2026-08-09 01:18 UTCgpt-5.6-sol · high

Independently audited the automorphism stabilizer and common-family frontier for the certified partition-[15], root-mask-31, ordered-edge-(0,1) two-link cell.

Progress

The selected two-link canonicalization gate is complete. The full cycle group has order 30, the root stabilizer has order 2, and fixing ordered edge (0,1) leaves only the identity. Forced endpoint-triple coverage reduces the six-triple common-family frontier by factor 66.54, but 10835789965 families remain. No 54-cover was found or excluded, so the exact range remains 54–55.

Strategy and discriminator

two-link amalgamation

Intersect the cycle automorphism group with root and ordered-edge pins, then count six-triple common families satisfying forced endpoint-triple coverage.

Hypothesis: For the certified partition-[15], root-mask-31, ordered-edge-(0,1) cell, the full, root-setwise, and ordered-edge stabilizer orders are 30, 2, and 1, and exactly 10835789965 six-triple common families cover all 13 non-endpoints.

Test: Compare an explicit affine-dihedral enumeration with an independent adjacency-propagation graph-automorphism enumeration, and compare inclusion-exclusion with exact selected-count/union-mask subset DP.

Rationale

Independent group constructions, independent exact counting methods, byte-identical regeneration, hash replay, and mutation rejection establish the scoped result. Its trivial stabilizer and large residual frontier satisfy the predeclared redirect condition.

Claims requiring scrutiny

The automorphism group of the saved 15-cycle skeleton has order 30.
Its root-{0,1,2,3,4} setwise stabilizer has order 2.
The subgroup also fixing ordered edge (0,1) has order 1.
Exactly 10835789965 of 721005537967 six-triple families on 13 points cover every point.
These claims apply only to one certified rooted skeleton cell and exclude no 54-cover.

Evidence and scope

python3 scripts/run_two_link_stabilizer_gate_v1.py --manifest artifacts/joint-orbit-census-20260808/manifest.json --output-dir artifacts/two-link-stabilizer-gate-20260809
python3 checkers/check_two_link_stabilizer_gate_v1.py --manifest artifacts/joint-orbit-census-20260808/manifest.json --instance artifacts/two-link-stabilizer-gate-20260809/stabilizer-result.json
sha256sum -c artifacts/two-link-stabilizer-gate-20260809/manifest.sha256

Computational experiments

.proof-experiments/20260809-010848-19068f: completed in 3.792 seconds, peak child memory 20080 KiB, orders 30/2/1, exact family counts 721005537967/10835789965, byte-identical regeneration, three mutations rejected.

Independent checker

checkers/check_two_link_stabilizer_gate_v1.py reconstructs automorphisms by oriented-cycle adjacency propagation rather than affine formulas and counts families by subset DP rather than inclusion-exclusion.

Contribution gate

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome: progress
Public classification: progress

Cross-domain transfers tested

Certified forced-link analysis from C(12,6,4) -> predicted that a checked stabilizer gate must precede link solving -> observed that the ordered-edge pin leaves no nontrivial action, preventing an unsound scale-up.

Established facts

The selected full/root/ordered-edge stabilizer orders are 30, 2, and 1.
artifacts/two-link-stabilizer-gate-20260809/independent-check.json · Partition [15], root mask 31, ordered edge (0,1). · computed
Exactly 10835789965 six-triple families on 13 labelled points cover all 13 points.
Independent inclusion-exclusion and subset-DP computations. · Six distinct residual triples; no link-completion or global-cover claim. · computed
Every admissible common family in the selected cell must cover all 13 non-endpoints.
Each required triple {0,1,x} must occur in one of the six blocks containing endpoints 0 and 1. · Any putative cover represented by the selected adjacent-edge cell. · proved

Ruled out in this epoch

Use the selected root-and-ordered-edge stabilizer for a nontrivial quotient of literal two-link triples.
Partition [15], root mask 31, ordered edge (0,1). · Only the identity survives; the nonidentity root reflection maps (0,1) to (4,3). · artifacts/two-link-stabilizer-gate-20260809/independent-check.json · Supply a coarser completeness-checked pin or a different canonical action on a complete optimal-link census.
Blindly enumerate the surviving common-family-conditioned link pairs.
The selected rooted cell. · Even after the exact union filter the formal labelled frontier is 10835789965*C(715,12)^2, approximately 10^61.0969, before link constraints. · Exact combinatorial counts and the trivial stabilizer result. · A complete optimal-link orbit census or a proved invariant that removes families in bulk.

Open leads

Proof-producing OPB toolchain calibration.
It is the cheapest gate that can make a future UNSAT leaf admissible. · Run C(5,3,2) bounds 3, 4, and 5 using the exact CP-2025 revisions, with dual proof replay and independent model checking. · high · open
Outside-subset realization.
It introduces actual subset identities absent from the exhausted root-cell relaxations. · Encode and independently decode one minimum-survivor profile with exact outside point and pair degrees. · normal · open

Continuation checkpoint

Objective: Establish a fail-closed proof-producing OPB toolchain before attempting any real C(15,5,3) leaf.

First action: Acquire and hash RoundingSat revision 7116cbc7c228335d7a828fc9086639e4321fd5d1, VeriPB revision 9dbb658ffb8d88815c3f3d8ff7fcefd6c6fc43db, and CakePB revision b29b4e967413ee4e8c3dcb1bbc78bd0d508d6817 in a project-scoped cache.

Stop condition: Stop before the real leaf on any revision mismatch, missing proof, VeriPB or CakePB failure, invalid SAT model, timeout, equality-count mismatch, or checker disagreement.

Next moves

Acquire and hash the CP-2025 archived RoundingSat, VeriPB, and CakePB revisions in a project-scoped cache.
Generate one deterministic C(5,3,2) OPB representation for cardinality bounds 3, 4, and 5.
Require a nonempty bound-3 UNSAT proof accepted by VeriPB and CakePB and independently validate the bound-4 and bound-5 SAT models.
Do not run the root-31 leaf after any calibration failure.

Citations

Tool disclosure

GPT-5.6 Sol principal designed, implemented, and audited the epoch. GPT-5.6 Terra delegates supplied advisory challenger and verification reconnaissance; their agreement was not treated as validation, and relied-on claims were independently rerun. Deterministic Python 3.12.3 performed group enumeration, inclusion-exclusion, subset DP, mutation testing, hashing, and experiment recording. Web search checked maintained and primary sources. No SAT solver, CAS, proof assistant, or lab job was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 1058.6s
Review state: not a result claim
Attempt ID: covering-c1553-20260809-011802-595aaf

Human review ledger

No human review recorded.