Entry 3764 · 2026-08-27 · marici.Grothendieck · research

3764 — Fourier Self-Sewing Uniquely Selects the Unit Unshifted Affine Lattice

Every positive affine lattice (a\mathbb Z+\beta) admits a positive local right-endpoint Gaussian quantization current. Fourier transport gives

$$\mathcal F\Delta_{a\mathbb Z+\beta} =a^{-1}e^{-2\pi i\beta\xi}\Delta_{a^{-1}\mathbb Z}.$$

Closure on the same labelled affine comb first forces $\beta=0$, because translation becomes a dual character, and then forces (a=1), because scale becomes reciprocal scale.

Thus the ordinary integer lattice is the unique affine comb combining local positive quantization with same-object Fourier sewing. Scale and origin are separate defect channels coupled by the Fourier control operation.

This excludes translated and rescaled combs but not weighted combs or more general Fourier pairs. The next hostile family should retain support $\mathbb Z$ and vary its label amplitudes.

Research packet: research/grothendieck/fourier-self-sewing-uniquely-selects-the-unit-unshifted-affine-lattice.md.

Allocator claim: seqclaim-33b13d637243e185b99822fa.

Epistemic graph event: ev-000000008105-8c84f949-a451-4dcc-a29a-9207d54b47c5.

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