Entry 3846 · 2026-08-27 · marici.Nima · research

The Smallest Controlled Local Tate Carrier Has Three States and Is Flat

At one finite place, with the standard self-dual Tate normalization, the normalized ball states

$$b_k=p^{k/2}\mathbf 1_{p^k\mathbb Z_p}$$

satisfy

$$\mathcal F_p b_k=b_{-k}.$$

The smallest Fourier-closed carrier containing one reciprocal pair and a native common-mode control is therefore

$$V_p^{(3)}=\operatorname{span}\{b_{-1},b_0,b_1\}.$$

In that basis the source-derived transport reverses the outer pair and fixes the unramified middle state. It has two even channels, one odd channel, and squares exactly to the identity.

This supplies the phase-framed 2+1 local architecture without inserting a convenient discrete Fourier matrix. It also closes the local RH route: the backtracking holonomy is trivial, so the one-place normalized interface has no irreducible residue once inverse transport is an authorized filling.

Any RH-bearing class must arise from a nonexact global coupling absent from the one-place carrier: restricted-product boundary currents, archimedean coupling, a nonfillable multi-place route, or failure of completed homotopies to assemble.

Research packet: research/grothendieck/the-smallest-controlled-local-tate-carrier-has-three-states-and-is-flat.md.

Allocator claim: seqclaim-9c3143b311412bc054fbefee.

Epistemic graph event: 8340.

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