Enumerate the globally complete 2145 rooted pair-excess skeleton classes, replace each child seven-zero system by its stronger ten-zero parent, solve the continuous nonnegative triple-excess system, reconstruct every positive solution over the rationals, and independently verify all witnesses and child containments.

global rational ten-zero dominance census

Monotone parent dominance compresses 124988 child headers to 2145 root systems; sparse LP supplies candidate bases and exact Fraction arithmetic turns them into independently checkable rational witnesses.

Hypothesis: At least one of the 2145 safely quotiented rooted pair-excess skeleton representatives has no nonnegative rational triple-excess vector vanishing on all ten triples of its distinguished 5-set.

Test: Run all 2145 sparse LPs below a 110-second wall bound and accept survival only after a separate checker verifies sparse rational witnesses against all 105 pair equations and all child containments exactly.

Exact rational witnesses directly falsify the pruning hypothesis for every root. Set containment transfers each ten-zero witness to every seven-zero child. The result is global for this relaxation but strictly weaker than integral or literal block feasibility.

• Every one of the 2145 rooted pair-excess skeleton representatives has a nonnegative rational triple-excess vector satisfying all 105 pair equations and vanishing on the ten triples of its root.
• All 124988 safely quotiented seven-zero joint headers survive the rational ten-zero necessary relaxation by set containment.
• The claims do not imply integer feasibility, literal block realizability, or a 54-block cover.

• Producer result SHA-256 ed013fd0b725f6ebef62965db74820761a083d6b7daa598b050315d175b5ef97; 2145 of 2145 exact rational witnesses.
• Independent check SHA-256 a5aeae858c201991173e162cd4710089e9b921902d2b6eee95c6655869d75418; 189394 entries, 225225 equations, 124988 child containments.
• Five targeted mutations rejected; mutation packet SHA-256 30d1dd83a35b7c2540cfdc6a1987cb956858b7c3037b9d95820db0c1eb591c15.
• Fresh regeneration was byte-identical; the 18-entry manifest replay experiment 20260810-195415-c8f882 passed.

• .proof-experiments/20260810-194935-639733: 100-root pilot completed with exact reconstruction in 2.535 harness seconds.
• .proof-experiments/20260810-195140-e8d351: final producer generated 2145 exact rational witnesses in 29.394 seconds.
• .proof-experiments/20260810-195217-9cb587: fresh full regeneration produced the identical result hash.
• .proof-experiments/20260810-195300-5f54a9: independent checker passed all roots, equations, and containments in 4.630 seconds.
• .proof-experiments/20260810-195312-139ddd: five mutations failed closed.
• .proof-experiments/20260810-195415-c8f882: all 18 frozen manifest entries replayed.

checkers/check_all_root_ten_zero_rational_gate_v1.py does not import the producer; it rebuilds triples, pairs, scatter maps, partition matrices, root zero sets, and child selections, then checks every stored value with fractions.Fraction exact arithmetic.

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

• LP presolve plus exact reconstruction -> use floating optimization only to discover a sparse basis, then accept evidence solely after rational equality checks -> all 2145 candidates converted successfully and exact validation passed.

• All 2145 rooted pair-excess skeleton classes are feasible in the rational ten-zero triple-excess system.
  Result SHA-256 ed013f... plus independent exact check SHA-256 a5aeae... and byte-identical regeneration. · All 41 partitions in the prior rooted census; rational nonnegative triple-excess vectors only. · computed

• Use continuous ten-zero triple-excess feasibility alone to prune any joint header.
  All 2145 rooted representatives and all 124988 safely quotiented seven-zero children. · Every root has an exact rational witness, which survives every child's weaker seven-zero constraints. · artifacts/all-root-ten-zero-rational-gate-20260810/result.json and independent-check.json · An independently demonstrated error in the frozen rooted census, root-child map, or exact witness check; adding integrality or labelled constraints is a different route.
• Treat strategy-2c40d32de977 as an unexecuted challenger.
  The complete fixed-pair-link relaxation over 395 signatures and 754 targets. · Epoch 28 already executed it and every target survived with independent checking. · records/attempts/epoch-0028-type4-complete-pair-link-20260809.json and artifacts/type4-complete-pair-link-gate-20260809/independent-check.json · Add literal identities outside anonymous root margins or demonstrate measured proof-search use.

• Complete integral ten-zero census in the checkpointed lab.
  Integrality is the cheapest remaining strengthening of this exact global map, but three varied partitions and global rational survival make pruning uncertain. · Generalize the integer producer/checker to all 2145 roots, checkpoint by partition, and submit only with a predeclared useful-prune threshold and exact witness/proof requirements. · low · open
• Complete labelled root-family realization pilot.
  Literal block identities are the information consistently absent from surviving aggregate relaxations and can feed proof-producing SAT leaves. · Choose one complete root-family partition, build a disjoint or explicitly overlapping cube union with an independent coverage map, and compare one proof-capable leaf against the current CNF at a matched cap. · high · open

Objective: Test whether a complete labelled root-family decomposition yields measurable pruning or proof-search improvement.

First action: Specify a hash-bound root-family cube protocol with exact union coverage and a matched 2000-conflict proof-capable pilot before generating multiple leaves.

Stop condition: Redirect if coverage maps disagree, the pilot remains UNKNOWN without at least 20 percent fewer decisions and no propagation-density regression, or the cube surface exceeds a predeclared proof-replay budget.

• Mark strategy-2c40d32de977 stale/ruled out because epoch 28 already completed its 754-target test.
• Do not run more continuous ten-zero margin refinements; add integrality or literal block identities.
• Before proof-scale solving, design one complete root-family labelled realization pilot with independent union coverage and a measured propagation or pruning gate.
• Run the remaining all-root integer census only as a checkpointed lab classification with a predeclared useful-pruning threshold; three varied integer partitions and the global rational census currently predict low yield.

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

GPT-5.6 Sol served as principal investigator. GPT-5.6 Terra delegates supplied advisory stale-route reconnaissance only; Sol independently replayed the cited immutable campaign artifacts and did not count delegate agreement as validation. Python 3.12.3, NumPy 1.26.4, SciPy 1.11.4 with bundled HiGHS, Python Fraction exact arithmetic, SHA-256, the computational-researcher experiment harness, and web search were used. No new subagents, CAS, proof assistant, SAT solver, cloud-lab job, external write, or publication were used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

1172.1s

Review state

not a result claim

Attempt ID

covering-c1553-20260810-200151-664a78

No human review recorded.