2026-08-09 06:38 UTCgpt-5.6-sol · high

Independently enumerate and verify the canonical four-block prefix forced by an ordered pair of multiplicity four in any hypothetical 30-block cover.

Progress

The forced multiplicity-four four-block prefix has exactly nine independently reproduced S13 orbits and 61311250 valid labelled unordered instances. This corrects a factor-six error in all six delegate [1,1,1] orbit sizes. The result is only a prefix classification; C(15,6,3) remains between 30 and 31.

Strategy and discriminator

ordered multiplicity-four edge-star canonical augmentation

Represent each of 13 residual points by its nonempty support among four common block positions, enumerate all feasible support counts, and quotient by residual-point relabelling and block-position permutations.

Hypothesis: The forced four-block prefix has exactly nine canonical orbits and the delegate-predicted labelled mass 303803500.

Test: Compare a direct excess-partition constructor with exhaustive recursion over all 15 nonempty support counts, then independently verify representatives, stabilizers, corruption controls, and total coverage by inclusion-exclusion.

Rationale

The two generators use materially different constructions, their canonical catalogues are identical, and a separate checker derives the same mass by orbit-stabilizer and inclusion-exclusion while rejecting targeted corruptions. This establishes the local quotient but supplies neither a cover nor an exhaustive global exclusion.

Claims requiring scrutiny

In every hypothetical 30-cover, an exact-multiplicity-four pair exists and may be fixed as an ordered root.
The four common residual 4-subsets of such a root pair have exactly 81 ordered support-count vectors.
Those vectors form exactly nine S13 orbits after treating the four blocks as unordered.
The nine orbits cover exactly 61311250 valid labelled unordered prefixes.
The delegate total 303803500 and its six [1,1,1] orbit sizes are incorrect by a factor of six on those six types.

Evidence and scope

python3 scripts/edge_star_prefix_structural_v1.py --output artifacts/epoch22-20260809/edge_star_prefix_structural_v1.json
python3 scripts/edge_star_prefix_support_v1.py --output artifacts/epoch22-20260809/edge_star_prefix_support_v1.json
python3 checkers/check_edge_star_prefix_catalogue_v1.py artifacts/epoch22-20260809/edge_star_prefix_structural_v1.json artifacts/epoch22-20260809/edge_star_prefix_support_v1.json --receipt artifacts/epoch22-20260809/edge_star_prefix_checker_receipt_v1.json
Final audit: PASS; 14 manifest files verified, nine orbits, 81 ordered vectors, 61311250 valid prefixes, and five rejected corruptions.
Hash manifest SHA-256 629440db272df7ceeeec346ab0064641b12da78107e493e7e022d96a80235687.

Computational experiments

.proof-experiments/20260809-062624-ee5d6a: structural construction produced nine orbits but falsified the predeclared 303803500 mass, returning 61311250.
.proof-experiments/20260809-062624-87ec3d: exhaustive support recursion produced 81 ordered vectors and the same nine-record catalogue digest.
.proof-experiments/20260809-062747-c7ef05: final independent checker confirmed mass 61311250, inclusion-exclusion equality, and five rejected corruptions.

Independent checker

checkers/check_edge_star_prefix_catalogue_v1.py independently reconstructs explicit blocks, repeat profiles, canonical images, full stabilizers, orbit sizes, affine point relabellings, and inclusion-exclusion coverage; it does not import either generator.

Contribution gate

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome: progress
Public classification: progress

Cross-domain transfers tested

C(12,6,4) certificate discipline -> require a hash-bound complete canonical frontier before any UNSAT tail claim -> applied as the stop rule; no tail search was started.

Established facts

Every hypothetical 30-cover contains an exact-multiplicity-four pair.
Point degree 12, lambda_xy >= 4, and sum_y(lambda_xy-4)=4. · All hypothetical 30-block C(15,6,3) covers. · proved
The fixed ordered-root prefix support equations have exactly 81 ordered solutions.
Exhaustive recursion in edge_star_prefix_support_v1.py and independent reconstruction from orbit stabilizers. · Four residual 4-subsets covering 13 points. · computed
The prefix family has exactly nine S13 orbits and 61311250 valid labelled unordered members.
Two matching catalogues, checker receipt, orbit-stabilizer sum, and inclusion-exclusion. · A fixed ordered multiplicity-four root pair. · computed

Ruled out in this epoch

Use the delegate orbit table or total 303803500 as the prefix coverage certificate.
The six [1,1,1] prefix types and their aggregate mass. · Each of those orbit sizes is six times too large, and the total contradicts direct inclusion-exclusion. · artifacts/epoch22-20260809/edge_star_prefix_checker_receipt_v1.json · A concrete flaw in both independent generators and the inclusion-exclusion derivation.
Treat the nine-prefix catalogue as evidence for C(15,6,3)=30 or 31.
The global covering-number target. · No prefix was extended to a complete cover and no extension family was excluded. · The checker receipt explicitly limits interpretation to the four-block prefix. · A checked 30-block cover or an exhaustive proof-bearing exclusion of every complete extension.

Open leads

Complete one-root-star canonical frontier above the nine prefixes.
It is the smallest next boundary and directly tests whether edge-star decomposition remains tractable. · Run two canonical augmenters under a 10000-node or 30-minute aggregate cap. · high · open
Proof-producing PB calibration on the retained sparse frozen leaf.
The 1998-variable OPB remains semantically audited, but only a proof-producing solver with independent replay would add useful evidence. · Calibrate a pinned PB proof format and verifier before contacting the leaf. · normal · open
Fresh nonisomorphic constructive starts.
A directly checked 30-cover remains the cheapest terminal certificate, while earlier local shells cover only two known anchors. · Design a start generator outside the exhausted U=5 components and validate every deficit-zero family directly. · normal · open

Continuation checkpoint

Objective: Determine whether all nine certified prefixes admit a manageable, independently checked complete one-root-star frontier.

First action: Implement canonical augmentation and canonical-parent replay for one root star, using the nine-record JSON as immutable input.

Stop condition: Stop on any cross-check mismatch, or after 10000 canonical nodes or 30 minutes without a complete frontier.

Next moves

Implement two independent canonical augmenters from each of the nine hash-bound prefixes to one complete 12-block root star.
Cap the aggregate pilot at 10000 canonical nodes or 30 minutes and preserve a complete canonical-parent frontier.
Stop on any parent, orbit, stabilizer, or frontier mismatch.
Only if the complete one-star frontier is manageable, enumerate two-root stars and then assess the conditional choose(13,6)=1716 tail universe.

Citations

Tool disclosure

GPT-5.6 Sol served as principal investigator. Two GPT-5.6 Terra delegates supplied advisory reconnaissance and experiment design; their orbit-mass table was audited and partially rejected. Deterministic evidence used Python 3.12.3, exact set and support-mask arithmetic, SHA-256, clean replay, the computational-researcher experiment recorder, and web status searches. No SAT/PB solver, CAS, proof assistant, external validator, or checkpointed lab job was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 1220.0s
Review state: not a result claim
Attempt ID: covering-c1563-20260809-063824-6e7c69

Human review ledger

No human review recorded.