Constructed and independently checked a lifted-Pasch-trade collision before launching the proposed symbolic state-DAG census.

completion-congruence specification gate

Two pair-balanced 5-uniform histories collide under the proposed state projection; a shared normalized-pair-avoiding suffix tests whether terminal triple coverage is preserved.

Hypothesis: Block count, all point degrees, all 105 labelled pair multiplicities, and coverage of the 169 triples meeting boundary {0,1} do not form a completion congruence for 15-point 5-block triple coverage.

Test: Find two fixed-common-family histories with the same projected state and a common suffix that preserves normalized-pair multiplicity five and all final pair counts but makes exactly one family cover all 455 triples.

The histories agree on every field of the declared state. Applying the same disjoint suffix, containing no additional block through the normalized pair, preserves final block-count and pair-vector equality but gives different terminal coverage. Thus acceptance is not a function of this state. The 211-block totals and non-target pair vector prohibit inference about the actual 54-block problem.

• The recorded 9-block histories have equal point degrees, equal labelled pair multiplicities, and equal coverage on all 169 triples meeting boundary {0,1}.
• Their complete triple-coverage sets differ.
• After the common 202-block suffix, both families have 211 blocks, normalized-pair multiplicity five, and identical pair vectors; family A covers 455 triples and family B covers 453.
• This does not settle or tighten 54 <= C(15,5,3) <= 55.

• python3 scripts/pair_summary_trade_collision_v1.py --protocol artifacts/pair-summary-trade-collision-20260810/protocol.json --output artifacts/pair-summary-trade-collision-20260810/result.json
• python3 scripts/check_pair_summary_trade_collision_v1.py --protocol artifacts/pair-summary-trade-collision-20260810/protocol.json --result artifacts/pair-summary-trade-collision-20260810/result.json --output /tmp/pair-summary-trade-check.json
• sha256sum -c artifacts/pair-summary-trade-collision-20260810/manifest.sha256
• Result SHA-256 20a1f5ba4d31d818cf65e290526f93015fd01211fe96cc09d112d03a75777169; independent-check SHA-256 e51e5615df8960aece31e403773cd49381539fd17c00fcbe7d9c697f379bab00

• .proof-experiments/20260810-100542-ac382d: producer PASS in 0.322 seconds; 202-block normalized-pair-avoiding suffix; terminal coverage 455 versus 453.
• .proof-experiments/20260810-100543-3a8627: checker PASS in 0.324 seconds; 20 semantic checks and five mutations passed.

scripts/check_pair_summary_trade_collision_v1.py uses an independent frozenset encoding, reconstructs all 105 pairs and 455 triples, and imports no 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

• Design trades -> predict equal lower-order incidence with changed higher-order coverage -> observed equal point/pair summaries and different terminal acceptance after a common suffix.

• The declared pair-summary/two-endpoint-boundary projection is not a completion congruence for unrestricted-length 15-point 5-block triple coverage, even when the normalized pair remains at multiplicity five.
  Explicit histories and common suffix in result.json; independent full-incidence reconstruction valid=true. · Protocol-v1 projection and boundary {0,1}; final non-normalized pair targets and block count are not the 54-cover targets. · computed

• Count a symbolic DAG using only block count, all point degrees, all labelled pair multiplicities, and coverage of triples meeting a two-point boundary.
  The protocol-v1 projection as a general exact triple-coverage quotient. · A checked common-suffix collision gives different terminal acceptance from the same projected state. · artifacts/pair-summary-trade-collision-20260810/result.json and independent-check.json · Provide an enriched state with a proof and independent tests of exact completion congruence for the actual 54-block constraints.

• Lifted-trade state-information lower bound
  A bulk census of boundary-hidden trades can quantify how much triple information an exact quotient must preserve. · Enumerate lifted Pasch trades under all four fixed-family stabilizers and proposed boundary depths; independently check hidden-triple coordinate coverage and orbit counts. · high · open
• Proof-core-weighted constructive repair
  A checked defect below 10 would revive the terminal constructive route. · Specify one deterministic proof-core-weighted move operator and require an independently replayed defect below 10. · normal · open

Objective: Determine whether pair-balanced trades force an essentially global triple-coverage state.

First action: Write a protocol enumerating lifted Pasch-trade orbits for each canonical common-family type and boundary, with a direct independent orbit/incidence checker.

Stop condition: Close symbolic aggregation if every small interface misses checked trade directions or requires essentially the full live triple bitset; advance only with a compact interface and a completion-equivalence proof.

• Do not run the proposed 2^20 census with the falsified state summary.
• Predeclare a lifted-trade span test across all four fixed common-family types and actual processing boundaries.
• Advance a symbolic DAG only after its enriched state has a proved and independently tested completion congruence.
• Keep the exact range unchanged until a checked 54-cover or complete proof-replayed exclusion exists.

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

GPT-5.6 Sol acted as principal investigator. Two GPT-5.6 Terra delegates supplied advisory prior-art challenge and verification design; their advice was audited and was not treated as validation. Deterministic evidence used Python 3.12.3, exact tuple/frozenset incidence enumeration, SHA-256, and the computational-researcher experiment harness. Web search checked current status and adjacent primary literature. No SAT solver, CAS, proof assistant, sub-agent, cloud lab, external write, or publication was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

973.2s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260810-100915-997ac1

No human review recorded.