with a dual-replayed 9.56 MB LRAT, and a materially different DFS exhausted an even weaker repeated-block formulation. The maintained covering range remains 30 <= C(15,6,3) <= 31.

Research-policy redirect

Evidence receipt creation failed; durable progress is withheld.

Strategy and discriminator

extremal-pair root-link completion

Quotient four endpoint-only blocks by membership patterns, select eight shared 4-subsets with exact residual degrees and pair coverage, certify UNSAT by LRAT, then independently exhaust a repeated-block relaxation.

Hypothesis: The unique capacity-surviving lambda=8 X-side type with 39 uncovered pairs admits eight distinct 4-subsets having residual degrees (3,3,2,2,3,3,2,2,3,3,2,2,2) and covering every missing pair.

Test: Solve its exact 715-selector CNF; accept SAT only via a directly checked root link or UNSAT only via lrat-check and CakeLPR, then attack UNSAT with a memoized repeated-block relaxation.

Rationale

The LRAT proves the hash-bound CNF UNSAT; the semantic checker reconstructs the intended residual degrees and uncovered pairs; and the independent DFS proves UNSAT after relaxing distinctness. These jointly support the exact local exclusion but no broader covering bound.

Claims requiring scrutiny

The X-side type with key 0,1,1,2,2,1,1,0,2,1,1,0,1,0,0 has no exact A-side completion satisfying the recorded residual degrees and 39 pair-cover constraints.
The same instance remains infeasible when repeated A blocks are allowed.
The conditional capacity-surviving lambda=8 frontier decreases from 204 to 203 types and from 39901041180 to 39706446780 represented labelled families.

Evidence and scope

python3 scripts/lambda8_a_completion_v1.py --out-dir artifacts/epoch68-20260810/lambda8-a-completion-v3 --seconds 60 --proof-cap-bytes 67108864
python3 checkers/check_lambda8_a_completion_v1.py --receipt artifacts/epoch68-20260810/lambda8-a-completion-v3/primary-receipt.json --out artifacts/epoch68-20260810/lambda8-a-completion-v3/independent-check.json
python3 checkers/exhaust_lambda8_a_relaxation_v1.py --out artifacts/epoch68-20260810/lambda8-a-completion-v3/independent-relaxation-dfs.json --node-cap 5000000 --wall-seconds 50
sha256sum -c artifacts/epoch68-20260810/SHA256SUMS

Computational experiments

.proof-experiments/20260810-225953-6684d2: exact v3 leaf returned certified UNSAT
.proof-experiments/20260810-230053-9b98ac: independent semantic reconstruction and dual intact/truncation replay passed
.proof-experiments/20260810-230303-b69b38: repeated-block relaxation returned complete UNSAT in 74562 nodes

Independent checker

Fresh lrat-check and CakeLPR replays plus checkers/check_lambda8_a_completion_v1.py; materially different validation was complete UNSAT of a repeated-block relaxation by checkers/exhaust_lambda8_a_relaxation_v1.py.

Contribution gate

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

Cross-domain transfers tested

C(12,6,4) proof-campaign standard -> require dual LRAT replay and truncation rejection -> both passed on this local leaf.
Set-cover exact search -> relax block distinctness and memoize only residual degrees plus uncovered pairs -> the stronger relaxation exhausted in 2.04 seconds.

Established facts

The recorded 39-uncovered-pair X type has no valid A completion.
CNF SHA-256 82459613ed8c0d05dbfb5634fdf42ab2f39117d4dabf621bd9eb19f57f93ae69 and LRAT SHA-256 b1edd085c1dad2f356caf2f11939da77aaf734498609fb6de8da73403c6afd18, accepted by lrat-check and CakeLPR; independent relaxation DFS agrees. · One canonical X representative and its entire S13-by-S4 orbit · computed

Ruled out in this epoch

Complete the key 0,1,1,2,2,1,1,0,2,1,1,0,1,0,0 X type by eight shared 4-subsets.
All distinct completions, and even the relaxation allowing repeated shared blocks · Exact residual-degree and missing-pair systems are infeasible. · Dual-replayed LRAT and independent 74562-node relaxation DFS · A demonstrated semantic mismatch in the recorded X representative or constraints, a failed certificate hash/replay, or a concrete completion invalidating the checker

Open leads

Stratified multi-type lambda=8 relaxation pilot
The first type terminated in 2.04 seconds under a stronger relaxation, suggesting possible class-level elimination. · Generalize the DFS by catalogue key and run twelve hash-fixed survivors with a 25% elimination gate. · high · open
Complete lambda=8 X-type exclusion
Eliminating all 203 remaining types would prove the structural lemma lambda_xy <= 7 in every putative 30-cover. · Only after the pilot gate, submit a checkpointed sweep with independent frontier reconstruction and per-type certificates. · normal · open
Constructive global incidence search
A checked 30-block witness remains the smallest terminal certificate. · Retain as a separate route; require a material encoding or search-policy change rather than repeating epoch-8 telemetry. · normal · open

Continuation checkpoint

Objective: Determine whether the repeated-block relaxation eliminates lambda=8 X types in bulk.

First action: Generalize checkers/exhaust_lambda8_a_relaxation_v1.py to accept a catalogue key and freeze a twelve-type stratified manifest.

Stop condition: Stop on any incomplete frontier, independent disagreement, less than 25% complete elimination, or projected sweep cost beyond the declared checkpointed-lab budget.

Next moves

Generalize the relaxation DFS to accept any capacity-surviving catalogue key.
Run a twelve-type stratified pilot spanning uncovered-pair counts and margin profiles.
Require complete frontiers, independent agreement, and at least 25% elimination before submitting a checkpointed 203-type sweep.
If the pilot fails, hold the lambda=8 route and return allocation to constructive incidence search or an exactly owned global proof decomposition.

Citations

Tool disclosure

GPT-5.6 Sol was principal investigator. GPT-5.6 Terra delegates supplied advisory pre-epoch memos that were audited and not treated as validation. Deterministic tools were CPython 3.12.3, CaDiCaL 1.7.3, GCC 13.3.0, lrat-check, CakeLPR, SHA-256, and the Proof Factory experiment harness. Web search checked the maintained repository, Gordon–Kuperberg–Patashnik, and the C(12,6,4) certificate standard. No CAS, cloud lab, external proof service, or human validator was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 1584.5s
Review state: evidence receipt failure; not durable progress
Attempt ID: covering-c1563-20260810-230911-c519b1

Human review ledger

No human review recorded.