Finite handles — degree-eight real schemes
Reconstructed on 2026-09-08 from PROBLEM, ATTACK, WALKTHROUGH and
q1–q5/q7. There is no q6 directory. Historical completed searches are
listed as residue: completion of a neighborhood did not improve the
classification record. They are not exclusions of algebraic schemes.
| Leaf | Status | Evidence / what survives |
|---|---|---|
| Seventeen schemes outside the published 2,367 | certified | certs/new_schemes.json; nonempty T-scheme count at least 2,384. |
| Maximal strata of all 184 census triangulations | residue | certs/census_span_classification.json; exactly the published 38 M-schemes. Other triangulations survive. |
| Radius-one thicken of all census maximal strata | residue | q1–q3; no scheme beyond census plus seventeen. Larger sign distance and other triangulations survive. |
| Up to three split changes around published deep-nest collections | residue | q4; no new M-scheme. More distant collections survive. |
| Purely odd collections through size five | residue | q3/q7; twelve known schemes. Larger odd collections and mixed odd/even collections survive; this does not bound the odd skeleton of all new M-schemes. |
| Even extensions of the published a=10 and a=17 odd skeletons | residue | q7; only the two known nests. Different odd skeletons survive. |
| Larger fixed-odd components and the all-even component | open | q7 counted some components but did not classify them all. A structural quotient is needed before further enumeration. |
| q8: escape the census by changing triangulation and signs together | certified | q8/CLAIM.md: number of nonempty degree-eight T-curve schemes ≥ 2,385. New scheme ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩; exact integer lifting and independent Python/Rust checks. Search stopped after 82/1,190 balls; the leftover 1,108 are the q9 row. |
| q9: remaining 1,108 balls of the q8 one-flip radius-three domain | certified | q9/CLAIM.md: number of nonempty degree-eight T-curve schemes ≥ 2,394. Nine schemes outside B ∪ A ∪ {⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩}. All 1,108 leftover balls finished; 115 one-flip balls of those certificates added four more; 21 flips of the q8 certificate added nothing. Neither open nest appeared. |
| q10: one-flip radius-three of the four q9 followup certificates | certified | q10/CLAIM.md: number of nonempty degree-eight T-curve schemes ≥ 2,407. Thirteen schemes outside B ∪ A ∪ Q ∪ N. All 106 leftover balls finished; 169 one-flip balls of those six certificates added seven more. Neither open nest appeared. |
| q11: one-flip radius-three of the seven q10 followup certificates | certified | q11/CLAIM.md: number of nonempty degree-eight T-curve schemes ≥ 2,415. Eight schemes outside B ∪ A ∪ Q ∪ N ∪ K. All 205 leftover balls finished; 150 one-flip balls of those five certificates added three more. Neither open nest appeared. |
| q12: one-flip radius-three of the three q11 followup certificates | residue | 93 balls, 1,416,018 evaluations, no scheme outside B ∪ A ∪ Q ∪ N ∪ K ∪ L. Plan hash 544498c92d5c94d04a0c087f8c6424080120da7420c4108a07a15691c744c912. Neither open nest appeared. |
| Algebraic obstruction outside combinatorial patchworking | open | Neither undecided (19,3) deep nest is decided. T-curve nonexistence alone would not decide algebraic nonexistence. |
The covering problem is frozen. One heavy process at a time; no unbounded BFS queue. The 1,190-ball one-flip domain is finished, as is the leftover one-flip of the four q9 followup certificates, the one-flip of the six leftover q10 certificates, the leftover one-flip of the seven q10 followup certificates, and the leftover one-flip of the three q11 followup certificates. Further progress is a scheme outside the 2,415, a decision of either open (19,3) nest, or a new recorded finite handle. The one-flip followup chain has no unused leftover certificate.