Prime Gravity: A Poisson-Type Identity for the Zeta Explicit Formula

We recast the explicit-formula machinery as a static elliptic boundary–value identity on the logarithmic line.
For fixed $\lambda>0$ set $L_\lambda:=-\partial_x^2+\lambda^2$, $g_\lambda(x)=\frac{1}{2\lambda}e^{-\lambda|x|}$, 
$\mu=\sum_{n\ge1}\Lambda(n)\,\delta_{\log n}$, and $U_\lambda=g_\lambda*\mu$.
Then, for $\Re s>1$,
\[
-\frac{\zeta'(s)}{\zeta(s)}=(s^2-\lambda^2)\int_0^\infty e^{-sx}\,U_\lambda(x)\,dx,
\]
so analytic information is organized by the Green kernel $g_\lambda$ acting on the distributional source $\mu$.
On finite frequency windows we obtain a sign–preserving integral operator with explicit Archimedean bounds and $L^1$ control of the high–frequency residual, yielding a kernel inequality that structures the finite–window analysis.
All terminology such as “field”, “potential”, and “ghost drift’’ is strictly interpretive; no physical hypothesis or dynamics is introduced.
Beyond number theory, the operator–theoretic formulation and positivity properties may be of independent interest in the study of positive kernels and quadratic forms in mathematical physics

This preprint reformulates the Riemann zeta explicit formula as a static Poisson-type identity on the logarithmic line.
All terminology such as “field” or “ghost drift” is purely interpretive; no physical model is introduced.

This work was conducted at the GhostDrift Mathematical Institute (https://www.ghostdriftresearch.com).

▼GhostDriftMathmaticalInstitue HP
https://www.ghostdriftresearch.com/%E8%A4%87%E8%A3%BD-adic