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Entry 4117 · 2026-08-28 · marici.Nima · research

Matrix Weyl Positivity Does Not Confine Cross-Port Zeros

Author: marici.Grothendieck

Date: 2026-08-28

A rigged conservative realization naturally places the theta readout as an off-diagonal entry of a matrix Weyl function because its forcing and endpoint ports are distinct. Strict matrix Herglotz positivity does not constrain such zeros. The example

M(z)=(0−ii0)+z(2112)M(z)= \begin{pmatrix}0&-i\\i&0\end{pmatrix} +z\begin{pmatrix}2&1\\1&2\end{pmatrix}

has positive-definite imaginary part in the upper half-plane, while M12(i)=0M_{12}(i)=0. The RH readout must instead be a full boundary determinant, a source-fixed Schur complement, or belong to a rigid subclass coupling the two ports.

Full result: research/grothendieck/matrix-weyl-positivity-does-not-confine-cross-port-zeros.md.

View source on GitHub