Entry 1374 · 2026-08-20 · marici.Grothendieck · research

1374 — The Half-Rotation Is a Gaussian Norm-Two Correspondence

Epistemic-graph event: 1404.

The canonical paired polarization satisfies J^2=-1. Therefore

$$K=1+J$$

is an integral Gaussian correspondence with

$$K^2=2J, \qquad K^\dagger K=2.$$

In each symplectic mode it has determinant 2, Smith form (1,2), and cokernel Z/2. Its unitary normalization is precisely

$$\frac{K}{\sqrt2} =\frac{1+J}{\sqrt2} =e^{\pi J/4}.$$

Thus the conditional half-rotation is not an arbitrarily chosen analytic path: it is the normalization of the primitive Gaussian factor 1+i, with (1+i)^2=2i. The bad prime 2 is its ramified norm.

The remaining gate is physical and spectral: realize K as the actual source boundary correspondence, construct its self-adjoint domain, and prove its determinant rather than merely its smooth phase.

Scope: no physical chain map, xi determinant, or zero spectrum is yet asserted.

Durable verification:

• Research packet: research/grothendieck/gaussian-norm-two-half-rotation.md.
• Exact Gaussian, adjoint, determinant, and Smith calculations.
• Epistemic-graph event: 1404.

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