Routes and residuals

Route	State	Test / surviving domain
One-flip radius-three of the seven q10 followup certificates	certified dent; domain finished	205 balls, five schemes outside B ∪ A ∪ Q ∪ N ∪ K. Plan and seed hashes match the leftover seven.
One-flip radius-three of those five certificates	certified dent; domain finished	150 balls, three further schemes.
One-flip of the three followup certificates	residue in q12	93 balls, no scheme outside the 2,415.
Two-flip, or radius four, of the same meshes	unsearched	Larger than this campaign.
Algebraic obstruction of either open (19,3) nest	open	Search failure here is not exclusion.

2026-09-22 — leftover followup seeds

q10 searched one-flip balls of its first six certificates (169 balls) and stopped. Seven further schemes appeared there and were not fed back as seeds. Each ball is one unimodular diagonal flip and every sign change of Hamming weight at most three. There is no symmetry reduction. Search continued through every ball even after a candidate appeared.

Discovery and lifting

Five schemes appeared in the leftover 205 balls:

scheme	ovals	(p,n)	first task	seed
⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩	18	(4,14)	17	⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩
⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩	19	(5,14)	17	same mesh
⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩	16	(3,13)	17	same mesh
⟨1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩	16	(3,13)	75	⟨2 ⊔ 1⟨1⟩ ⊔ 1⟨2⟩ ⊔ 1⟨10⟩⟩
⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨5⟩ ⊔ 1⟨9⟩⟩	19	(4,15)	167	⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩

⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨9⟩⟩ fills the a=0 hole of the family q9/q10 had at a=1,2,3,4. One-flip balls of those five certificates then produced three more: ⟨1 ⊔ 1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨10⟩⟩, ⟨1⟨1⟩ ⊔ 1⟨4⟩ ⊔ 1⟨9⟩⟩, and ⟨1⟨1⟩ ⊔ 1⟨3⟩ ⊔ 1⟨10⟩⟩.

SciPy 1.15.3 / HiGHS produced an integer lifting in each case. Exact arithmetic then checked every global strict inequality. lift_candidate.py records the optional solve; replaying the certificates needs no numerical package.

Neither preferred 22-oval nest appeared. No algebraic exclusion is claimed.