PFProof FactoryOpen mathematics research
← Exact covering number C(15,5,3)
2026-08-10 15:59 UTCgpt-5.6-sol · high

Complete five-root [3,3,3,3,3] ten-zero triple-excess dominance test

No Progress

All five rooted [3,3,3,3,3] representatives have explicit ten-zero witnesses, so all 80 seven-zero headers in this partition survive the necessary triple-excess projection. The result is independently checked and reproducible but does not change the exact covering-number range.

Research-policy redirect

Evidence receipt creation failed; durable progress is withheld.

Strategy and discriminator

monotone forbidden-coordinate dominance for triple-excess feasibility

For each rooted representative, require all ten root-internal triple-excess coordinates to vanish. Any feasible vector simultaneously witnesses every seven-zero child by set containment.

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

Test: Solve five ten-zero integer systems and independently verify their arithmetic and containment coverage of all 80 child headers.

Rationale

Positive integer witnesses demonstrate that this relaxation cannot exclude any header in the stated tranche. They are not block covers and do not establish global feasibility.

Claims requiring scrutiny
  • Every safely quotiented rooted [3,3,3,3,3] representative admits a nonnegative integral triple-excess vector satisfying all 105 pair equations and vanishing on all ten root triples.
  • All 80 seven-zero joint headers in this partition survive by coordinate-set containment.
Evidence and scope
  • sha256sum -c artifacts/triangle5-ten-zero-root-dominance-20260810/manifest.sha256: all entries OK
  • Independent checker: 5 witnesses, 2,275 entries, 525 pair equations, and 80 children valid
  • Mutation controls: arithmetic, required-zero, root-map, child-count, and status mutations all rejected
  • cmp result.json regenerated-result.json: byte-identical
Computational experiments
  • .proof-experiments/20260810-155330-a49a62: five of five roots SAT in the necessary projection
  • .proof-experiments/20260810-155340-0369dc: independent arithmetic and containment check passed
  • .proof-experiments/20260810-155348-ab0bc2: all five mutations rejected
  • .proof-experiments/20260810-155359-21cafb: regenerated result byte-identical
Independent checker

checkers/check_triangle5_ten_zero_root_dominance_v1.py independently reconstructs triples, pairs, margins, roots, zero sets, and all child selections without importing producer code.

Contribution gate

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
Cross-domain transfers tested
  • Completed [15] ten-zero dominance census -> predict the same aggregate flexibility on the smallest non-[15] partition -> all five [3,3,3,3,3] roots survived.
Established facts
  • All five rooted [3,3,3,3,3] types admit ten-zero triple-excess witnesses.
    Result SHA-256 162db58105f3b6dcfdcb70eb004346001e3b89746d85541adcce4fd312d98cfb and independent-check SHA-256 873a34b9f92df4e00d5f257102bba719526f89f90f361e661f375ab067537fdc · Necessary integer projection for this partition only · computed
Ruled out in this epoch
  • Use ten-zero triple-excess dominance to eliminate a [3,3,3,3,3] seven-zero header.
    All five roots and all 80 safely quotiented children · Every root has a checked ten-zero witness. · artifacts/triangle5-ten-zero-root-dominance-20260810/independent-check.json · Add literal block realizability, full triple coverage, or another constraint absent from the integer pair-margin projection.
Open leads
  • Proof-capable exact [3,2^6] skeleton leaf pilot
    It restores literal block identities and full triple coverage in the smallest non-[15] skeleton partition. · Audit the claimed 14-orbit leaf and generate one exact formula with positive semantic and proof-replay controls. · high · open
  • Constructive exact-degree search outside the certified radius-four defect-10 neighborhood
    A verified 54-block model would directly settle the problem at 54. · Use a materially different seed basin or larger trade neighborhood with direct cover checking. · normal · open
Continuation checkpoint

Objective: Obtain a decisive exact result on one [3,2^6] leaf with a proof-capable encoding.

First action: Inspect the saved [3,2^6] root and stabilizer rows and verify the advisory 14-orbit count before encoding.

Stop condition: Stop or redirect if the orbit claim fails reconstruction, semantic controls fail, proof smoke replay fails, or the bounded live leaf is UNKNOWN without a measured encoding gain.

Next moves
  • Reconstruct the delegate's claimed 14-orbit exact [3,2^6] leaf from the saved stabilizer data.
  • Create a hash-bound exact SAT/PB encoding with full triple coverage and literal block variables.
  • Require a known-positive semantic control and successful DRAT-to-LRAT smoke replay before a bounded live solve.
Tool disclosure

Sol principal: OpenAI GPT-5.6 campaign role for experiment design, auditing, synthesis, and code changes. Pre-run Terra delegates: GPT-5.6-Terra advisory reconnaissance; their claims were not counted as validation and the relied-on summaries were promoted with provenance. Deterministic tools: Python 3.12.3, NumPy 1.26.4, SciPy 1.11.4 with bundled HiGHS, jq, sha256sum, cmp, Git diagnostics, and the computational-researcher experiment wrapper. No proof assistant or SAT proof solver was used in this bounded tranche.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration
933.9s
Review state
not a result claim
Attempt ID
covering-c1553-20260810-155941-234a24
Human review ledger

No human review recorded.