Exact solver-free classification and parent ownership of two-row prefixes of the forced loopless 4-regular pair-surplus multigraph.

canonical weighted pair-surplus multigraph prefix

Represent two processed surplus rows by a count table of the fifteen allowed incident multiplicity pairs, quotient by transpose, and assign each canonical key to one depth-one partition parent.

Hypothesis: The globally necessary pair-surplus family admits an exact duplicate-free depth-two owner map under S2 x S13.

Test: Enumerate all joint count-table orbits, then require a coefficient-DP and row-stabilizer checker to reproduce every key, labelled total, parent, owner, and digest while rejecting five mutations.

The pair-surplus equations are unconditional for any putative 30-cover. Exact count-table enumeration, a materially different stabilizer/DP checker, deterministic rerun, and fail-closed mutations establish the 82-profile statement at its declared scope. The absent completed-H catalogue and proof leaves prevent any stronger claim.

• Every hypothetical 30-block C(15,6,3) cover induces a loopless weighted 4-regular pair-surplus multigraph h_ij=lambda_ij-4.
• The complete two-row prefix domain of that multigraph has exactly 3,489,526 labelled assignments, 126 S13 ordered-root orbits, and 82 S2 x S13 unordered-root orbits.
• The 82 prefix orbits have an exact unique parent assignment with counts 6,23,15,29,9 across the five depth-one partition types.
• A completed H has one canonical unordered-pair automorphism orbit under the minimum rooted colored-graph code rule, so its owner-pair prefix has one of the 82 keys.
• The maintained exact range remains 30 <= C(15,6,3) <= 31.

• python3 scripts/pair_surplus_depth2_ownership_v1.py --out artifacts/epoch98-20260811/pair-surplus-depth2-ownership-v1/producer-receipt.json
• python3 checkers/check_pair_surplus_depth2_ownership_v1.py --receipt artifacts/epoch98-20260811/pair-surplus-depth2-ownership-v1/producer-receipt.json --out artifacts/epoch98-20260811/pair-surplus-depth2-ownership-v1/independent-check-v2.json
• The producer and rerun receipts are byte-identical with SHA-256 bdaceabb5a5d7eaa366b4e0d472edc4c8328d968115c0e02534f0521f28a8a50.
• The independent receipt has SHA-256 a1cc8ca8a3fa16530c56e751209a4b06e4ac2781ba199008f6b9a5eee66289b6 and records five rejected mutations.

• .proof-experiments/20260811-211856-b4f6eb: producer PASS in 0.122 seconds; 3,489,526 labelled prefixes, 126 ordered orbits, 82 unordered orbits.
• .proof-experiments/20260811-212223-69f500: independent checker PASS in 0.122 seconds; exact key/count/parent agreement and five rejected mutations.
• .proof-experiments/20260811-212234-fe0d77: producer rerun PASS in 0.123 seconds and byte-identical receipt.
• .proof-experiments/20260811-211912-07514b: first allocation-heavy checker implementation ended without experiment metadata or a result artifact and is not evidence.

checkers/check_pair_surplus_depth2_ownership_v1.py uses an x-row partition plus y-stabilizer histogram encoding and coefficient DP, rather than the producer's recursion over joint pair-count types; it reconstructs parents and rejects five 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 orbit case splits for C(12,6,4) -> predict that local UNSAT leaves require a complete owner layer -> observed an exact 82-key two-row owner layer, while complete-H ownership compilation remains open.

• For a putative 30-block cover, h_ij=lambda_ij-4 is a loopless weighted 4-regular multigraph with 30 edge units.
  Pair loads are at least 4; exact point degree 12 gives row surplus 5*12-14*4=4. · Every hypothetical 30-block C(15,6,3) cover. · proved
• The fixed-pair two-row prefix domain has exactly 3,489,526 labelled members and 82 S2 x S13 orbits.
  Count-table producer plus independent coefficient DP and row-stabilizer checker. · All two-row prefixes satisfying the forced row equations and residual degree nonnegativity. · computed
• All 82 prefix orbits have exactly one declared depth-one parent owner.
  Exact parent ledger, independent reconstruction, stable key digest, and rejected deletion, duplication, and owner mutations. · The canonical 82-key prefix catalogue. · computed
• Minimum rooted colored-graph code selects one unordered-pair automorphism orbit in every completed H.
  Equality of two rooted canonical codes supplies an automorphism mapping their colored pairs. · Every finite completed weighted multigraph H. · proved

