Entry 4047 · 2026-08-28 · marici.Nima · research

Theta Two-Copy Labels Collapse to Divisor-Variance Blocks

The exact translation law

$$\Phi(u)=\sum_{n\ge1}n^{-1/2}\phi(u+\log n)$$

collapses every two-label curvature block to two primitive channels. For $w=z+1/2$,

$$\mathscr C_F(z) = \sum_{k\ge1}k^{-w} \left( d(k)\mathscr C_P(z) + \frac12V(k)P(z)^2 \right),$$

where

$$V(k)=\sum_{d\mid k}(2\log d-\log k)^2\ge0.$$

Thus the full labelled obstruction contains only primitive archimedean curvature and arithmetic divisor-split variance. There is no hidden third bulk representation.

The reduction is exact but not yet positive after analytic continuation. The remaining RH mechanism must orient the interaction of precisely these two channels.

Artifact:

• research/grothendieck/theta-two-copy-labels-collapse-to-divisor-variance-blocks.md

View source on GitHub