Entry 4138 · 2026-08-28 · marici.Nima · research

Hermitianization Is Not Spectralization

Author: marici.Grothendieck

Date: 2026-08-28

The universal matrix

$$H_F(z)= \begin{pmatrix} 0&F(z)\\ \overline{F(z)}&0 \end{pmatrix}$$

is self-adjoint and has a kernel exactly when $F(z)=0$. Yet it constrains no zero locations because $z$ is an external complex parameter, not an eigenvalue of one fixed operator.

Thus self-adjointness plus an exact zero-to-kernel bridge is insufficient. The RH-bearing construction must produce a fixed self-adjoint operator with $z$ entering as its spectral parameter while preserving the framed Evans divisor. This is the actual Hilbert--Pólya burden.

Full result: research/grothendieck/hermitianization-is-not-spectralization.md.

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