Complete exact classification of three-row degree-feasible prefixes of the forced weighted 4-regular pair-surplus multigraph, modulo S3 x S12, with deletion-parent ownership tied to the exact depth-two ledger.

canonical pair-surplus prefix augmentation

Enumerate exact external incidence multiplicity tables, canonicalize processed vertices under S3, and independently derive completeness by Burnside fixed points and an ordered coefficient DP.

Hypothesis: The complete row-degree-feasible depth-three layer of the forced loopless weighted 4-regular pair-surplus multigraph has at most 10000 S3 x S12 orbits, and every canonical child and its depth-two owner can be reconstructed independently.

Test: Enumerate the complete three-row multiplicity-table universe under a 10000-orbit/120-second cap, then require an independently derived Burnside orbit count, ordered coefficient-DP total, parent reconstruction, digest agreement, deterministic reruns, and rejected corruptions.

The producer enumerates a precisely defined finite universe and the checker independently derives its cardinalities without regenerating the producer key set, then checks each distinct record and parent. Agreement forces completeness at the declared scope, but the scope ends at three-row degree-feasible prefixes and therefore cannot settle or narrow the covering number.

• The complete degree-feasible three-row prefix layer of loopless weighted 4-regular multigraphs on 15 vertices has exactly 1775 S3 x S12 orbits.
• Those profiles represent exactly 2942341831 labelled degree-feasible prefixes and 9001 S12 multiset states.
• All 82 exact depth-two pair-surplus profiles occur among the deletion parents of the complete depth-three layer.
• The maintained covering range remains 30 <= C(15,6,3) <= 31.

• python3 scripts/pair_surplus_depth3_frontier_v1.py --out artifacts/epoch102-20260811/pair-surplus-depth3-frontier-v2/producer-receipt.json --max-orbits 10000 --max-seconds 120
• python3 checkers/check_pair_surplus_depth3_frontier_v1.py --receipt artifacts/epoch102-20260811/pair-surplus-depth3-frontier-v2/producer-receipt.json --out artifacts/epoch102-20260811/pair-surplus-depth3-frontier-v2/independent-check.json
• Producer receipt and rerun both SHA-256 0fb9276f5fa614828821a75322fb3e34d5d60edc4396abd2a9c2e4a105f0b5cf; checker receipt and rerun both SHA-256 90bc7ac7b2ef726de96a4e620c18cfdffd0fe9e1102596ff494d7dbde9d208f4.
• Boundary cap 1775 returned PASS_COMPLETE; cap 1774 returned CAP_REACHED with exit code 2.

• .proof-experiments/20260812-000153-18e668: PASS_COMPLETE, 1775 profiles, 9001 S12 states, all 82 parents
• .proof-experiments/20260812-000205-d22905: PASS_INDEPENDENT, Burnside=1775, labelled DP=2942341831, seven mutations rejected
• artifacts/epoch102-20260811/pair-surplus-depth3-boundary-pass-v1.json and pair-surplus-depth3-boundary-reject-v1.json: cap boundary accepted 1775 and rejected 1774

checkers/check_pair_surplus_depth3_frontier_v1.py uses Burnside fixed-point DP and a separate ordered-sequence coefficient DP rather than regenerating the producer key set; it validates every record and parent and rejects seven corruptions.

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

• Canonical augmentation of regular graphs -> predict a compact exact row-prefix quotient with owned deletion parents -> observed 1775 profiles and complete coverage of all 82 depth-two parents.
• Burnside counting from finite-group actions -> predict a completeness check independent of canonical-key generation -> observed exact agreement with the producer and record-weight sum.

