Exact Burnside-only count of the complete four-row degree-feasible prefix universe of the forced weighted 4-regular pair-surplus multigraph, modulo S4 x S11.

canonical pair-surplus prefix augmentation

Count invariant external-vector multisets for every admissible internal weight matrix and processed-vertex permutation, then apply Burnside; independently reproduce with conjugacy-class polynomial DP.

Hypothesis: The complete degree-feasible four-row pair-surplus prefix layer has at most 100000 S4 x S11 orbits.

Test: Compute the exact Burnside count under 120 seconds and 1 GiB, then require a separate conjugacy-class polynomial implementation to reproduce internal, raw, labelled, class-fixed, Burnside, and canonical totals and reject mutations.

Two materially different exact implementations agree on every fixed-point and aggregate total, and fail-closed controls passed. This proves the finite count at the declared degree-feasible prefix scope, but that scope omits later pair-surplus rows and all cover-completion constraints.

• The complete degree-feasible four-row prefix layer of loopless weighted 4-regular multigraphs on 15 vertices has exactly 62437 S4 x S11 orbits.
• Those profiles represent exactly 1341414 S11 external-multiset states and 1292182981632 labelled degree-feasible prefixes.
• The five S4 conjugacy-class fixed counts are 1341414, 21828, 7810, 255, and 106 per permutation for cycle types 1+1+1+1, 2+1+1, 2+2, 3+1, and 4 respectively.
• The maintained covering range remains 30 <= C(15,6,3) <= 31.

• python3 scripts/pair_surplus_depth4_burnside_v1.py --out artifacts/epoch105-20260812/pair-surplus-depth4-burnside-v1/producer-receipt.json --max-orbits 100000 --max-seconds 120
• python3 checkers/check_pair_surplus_depth4_burnside_v1.py --receipt artifacts/epoch105-20260812/pair-surplus-depth4-burnside-v1/producer-receipt.json --out artifacts/epoch105-20260812/pair-surplus-depth4-burnside-v1/independent-check.json
• Producer and rerun are byte-identical with SHA-256 941d5589e59a505a582b897e9bdd2d9486912c85e18cc1416e7fb06b0eaf55e7; independent receipt SHA-256 is 56a977f4bbe37ec8666ec53b883fb1cd3d5245d533dac5f728d4a31b65526ec1.

• .proof-experiments/20260812-021556-dc8d0a: producer returned PASS_BELOW_GATE with 62437 profiles in 118.075 seconds and 61320 KiB peak child memory.
• .proof-experiments/20260812-021820-92d870: independent checker returned PASS_INDEPENDENT in 52.081 seconds and rejected six mutations.
• .proof-experiments/20260812-021935-7938a8: producer rerun returned the byte-identical receipt in 115.617 seconds.

checkers/check_pair_surplus_depth4_burnside_v1.py recursively generates capacity-admissible internal matrices and uses conjugacy-class polynomial multiplication, materially different from the producer's flat all-permutation recursive fixed-point scan.

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 link and orbit decomposition for C(12,6,4) -> predict that forced local structure plus exact orbit ownership can yield manageable certificate cases here -> depth four has 62437 profiles, below the declared 100000 materialization gate but not yet a proof frontier.

• The complete four-row degree-feasible pair-surplus layer has exactly 62437 S4 x S11 profiles.
  Hash-bound producer, independent conjugacy-class polynomial reconstruction, six rejected mutations, and byte-identical rerun. · Four processed vertices and eleven external vertices in loopless weighted degree-4 graph prefixes on 15 vertices. · computed
• The same layer contains exactly 1292182981632 labelled degree-feasible prefixes and 1341414 S11 multiset states.
  Independent ordered-sequence DP and identity fixed-point polynomial coefficient agree with the producer. · The exact depth-four degree-feasible prefix universe. · computed

• Depth-four pair-surplus growth exceeds the predeclared 100000-profile materialization cap.
  The complete degree-feasible S4 x S11 depth-four layer. · The exact independently checked count is 62437. · Producer and independent fixed-point receipts. · A demonstrated semantic mismatch or independent checker counterexample.
• Treat the 62437 count as a complete quotient or exclusion of weighted H or 30-covers.
  Complete multigraphs and cover completions beyond four processed vertices. · Later weights and all triple-cover variables are absent, and no profile manifest or completion certificate exists. · Receipt scope and technical report. · A complete independently checked H catalogue plus terminal cover-completion evidence for every owned class.
• Repeat the 10000-node unfiltered canonical root-link pilot.
  The unchanged epoch-19 root-link mechanism. · It already exceeded its cap before completing the first degree type; no reopen condition was met. · records/attempts/covering-c1563-20260809-042813-d089c5.json · A new sound bulk filter, materially different ownership mechanism, or complete terminal evidence.

• Complete depth-four pair-surplus record and parent ledger.
  The exact count passed the gate and this is the smallest missing ownership layer before later growth and certificate forecasts. · Checkpointed materialization of exactly 62437 records with an independent deletion-parent and coverage checker. · high · open
• Materially different exact-degree-12 constructive cover search.
  A directly checked 30-block list remains the shortest terminal certificate and avoids exhaustive negative proof storage. · Specify a new global encoding or neighborhood outside the closed local-shell and six-representative protocols, then require a short direct-model or measured throughput gate. · normal · open

Objective: Materialize and independently validate the complete depth-four pair-surplus owner ledger.

First action: Implement a checkpointable 62437-record generator seeded by artifacts/epoch102-20260811/pair-surplus-depth3-frontier-v2/producer-receipt.json, then submit it with submit_lab.py.

Stop condition: Stop or redirect on any total mismatch, duplicate or missing profile, foreign deletion parent, independent checker failure, or infeasible runtime/storage projection.

• Implement a checkpointable generator for all 62437 canonical depth-four records and their deletion-parent keys.
• Submit it through submit_lab.py with a 100000-record cap, 1-GiB limit, immutable inputs, progress and checkpoint paths, and a separately written parent checker.
• Forecast later complete-H and replayable proof volume before compiling any SAT leaves.
• Keep a materially different exact-degree-12 constructive route active as the shortest terminal-certificate path.

• 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://doi.org/10.5281/zenodo.21572069 ↗

GPT-5.6 Sol principal investigator; two GPT-5.6 Terra delegates supplied advisory reconnaissance in the injected packet but no mathematical evidence or independent validation. Python 3.12.3, exact integer arithmetic, SHA-256, Burnside fixed-point counting, coefficient and polynomial dynamic programming, and the computational-researcher experiment harness were used. Web access attempted source retrieval but returned no page body, so the pre-acquired source baseline was retained. No SAT/PB solver, CAS, proof assistant, cloud lab, external proof service, publication, or human validator produced evidence.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1346.6s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260812-022937-c43d35

No human review recorded.