Freeze one surviving canonical type-4 endpoint header, encode its 26 endpoint pair counts identically in a baseline and challenger, add only the 78 internal pair-threshold definitions and 13 residual excess-degree rows to the challenger, independently reconstruct both formulas, and run a matched two-seed 25000-conflict CaDiCaL gate.

header-conditioned lazy pair propagation

Condition the literal block formula on endpoint header [0,2], represent every internal pair excess by threshold atoms linked to block incidences, and enforce the header-specialized residual degree equations without materializing weighted skeletons.

Hypothesis: On the canonical type-4 endpoint-header branch [0,2], adding literal internal-pair threshold definitions and the residual excess-degree equations either decides the branch or reduces both decisions and propagations by at least 20 percent at 25000 conflicts for seeds 0 and 1553.

Test: Require exact independent reconstruction, nonempty non-subsumed rows, primary-model equivalence, byte-identical regeneration, and mutation rejection; then compare fresh baseline/coupled CaDiCaL 1.7.3 processes at 25000 conflicts for seeds 0 and 1553.

The semantic gate passed, so the solver comparison tested the intended mechanism. The raw counters fail the predeclared threshold: propagation increased on both seeds, decisions worsened on seed 0, and the 19.622205-percent decision reduction on seed 1553 was below threshold. UNKNOWN runs and incomplete DRAT streams have no exclusion value, so the only justified outcome is a scoped negative routing result.

• For source canonical type-4 header 2 with header indices [0,2], the recorded coupled CNF is a definitional extension of the recorded exact header baseline and adds 428,642 non-subsumed exact rows.
• At 25000 conflicts with CaDiCaL 1.7.3, both formulas remained UNKNOWN for seeds 0 and 1553; the challenger failed the predeclared two-metric gate on both seeds.
• This computation excludes no header, primary assignment, 54-block cover, or covering-number value.

• sha256sum -c artifacts/type4-header-lazy-pair-coupling-20260812/manifest.sha256 passes.
• check_type4_header_lazy_pair_coupling_v1.py reconstructed 682,332 baseline clauses and 1,110,974 coupled clauses with valid=true.
• test_type4_header_lazy_pair_coupling_controls_v1.py reproduced all generated files byte-for-byte and rejected three activated mutations.
• check_type4_header_lazy_pair_coupling_result_v1.py reparsed four raw CaDiCaL logs and returned valid=true, decisive=false, telemetry_gate=false.

• .proof-experiments/20260812-110817-18eade: generated the baseline and coupled formulas in 3.634 seconds.
• .proof-experiments/20260812-110832-c729dc: independently reconstructed both formulas with valid=true.
• .proof-experiments/20260812-111034-12a525: byte-identical regeneration and three mutation rejections passed.
• .proof-experiments/20260812-111152-3d770b and -eeaaa7: seed-0 baseline/coupled UNKNOWN at the cap; challenger decisions +9.886617 percent, propagations +38.038487 percent.
• .proof-experiments/20260812-111152-a02313 and -70922c: seed-1553 baseline/coupled UNKNOWN at the cap; challenger decisions -19.622205 percent, propagations +30.639962 percent.
• .proof-experiments/20260812-111402-4f050f: independent raw-log result check returned valid=true and gate=false.

checkers/check_type4_header_lazy_pair_coupling_v1.py independently reconstructs source selection, variable allocation, every totalizer clause, algebra, and formula prefix; checkers/check_type4_header_lazy_pair_coupling_result_v1.py independently reparses all four raw logs and recomputes the 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

• Symbolic factor-graph encoding of a combinatorial object family -> predict that avoiding 5,238,370 explicit skeletons preserves useful local propagation -> object materialization was avoided exactly, but the totalizer representation increased propagations on both seeds.

