q12 claim — leftover one-flip of three q11 followup certificates

Status: residue (2026-09-26). Selected leaf: q12: one-flip radius-three of the three q11 followup certificates in ../LEAVES.md. The existence predicate below is not established.

Let B be the set of 2,367 nonempty schemes in the published deg8.pcoms.txz archive of Geiselmann et al., Limits of combinatorial patchworking, arXiv:2602.06888v4, §4.3. Let A be the seventeen independently verified schemes in ../certs/new_schemes.json. Let Q be the q8 scheme ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩. Let N be the nine q9 schemes in ../q9/certs/new_schemes.json. Let K be the thirteen q10 schemes in ../q10/certs/new_schemes.json. Let L be the eight q11 schemes in ../q11/certs/new_schemes.json. The baseline is |B ∪ A ∪ Q ∪ N ∪ K ∪ L| = 2,415.

The attempted predicate was the existence of certificates (T, h, s) such that T is a primitive triangulation of the 45 lattice points of 8Δ₂, h is an exact strict convex lifting for T, s assigns a sign ±1 to every lattice point, and the recomputed real scheme S(T,s) is not in B ∪ A ∪ Q ∪ N ∪ K ∪ L. No such certificate is in certs/new_schemes.json. This does not change the inequality

number of nonempty degree-eight T-curve schemes ≥ 2,415.

The two preferred targets remain ⟨4 ⊔ 1⟨2 ⊔ 1⟨14⟩⟩⟩ and ⟨14 ⊔ 1⟨2 ⊔ 1⟨4⟩⟩⟩. A regular patchwork realizing either would also decide its algebraic realizability. No search failure is reported as an algebraic exclusion. Neither preferred target is decided.

Finite search domain:

1. One-flip radius-three balls of the three q11 followup certificates ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨10⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩. These three were produced by q11 followup and were never used as one-flip seeds. Plan hash 544498c92d5c94d04a0c087f8c6424080120da7420c4108a07a15691c744c912. Each ball is one unimodular diagonal flip followed by all sign changes of Hamming weight at most three. All 93 finished (1,416,018 evaluations) and produced no scheme outside the baseline (certs/coverage_leftover.json).
2. One-flip of a new leftover certificate was not started: the leftover produced none.

Falsifiers of a proposed certificate: a missing/extra lattice point, nonprimitive or overlapping cells, a failed strict lifting inequality, invalid sign, disagreement between two independently implemented topology computations, or membership of the actual scheme in B ∪ A ∪ Q ∪ N ∪ K ∪ L. A candidate rejected by either implementation does not improve the bound. This finished leftover with no new scheme is residue, not a lower bound.

run_all.sh must exit zero only when the existence predicate has been verified. Absence of candidates must not produce exit zero. Search coverage has a separate collect entry point: python3 collect.py then python3 coverage.py.