Exhaustively scan the frozen 2258-node type-(8,4^13) depth-four root-link frontier using exact individual-degree subset and cut capacities, then replay every blocker with a separately written row-load dynamic program.

exact individual-degree subset-capacity filtering

For every subset S, enumerate the eight residual block intersection-size profiles compatible with every point degree on S and its complement; reject a prefix when its uncovered internal or crossing pair demand exceeds the exact degree-profile capacity.

Hypothesis: At least 2033 of the 2258 frozen type-(8,4^13) depth-four root-link nodes violate an exact individual-degree subset or cut capacity inequality.

Test: Scan all proper subsets of all 2258 nodes, require at least 2033 distinct blockers, and independently replay every claimed witness with a non-Gale-Ryser row-load DP.

Every actual residual link completion induces one of the enumerated row-load profiles, so each violated capacity is a sound non-completion certificate. Independent reconstruction validates the three incremental exclusions. The small observed increment does not justify catalogue scale-up or a global inference.

• The depth-four key Q???????????????@wG@]?FoFo? cannot extend to a type-(8,4^13) 12-block root link.
• The depth-four key Q???????????????^??N_Oy@{?? cannot extend to a type-(8,4^13) 12-block root link.
• The depth-four key Q???????????????^??WMA`pu?? cannot extend to a type-(8,4^13) 12-block root link.
• At least 167 of the frozen 2258 type-(8,4^13) depth-four root-link orbits are non-completable.
• The maintained global range remains 30 <= C(15,6,3) <= 31.

• Primary: python3 scripts/exact_subset_capacity_v1.py --source artifacts/epoch19-20260809/root_link_canonical_primary_receipt.json --old-receipt artifacts/epoch62-20260810/root_link_subset_capacity_primary_receipt.json --out artifacts/epoch107-20260812/exact-subset-capacity-primary.json
• Independent: python3 checkers/check_exact_subset_capacity_v1.py --source artifacts/epoch19-20260809/root_link_canonical_primary_receipt.json --old-receipt artifacts/epoch62-20260810/root_link_subset_capacity_primary_receipt.json --result artifacts/epoch107-20260812/exact-subset-capacity-primary.json --rerun artifacts/epoch107-20260812/exact-subset-capacity-rerun.json --out artifacts/epoch107-20260812/exact-subset-capacity-independent-check-v2.json
• Rejected-key SHA-256 4de42dd1801958b008d2f6a9a4d79dedcdcf859c719cff6d835b2bd04baeeec9.
• sha256sum -c artifacts/epoch107-20260812/SHA256SUMS passed for every bound artifact.

• .proof-experiments/20260812-040423-97a685: primary complete scan, 167 blockers, 66.484 seconds, 30,736 KiB peak, gate failed.
• .proof-experiments/20260812-040608-3dc5da: fresh producer rerun, all stable fields identical, 64.768 seconds.
• .proof-experiments/20260812-041112-f996b2: final independent replay, 167 witnesses reconstructed, five mutations rejected, 1.588 seconds.

checkers/check_exact_subset_capacity_v1.py uses exact labelled-row reachability DP and complementary 5-s matching; it does not call Gale-Ryser or import producer code.

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-orbit decomposition for C(12,6,4) -> predict that a sound bulk root-link filter could make canonical ownership tractable -> exact individual-degree capacity added only three blockers, so the bulk-filter prediction failed and unfiltered augmentation remains closed.

• The exact individual-degree subset-capacity inequalities hold for every residual block completion.
  Incidence-matrix proof in exact-subset-capacity-lemma.md and independent DP equivalence checks. · Any R-block b-uniform residual system with prescribed point degrees; instantiated here at R=8 and b=5. · proved
• Exactly three additional frozen prefix keys are excluded beyond the prior 164-key filter.
  Primary rejected-key comparison and independent replay of all 167 witnesses. · The complete frozen 2258-node type-(8,4^13) depth-four frontier. · computed
• The represented local survivor set has 2091 isomorphism classes after both filters.
  2258 source keys minus 167 distinct independently replayed blockers. · The frozen type-(8,4^13) depth-four frontier only. · computed
• The currently maintained range is 30 <= C(15,6,3) <= 31.
  Maintained LJCR covering page checked on 2026-08-12. · Global covering number C(15,6,3). · conditional

• Scale the unchanged exact individual-degree subset-capacity filter as the missing bulk root-link filter.
  All 2258 frozen type-(8,4^13) depth-four nodes. · It blocked only 167 nodes versus the predeclared 2033-node gate and added only three over the old filter. · Primary result, stable rerun, and independent replay. · A materially stronger multi-subset or ownership coupling that independently blocks at least 10% of the 2091 survivors on a fresh bounded sample.
• Reopen hash-owned complete-link enumeration using this filter.
  The previously capped canonical full-link catalogue route. · Its reopen condition required a measured bulk filter below the 10,000-node cap; 7.3959% rejection does not satisfy it. · Epoch-19 route closure and epoch-107 gate failure. · A sound substantially stronger filter or decomposition meeting the recorded node-cap pilot.
• Infer C(15,6,3)=31 from the three new local blockers.
  The global covering-number target. · Four other root-degree types and 2091 local type-0 prefix orbits remain, and no exhaustive global ownership/certificate exists. · Explicit scope of the source frontier and independent checker. · A complete independently checked exclusion of every 30-block case.

• Checkpointed complete depth-four pair-surplus ownership ledger
  A 2548-profile internal-edge bucket and the 62437-profile total have already passed independent finite checks; restart qualification is the next cheap safety gate. · Compare a two-orbit uninterrupted output stream with an interrupted/resumed stream byte-for-byte. · high · open
• Materially changed exact-degree-12 constructive witness search
  A directly checked 30-block list remains the smallest terminal certificate. · Specify one new global move or encoding-level mechanism and compare it against the frozen incidence baseline before solver scale-up. · normal · open
• Surplus-graph-conditioned root-link completion
  A complete pair-surplus graph fixes every target pair multiplicity and may provide the global coupling absent from subset capacities. · After one complete-H class is available, compile its canonical root-link compatibility constraints and compare formula size with the direct root-link leaf. · low · open

Objective: Qualify checkpoint/restart semantics for a complete depth-four pair-surplus ownership ledger.

First action: sed -n '1,320p' scripts/pair_surplus_depth4_matching_bucket_v1.py

Stop condition: Redirect on restart nondeterminism, duplicate profile, parent mismatch, source/hash drift, resource projection beyond the declared tranche, or absent human approval for a full lab submission.

• Inspect and refactor the pair-surplus depth-four producer into deterministic internal-edge-orbit chunks.
• Run a two-orbit uninterrupted versus interrupted/resumed byte-identity control.
• Do not submit the full 62,437-profile lab tranche without human approval of its exact scope.
• Retain constructive witness search only after a material move or encoding change.

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

GPT-5.6 Sol principal investigator; two pre-completed GPT-5.6 Terra delegates for advisory prior-art and verification design only. Deterministic evidence used CPython 3.12.3, exact integer/set/bitmask arithmetic, Gale-Ryser inequalities, a separately written row-load DP, SHA-256, and the computational-researcher experiment harness. Web search checked LJCR, the official Covering Repository endpoint, Gordon-Kuperberg-Patashnik, and arXiv:2607.23766. No SAT/PB solver, CAS, proof assistant, cloud lab, external proof service, human validator, system installation, publication, or external write produced evidence this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1393.1s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260812-041458-f1c086

No human review recorded.