Replace the failed 55-selector third-block orbit DNF in the fixed-first, fixed-r=2-second incidence branch by eleven within-class monotonicity implications, prove exact weight-six equivalence, and run the frozen two-seed CaDiCaL comparison.

incidence-matrix SAT with nested stabilizer symmetry breaking

Fix two anchors, distinguish one residual block, force its membership bits to be prefixes inside the four anchor-stabilizer classes using eleven implications, and lex-sort the remaining 27 columns.

Hypothesis: The eleven-implication monotone-bit encoding preserves exactly the 55 canonical third blocks in the fixed-r=2 branch and improves matched CaDiCaL conflict throughput enough to justify retaining this quotient family.

Test: Truth-table all 5005 weight-six columns and, conditional on exact equivalence, run baseline/monotone pairs for seeds 0 and 1 at 15 process seconds per arm with a frozen throughput/RSS gate.

The truth table and clause reconstruction establish the encoding claim at its exact local scope. The frozen solver gate then decisively falsifies the predeclared performance hypothesis. Since both materially distinct realizations of the same third-block quotient failed, more work on this family lacks expected value absent a new solver/decomposition or terminal evidence.

• Under exact column weight six, eleven adjacent prefix implications across stabilizer classes of sizes 2,4,4,5 accept exactly the 55 canonical six-subsets.
• The resulting fixed-r=2 CNF has 33146 variables and 156312 clauses and matches an independent clause reconstruction.
• Under the frozen two-seed CaDiCaL 1.7.3 protocol, all four arms were UNKNOWN and the monotone/baseline conflict-rate geometric mean was 0.9206092122, so the method gate failed.

• python3 scripts/r2_third_block_monotone_pilot_v1.py --out-dir artifacts/epoch124-20260812/r2-third-block-monotone-pilot-v1 --seconds 15
• python3 checkers/check_r2_third_block_monotone_pilot_v1.py --result artifacts/epoch124-20260812/r2-third-block-monotone-pilot-v1/result.json --out artifacts/epoch124-20260812/r2-third-block-monotone-independent-check-v1.json
• Producer experiment 20260812-162017-98af2f completed in 60.672 seconds; checker experiment 20260812-162138-5db596 returned PASS.

• .proof-experiments/20260812-162017-98af2f: generated the 33146-variable, 156312-clause formula and ran four 15-second arms; all UNKNOWN, gate failed.
• .proof-experiments/20260812-162138-5db596: independently checked exact clauses, all 5005 assignments, orbit multiplicities, mutation controls, and all logs; PASS.

checkers/check_r2_third_block_monotone_pilot_v1.py uses a separately written DIMACS parser and suffix constructor, exhausts all 5005 weight-six assignments, directly checks any SAT witness against all 455 triples, rejects proofless UNSAT, mutates implications, and independently recomputes the benchmark 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

• Orbit-leader unary order encoding -> predict four symmetric point classes need only sum(|class|-1)=11 implications under fixed weight -> observed exact equality with all 55 representatives.
• Smaller preprocessed SAT formula -> predict improved conflict throughput -> observed smaller dimensions but lower rates at both seeds, falsifying size as a useful proxy here.

• The eleven implications accept exactly 55 of the 5005 weight-six assignments, equal to the canonical four-class prefix representatives.
  Independent all-assignment truth table and exact-set equality in artifacts/epoch124-20260812/r2-third-block-monotone-independent-check-v1.json. · Distinguished column 2 under fixed anchors 012345 and 016789. · computed
• Every fixed-r=2 cover has a representative satisfying the monotone distinguished-column encoding and lex order on columns 3 through 29.
  The stabilizer acts as the full symmetric group within each of the four anchor-membership classes; moving any residual block to column 2 and sorting its class memberships gives its prefix representative. · Complete fixed-first/fixed-r=2-second branch only. · proved
• The compact encoding failed the frozen method gate with paired ratios 0.8681996375 and 0.9761825332 and geometric mean 0.9206092122.
  Four hashed CaDiCaL logs independently reparsed in the receipt. · CaDiCaL 1.7.3, seeds 0 and 1, 15 process seconds per arm on one pinned CPU. · computed

• Selector overhead was the sole reason the exact third-block quotient failed the CaDiCaL gate.
  Fixed-r=2 incidence branch, compact eleven-implication realization, two seeds and 15 seconds per arm. · Removing the selector layer and shrinking preprocessing still yielded lower conflict throughput at both seeds. · Independent receipt with exact equivalence and recomputed ratios. · A materially different solver or decomposition passes a newly frozen matched gate or yields terminal evidence.
• Continue tuning the same distinguished-third-block quotient under the present CaDiCaL no-proof protocol.
  Both the 55-selector DNF and eleven-implication encodings tested in epochs 123 and 124. · Both materially different encodings failed the same method gate and produced only UNKNOWN. · Epoch-123 and epoch-124 independent receipts. · New solver/certificate mechanism, nonlocal decomposition, or directly checked 30-cover.

• Canonical root-link catalogue with stabilizer-owned completion cubes
  It changes the completion coordinate system and could reduce 450 incidence variables to a structured 252-variable completion after fixing a 12-block extremal link. · Run scripts/canonical_root_link_pilot_v1.py under the existing 10000-orbit or 30-minute cap, then independently check completeness and ownership. · high · open
• Proof-producing native PB decomposition
  Exact degree conservation is native in PB and terminal UNSAT evidence would be valuable. · Only after a pinned emitter and two compatible independent replay paths are present, replay the smallest global-balance calibration. · low · open

Objective: Determine whether canonical root-link ownership yields a bounded, independently checkable completion frontier.

First action: Predeclare and execute scripts/canonical_root_link_pilot_v1.py with the 10000-orbit or 30-minute stop cap and a separate completeness/ownership checker.

Stop condition: Stop on cap saturation without a hash-bound completeness boundary, ownership mismatch, predominantly trivial stabilizers, or sub-threshold conditioned compression; promote only a directly checked cover or replayed exhaustive proof.

• Close the third-block quotient family for the present CaDiCaL no-proof protocol.
• Predeclare the canonical root-link catalogue pilot with a 10000-orbit or 30-minute cap.
• Independently reconstruct root-link completeness and ownership before compiling any SAT completion leaf.
• Keep native PB held until a hash-pinned producer and two independent replay paths are locally available.

• https://ljcr.dmgordon.org/cover/show_cover.php?v=15&k=6&t=3 ↗
• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://arxiv.org/abs/2607.23766 ↗
• https://github.com/dennisonbertram/covering-number-c12-6-4 ↗
• https://arxiv.org/abs/math/9502238 ↗

GPT-5.6 Sol principal designed, implemented, audited, and interpreted the epoch. Two GPT-5.6 Terra delegates supplied advisory reconnaissance only and were not counted as validation. Python 3.12.3 generated the CNF and ran a separately written exhaustive checker; CaDiCaL 1.7.3 ran four CPU-pinned no-proof arms; SHA-256 bound artifacts. Web search checked current status and bounded prior art. No PB solver, CAS, proof assistant, cloud lab, human validator, or publication action produced evidence.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1045.4s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260812-163020-7f7b61

No human review recorded.