Close the three-line F-star census remainder with a correct joint quotient

[Arham-Tahir64]

Change and mathematical scope

PR #3031 leaves 20 shareable three-line patterns not found in its realization cap. All 20 violate the necessary zero-arc anchor-parity condition, so none can be realized over the rationals.

This scoped successor audit finds a second defect in PR #2978's quotient: canon_config pulls back line data but pushes forward zero arcs. A three-cycle turns valid input into an inadmissible “canonical” configuration. The package supplies the minimal counterexample and a corrected joint census using one global total permutation and a consistent line permutation.

The complete domain has 450,028 ordered configurations and 13,449 joint orbits: 13,442 have box=3; seven have box=2. Rank/anchor attainment excludes four of the seven; the other three are Aaron Wang's existing explicit realizations, independently replayed here. This closes the three-line mask/zero-arc realization remainder. It does not prove general F-star or change a supplier, paid moment, kappa, or the selected frontier.

Task and overlap

Started from canonical main 18aacb215fc3717b06be8ceb9fca62aeca0d3e99, priority 6; the pre-submission fetch returned the same head.

Handoff-first override, changed scope from census-pruning-audit: the F7 starting case already had follow-ups, while #3031 explicitly requested a joint-sound three-line census. Task, proof, exact domain and next questions.

Exact predecessors:

• F-star family gap: Lemma BPP's without-loss question answered NO - box-completeness fails at tau >= 1, minimal counterexample v = 3 (no kappa) #2978: 12eb7b832de9239a314a7cddf6c6033d618cd8f2 — two byte-pinned source files retained.
• KB5L: tau >= 1 box floor PROVEN at v = 5 - shareable class narrowed - no kappa #3031: e75906fc4e8b9f756a9877b2e467f4e564851c7d — canonicalization warning and explicit follow-up; its separate v=5 proof is not replayed.

Fresh overlap scan: 256 open PRs and 1,862 discussion records, complete remote pagination, no new competing scope found. Local sparse coverage is recorded. Canonical claim.

Validation

• Independent Burnside fixed-point count agrees with the canonical census.
• Definition-level exhaustive label/order enumeration agrees on all 13,449 orbits.
• Exact legacy replay reproduces 3,023 keys and 27 candidates; 1,370 keys and 20 candidates violate scope parity.
• All 27 predecessor candidates are accounted for; all three credited integer P,Q witnesses are recomputed.
• Direct Python 3.14.7, optimized replay, and registered Python 3.12 replay pass with identical receipt.
• Positive/rejecting witness controls, both quotient counterexamples, and temporary source/receipt corruption controls pass.
• Metadata/source pins, all 10 committed publication files, and an optimized Git-free export of the actual commit pass.

Reproduce:

python3 -B research/fstar-joint-label-census/verify.py
python3.12 scripts/research_package.py --package research/fstar-joint-label-census
python3 scripts/check_research_commit.py --package research/fstar-joint-label-census

Receipt SHA-256: 391852c830995818f55d8a1de6dfdcc361e397d8be3a7b7ffbc6d250e59b3c6f.

Not checked: v>=4 classifications, PR3031's separate v=5 proof, full suppliers/native replay, physical costs, moments, or Lean. Hosted CI and independent human review are separate.

Credit

DreamingOfClouds / #1590 supplied F-star, L8 and cert. Aaron Wang (navinor), with his stated opencode assistance, supplied the anchor/box lineage, #2978's three realized examples, and #3031's per-line-canonicalization warning and rank-two exclusion. Those results are credited rather than claimed as new. The consistent census, line-action counterexample and exact disposition of the remaining 20 patterns are new here.

Prepared by Arham Tahir with substantial OpenAI Codex assistance. Apache-2.0; predecessor source bytes and attribution preserved.

Open next questions

The package names a separate audit of #3031's raw-type v=5 proof and a bounded larger-v joint census. Both need their own overlap check and budget; the three-line result does not settle those domains.