Exact marked-automorphism quotient of every rooted [3^5] pair-excess skeleton placement against all 353 surviving coincident-link interfaces.

triangle-rooted skeleton/interface orbit compatibility

Incidence-graph automorphisms generate placement orbits of four unordered triangles on the 12 nonroot points; a signature-based implementation independently reconstructs each orbit and checks its count by Burnside's lemma.

Hypothesis: Quotienting the 15,400 rooted [3^5] skeleton placements by each surviving interface's exact marked automorphism group reduces the complete 353-interface attachment frontier by at least a factor of two.

Test: Generate one least-index representative for every placement orbit, then independently reproduce all representatives from six-triple membership signatures and verify every row with Burnside's lemma.

Producer/checker agreement, rowwise Burnside identities, mutation rejection, byte-identical regeneration, and hash-bound upstream inputs make the quotient independently reproducible. Quotienting removes only relabelling-equivalent placements and supplies no extendability conclusion by itself.

• The 5,436,200 rooted [3^5] relative placements across all 353 saved B interfaces form exactly 471,270 orbits under the respective marked interface automorphism groups.
• The exact representation reduction factor is 11.535213359645214, with per-interface orbit counts between 8 and 9,100.
• The maintained covering-number range remains 54 <= C(15,5,3) <= 55.

• Producer experiment 20260812-173456-fbc379 completed in 61.586 seconds with 353 rows and 471,270 representatives.
• Independent experiment 20260812-173612-a743c6 reconstructed 78,850 signature-group elements, matched 471,270 representatives, and checked 3,451 Burnside terms.
• Mutation experiment 20260812-173741-155af0 rejected changed-representative, changed-source-binding, and dropped-row ledgers.
• Regeneration experiment 20260812-173754-b07147 reproduced attachment-orbits.jsonl and result.json byte-identically.
• sha256sum -c artifacts/triangle5-interface-attachment-orbits-20260812/manifest.sha256 passes.

• .proof-experiments/20260812-173456-fbc379: producer PASS, 5,436,200 placements to 471,270 orbits in 61.586 seconds.
• .proof-experiments/20260812-173612-a743c6: independent checker PASS on all representatives and Burnside rows in 45.715 seconds.
• .proof-experiments/20260812-173741-155af0: three semantic/coverage mutations rejected.
• .proof-experiments/20260812-173754-b07147: byte-identical producer regeneration in 64.140 seconds.

scripts/check_triangle5_interface_attachment_orbits_v1.py uses six-triple membership signatures and triple-coordinate permutations rather than the producer's NetworkX incidence-graph isomorphism engine; it reconstructs every orbit and separately verifies counts via Burnside cycle types.

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 covering search for C(12,6,4) -> forced-link orbit reduction should compress rooted C(15,5,3) attachment cases by at least twofold -> the complete [3^5] attachment family compressed by factor 11.535213359645214.

• Exactly 471,270 marked-automorphism orbits cover all 5,436,200 rooted [3^5] relative placements for the 353 saved B interfaces.
  Receipt SHA-256 b06b90991f05e829de1b3ec2bca1a3a247ccf12178e7ea4d6109a3097f6a5c21 and independent-check status PASS. · The complete local [3^5] interface/placement product only. · computed

• Treat all 15,400 [3^5] placements per interface as inequivalent completion cases.
  All 353 saved B interfaces under their exact marked automorphism groups. · The exact orbit ledger identifies 4,964,930 duplicate representations and leaves 471,270 canonical cases. · artifacts/triangle5-interface-attachment-orbits-20260812/independent-check.json · A future test that is not invariant under marked point relabelling.
• Repeat the profile-level pair/point count relaxation as the next triangle-compatibility test.
  All 1,059 interface/profile cells from epoch 124. · Every cell already has an independently checked feasible integral witness. · artifacts/triangle-skeleton-count-compatibility-20260812/independent-check.json · A proved higher-order constraint not implied by the same pair and point marginals.

• Block-decomposition-aware filter on canonical [3^5] attachments.
  It adds actual 4-set identities absent from the closed count-only relaxation while operating on an exact 11.535-times smaller frontier. · Pre-count candidate residual-side decompositions, then run a bitset-batched feasibility filter with an independent map checker if bounded. · high · open
• Coverage-aware 3-for-3 census at the defect-nine constructive seed.
  A lower-defect family or 54-cover is terminally useful and materially distinct from the exclusion route. · Build two independent candidate counters and proceed only if exact coverage-aware candidates are at most 2^22. · normal · open
• Quotient [3^3,2^3] and [3,2^6] attachments.
  The verified group action may compress the remaining profiles, but their value should be measured only after the [3^5] higher-order filter shows benefit. · Derive Burnside counts without materializing representatives and compare projected frontier sizes. · low · open

Objective: Convert the exact [3^5] attachment quotient into a genuine whole-orbit elimination using actual residual 4-set decomposition information.

First action: Write an efficiency protocol that pre-counts residual 4-set candidates for all 471,270 canonical representatives and proves the filter is stronger than the closed pair/point marginal system.

Stop condition: Stop or redirect if the pre-count exceeds the approved bound, the filter is equivalent to prior marginals, no orbit is rejected, producer/checker maps disagree, or any exclusion lacks a replayable certificate.

• Derive and pre-count a placement-specific residual 4-set decomposition filter on the 471,270 canonical representatives.
• Implement the filter with bitset batching and an independent rejection-map reconstruction; stop if it is only a restatement of pair/point marginals or rejects zero orbits.
• Retain the constructive coverage-aware 3-for-3 census around the defect-nine seed as the next distinct route if the attachment filter fails its gate.

• https://www.coveringrepository.com/systems.aspx?k=5&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=5&t=3&v=15 ↗
• https://arxiv.org/abs/2607.23766 ↗

GPT-5.6 Sol was the principal investigator. Two GPT-5.6 Terra delegates supplied advisory reconnaissance; their agreement was not treated as evidence, and the fixed-pair rerun, repeated count-only relaxation, and unbounded constructive census were rejected after audit. Deterministic work used Python 3.12.3, NetworkX 3.3, exact integer/group algorithms, the computational-researcher experiment harness, SHA-256, a separate membership-signature checker, Burnside's lemma, mutation controls, and byte-identical regeneration. No SAT solver, CAS, proof assistant, cloud lab, package installation, system change, external write, or subagent was used in the decisive result.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1751.4s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260812-174704-04afec

No human review recorded.