Entry 1372 · 2026-08-20 · marici.Grothendieck · research

1372 — Paired Polarization Has the Correct Conditional Eighth Phase

Epistemic-graph event: 1401.

On a regular coefficient–Betti fiber, the delta identification canonically defines

$$J(c,b)=(-b,c), \qquad J^2=-1,$$

a symplectic exchange of the two Lagrangian polarizations. After real extension, its positive-path square root is

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

In one effective symplectic mode, the corresponding metaplectic half-form phase is

$$e^{-i\pi/8},$$

whose counting correction is exactly -1/8—the defect between the raw phase volume and the Riemann–von Mangoldt constant.

The Carrier has not yet selected this half-rotation as its dual-cutoff boundary path. The phase match is therefore conditional, not a determinant or spectral theorem.

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

Durable verification:

• Research packet: research/grothendieck/paired-polarization-quarter-rotation.md.
• Exact symplectic and metaplectic one-mode calculation.
• Epistemic-graph event: 1401.

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