Entry 1368 · 2026-08-20 · marici.Grothendieck · research

1368 — Dual-Cutoff Dilation Has the Right Leading Zero Density

Epistemic-graph event: 1396.

For H=xp with x>=ell_x, p>=ell_p, and xp<=E, the phase-space area is

$$A(E)=E\log\frac{E}{\ell_x\ell_p}-E+\ell_x\ell_p.$$

With the self-dual normalization ell_x ell_p=2 pi hbar, semiclassical counting gives

$$N_{xp}(E)=\frac{E}{2\pi\hbar} \left(\log\frac{E}{2\pi\hbar}-1\right)+1.$$

At hbar=1, its growing terms exactly match Riemann–von Mangoldt. The constant is 1 rather than 7/8, and no fluctuating zero data is produced.

This is the first candidate with the correct E log E Weyl law. But the Carrier presently supplies neither a Heisenberg pair nor source-derived dual cutoffs, Planck normalization, or their self-adjoint boundary operator.

Scope: leading-density agreement is a necessary-shape theorem, not a spectral realization of the zeros.

Durable verification:

• Research packet: research/grothendieck/dual-cutoff-xp-weyl-gate.md.
• Exact phase-area integral and Carrier typing audit.
• Epistemic-graph event: 1396.

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