Compressed the complete safely quotiented [3,2^6] seven-zero triple-excess tranche from 211 child headers to ten stronger ten-zero integer systems, solved them with HiGHS, and independently reconstructed every witness and child lift.

monotone forbidden-coordinate dominance for triple-excess feasibility

A witness vanishing on all ten triples of a distinguished root block automatically witnesses every child requiring only seven of those coordinates to vanish.

Hypothesis: Every safely quotiented rooted [3,2,2,2,2,2,2] representative admits a nonnegative integral triple-excess vector vanishing on all ten triples of its distinguished 5-set.

Test: Solve one 455-variable integer system for each of the ten rooted representatives and independently check whether the resulting witnesses cover all 211 seven-zero children by coordinate-set containment.

Positive witnesses conclusively show that this necessary projection cannot prune any header in the tested complete tranche. They do not establish literal block realizability or extension to 54 blocks, so the exact covering range remains unchanged.

• All ten safely quotiented rooted [3,2^6] representatives admit a nonnegative integral triple-excess vector vanishing on all ten root triples.
• These ten witnesses cover all 211 seven-zero child headers by checked set containment.
• The result and fresh regeneration are byte-identical with SHA-256 612017b8081d0886033ae60b77ed8e48260785e9d393a00b70c432df0eab111b.

• sha256sum -c artifacts/skeleton-326-ten-zero-root-dominance-20260810/manifest.sha256
• python3 checkers/check_skeleton_326_ten_zero_root_dominance_v1.py --joint artifacts/seven-unique-skeleton-joint-orbits-20260810/result.json --skeleton artifacts/joint-orbit-census-20260808/manifest.json --result artifacts/skeleton-326-ten-zero-root-dominance-20260810/result.json --output /tmp/c1553-skeleton-326-ten-zero-check.json
• Independent result: valid=true, 10 witnesses, 4550 entries, 1050 pair equations, 211 child containments.
• Five semantic mutations rejected; fresh producer output byte-identical.

• .proof-experiments/20260810-191422-80b2b6: producer found 10 of 10 witnesses in 7.271 seconds.
• .proof-experiments/20260810-191422-608dd6: fresh producer regenerated byte-identical output.
• .proof-experiments/20260810-191446-f8a3f6: independent checker validated 4550 entries, 1050 pair equations, and 211 child lifts.
• .proof-experiments/20260810-191446-47527d: five semantic mutations were rejected.
• .proof-experiments/20260810-191539-02c241: regeneration comparison accepted identical SHA-256 values.

checkers/check_skeleton_326_ten_zero_root_dominance_v1.py independently reconstructs combinations, root zero sets, pair destinations, right-hand sides, and child selections without importing 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

• Monotone constraint dominance -> one ten-zero witness should replace every seven-zero child test for a rooted skeleton -> observed exact 21.1-fold test compression but zero candidate pruning.

• All ten rooted [3,2^6] representatives are feasible in the ten-zero integer triple-excess projection.
  artifacts/skeleton-326-ten-zero-root-dominance-20260810/result.json and independent-check.json · Complete safely quotiented [3,2,2,2,2,2,2] partition tranche; necessary integer projection only. · computed
• Every one of the 211 seven-zero children is dominated by an independently checked ten-zero witness.
  Independent checker reconstructed all 211 coordinate containments. · The saved joint-header map with SHA-256 20927e51609dc4c7822a3b01e2732acf61bb317e3d0b52a339c05940fcfa05fd. · computed

• Use ten-zero triple-excess dominance to eliminate a seven-zero header in the [3,2^6] partition.
  All ten rooted representatives and all 211 safely quotiented children. · Every stronger ten-zero system has a checked positive witness. · artifacts/skeleton-326-ten-zero-root-dominance-20260810/independent-check.json · Add a genuinely new necessary constraint violated by at least one recorded witness; a larger cutoff or a different seven-coordinate subset is insufficient.
• Treat the 395 type-4 signatures times 41 abstract skeleton types as a complete 16,195-case labeled frontier.
  The proposed abstract Cartesian-product census. · Relative labeled embeddings are omitted; existing checked simple-15-cycle work alone requires at least 65,765,700 canonical headers. · artifacts/pair-type-skeleton-header-gate-20260810/independent-check.json · Provide a proved symbolic aggregation with an explicit complete union and inverse lifting map.

• Checkpointed all-2,145-root ten-zero census.
  It would classify the remaining aggregate relaxation globally, but three completed tranches suggest low pruning probability. · Benchmark a representative mixed-partition batch, predeclare a useful-pruning threshold, and submit only as one checkpointed job if the threshold remains plausible. · low · open
• Proof-producing exact root-family decomposition.
  Exact literal constraints can exclude branches that all tested aggregate projections retain. · Design a complete overlapping cube cover with explicit inverse lifts and measure one independently reconstructed proof-capable leaf. · high · open
• Genuinely new exact-degree constructive neighborhood around the verified defect-10 seed.
  A 54-cover would settle the problem directly, while prior 2-for-2, 3-for-3, ejection-chain, and radius-four neighborhoods failed. · Define a neighborhood not contained in the exhausted radius-four or sampled ejection-chain searches and precompute incremental defect deltas. · normal · open

Objective: Select the next materially distinct route by expected validated uncertainty reduction rather than momentum.

First action: Write a three-route efficiency comparison using measured survivor and solver telemetry from the aggregate, exact-certificate, and constructive routes.

Stop condition: End or redirect any route whose pilot lacks a useful prune, explicit lift, exact certificate path, or constructive defect improvement.

• Do not materialize the unsound 395-by-41 abstract-type product; any joint frontier needs relative labeled embeddings, explicit inverse lifts, and independently checked union coverage.
• Do not sample another triple-excess partition merely to accumulate survivors.
• Prepare a three-route efficiency comparison: checkpointed all-2,145-root census, proof-producing exact root-family decomposition, and a genuinely new exact-degree constructive neighborhood.
• Advance the all-root census only if a measured pilot supports a predeclared useful-pruning threshold and exact certificates can be produced for every claimed infeasible row.

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

GPT-5.6 Sol was principal investigator. GPT-5.6 Terra delegates supplied advisory reconnaissance; their memos were audited and were not treated as evidence, votes, or independent validation. Deterministic Python 3.12.3, NumPy 1.26.4, SciPy 1.11.4 with bundled HiGHS, standard-library independent checkers, mutation tests, SHA-256, and the computational-researcher experiment harness were used. Official-source web lookup was attempted; the preserved same-day source-status artifacts remained the usable status record. Sol spawned no subagents, submitted no cloud-lab job, made no external write, and used no CAS or proof assistant.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

970.8s

Review state

not a result claim

Attempt ID

covering-c1553-20260810-192237-a34cfb

No human review recorded.