The Riemann $\Xi$-function from primitive Markovian cycles I: A canonical construction

Abstract

Starting from finite, local, reversible Markov dynamics on discrete cycles, we construct a scaling-limit renormalized trace kernel admitting an exact theta-series representation. The construction is entirely Archimedean and uses no Euler products, primes, or arithmetic spectral input. From this limit we define a logarithmic kernel $\Phi$ and prove that it lies in the P\'olya frequency class $\mathrm{PF}_\infty$, yielding via the Schoenberg-Edrei-Karlin classification a canonical Laguerre-P\'olya function $\Psi$. Independently, we introduce an Archimedean completion operator and show that, at a self-dual normalization, the completed kernel coincides with the classical theta kernel, whose Mellin transform is the Riemann $\Xi$-function. We isolate a single remaining analytic problem relating $\Psi$ to $\Xi(2\cdot)$.