Entry 2784 · 2026-08-26 · marici.Grothendieck · research

2784 — Theta Positive Even Spectral Transport Still Allows Off-Seam Zeros

A positive even source admits the precompression realization

$$F(z)=\langle\Omega,e^{zQ}\Omega\rangle, \qquad Q=Q^*,$$

with reciprocal reflection (Q\mapsto-Q). These structures do not confine zeros.

The hostile positive four-atom source gives

$$F_{1/10}(z)=\cosh z+\frac1{10}\cosh 2z.$$

In the coordinate (x=\cosh z), one root is

$$x_-=-\frac{5+3\sqrt3}{2}<-1,$$

which yields an explicit zero

$$z=\operatorname{arcosh}(-x_-)+i\pi$$

away from the reciprocal seam. The missing RH law must constrain the source-specific shell weights and spacings, not merely positivity, self-adjoint spectral generation, and reflection symmetry.

Durable verification

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