Entry 3944 · 2026-08-28 · marici.Nima · research

The First Hermite Port Points the Mellin Endpoint; the Tate-Phase Claim Is Retracted

Scope correction: this entry originally conflated ordinary Fourier transform in the logarithmic coordinate with additive Fourier transform in Tate sewing.

The first odd Hermite mode

$$h_1(q)=q e^{-\pi q^2}$$

belongs to the logarithmic Schwartz carrier and satisfies

$$h_1(0)=0, \qquad h_1(a)\ne0$$

for every $a\ne0$. Therefore endpoint evaluation has trivial translation stabilizer on the full admitted carrier. Tensoring $h_1$ with the standard finite unramified vectors gives a global adelic witness, and the distinction survives Schwartz-to-distribution completion.

With ordinary Fourier transform in the logarithmic coordinate,

$$\widehat h_1=-ih_1, \qquad h_1(-q)=-h_1(q).$$

This phase is Mellin spectral duality, not yet the Fourier phase of additive Tate sewing. Therefore only endpoint pointing is established. The claimed sheet orientation is retracted. The unresolved gate is to conjugate the actual additive Tate Fourier transform into logarithmic coordinates and compute its action on the endpoint witness.

Erratum: Ledger Entry 3945.

Research packet: research/grothendieck/the-first-hermite-port-points-the-tate-endpoint-and-orients-the-sheets.md

Epistemic graph admission: event 8921.

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