• The selected header has residual internal excess-degree profile 0^2 2^11 and 5,238,370 labelled weighted skeletons.
  The prior weighted-frontier result was hash-checked and the selected row was independently reconstructed in independent-check.json. · Source canonical type-4 header index 2 only. · computed
• The recorded coupled formula preserves exactly the recorded header-baseline primary models.
  Exact prefix reconstruction plus the checked pair lower bound, point excess sum, threshold identity, regeneration, and mutations. · The two hash-bound formulas with mutation=none. · proved
• The coupled formula failed the two-seed 20-percent decision-and-propagation gate.
  Four raw CaDiCaL logs independently reparsed in result.json. · CaDiCaL 1.7.3, seeds 0 and 1553, 25000 conflicts, recorded formula hashes. · computed

• Scale the exact header-conditioned balanced-totalizer internal-pair coupling solely by increasing its conflict cap.
  Source header 2, baseline SHA-256 29dcf0df..., coupled SHA-256 9289637f..., CaDiCaL 1.7.3, the fixed two-seed gate. · Both runs remained UNKNOWN, propagation increased by 30.64 to 38.04 percent, and the decision gate failed on both seeds. · artifacts/type4-header-lazy-pair-coupling-20260812/result.json · A materially different encoding or solver with exact semantic checks and a fresh predeclared gain, or a checked SAT model/replayed UNSAT proof.
• Treat non-subsumed redundant clauses, one-seed decision improvement, or incomplete DRAT output as mathematical progress.
  This selected-header experiment. · The extension is primary-model equivalent, the complete gate failed, and all statuses are UNKNOWN. · independent-check.json and result.json · A direct 54-cover check or complete replayed proof for an explicitly covered branch.

• Proof-producing auxiliary-free/native-PB pair-upper route.
  The earlier all-type native-PB pair-upper gate passed, unlike totalizer CNFs, and it could satisfy the negative verification contract if proof production and dual replay are available. · Only after a hash-pinned RoundingSat plus VeriPB/CakePB bundle is present, run the saved C(5,3,2) SAT/UNSAT proof-model calibration before any C(15,5,3) leaf. · high · open
• Constructive move-family containment gate from saved defect-ten states.
  A 54-cover has a tiny direct certificate, but further radius growth is unjustified until a new move family is proved not to duplicate exhausted shells. · Build a deterministic containment checker over the saved 2-for-2, four-delete/three-add, and radius-six move definitions; admit a proposed move family only if it has an explicit non-contained witness and an exact below-ten defect discriminator. · normal · open

Objective: Create an admissible proof-producing route or a genuinely new constructive neighborhood without repeating exhausted encodings or arbitrary radius growth.

First action: Do not launch compute. First either record hashes and successful tiny controls for a newly acquired RoundingSat plus VeriPB/CakePB stack, or implement a move-family containment checker against the three saved exhausted neighborhood definitions.

Stop condition: Hold immediately on unchanged proof-tool absence, containment in an exhausted neighborhood, lack of an exact below-ten discriminator, failed proof replay, or invalid decoded model.

• Do not rerun or scale the exact totalizer CNF SHA-256 9289637f... without a materially different encoding and a fresh gate.
• Reopen proof-producing pair search only after a project-scoped hash-pinned RoundingSat plus VeriPB/CakePB stack passes the saved tiny proof/model controls.
• Before new constructive compute, derive and check a move-family containment/dominance lemma proving that the proposed move is not contained in the exhausted 2-for-2, four-delete/three-add, or radius-six neighborhoods.

• https://www.coveringrepository.com/systems.aspx?k=5&t=3&m=-1 ↗
• https://ljcr.dmgordon.org/cover/show_cover.php?k=5&t=3&v=15 ↗
• https://arxiv.org/abs/2607.23766 ↗
• https://arxiv.org/abs/math/9502238 ↗

GPT-5.6 Sol was principal investigator and independently audited, designed, implemented, executed, and interpreted the epoch. Two GPT-5.6 Terra delegates supplied bounded advisory reconnaissance; their roles and memo hashes are preserved in sources/advisory/terra-epoch117-header-lazy-pair-coupling-20260812.md, and model agreement was not evidence. Python 3.12.3 generated and independently reconstructed formulas, ran mutation/regeneration controls, parsed logs, and captured experiments. CaDiCaL 1.7.3 produced conflict-capped telemetry and incomplete non-evidentiary DRAT streams. SHA-256 bound the packet. The web reader checked the maintained repository and primary methodology/construction sources. No CAS, proof assistant, cloud lab, external publication, Git commit, or remote write was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1477.9s

Review state

not a result claim

Attempt ID

covering-c1553-20260812-112351-0066ed

No human review recorded.