Exact third-block stabilizer quotient inside the fixed-first, fixed-r=2-second incidence branch, realized as one selector-DNF CNF and compared against the retained r=2 baseline.

incidence-matrix SAT with nested stabilizer symmetry breaking

Fix two anchor blocks, distinguish a third block, quotient it under the S2 x S4 x S4 x S5 anchor stabilizer, and measure the resulting one-formula constructive search against the original r=2 encoding.

Hypothesis: A single third-block stabilizer-orbit disjunction preserves every fixed-r=2 cover while improving matched CaDiCaL conflict throughput enough to justify proof-oriented scale-up.

Test: Independently reconstruct the formula and all 5005 orbit assignments, then run seed-paired 15-second no-proof CaDiCaL arms; continue only on a checked witness or both rate ratios at least 1.10, geometric mean at least 1.15, and RSS ratio at most 1.10.

The exhaustive orbit checker establishes the local reduction, while four UNKNOWN runs and rate ratios below every continuation threshold reject only this encoding realization. Neither observation bears on global SAT or UNSAT.

• Under the stabilizer of fixed blocks 012345 and 016789, the 5005 six-subsets form exactly 55 orbits indexed by intersections with point classes of sizes 2,4,4,5.
• Every fixed-r=2 cover has a representative with one distinguished nonanchor block equal to one of the 55 canonical blocks and its other 27 nonanchor blocks lex sorted.
• The recorded selector-DNF quotient formula failed the predeclared matched CaDiCaL gate; all four arms were UNKNOWN.

• python3 scripts/r2_third_block_orbit_pilot_v1.py --out-dir artifacts/epoch123-20260812/r2-third-block-orbit-pilot-v1 --seconds 15
• python3 checkers/check_r2_third_block_orbit_pilot_v1.py --result artifacts/epoch123-20260812/r2-third-block-orbit-pilot-v1/result.json --out artifacts/epoch123-20260812/r2-third-block-orbit-independent-check-v1.json
• Producer experiment 20260812-154209-fe09ab completed in 60.675 seconds; checker experiment 20260812-154337-8c33e9 returned PASS.

• .proof-experiments/20260812-154209-fe09ab: four 15-second CaDiCaL arms all UNKNOWN; quotient/baseline conflict-rate ratios 0.778 and 0.969.
• .proof-experiments/20260812-154337-8c33e9: independent checker PASS on complete CNF structure, 5005-to-55 orbit coverage, exact multiplicities, logs, and controls.

checkers/check_r2_third_block_orbit_pilot_v1.py independently parses the DIMACS, matches the retained core, reconstructs selectors and lex suffix, enumerates all 5005 blocks, checks exact orbit sizes, reparses logs, and tests two invalid shortcuts.

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

• two-anchor stabilizer orbits -> predict a 91-fold distinguished-coordinate quotient -> exactly 55 orbits were independently observed, but selector-DNF CDCL throughput decreased
• canonical binary necklaces within colour classes -> predict the same 55 representatives via monotone bits -> not yet tested; retained as the next discriminator

• The two-anchor stabilizer has exactly 55 orbits on six-subsets.
  Exhaustive four-class intersection enumeration and multiplicities summing to 5005 in the independent receipt. · Anchors 012345 and 016789 on the 15-point set. · computed
• The distinguished-third-block quotient preserves every solution of the fixed-r=2 branch.
  Explicit stabilizer mapping argument plus complete formula reconstruction. · The fixed-first, fixed-r=2-second incidence formula only. · proved

• Scale the 55-selector, 826-clause third-block DNF under the current CaDiCaL incidence route.
  Two seeds, 15 seconds per matched arm, fixed-r=2 formula. · Both rate ratios and their geometric mean failed the gate; preprocessing retained more variables and clauses, and no terminal result occurred. · artifacts/epoch123-20260812/r2-third-block-orbit-pilot-v1/result.json · A material encoding or solver change passing the same matched gate, or a directly checked witness.
• Treat the exact 91-fold coordinate quotient as a 91-fold reduction of global covers or isomorphism classes.
  The fixed-r=2 distinguished-column representation. · The factor concerns one selected coordinate only; the other 27 block columns and five other r branches remain. · Independent completeness argument and explicit scope in the receipt. · An independently checked full-cover orbit count or exhaustive union establishing such a factor.

• Compact monotone-bit encoding of the same 55 third-block representatives.
  Eleven implications replace 55 variables and 826 clauses, so the exact quotient can be tested without the observed DNF overhead. · Build the implication suffix, independently truth-table equivalence under weight six, and repeat the frozen matched gate. · high · open
• Pinned native-PB proof qualification.
  Global exact-degree conservation remains difficult for direct LRAT but is native in PB. · Provision exact compatible source revisions only when hashes and two replay paths can be recorded, then replay the smallest global-balance calibration. · low · open

Objective: Determine whether the exact 55-orbit third-block quotient helps when expressed without selector overhead.

First action: Add eleven clauses (-x[g[i+1],2] or x[g[i],2]) across the four anchor colour classes, remove the 55 selectors, sort columns 3..29, and independently verify equality with the 55 representatives under the weight-six equation.

Stop condition: Close this quotient family on any semantic mismatch or a second failure of the same paired throughput/memory gate; promote only a checked cover or replayed proof.

• Encode the 55 canonical blocks with eleven within-colour adjacent implications rather than 55 selectors and independently prove equivalence using exact column weight six.
• Run the same two-seed, four-arm, 15-second gate; close the third-block quotient family if it again fails.
• Keep native PB held until a hash-pinned producer plus two independent replay paths are available.

• https://www.coveringrepository.com/systems.aspx?k=6&t=3&m=-1 ↗
• https://arxiv.org/abs/2607.23766 ↗
• https://github.com/dennisonbertram/covering-number-c12-6-4 ↗

GPT-5.6 Sol principal designed, implemented, audited, and interpreted the epoch. Two GPT-5.6 Terra delegates supplied advisory reconnaissance only and were not counted as validation. Python 3.12.3 generated and independently reconstructed exact CNF/orbit artifacts; CaDiCaL 1.7.3 ran four CPU-pinned no-proof arms; SHA-256 bound files. No PB solver, CAS, proof assistant, cloud lab, external proof service, human validator, or publication action produced evidence.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1147.6s

Review state

evidence receipt failure; not durable progress

Attempt ID

covering-c1563-20260812-155023-ccf5bc

No human review recorded.