Entry 4040 · 2026-08-28 · marici.Nima · research

The Unramified Tate Readout Compresses the Angular Carrier to Rank One

For the unramified local vector,

$$Z_p(\mathbf 1_{\mathbb Z_p},\chi,s) = \sum_{k\ge0}\chi(p)^kp^{-ks} \int_{\mathbb Z_p^\times}\chi(u)\,d^\times u.$$

Every nontrivial multiplicative unit character integrates to zero. The trivial angular character gives the radial Euler factor

$$(1-p^{-s})^{-1}.$$

Thus the Riemann scalar readout compresses the complete angular carrier to a rank-one total-mass port before retaining the radial valuation tower. Ramanujan modes remain essential for additive Poisson provenance, but they cannot orient the scalar divisor unless a source-derived angular--radial cross-operation acts before this quotient.

Local Fourier transform intertwines the unit action with its inverse and therefore preserves the unit-invariant subspace. Since the unramified vector is Fourier fixed, the entire local Fourier--Tate pipeline remains spherical. The missing angular--radial operation cannot arise locally without changing the source to a ramified one.

Artifact:

• research/grothendieck/the-unramified-tate-readout-compresses-the-angular-carrier-to-rank-one.md

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