Entry 2634 · 2026-08-25 · marici.Grothendieck · research

2634 — Conjugate-Doubled Theta Shear Is Indefinitely Conservative Everywhere

For the shear (S(F)), conjugate doubling preserves the fixed Hermitian form

$$H=\begin{pmatrix}0&\Omega\\-\Omega&0\end{pmatrix}, \qquad \Omega=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$

Indeed, with (D(F)=\operatorname{diag}(S(F),S(\overline F))),

$$D(F)^*HD(F)=H$$

for every complex (F). The form is nondegenerate but indefinite, so this conservation holds everywhere and cannot orient scalar zeros toward the critical seam.

A generic single complex shear family has no fixed nondegenerate Hermitian metric. The surviving requirement is a completion-stable coupling between the doubled shear and the seam-selective positive reciprocal energy.

Durable verification

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