Nine more degree-eight T-curve schemes

Number of nonempty degree-eight T-curve schemes ≥ 2,394. Nine schemes lie outside the 2,385 baseline formed by the 2,367 certificates of Geiselmann et al., arXiv:2602.06888v4, the notebook's seventeen prior additions, and the q8 scheme ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩:

• ⟨3 ⊔ 1⟨5⟩ ⊔ 1⟨10⟩⟩ (20 ovals)
• ⟨5 ⊔ 1⟨2⟩ ⊔ 1⟨3⟩ ⊔ 1⟨7⟩⟩ (20 ovals)
• ⟨5 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨9⟩⟩ (20 ovals)
• ⟨4 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨9⟩⟩ (19 ovals)
• ⟨3 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨9⟩⟩ (18 ovals)
• ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨9⟩⟩ (17 ovals)
• ⟨4 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩ (20 ovals)
• ⟨3 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩ (19 ovals)
• ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩ (18 ovals)

Neither of the two open maximal deep nests is decided.

The claim is the existence of these nine certificates. From the repository root, replay with:

sh problems/hilbert16-degree-8/compute/q9/run_all.sh

This needs Python 3, a C compiler and rustc, with no Python packages to install. It runs positive and negative controls, checks each integer lifting, independently computes the nesting tree in Python and Rust, and checks absence from the 2,385-scheme baseline. Exit zero means the claim holds.

The remaining 1,108 balls of the q8 one-flip radius-three plan all finished: 16,870,408 evaluations, five of the nine schemes. One-flip balls of those five certificates then produced the other four (115 balls, 1,750,990 evaluations). One-flip balls of the q8 certificate itself (21 balls, 319,746 evaluations) added nothing. The published (19,3) M-certificate neighbourhood in the leftover plan (123 balls) hit neither open nest.

python3 problems/hilbert16-degree-8/compute/q9/collect.py
python3 problems/hilbert16-degree-8/compute/q9/coverage.py

lift_candidate.py can repeat a lifting solve with SciPy; those packages are unnecessary for verification.