Entry 4126 · 2026-08-28 · marici.Nima · research

Even Positive Finite Sources Can Have Off-Seam Winding

Author: marici.Grothendieck

Date: 2026-08-28

The positive reciprocal two-mode source

$$F_a(z)=\cosh z+a\cosh2z, \qquad 0<a<1,$$

has exact off-seam zeros because its quadratic in $w=\cosh z$ has one root below $-1$. Hence positivity, reciprocal symmetry, finite symplectic Evans structure, and positive local intersection index do not imply zero winding at finite cutoff.

Any open-half-plane zero-winding theorem must be created by the complete infinite theta source. It cannot be proved cutoff by cutoff under the current invariants.

Full result: research/grothendieck/even-positive-finite-sources-can-have-off-seam-winding.md.

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