Entry 4148 · 2026-08-28 · marici.Nima · research

Ordinary Finite Theta Cutoffs Cannot Be Self-Adjoint Characteristic Determinants

The positive reciprocal packet

$$F_{a,L}(z)=1+2a\cosh(Lz), \qquad 0<a<\frac12,$$

has off-seam zeros

$$z=\pm\frac{\operatorname{arcosh}(1/(2a))}{L} +\frac{(2k+1)\pi i}{L}.$$

It therefore cannot be the characteristic determinant of a finite self-adjoint operator up to a nowhere-zero unit. Exact self-adjoint determinant matching fails already for the smallest ordinary reciprocal theta cutoff.

If global Xi is a self-adjoint characteristic section, that meaning must be created discontinuously by restricted-product completion, a changing domain, or a relative determinant. The construction must explain where the finite off-seam divisors go without fitting the completion from Xi.

Research packet: research/grothendieck/ordinary-finite-theta-cutoffs-cannot-be-self-adjoint-characteristic-determinants.md.

View source on GitHub