q8 claim — a scheme beyond the current census

Status: certified dent (2026-09-08). Selected leaf: q8: escape the census by changing triangulation and signs together 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. The baseline is |B ∪ A| = 2,384.

The verified predicate is the existence of a certificate (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. This establishes the exact inequality

number of nonempty degree-eight T-curve schemes ≥ 2,385.

Witness: certs/new_schemes.json, realizing ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩ (20 ovals, with (p,n) = (5,15)). All 2,688 global strict lifting inequalities pass with minimum slack 4. Python's curve-segment computation and Rust's monochromatic-region graph independently recover the same nesting forest. The scheme is absent from the archive's TYPE fields and filenames, the replayed B, and the seventeen additions A.

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.

Initial finite search domain: one unimodular diagonal flip from each of the 38 published M-certificates and the seventeen additions, followed by all sign changes of Hamming weight at most three. Regularity is a separate exact check. Duplicate triangulations and equivalent sign distributions may be identified only with recorded maps. Further routes, reductions, controls and residual domains will be recorded in ROUTES.md before they are used. Search stopped on the certified dent after 82 of 1,190 balls; the 1,108 unsearched balls are not needed for this existence claim.

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. 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 replay entry point.