• Treat the 82 depth-two keys as a complete enumeration or exclusion of pair-surplus multigraphs or covers.
  Any conclusion beyond two processed rows. · The remaining thirteen-by-thirteen weighted submatrix and every cover-completion variable remain unenumerated. · Producer scope limit and technical report. · A complete independently checked canonical H catalogue and terminal completion certificate for every owned family.
• Use allocation-heavy explicit Python materialization of all labelled prefix sequences as the independent checker.
  This depth-two verification task under the current execution transport. · Two invocations left no result artifact; the exact DP/stabilizer method gives the same stronger gate in 0.12 seconds. · artifacts/epoch98-20260811/independent-checker-efficiency-correction-v1.json and the successful independent experiment. · A need for per-labelled-sequence certificates not supplied by exact coefficient and key coverage, plus a bounded streaming implementation.
• Run more isolated fixed-H SAT cylinders before ownership qualification.
  Unowned negative-search leaves analogous to epochs 95 and 97. · They do not aggregate to a global exclusion; this epoch identified full-H owner compilation as the next missing premise. · Prior fixed-H UNKNOWN results and the current scope analysis. · A directly checked SAT witness protocol or a complete independently checked owner manifest.

• Canonical complete-H owner-pair orbit and depth-three augmentation.
  It is the smallest extension that tests whether the exact 82-key layer can become a practical global decomposition. · Dual-code H=3K5 and circulant controls, then enumerate at most 10,000 depth-three nodes with exact parents. · high · open
• Direct constructive exact-degree search with a materially different global mechanism.
  One independently checked 30-block cover settles the target and avoids complete-H proof storage. · Predeclare a new neighborhood or solver encoding and require a five-second direct-model or measured mechanism-level gate. · normal · open
• Owned fixed-circulant 70-root frontier.
  Its local root catalogue is complete and can serve as a canonical-owner compiler control. · Compile ownership predicates only and compare them against the full-H rooted-pair owner on this H. · normal · open

Objective: Qualify a practical canonical owner-pair orbit and measure depth-three growth.

First action: Reproduce the depth-two receipts, then implement rooted colored-multigraph codes on the saved H=3K5 and circulant-H controls before enumerating any new frontier.

Stop condition: Any non-automorphic owner tie, canonicalizer disagreement, missing or duplicate parent, 10,000-node/120-second cap exceedance, or lack of independent reconstruction redirects the route.

• Implement the minimum rooted colored-graph owner-pair orbit with two canonicalizers on H=3K5 and the saved circulant H.
• Generate a depth-three canonical frontier under a 10,000-node or 120-second cap and independently verify every parent and digest.
• Only if that gate passes, estimate complete-H catalogue and certificate volume before compiling SAT leaves.
• Keep a materially different direct constructive route active because one checked 30-block witness bypasses the exclusion portfolio.

• 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://www.math.ucdavis.edu/~deloera/MISC/LA-BIBLIO/trunk/Kuperberg/Kuperberg5.pdf ↗
• https://arxiv.org/abs/2607.23766 ↗

GPT-5.6 Sol principal synthesized and audited the route; two GPT-5.6 Terra delegates supplied advisory prior-art and verification memos, promoted with provenance but not treated as evidence. Python 3.12.3 ran the deterministic producer, coefficient-DP/stabilizer checker, SHA-256 checks, and project computational-researcher experiment harness. Web search checked the maintained LJCR page, Covering Repository endpoint, historical Gordon-Kuperberg-Patashnik source, and arXiv:2607.23766. No SAT solver, PB solver, CAS, proof assistant, cloud lab, external proof service, or human validator was used this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1259.7s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260811-213126-5a7ad3

No human review recorded.