Eight more degree-eight T-curve schemes

Number of nonempty degree-eight T-curve schemes ≥ 2,415. Eight schemes lie outside the 2,407 baseline formed by the 2,367 certificates of Geiselmann et al., arXiv:2602.06888v4, the notebook's seventeen prior additions, the q8 scheme ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩, the nine q9 schemes, and the thirteen q10 schemes:

• ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩ (18 ovals)
• ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩ (19 ovals)
• ⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩ (16 ovals)
• ⟨1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩ (16 ovals)
• ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨5⟩ ⊔ 1⟨9⟩⟩ (19 ovals)
• ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨10⟩⟩ (19 ovals)
• ⟨1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩ (17 ovals)
• ⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩ (17 ovals)

Neither of the two open maximal deep nests is decided.

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

sh problems/hilbert16-degree-8/compute/q11/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 2,407-scheme baseline. Exit zero means the claim holds.

The 205 leftover one-flip balls of the seven q10 followup certificates all finished: 3,121,330 evaluations, five of the eight schemes. One-flip balls of those five certificates then produced the other three (150 balls, 2,283,900 evaluations).

python3 problems/hilbert16-degree-8/compute/q11/collect.py
python3 problems/hilbert16-degree-8/compute/q11/coverage.py

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