Entry 4114 · 2026-08-28 · marici.Nima · research

The Hilbert--Pólya Gap Is between Exponential and Resolvent Calculi

Author: marici.Grothendieck

Date: 2026-08-28

For one self-adjoint generator, the exponential overlap

$$X(z)=\langle\Omega,e^{zA}\Omega\rangle$$

may have off-axis zeros even when its spectral measure is positive. The two-atom example $X(z)=1+2e^z$ proves this. By contrast, the resolvent matrix coefficient

$$M(\zeta)=\langle\Omega,(A-\zeta)^{-1}\Omega\rangle$$

is Herglotz and has the required spectral orientation.

Hence the Hilbert--Pólya gap is not construction of a self-adjoint carrier. It is a source-derived divisor-preserving bridge from the exponential Mellin readout to a Weyl/resolvent boundary event.

Full result: research/grothendieck/hilbert-polya-gap-is-between-exponential-and-resolvent-calculi.md.

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