Routes and residuals

Route	State	Test / surviving domain
Another maximal-sign sweep on a census triangulation	dead in the prior record	All 184 were swept; neither open nest occurs. Other triangulations survive.
Another small even-split walk from the two completed nest components	dead in the prior record	q7 completed both. Different odd skeletons survive.
Structural quotient of the enormous remaining collection graph	live, exploratory	Any proposed deletion rule must preserve the full nesting tree, not merely (p,n).
One diagonal flip plus radius-three sign changes from the 55 seed certificates	certified dent; hunt stopped	Ball 82 gives ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩ outside B ∪ A. Exact lifting and both topology checks pass. 1,108 balls remain unsearched.

The first two rows rely on the recorded completed searches; they are not independently rerun billion-evaluation claims of this campaign. The published certificates themselves are replayed before searching.

2026-09-08 — resumed finite domain

The recovered source files had no saved search outputs. There are 1,190 seed/flip pairs from the 55 seeds. Each sign ball has 1 + 45 + 990 + 14,190 = 15,226 evaluations, for 18,118,940 in all, counting repetitions between seeds. There is no symmetry reduction.

Search every combinatorial flip, including any whose regularity is not yet known. This is a superset of the selected regular-flip domain. If a new topology appears, solve for a strict lifting and verify it exactly before claiming a T-curve. No failed lifting attempt is an algebraic exclusion. The empty curve is excluded from novelty.

search.py writes an atomic coverage checkpoint after each complete ball and stops on a candidate. The checkpoint records source hashes, the seed and plan hashes, each removed/added diagonal, the resulting triangulation hash, full scheme frequencies and any candidate signs. run_all.sh is reserved for the existence claim; coverage has a separate replay command.

Discovery and lifting

The search stopped after 82 complete balls, 1,248,532 evaluations and 807 distinct observed schemes. Task 81 (zero-based) uses deg8/o22-p07-n15/(5v1(3)v1(12)).pcom as its seed. Replace the diagonal [(2,3),(6,0)] with [(3,2),(5,1)] and change the signs at (0,0), (0,1) and (7,0). Python independently returns ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩. This scheme occurs 79 times in that ball.

The old lifting fails two strict inequalities on the changed mesh. The existing 200,000-step projection search (haas.regularize, seed 0) obtained no new lifting. That was a failed numerical search, not evidence of nonregularity. A linear feasibility solve with SciPy 1.18.1 / HiGHS succeeded. Clearing the rational denominators gave integer heights between -1,290 and 0; all 2,688 global strict inequalities have slack at least 4. The committed certificate is checked without SciPy. lift_candidate.py records the optional discovery calculation.

The 1,108 remaining balls and the larger structural routes remain unsearched. The new scheme has 20 ovals and decides neither open 22-oval nest. There is no algebraic exclusion claim.