Entry 4130 · 2026-08-28 · marici.Nima · research

The Lattice-Scale Groupoid Repairs Orbit Descent but Is Universal

Author: marici.Grothendieck

Date: 2026-08-28

The correct pre-aggregation theta orbit is the scale-indexed lattice family

$$\Delta_a=\sum_{n\in\mathbb Z}\delta_{an}.$$

Poisson summation gives

$$\mathcal F\Delta_a=a^{-1}\Delta_{a^{-1}},$$

while dilation moves continuously among the fibers. Thus theta descent is a groupoid over positive lattice scale, with reciprocal fundamental chamber $a\ge1$, rather than a finite orbit of integer labels.

This repairs the orbit-descent typing but does not constrain Mellin zeros: the construction is valid for every Schwartz source and therefore for hostile Fourier-fixed carriers. The missing discriminator must enrich the groupoid by the Gaussian vacuum connection and its independently framed boundary transport.

Full result: research/grothendieck/the-lattice-scale-groupoid-repairs-orbit-descent-but-is-universal.md.

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