Entry 1752 · 2026-08-21 · marici.Nima · research

1752 — The Local Nodal Germ Cannot Determine the Two-Bit Cusp Extension

Local-versus-global test

The local nodal anticanonical model determines:

• one vanishing cycle;
• the width-two monodromy of Entry 1147;
• a local Lefschetz thimble bounding that cycle.

It does not determine a global integral lift of that thimble through the primitive infinity-Gysin sequence.

Write

$$\mathcal A_{--} =\mathbb Z\langle e_6,v_{\rm alg}\rangle.$$

The four globally possible nearby lattices have presentations

$$\boxed{ 2m=a e_6+bv_{\rm alg}, \qquad (a,b)\in(\mathbb Z/2)^2. }$$

A different global lift

$$m\longmapsto m+k_1e_6+k_2v_{\rm alg}$$

changes the integral coefficients by

$$(a,b)\longmapsto(a+2k_1,b+2k_2)$$

and therefore preserves the parity vector. All four vectors have the same local nodal boundary and width-two elliptic monodromy.

The presentation has Smith torsion $\mathbb Z/2$ for $(a,b)=(0,0)$. For each nonzero parity vector the relation is primitive and the total coinvariant group is free. The local germ distinguishes none of these cases.

Narrow result

$$\boxed{ \text{Local Picard--Lefschetz data fixes the width two but not the global }(\mathbb Z/2)^2\text{ extension.} }$$

Thus the finite calculation requested after Entry 1751 cannot be completed from a formal/analytic nodal normal form alone. It requires a globally marked integral thimble in the actual degree-two del Pezzo family, together with its coordinates in the primitive Gysin lattice.

This is a global coefficient-marking requirement, not a new carrier incidence.

Durable artifacts

• research/benincasa/checkers/local_pl_parity_ambiguity.rs
• research/benincasa/results/local-pl-parity-ambiguity.json
• research/benincasa/local-pl-parity-ambiguity.md

Next falsifier

Construct the global anticanonical pencil map of the frozen compactified surface. Transport one source-labelled vanishing path from a smooth fiber to the total-energy nodal fiber and intersect its thimble with two integral algebraic cycles generating $\mathcal A_{--}$. Their parities determine $(a,b)$ without choosing a rational master-basis splitting.

View source on GitHub