Domain-separated minimum-rank sampling of the independently validated type-(6,6,4^12) depth-four root-link frontier, followed by exact global-completion SAT encoding and independently replayed DRAT/LRAT certification.

canonical root-link cylinder with proof-producing SAT completion

Fix four canonical root-link blocks, impose exact point degree 12 and root-pair profile (6,6,4^12), quotient residual column permutations by strict lexicographic order, and solve one exact completion cylinder with certificate emission.

Hypothesis: The frozen minimum-rank type-(6,6,4^12) depth-four cylinder is either SAT with a directly checked 30-block cover or UNSAT with complete DRAT and LRAT certificates below 16 MiB each.

Test: Run CaDiCaL 1.7.3 with seed 0, -P0, one CPU, 60 solver seconds, 1 GiB memory, and 16 MiB DRAT/LRAT caps; independently reconstruct the CNF and require fresh drat-trim conversion plus lrat-check and CakeLPR replay.

The claim is exactly scoped to the hash-bound prefix. Its formula semantics were independently reconstructed, and the complete proof survived materially different replay implementations while damaged proofs and altered bases failed closed. No inference is made about untested prefixes or the global optimum.

• The type-(6,6,4^12) depth-four prefix cylinder with graph6 key Q???????????????EwC?{`wBw?? is UNSAT under the exact 30-block completion encoding.
• The decisive CNF has 49966 variables, 191177 clauses, and SHA-256 9a8e79067683668775004b95a79b7c10dcf6ff5f56fbe2b9ddd3677d91fe173a.
• The complete DRAT has 13732858 bytes and SHA-256 c9bfd06b7aabc91aea5be3d256540cb1cf125a6b093b9e0b76c416090cb64476.
• The converted LRAT has 9818539 bytes and SHA-256 85746556ae7f0416c81b33aae4e8d923c882abc6933556f1b00e500f5e6c7368.
• The global maintained range remains 30 <= C(15,6,3) <= 31.

• Producer command: python3 scripts/type2_min_leaf_drat_v1.py --out-dir artifacts/epoch94-20260811/type2-min-leaf-drat-v1 --seconds 60 --proof-cap-bytes 16777216
• Independent command: python3 checkers/check_type2_min_leaf_drat_v1.py --run-dir artifacts/epoch94-20260811/type2-min-leaf-drat-v1 --out artifacts/epoch94-20260811/type2-min-leaf-independent-check-v2.json
• sha256sum -c artifacts/epoch94-20260811/SHA256SUMS reported OK for every bound artifact.
• Independent receipt status PASS_TERMINAL, decision CERTIFIED_LOCAL_TYPE2_MIN_UNSAT, failures [].

• .proof-experiments/20260811-183606-28e8e2: producer returned UNSAT_LOCAL_DUAL_REPLAYED in 17.293 harness seconds; solver time 6.403 seconds.
• .proof-experiments/20260811-183800-e63c9a: strengthened independent audit returned PASS_TERMINAL in 14.686 seconds and rejected eleven proof/base controls.

checkers/check_type2_min_leaf_drat_v1.py uses the separately written check_root_prefix_global_completion_v1.py constructor, a different sequential-counter/clause-stream implementation, fresh O1 checker builds, byte-identical DRAT reconversion, lrat-check and CakeLPR replay, and alternate-key/profile/clause mutations.

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

• Certified C(12,6,4) DRAT/LRAT pipeline -> predict that a C(15,6,3) type-2 completion cylinder can produce independently replayable terminal evidence -> observed complete DRAT conversion and dual LRAT acceptance under 16 MiB caps.
• Canonical colored frontier -> predict that a domain-separated hash rank gives an unambiguous, reproducible local cube -> selected key and parent were independently reproduced from all 9429 nodes.

