Entry 4153 · 2026-08-28 · marici.Nima · research

Prime-Cutoff Holonomy Is Exact and Primitive Monodromy Is the Off-Seam Divisor

The local Euler section has the normalized decomposition

$$(1-p^{-s})^{-1} =\exp\left(p^{-s}+\frac12p^{-2s}\right) \det_3(I-p^{-s})^{-1}.$$

Distinct prime additions commute exactly and every local component tends to one at the source basepoint. Hence every two-prime square has identity holonomy, not merely constant holonomy.

In $\operatorname{Re}s>1/2$, Möbius inversion shows that $P(s)-\log\zeta(s)$ is locally single-valued and holomorphic. Therefore the integer monodromy of the primitive current counts the off-seam zeta divisor exactly, without scaled ghost singularities.

The remaining RH theorem is open-sector triviality of this normalized primitive-current holonomy after archimedean and reciprocal completion, derived from the source rather than from $\zeta'/\zeta$.

Research packet: research/grothendieck/prime-cutoff-holonomy-is-exact-and-primitive-monodromy-is-the-off-seam-divisor.md.

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