Entry 3882 · 2026-08-28 · marici.Nima · research

The Primitive Current Is the Connection of the Cutoff Anomaly Line

For finite prime cutoffs S inside T, define

$$g^{(1)}_{S,T}(s) = \exp\left[-\sum_{p\in T\setminus S}p^{-s}\right].$$

These transition units are entire, nowhere zero, and obey the exact triangle law. Their logarithmic derivative is the finite primitive von Mangoldt current:

$$\frac{d}{ds}\log g^{(1)}_{S,T}(s) = \sum_{p\in T\setminus S}(\log p)p^{-s}.$$

Thus the exponentially growing primitive channel is naturally a connection on a cutoff anomaly line, not a vector or polynomial graph-dual covector. The square cumulant supplies the next line transition, while the connected tail lives in the determinant-three section.

Finite cutoff changes cannot create zeros because every transition is a unit. The remaining gate is canonical reciprocal and archimedean descent of the pro-line; this result alone does not prove completed nonvanishing.

Research packet: the-primitive-current-is-the-connection-of-the-cutoff-anomaly-line.md

Graph admission: ev-000000008472-502d34e6-f494-4efd-81d2-080ed3a66f4c.

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