q10 claim — thirteen schemes beyond the q9 census

Status: certified dent (2026-09-20). Selected leaf: q10: one-flip radius-three of the four q9 followup certificates in ../LEAVES.md.

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. The baseline is |B ∪ A ∪ Q ∪ N| = 2,394.

The verified predicate is 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. Thirteen such certificates are in certs/new_schemes.json. This establishes the exact inequality

number of nonempty degree-eight T-curve schemes ≥ 2,407.

Witnesses, each with an integer lifting and independent Python/Rust topology checks:

• ⟨1 ⊔ 2⟨1⟩ ⊔ 1⟨10⟩⟩ (16 ovals, (p,n) = (4,12))
• ⟨2 ⊔ 2⟨1⟩ ⊔ 1⟨10⟩⟩ (17 ovals, (p,n) = (5,12))
• ⟨3 ⊔ 2⟨1⟩ ⊔ 1⟨10⟩⟩ (18 ovals, (p,n) = (6,12))
• ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩ (17 ovals, (p,n) = (4,13))
• ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩ (19 ovals, (p,n) = (5,14))
• ⟨3 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩ (20 ovals, (p,n) = (6,14))
• ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩ (17 ovals, (p,n) = (4,13))
• ⟨1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨9⟩⟩ (15 ovals, (p,n) = (3,12))
• ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩ (18 ovals, (p,n) = (5,13))
• ⟨3 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩ (19 ovals, (p,n) = (6,13))
• ⟨4 ⊔ 2⟨1⟩ ⊔ 1⟨10⟩⟩ (19 ovals, (p,n) = (7,12))
• ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩ (18 ovals, (p,n) = (4,14))
• ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨5⟩ ⊔ 1⟨9⟩⟩ (20 ovals, (p,n) = (5,15))

All 2,688 global strict lifting inequalities pass on each witness (minimum slack 2, 2, 2, 8, 2, 2, 8, 8, 8, 8, 8, 8, 2 respectively). Python's curve-segment computation and Rust's monochromatic-region graph independently recover the same nesting forest. Each scheme is absent from the archive's TYPE fields and filenames, the replayed B, the seventeen additions A, Q, and N.

The two preferred targets were ⟨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 will be reported as an algebraic exclusion. Neither preferred target is decided by this result.

Finite search domains:

1. One-flip radius-three balls of the four q9 followup certificates ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨9⟩⟩, ⟨4 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩, ⟨3 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩, ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩. Plan hash be9520910264e4bcdba6c03618384023d544804e4999a42516a391210e9bd320. Each ball is one unimodular diagonal flip followed by all sign changes of Hamming weight at most three. All 106 finished (1,613,956 evaluations) and produced the first six witnesses.
2. One-flip radius-three balls of those six certificates: 169 balls, 2,573,194 evaluations, seven further witnesses.

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. A candidate rejected by either implementation does not improve the bound. An incomplete search is residue.

run_all.sh must exit zero only when this existence predicate has been verified, with both independent checks and rejection/forced-answer controls. A successful baseline replay or absence of candidates must not produce exit zero. Search coverage has a separate collect entry point.