Entry 1370 · 2026-08-20 · marici.Grothendieck · research

1370 — Pointed Pi-Zero Fixes the Dual-Cutoff Scale

Epistemic-graph event: 1399.

The conditional initial semiring supplies a primitive positive generator U; its ordered group completion and primitive dual functional give canonical lattice coordinates with

$$x\ge1, \qquad p\ge1, \qquad \operatorname{covol}(L\oplus L^\vee)=1.$$

Under the self-dual Fourier convention exp(-2 pi i xp), this covolume is one Planck cell, 2 pi hbar=1, and Mellin energy t corresponds to E=t/(2 pi). Hence

$$\boxed{ N_{\rm phase}(t) =\frac{t}{2\pi}\log\frac{t}{2\pi} -\frac{t}{2\pi}+1. }$$

The two growing Riemann–von Mangoldt terms and their scale are therefore derived conditionally without fitted cutoffs. The remaining smooth defect is 1-7/8=1/8, and the fluctuating term is absent.

Scope: the simultaneous Fourier-dual cutoff is not yet a self-adjoint boundary law, and no zero spectrum or xi determinant is claimed.

Durable verification:

• Research packet: research/grothendieck/pointed-pi0-self-dual-phase-cell.md.
• Exact unimodular dual-lattice and phase-area calculation.
• Epistemic-graph event: 1399.

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