Exhaustively screen all lifted Pasch trades against every two-point boundary of canonical fixed-family type 1, quotienting boundaries by the exact fixed-family stabilizer and independently reconstructing all labelled counts.

completion-congruence boundary stress test

Pair-balanced lifted Pasch trades preserve block count and all point and pair incidences while exposing triple-coverage information hidden from a selected boundary.

Hypothesis: For canonical fixed-family type 1, every two-point boundary admits a compatible lifted Pasch trade whose two histories agree on block count, point degrees, pair multiplicities, and all boundary-touching triple-coverage bits, but disagree on full triple coverage.

Test: Enumerate all C(15,6)C(9,2)15 = 2,702,700 labelled lifted Pasch trades and require a positive independently reconstructed hidden-trade count in every exact boundary orbit.

The two trade sides have equal lower-order incidence. For each boundary, the census found trades whose remaining coverage difference avoids that boundary after union with the fixed family. Exact group coverage transfers representative results to all labels, while the independent checker directly confirms every count and witness.

• The canonical type-1 fixed family has an exact stabilizer of order 6,912 and 11 orbits on the 105 two-point boundaries.
• Exactly 2,059,200 of the 2,702,700 labelled lifted Pasch trades are compatible and coverage-changing under the tested definition.
• Every labelled two-point boundary has between 666,270 and 900,900 compatible hidden trades.
• For every two-point boundary, there exist two nine-block histories with equal block count, point-degree vector, pair-multiplicity vector, and boundary-touching triple coverage, but unequal full triple coverage.
• These claims do not construct or exclude a 54-block cover.

• Producer experiment 20260810-104539-75670b completed in 48.818 seconds and wrote result SHA-256 72d6cece2dc8c7fa0dbebbd77d00ba46ad828838cc31e52144f92d4cd2a07a1d.
• Independent checker experiment 20260810-104843-b44029 completed in 56.486 seconds with valid=true and output SHA-256 aea7328da8617c380140e90aadaf01adb98514b07eadacc18d61b41c196bcd23.
• Regeneration experiment 20260810-104957-10f523 was byte-identical to the original result.
• sha256sum -c artifacts/type1-lifted-pasch-boundary-screen-20260810/manifest.sha256 passed for every listed artifact.

• .proof-experiments/20260810-104235-a121d5: stronger trade-orbit BFS reached the predeclared 120-second cap and produced no result artifact.
• .proof-experiments/20260810-104539-75670b: exact labelled screen PASS; 2,059,200 compatible trades and all 11 boundary orbits positive.
• .proof-experiments/20260810-104843-b44029: independent checker PASS; all counts, witnesses, group checks, and mutations passed.
• .proof-experiments/20260810-104957-10f523: deterministic regeneration PASS; byte-identical result.

checkers/check_type1_lifted_pasch_boundary_screen_v2.py imports no producer code, derives Pasch trades as paired linear 2-regular triple systems, constructs the stabilizer explicitly, recounts all 105 labels, and directly reconstructs witness incidences.

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 -> equal lower-order incidence with changed higher-order coverage -> observed hidden collisions for every two-point boundary.
• Hypergraph transversals -> boundary points must hit every hidden-difference vertex mask -> two-point sets are now exhaustively falsified as transversals.

• For canonical fixed-family type 1 and every two-point boundary B, a compatible lifted Pasch trade yields two partial histories identical on the tested B-interface and all lower-order incidences but different in full triple coverage.
  Exact exhaustive result plus independently reconstructed all-label counts and representative incidence witnesses · All 105 two-point boundaries; canonical type 1; unrestricted partial histories · computed

• Repair the tested type-1 pair-summary projection merely by choosing a different single two-point boundary.
  Every one of the 105 labelled two-point boundaries for canonical fixed-family type 1 · Every boundary has at least 666,270 compatible hidden lifted-Pasch trades. · artifacts/type1-lifted-pasch-boundary-screen-20260810/result.json and independent-check.json · Prove that exact 54-block constraints forbid all recorded partial-history collisions, or enrich the interface beyond a single two-point boundary.
• Run the unoptimized inner trade-orbit BFS under the 120-second local protocol.
  The recorded v1 Python BFS implementation only · It reached the predeclared timeout without emitting a result. · .proof-experiments/20260810-104235-a121d5/experiment.json · Provide a materially faster canonicalization or Burnside implementation with a measured pilot below the cap.

• Minimum boundary transversal for type-1 hidden trades
  It converts millions of history collisions into a tiny exact set-family problem and quantifies how global any safe interface must be. · Enumerate distinct hidden-vertex masks, then solve and independently check their minimum hitting-set size. · high · open
• Proof-core-weighted constructive repair
  A directly checked defect below 10 would revive the terminal constructive route after the existing plateau. · Run one predeclared deterministic operator tranche with a below-10 acceptance gate. · normal · open
• Proof-producing canonical SAT decomposition
  It retains terminal value but requires a measured proof-producing leaf before global scale-up. · Generate one enriched canonical leaf and require complete LRAT replay under a fixed cap. · normal · open

Objective: Determine the minimum boundary size required to intersect every compatible type-1 hidden trade direction.

First action: Extract distinct hidden-vertex masks from the exact trade enumeration and solve the resulting minimum-transversal instance.

Stop condition: Redirect if the minimum safe interface is essentially global, if producer and checker disagree, or if no separator theorem follows from the computed threshold.

• Enumerate distinct hidden-vertex masks and solve their exact minimum-transversal problem.
• Independently reconstruct the boundary-size threshold using a different set-family encoding.
• Close symbolic boundary aggregation if the required interface is essentially global.
• Preserve constructive repair and proof-producing canonical SAT as distinct alternatives.

• 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/math/9502238 ↗

GPT-5.6 Sol acted as principal investigator. GPT-5.6 Terra delegates supplied advisory reconnaissance before this pass; their relied-on memo was promoted with provenance and was not treated as evidence or validation. Deterministic work used Python 3.12.3, NetworkX 3.3 for producer-side graph automorphisms, a separately implemented pure-Python combinatorial checker, SHA-256, and the computational-researcher experiment harness. Web search checked the maintained status and prior-art trail. No SAT solver, CAS, proof assistant, cloud lab, external write, or publication was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1408.2s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1553-20260810-105731-bf0f24

No human review recorded.