Test the exact joint point/pair/triple-cover Venn-cell projection for the fixed labelled complete surplus graph H=3K5 and overlapping subsets S={0,1,2}, T={2,3,4}.

two-subset Venn-cell integer projection

Aggregate all six-subsets by four Venn-cell counts, impose exact cell-resolved block, point, pair, and triple-cover moments, and test integer feasibility.

Hypothesis: The exact joint Venn-cell moments for S={0,1,2}, T={2,3,4} exclude the fixed labelled complete pair-surplus graph H=3K5.

Test: Solve the 18-variable integer projection and independently scan all 5005 labelled blocks to reconstruct and exactly check its coefficient table and witness.

A checked feasible vector falsifies the exclusion hypothesis for the exact projection. Because the projection discards point identities, feasibility cannot be lifted to a cover, so it supplies route information but no field-level result.

• For H=3K5 and every ordered pair of 3-subsets contained in one K5 and intersecting in one point, the 18-type Venn-cell point/pair/triple-cover moment projection is integer feasible.
• The coefficient catalogue has exactly 18 types covering all 5005 labelled blocks for the frozen partition.
• No cover, complete-H exclusion, or covering-number improvement is obtained.

• python3 scripts/venn_projection_pilot_v1.py --input artifacts/epoch127-20260812/h-3k5-venn-input.json --output artifacts/epoch127-20260812/venn-projection-result-v1.json
• python3 checkers/check_venn_projection_pilot_v1.py --input artifacts/epoch127-20260812/h-3k5-venn-input.json --receipt artifacts/epoch127-20260812/venn-projection-result-v1.json --output artifacts/epoch127-20260812/venn-projection-independent-check-v2.json
• cmp returned 0 for result-v1.json and result-rerun-v1.json.

• .proof-experiments/20260812-185258-e75e4b: producer returned PASS_FEASIBLE_PROJECTION with 18 types and 29 rows in 0.422 seconds.
• .proof-experiments/20260812-185434-253c32: independent checker reconstructed all 5005 blocks and rejected five mutations in 0.476 seconds.
• .proof-experiments/20260812-185441-c18029: producer rerun generated a byte-identical receipt in 0.426 seconds.

checkers/check_venn_projection_pilot_v1.py uses a separate all-5005 labelled-block enumeration, derives pair/triple coefficients from each selected block, uses no SciPy, and rejects five mutations.

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

• Classical intersection-number equations -> predict that two overlapping subsets may couple separate H(S) histograms -> the exact overlap-one 3K5 orbit still has a feasible aggregate vector.

• The frozen Venn projection has exactly 18 coefficient types, 15 equalities, 14 inequalities, and a feasible integer vector of total 30.
  Producer receipt a8f44acf... plus independent receipt 2f06f8f7.... · Labelled H=3K5, S={0,1,2}, T={2,3,4}, aggregate projection only. · computed
• The same feasibility result holds for the same-K5, intersection-one ordered subset-pair orbit under Aut(3K5).
  Within-component S5 maps the four cells of sizes 1,2,2 and the remaining two K5 components accordingly. · That single automorphism orbit only. · proved

• Exclude H=3K5 using only the tested same-K5, overlap-one two-subset Venn-cell block/point/pair/triple moments.
  The exact 18-variable projection and its one Aut(3K5) subset-pair orbit. · A checked integer feasible vector satisfies every necessary row. · venn-projection-result-v1.json and venn-projection-independent-check-v2.json. · Add a genuinely edge-resolved or block-ownership constraint not determined by the four cell counts.

• Direct CaDiCaL PBP calibration with two independent replayers.
  Could restore proof-producing incidence cubes without the failed direct-LRAT size path. · Pin VeriPB and CakePB; require acceptance of the complete two-clause contradiction and rejection of final-line, input-clause, and base-mismatch mutations. · high · open
• Complete parent-rich depth-four ownership union.
  Would create a hash-bound global frontier prerequisite for exhaustive negative search. · After explicit approval, materialize exactly 38705 remaining profiles across the qualified 82-chunk geometry and independently reconcile all 62437 profiles. · high · open
• Edge-resolved complete-H orbit invariant.
  It targets the information discarded by all one- and two-subset aggregate moment routes. · For one complete H orbit, add the smallest edge-coloured block type and require whole-orbit exclusion under an independently checked ownership map. · normal · open

Objective: Unlock a route capable of whole-profile or whole-orbit elimination rather than another aggregate feasibility sample.

First action: Request approval for the exact 38705-profile scope or pin both PBP replayers and run the synthetic calibration; otherwise formulate one edge-resolved whole-orbit discriminator.

Stop condition: Hold on absent authority/tools, any checker mutation acceptance, or another relaxation that only returns a feasible aggregate vector.

• Do not run another arbitrary subset-pair Venn sample.
• Obtain explicit human approval before parent-materializing the remaining 38705 depth-four profiles.
• Alternatively acquire and pin VeriPB and CakePB, then execute only the four-control synthetic CaDiCaL PBP calibration before target solving.
• Without either gate, design an edge-resolved invariant with a predeclared whole-complete-H-orbit exclusion certificate.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=6&t=3&v=15 ↗

GPT-5.6 Sol served as principal investigator. Two supplied GPT-5.6 Terra delegates provided advisory prior-art and experiment-design memos; their agreement was not treated as evidence. Deterministic work used Python 3.12.3, SciPy 1.11.4/HiGHS MILP, exact Python integer arithmetic, exhaustive itertools enumeration, SHA-256, cmp, and the project run_experiment harness. No external solver proof was claimed; local command audit found CaDiCaL 1.7.3 but no VeriPB or CakePB replayer.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1092.7s

Review state

not a result claim

Attempt ID

covering-c1563-20260812-190156-d87f75

No human review recorded.