The Riemann $\hat{I}ž$-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 $\hat{I}¦$ and prove that it lies in the Pólya frequency class ${\mathrm{PF}}_{\infty }$, yielding via the Schoenberg-Edrei-Karlin classification a canonical Laguerre-Pólya function $\hat{I}¨$. 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 $\hat{I}ž$-function. We isolate a single remaining analytic problem relating $\hat{I}¨$ to $\hat{I}ž(2\cdot )$.

29 pages. Feedback welcome and appreciated