files / problems / hilbert16-degree-8 / compute / q9 / CLAIM.md

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q9 claim — nine schemes beyond the q8 census

Status: certified dent (2026-09-12). Selected leaf: q9: remaining 1,108 balls of the q8 one-flip radius-three domain 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⟩⟩ in ../q8/certs/new_schemes.json. The baseline is |B ∪ A ∪ Q| = 2,385.

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. Nine such certificates are in certs/new_schemes.json. This establishes the exact inequality

number of nonempty degree-eight T-curve schemes ≥ 2,394.

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

All 2,688 global strict lifting inequalities pass on each witness (minimum slack 16, 8, 2, 2, 2, 8, 2, 2, 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, and Q.

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. The 1,108 unsearched balls of the q8 plan (tasks 82 through 1,189). Plan and seed hashes match ../q8/certs/coverage.json. Each ball is one unimodular diagonal flip followed by all sign changes of Hamming weight at most three. All 1,108 finished (16,870,408 evaluations) and produced the first five witnesses.
  2. One-flip radius-three balls of the q8 certificate: 21 balls, 319,746 evaluations, no further scheme.
  3. One-flip radius-three balls of the five certificates from (1): 115 balls, 1,750,990 evaluations, four 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. 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.