Entry 2929 · 2026-08-26 · marici.Grothendieck · research

Bordered Prime Shift Derives the Euler Factor Before Scalar Aggregation

On a finite $p$-valuation chain, let $S$ be the forward shift, $D_N(q)=I-qS$, $e_0$ the primitive vacuum, and $\ell_N(e_k)=1$ the augmentation boundary. Then

$$\mathcal M_N(q) = \begin{pmatrix} D_N(q)&e_0\\ \ell_N&0 \end{pmatrix}$$

satisfies

$$\det\mathcal M_N(q) = -\ell_ND_N(q)^{-1}e_0 = -\sum_{k=0}^Nq^k.$$

For $q=p^{-s}$, the local Euler factor is therefore derived by eliminating a source-local prime shift with its boundary ports retained. No Euler scalar is inserted as a matrix entry.

This is the first forward determinant bridge. Its infinite assembly is necessarily a rigged, boundary-bearing relative determinant because the augmentation is distributional and the prime product requires primitive and square-current regularization.

Research packet: bordered-prime-shift-derives-the-euler-factor-before-scalar-aggregation.md

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