Thirteen more degree-eight T-curve schemes

Number of nonempty degree-eight T-curve schemes ≥ 2,407. Thirteen schemes lie outside the 2,394 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⟩⟩, and the nine q9 schemes:

• ⟨1 ⊔ 2⟨1⟩ ⊔ 1⟨10⟩⟩ (16 ovals)
• ⟨2 ⊔ 2⟨1⟩ ⊔ 1⟨10⟩⟩ (17 ovals)
• ⟨3 ⊔ 2⟨1⟩ ⊔ 1⟨10⟩⟩ (18 ovals)
• ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩ (17 ovals)
• ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩ (19 ovals)
• ⟨3 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩ (20 ovals)
• ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩ (17 ovals)
• ⟨1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨9⟩⟩ (15 ovals)
• ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩ (18 ovals)
• ⟨3 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩ (19 ovals)
• ⟨4 ⊔ 2⟨1⟩ ⊔ 1⟨10⟩⟩ (19 ovals)
• ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩ (18 ovals)
• ⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨5⟩ ⊔ 1⟨9⟩⟩ (20 ovals)

Neither of the two open maximal deep nests is decided.

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

sh problems/hilbert16-degree-8/compute/q10/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 2,394-scheme baseline. Exit zero means the claim holds.

The 106 leftover one-flip balls of the four q9 followup certificates all finished: 1,613,956 evaluations, six of the thirteen schemes. One-flip balls of those six certificates then produced the other seven (169 balls, 2,573,194 evaluations).

python3 problems/hilbert16-degree-8/compute/q10/collect.py
python3 problems/hilbert16-degree-8/compute/q10/coverage.py

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