We introduce a rigorous mathematical framework for the Hilbert-Pólya spectral programbased on a regularized integral operator Hη,c coupled with the modified Jacobi theta functionΦ(u). By incorporating a Gaussian confinement envelope parameter η > 0 and an arithmeticshifting coupling constant c > 12, we prove that the kernel belongs to the Hilbert-Schmidtclass over a weighted Lebesgue space Hη,c, which strictly ensures compactness. We definethe 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 thespectral fluctuations of its Fredholm determinant asymptotically encode the von Mangoldtarithmetic function Λ(n) and match the explicit formula of Riemann-von Mangoldt. Underthis topological framework, the self-adjointness of the regularized boundary conditions impliesthat the non-trivial zeroes of the Riemann zeta function ζ(s) must reside exclusively on thecritical line Re(s) = 1/2 "Includes a Python file with solved exercises: riemann.zip "