Provisional confirmation cards: normal-sector extension

Discussion rubric v0.1, 8 September 2026. Not a jointly adopted standard. M measures mathematical confirmation, not probability, novelty, difficulty or significance. No automatic change to legacy site levels or publication ordering.

NS-1efficient-v1 — supported conditional theorem

Exact claim: For a connected oriented torus-boundary manifold other than the solid torus and T^2 x I, every 1-efficient ideal triangulation has the native generic trace-cokernel bound dim(Q^6/im tau)>=2^r-1, with the retained native dictionary. The note separately proves the bound for the two specified exceptional triangulations.

Provisional band: M10–19. Complete candidate proof and internally reproduced controls; no documented independent reconstruction of the complete argument. Publication of a dependency does not supply independent checking of the new translation.

Evidence: Exact integer lattice identification; complete native phase computation; published normal-surface positivity; prior correct-sided convergence and cohomological grading; Gaussian-integer series checks on the exceptional data. No physical measurements are relevant to this mathematical claim. No formal-language proof is claimed.

Independent lines: The two exceptional local evaluations are separate cases, not independent proofs of the general bound. Lattice, phase and topology arguments are complementary dependencies, not three confirmations. No other-model or external review of the full synthesis is recorded.

Source-to-claim distance: On the main route, (i) identify the actual root unit lattice with all closed integer normal sectors, including phases; (ii) extend the published irreducible positivity premise to every reduced term by monoid decomposition; (iii) apply the existing Lab image-grading detector. These are substantive changes, not text edits. The accepted/published-source branch and internally audited Lab branches have different anchor burdens. An exact weighted burden is withheld until the full essential dependency graph and draft weighting conventions are audited; do not sum only the shortest branch.

Next maturity gate: Independent reconstruction of (4)-(6) against the native root conventions and of the entire-image grading chain. A Lean/Rocq/Isabelle effort must separately formalize the lattice/phase statements, declare the imported topology axioms, and check alignment with the mathematical statement. Formalizing only the finite matrices would not establish M50.

NS-exceptional-all-v1 — unresolved strengthening

Exact claim under investigation: F is not identically zero for every 1-efficient triangulation of the solid torus or T^2 x I, with its actual native character and image lattice.

Provisional band: M0–9. Precise and testable, with two verified instance formulas but no complete general proof.

Evidence boundary: The fixed controls establish F(1)=0 with a nonzero nonperipheral value. Published scalar boundary indices can vanish and cannot transport that detection. No statement about all exceptional triangulations follows from those instances.

Next decisive gate: Construct a correct-sided transport of a detecting insertion through a local triangulation move, stating any connectivity restriction, or construct a detecting input intrinsically from the manifold’s compressing-disk/product structure. Do not replace this by a larger example census.

Change from the preceding pilot

The old signed-angle nonvanishing question is now split: the nonexceptional 1-efficient class has a complete conditional argument, while the universal exceptional case remains open. The higher band on the proved subclaim does not promote the unresolved universal claim.