2026-08-09 05:52 UTCgpt-5.6-sol · high

Re-encode the frozen epoch-20 root-link completion leaf as sparse native pseudo-Boolean constraints, independently reconstruct its semantics, and run one predeclared ten-second Z3 discriminator.

Progress

A hash-bound sparse PB formulation and independent checker were completed. They exactly compress the frozen leaf from 38727 to 1998 variables, but the single predeclared Z3 run timed out. No root-link node was eliminated and the covering range remains 30 through 31.

Strategy and discriminator

sparse exact-cover or pseudo-Boolean root-link completion

One Boolean variable per unused candidate 5-subset, fourteen exact residual-degree equations, fifty-one uncovered-pair inequalities, and one implied exact-eight equation.

Hypothesis: The frozen leaf-00-u51 admits an independently reconstructible 1998-variable, 66-row sparse PB representation, and a deterministic ten-second native-PB run either returns a directly checkable SAT completion or redirects this solver variant.

Test: Audit every OPB coefficient and RHS using a separately written recursive-mask incidence construction, reject three semantic mutations, then run Z3 4.13.0 once with timeout 10000 ms and random seed 0.

Rationale

The independent checker establishes the encoding reduction and semantic identity, which is reusable progress. The UNKNOWN solver response supplies no satisfiability inference, so it cannot support a covering bound or candidate promotion.

Claims requiring scrutiny

The frozen leaf has an independently reconstructed 1998-variable, 66-row sparse PB encoding with 23208 incidence terms.
Z3 4.13.0 returned UNKNOWN due to timeout after 10.017504 seconds on the exact hash-bound sparse instance.
Zero of the 2258 recorded depth-four root-link nodes were eliminated this epoch.

Evidence and scope

python3 scripts/root_link_sparse_pb_v1.py prepare ... emitted OPB SHA-256 6ea46187049b2dd3ef44675b860c8a239edca779be7adb6a014f6007b1f4df1e
python3 checkers/check_root_link_sparse_pb_v1.py ... accepted 1998 variables, 66 rows, 23208 terms, and rejected three mutations
python3 scripts/root_link_sparse_pb_v1.py solve ... --timeout-ms 10000 --seed 0 returned UNKNOWN(timeout)
Hash manifest verification succeeded for all 19 listed files

Computational experiments

.proof-experiments/20260809-054540-4dd0f3: generated 1998-variable, 66-row OPB
.proof-experiments/20260809-054557-8b98e1: independent pre-solver semantic audit passed
.proof-experiments/20260809-054611-3deda4: Z3 returned UNKNOWN(timeout) after 10.017504 seconds
.proof-experiments/20260809-054639-49419a: final hash-bound result audit passed

Independent checker

checkers/check_root_link_sparse_pb_v1.py independently generates recursive masks and scatters incidences rather than using the generator's tuple scans; it reparses the OPB and validates any SAT completion through both sets and bitmasks.

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

Certified C(12,6,4) link-orbit solving -> require independently replayable UNSAT certificates -> no UNSAT claim was accepted from Z3.
Sparse exact-cover modelling -> removing cardinality auxiliaries should reduce formula overhead -> variables fell 19.3829-fold but terminal search did not result.
Two-root star decomposition -> a multiplicity-four pair should leave a smaller ten-block tail -> retained as the next measurable structural test, not yet executed.

Established facts

The sparse OPB contains 1998 variables, 14 degree equalities, 51 pair inequalities, one cardinality equality, and 23208 terms.
artifacts/epoch21-20260809/sparse_pb_checker_receipt.json and OPB SHA-256 6ea46187049b2dd3ef44675b860c8a239edca779be7adb6a014f6007b1f4df1e · Frozen root-link leaf Q???????????????^?@KKCXOGZ? only. · computed
The sparse OPB is semantically identical to the declared residual-degree and pair-coverage completion problem.
Independent recursive-mask reconstruction and three rejected mutations. · The declared candidate ordering and frozen leaf. · computed
Z3 did not decide the sparse PB instance within 10000 ms.
artifacts/epoch21-20260809/sparse_pb_z3_result.json · Z3 4.13.0, seed 0, exact recorded instance and limits. · computed

Ruled out in this epoch

Scale the Z3-native sparse PB solver on this leaf or additional leaves under the current protocol.
Z3 4.13.0 native PB, seed 0, ten-second discriminator on the frozen leaf. · The exact run returned UNKNOWN at the full limit. · artifacts/epoch21-20260809/sparse_pb_z3_result.json · A material constraint decomposition, measured dominance filter, terminal evidence, or proof-producing PB solver with independent replay.
Rerun the six incidence branches with only a fresh seed.
The unchanged epoch-9 incidence encoding and six canonical second-block branches. · Epoch 9 already failed the frozen conflict and decision gates; seed change is replication, not the recorded material reopen condition. · artifacts/epoch9-20260808/incidence_orbit_receipt.json and artifacts/epoch21-20260809/source_and_prior_art_audit.md · A material encoding change, proof-prefix reuse design, or terminal solver result.

Open leads

Forced multiplicity-four ordered edge-star prefix catalogue
Every point has at least ten multiplicity-four neighbours; the four shared blocks have only three excess profiles and conditionally expose a ten-block tail on 13 points. · Write two independent canonical prefix enumerators and compare complete orbit/stabilizer outputs. · high · open
Proof-producing PB calibration on the frozen OPB
The semantics are already independently audited and the input is 19.3829-fold smaller in variables than the totalizer CNF. · Provision a pinned project-scoped PB solver and independent proof verifier, then run a small proof-format calibration before leaf contact. · normal · open
Fresh nonisomorphic constructive search
A validated 30-block witness would settle the problem directly and remains symmetric with exclusion routes. · Design a bounded constructive start generator that excludes the two exhausted U=5 local components and directly validates any cover. · normal · open

Continuation checkpoint

Objective: Establish or reject a complete canonical quotient for the forced multiplicity-four edge-star prefix.

First action: Implement two independent enumerators for four distinct 4-subsets covering 13 residual points and classify them under ordered-root point relabelling.

Stop condition: Stop on any disagreement in representative set, stabilizer, profile, or labelled-prefix coverage; do not extend to complete stars until all checks pass.

Next moves

Implement independent set-based and bitmask-based canonical enumerators for the four common blocks of an ordered multiplicity-four pair.
Require agreement on canonical labels, stabilizer sizes, orbit counts, and coverage of every labelled prefix in profiles 3, 2+1, and 1+1+1.
Only after that audit, measure complete two-root star catalogue size before encoding any ten-block tail.
Retain the OPB instance for a proof-producing PB solver only when an independently replayable proof toolchain is available.

Citations

Tool disclosure

GPT-5.6 Sol served as principal investigator. Two pre-existing GPT-5.6 Terra delegate memos supplied advisory reconnaissance and were audited rather than treated as evidence. Deterministic work used Python 3.12.3, Z3 4.13.0 native pseudo-Boolean constraints, exact set/bitmask checkers, SHA-256, the computational-researcher experiment recorder, and web searches of the Covering Repository, La Jolla tables, and arXiv. CaDiCaL 1.7.3 artifacts from epoch 20 were inspected for comparison but CaDiCaL was not rerun this epoch.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 876.4s
Review state: not a result claim
Attempt ID: covering-c1563-20260809-055240-03787f

Human review ledger

No human review recorded.