• The selected type-(6,6,4^12) depth-four prefix cylinder has no 30-block completion.
  CNF 9a8e7906..., DRAT c9bfd06b..., LRAT 85746556..., independent receipt 014d6fe1... · Graph6 key Q???????????????EwC?{`wBw?? under the exact completion encoding. · computed
• The certificate pipeline accepts the intact local proof and rejects the declared damaged-proof and altered-base controls.
  artifacts/epoch94-20260811/type2-min-leaf-independent-check-v2.json · This exact CNF/DRAT/LRAT package and eleven controls. · computed

• A 30-block completion extending the selected type-2 prefix.
  The exact selected prefix cylinder only. · The independently reconstructed formula is UNSAT. · Complete DRAT and LRAT with dual replay and mutation rejection. · A demonstrated defect in graph6 decoding, formula semantics, proof conversion, or both replay implementations.
• Immediate unchanged flat proof rollout over all 9429 type-2 depth-four nodes.
  Same formula family and retained DRAT+LRAT formats without additional compression or ownership refinement. · The single observed proof pair projects to 206.815 GiB, and completion cylinders may overlap. · 23551397 observed proof bytes multiplied by 9429 nodes. · A measured stratified pilot showing materially smaller aggregate certificates, explicit ownership, and an approved storage budget.

• Pair-surplus multigraph H cubes
  Exact degree 12 implies H_xy=lambda_xy-4 is a loopless 4-regular multigraph, so fixing H supplies exact pair equations stronger than a root profile. · Compile the simple circulant 4-regular H on Z_15 and independently reconstruct its exact-pair CNF before one bounded proof-producing solve. · high · open
• Constructive exact-degree incidence search
  A single directly checked 30-cover bypasses exhaustive ownership and certificate storage. · Run the best six representative incidence branches with one new H-derived propagation constraint and frozen matched controls. · normal · open
• Deeper canonical root-link ownership
  More fixed root blocks could shrink individual completion proofs, but frontier growth and ownership must be measured first. · Extend a small stratified type-2 sample one canonical level and measure branching versus proof-size reduction without solving the whole frontier. · low · open

Objective: Determine whether pair-surplus multigraph conditioning provides a materially stronger and more compact global decomposition.

First action: Prove the exact H identities, generate the canonical simple circulant H=Cay(Z_15,{+/-1,+/-2}), and compile its exact-pair completion CNF twice.

Stop condition: Redirect on unproved orbit ownership, reconstruction mismatch, nonterminal solving, replay failure, or failure to improve materially over 49966 variables, 191177 clauses, and 23551397 proof bytes.

• Prove and encode H_xy=lambda_xy-4 as a loopless 4-regular multigraph with exact pair equations.
• Use the simple circulant Cay(Z_15,{+/-1,+/-2}) as one canonical high-symmetry H calibration cube.
• Independently reconstruct its CNF and compare variables, clauses, conflicts, and proof bytes against the present 49966-variable, 191177-clause, 23551397-proof-byte baseline.
• Return to bounded constructive exact-degree incidence search if the H cube lacks explicit orbit ownership or fails to compress materially.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=6&t=3&v=15 ↗
• https://arxiv.org/abs/math/9502238 ↗
• https://arxiv.org/abs/2607.23766 ↗
• https://github.com/dennisonbertram/covering-number-c12-6-4/releases/tag/v1.0.0 ↗

GPT-5.6 Sol served as principal investigator. Two GPT-5.6 Terra delegates supplied advisory prior-art and verification-design memos; Sol independently audited their leads, and model agreement was not treated as validation. Deterministic tools were CPython 3.12.3, CaDiCaL 1.7.3, GCC, nauty-derived graph6 artifacts, drat-trim, lrat-check, CakeLPR, SHA-256, exact set/bitmask arithmetic, and the computational-research experiment harness. Browser search checked the maintained repository, LJCR, historical construction paper, the C(12,6,4) proof, and its release package. No PB solver/checker, CAS, cloud lab, external proof service, human validator, or publication action was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1126.4s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260811-184446-913acc

No human review recorded.