Audit the proposed 2380-to-5-to-12 pair-excess two-row canonical prefix using an orbit generator, exhaustive labelled reconstruction, and overlap controls.

canonical excess-multigraph prefix partition

Generate degree-feasible two-row orbit representatives under first-row stabilizers, then independently enumerate every labelled pointed two-row state and compare exact orbit keys and multiplicities.

Hypothesis: The 2380 labelled root rows reduce to five root types and twelve disjoint canonical two-row branches with a frontier of at most 126 materialized prefix orbits.

Test: Generate the stabilizer frontier under the predeclared cap 126, independently reconstruct all labelled degree-feasible two-row states, and test whether a single full 4-regular multigraph can reach multiple endpoint branches.

The complete Python frontier and an independent exhaustive C enumeration agree exactly, while the overlap counterexample directly violates the disjointness required for a twelve-leaf full-graph partition.

• The 2380 labelled weak compositions of four into fourteen coordinates have exactly five S_14 orbits.
• The 33320 labelled root-row/endpoint states have twelve coarse root-partition/endpoint-excess classes.
• The 48,853,364 degree-feasible labelled pointed two-row states have exactly 126 orbits under the first-row-and-endpoint stabilizers.
• The twelve coarse endpoint classes are not disjoint after projection to full 4-regular multigraphs.
• No conclusion about existence or nonexistence of a 30-block cover follows.

• python3 scripts/generate_two_row_prefix_v1.py --frontier artifacts/epoch12-20260808/two_row_frontier_v1.jsonl --receipt artifacts/epoch12-20260808/two_row_prefix_generator_receipt.json
• artifacts/epoch12-20260808/reconstruct_two_row_prefix_v1 artifacts/epoch12-20260808/two_row_c_reconstruction_v1.json
• python3 checkers/check_two_row_c_reconstruction_v1.py artifacts/epoch12-20260808/two_row_frontier_v1.jsonl artifacts/epoch12-20260808/two_row_prefix_generator_receipt.json artifacts/epoch12-20260808/two_row_c_reconstruction_v1.json --c-source checkers/reconstruct_two_row_prefix_v1.c --c-binary artifacts/epoch12-20260808/reconstruct_two_row_prefix_v1 --receipt artifacts/epoch12-20260808/two_row_prefix_checker_receipt.json
• Clean replay reproduced the frontier, C reconstruction, and checker receipt byte-for-byte and verified all 14 hashes in artifacts/epoch12-20260808/hash_manifest.json.

• .proof-experiments/20260808-225819-9f0ff9: Python generator emitted exactly 126 orbit records in 0.124 seconds.
• .proof-experiments/20260808-225853-fae77a: literal Python labelled reconstruction timed out at 120 seconds and has no mathematical force.
• .proof-experiments/20260808-230306-a9813b: first C implementation timed out because of string construction and linear orbit lookup.
• .proof-experiments/20260808-230608-bfdd33: optimized exhaustive C replay visited 48,853,364 states in 7.803 seconds and emitted 126 orbit codes.
• .proof-experiments/20260808-230644-0b582f: final set, multiplicity, permutation, overlap, and corruption validator passed.

checkers/reconstruct_two_row_prefix_v1.c is a separately written exhaustive labelled enumerator; checkers/check_two_row_c_reconstruction_v1.py compares its 126 codes and multiplicities against the Python frontier and runs adversarial controls.

not_requested

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

Original model outcome

progress

Public classification

progress

• Certified C(12,6,4) frontier discipline -> every proof leaf needs a unique, complete branch identifier -> the chosen-endpoint twelve-way split fails because one full graph receives multiple identifiers.

• In any hypothetical 30-block cover, E_xy=lambda_xy-4 is a loopless 4-regular multigraph.
  lambda_xy>=4 and sum_(y!=x)lambda_xy=12*5=60, hence every excess row sums to four. · Every hypothetical 30-block C(15,6,3) cover. · proved
• There are exactly 126 degree-feasible pointed two-row stabilizer orbits.
  Exact agreement of the Python orbit generator and independent C reconstruction over 48,853,364 labelled states. · Fixed-root pointed prefixes satisfying sum(a)=4, sum(b)=4-a_i, and b_j<=4-a_j. · computed
• The twelve endpoint classes are not a disjoint partition of rooted full excess multigraphs.
  Multiplicity-four edge 0--1 plus doubled 13-cycle reaches q=4 and q=0 at root 0. · The advertised classification by root partition and an arbitrarily chosen second vertex. · proved

• Use the twelve root-partition/endpoint-excess labels as a disjoint canonical two-row SAT frontier.
  All rooted loopless 4-regular excess multigraphs when the chosen endpoint is not part of the target object. · The labels classify pointed states; one full graph can occur in multiple endpoint classes, and materialized pointed prefixes refine to 126 orbits. · artifacts/epoch12-20260808/two_row_prefix_checker_receipt.json and artifacts/epoch12-20260808/two_row_prefix_scope_lemma.md · Supply an explicit canonical endpoint or full canonical-augmentation rule and independently prove that every full graph maps to exactly one hash-bound leaf.

• Degree-preserving constructive local search
  A 30-block hit settles the target with a compact direct certificate, and exact point degree twelve is free structural guidance. · Compare fixed-seed two-block degree-preserving exchanges with block-count-only replacements under identical proposal counts. · high · open
• Canonical endpoint/full-graph augmentation
  It is the missing completeness mechanism needed to convert pair-excess structure into replayable UNSAT leaves. · Define a lexicographic endpoint-selection rule and exhaustively test uniqueness and coverage on a bounded complete 4-regular multigraph catalogue. · normal · open
• Prefix extendability filters for the 126 pointed types
  Sound residual-degree or graphicality filters could eliminate whole prefix classes before SAT. · Apply an independently checked loopless-multigraph residual-degree feasibility test to all 126 records. · low · open

Objective: Measure whether exact-degree-preserving constructive moves improve proximity to a 30-block cover.

First action: Implement scripts/degree_preserving_local_search_v1.py, then invoke it through run_experiment.py for matched fixed seeds and proposal counts against a block-count-only control.

Stop condition: Stop immediately on a dual-validated 30-block cover; otherwise redirect if degree-preserving search does not improve the predeclared best and distributional uncovered-triple metrics.

• Implement a deterministic degree-preserving local search maintaining all point degrees at twelve and incrementally updating 455 triple deficits.
• Run matched fixed-seed degree-preserving and block-count-only controls with identical proposal counts.
• Validate any 30-block witness with the existing independent Python and C cover checkers.
• Reopen excess-profile SAT leaves only after a canonical endpoint/full-graph rule has a complete independently checked frontier.

• 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/abs/2607.23766 ↗
• https://arxiv.org/abs/math/9502238 ↗

GPT-5.6 Sol served as principal investigator. GPT-5.6 Terra delegates supplied advisory challenge and experiment-design memos; their agreement was not counted as validation. Deterministic tools were Python 3.12.3, GCC 13.3.0, SHA-256, and the computational-researcher experiment harness. No SAT/SMT solver, CAS, proof assistant, or proof-log checker was used in the decisive result.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1369.4s

Review state

not a result claim

Attempt ID

covering-c1563-20260808-231300-9d5b0b

No human review recorded.