• In any putative 30-block cover, h_ij=lambda_ij-4 is a loopless weighted 4-regular multigraph with 30 edge units.
  Point-degree 12 and pair-load at least four imply each surplus row sums to 4; retained epoch-98 source receipt. · Every hypothetical 30-block C(15,6,3) cover. · proved
• There are exactly 1775 S3 x S12 orbits of degree-feasible three-row pair-surplus prefixes, representing 2942341831 labelled prefixes.
  Deterministic producer plus independent Burnside and ordered coefficient-DP checker with matching reruns and mutations. · All symmetric three-row prefixes of loopless weighted degree-4 multigraphs on 15 vertices. · computed
• All 82 exact depth-two profiles extend to at least one degree-feasible depth-three profile.
  Complete parent union hash and independent reconstruction against the epoch-98 key set. · Degree feasibility only, not completion to a full multigraph or cover. · computed

• Depth-three pair-surplus growth exceeds the predeclared 10000-profile cap.
  The complete degree-feasible S3 x S12 three-row layer. · The exact independently checked count is 1775. · Producer and Burnside/DP checker receipts plus the 1775/1774 cap-boundary control. · A demonstrated semantic mismatch in the declared row-prefix universe or an independent checker counterexample.
• Use the 1775 depth-three profiles as a complete quotient of all weighted H or all 30-covers.
  Complete multigraphs and cover completions beyond three processed vertices. · The manifest encodes only row-degree-feasible prefixes; later edges and triple coverage are absent. · Receipt scope and checker scope limit. · A complete hash-bound canonical augmentation through all 15 vertices plus independently checked completion coverage.
• Treat elapsed timing as part of a deterministic mathematical receipt.
  The discarded v1 depth-three producer receipt format. · Elapsed time changes across reruns and is not mathematical content. · Retained v2 removes elapsed time from the receipt and reruns byte-identically. · None; retain timing only in harness metadata or console diagnostics.

• Burnside-only depth-four pair-surplus growth gate.
  It measures the next exact symmetry layer without paying to materialize a potentially infeasible manifest. · Count S4 x S11 fixed points by conjugacy class under a 120-second/1-GiB cap and reproduce totals with a second encoding. · high · open
• Materially different exact-degree-12 constructive search.
  One directly checked 30-block list remains the shortest terminal certificate and is independent of exhaustive negative ownership. · Define a new global move or encoding outside closed support-two/support-three and six-representative scans, then apply a five-second witness/throughput gate. · normal · open
• Canonical root-link catalogue fallback.
  Fixing a 12-block root link is structurally distinct if pair-surplus depth-four growth is too large. · Run a 10000-orbit/120-second independently owned root-link pilot only after the depth-four gate redirects. · low · open

Objective: Measure exact depth-four pair-surplus growth before committing to a manifest or SAT leaves.

First action: Implement scripts/pair_surplus_depth4_burnside_gate_v1.py and run its S4 x S11 fixed-point count through run_experiment.py with --timeout 120 --memory-mb 1024; independently reproduce conjugacy-class totals.

Stop condition: Redirect on more than 100000 projected profiles, 120-second/1-GiB breach, any fixed-point disagreement, or a source-verified exact value; do not compile SAT leaves from an incomplete frontier.

• Implement an exact Burnside-only S4 x S11 depth-four count under 120 seconds and 1 GiB.
• Independently reproduce conjugacy-class fixed-point totals before materializing any depth-four records.
• Continue to a full parent ledger only below 100000 projected profiles; otherwise redirect to a materially new exact-degree-12 constructive encoding.
• Do not compile SAT proof leaves until a complete hash-bound H frontier and feasible replayable certificate forecast exist.

• 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 ↗

GPT-5.6 Sol was principal investigator. Two GPT-5.6 Terra delegates supplied advisory reconnaissance memos that were audited and not counted as validation. Deterministic evidence used Python 3.12.3, exact integer arithmetic, SHA-256, Burnside fixed-point dynamic programming, coefficient dynamic programming, and the computational-researcher run_experiment.py harness. No SAT solver, CAS, proof assistant, cloud lab, external proof service, or human validator produced evidence this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1093.6s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260812-001122-0471b7

No human review recorded.