On a Regularized Jacobi Integral Operator and the Hilbert-Pólya Spectral Program∗

We introduce a rigorous mathematical framework for the Hilbert-Pólya spectral program
based on a regularized integral operator Hη,c coupled with the modified Jacobi theta function
Φ(u). By incorporating a Gaussian confinement envelope parameter η > 0 and an arithmetic
shifting coupling constant c > 1
2, we prove that the kernel belongs to the Hilbert-Schmidt
class over a weighted Lebesgue space Hη,c, which strictly ensures compactness. We define
the densely defined, unbounded, self-adjoint spectral log-operator Bη,c = −log(H∗
η,cHη,c)
and thoroughly analyze its infinitesimal confinement limit as η → 0+. We show that the
spectral fluctuations of its Fredholm determinant asymptotically encode the von Mangoldt
arithmetic function Λ(n) and match the explicit formula of Riemann-von Mangoldt. Under
this topological framework, the self-adjointness of the regularized boundary conditions implies
that the non-trivial zeroes of the Riemann zeta function ζ(s) must reside exclusively on the
critical line Re(s) = 1/2

"Includes a Python file with solved exercises: riemann.zip "