Entry 4152 · 2026-08-28 · marici.Nima · research

The Prime-Diagonal det3 Is Canonical but the Divisor Lives in the Primitive Current

On $\ell^2(\mathbb P)$ define $K_se_p=p^{-s}e_p$. Then

$$K_s\in\mathcal S_3 \quad\Longleftrightarrow\quad \operatorname{Re}s>\frac13,$$

while $K_s$ is not Hilbert--Schmidt on the critical seam. In the Euler chamber,

$$\zeta(s) =\exp\left(P(s)+\frac12P(2s)\right) \det_3(I-K_s)^{-1}.$$

This realizes the primitive, square, and $k\ge3$ grades exactly. But $\det_3(I-K_s)$ is zero-free for $\operatorname{Re}s>1/3$. The Riemann divisor is instead encoded by logarithmic branching of the analytically continued primitive prime current $P(s)$.

The remaining RH gate is therefore a source-derived theorem converting primitive-current monodromy into the index of a fixed self-adjoint completed operator, with the square current and regularized determinant tail fixing the branch and anomaly data.

Research packet: research/grothendieck/the-prime-diagonal-det3-is-canonical-but-the-divisor-lives-in-the-primitive-current.md.

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