An Explicit Hilbert-Pólya Operator from Symmetrized Prime Factorization: Construction, Proof of Hermiticity, and Numerical Evidence

The Hilbert-Pólya conjecture proposes the existence of a Hermitian operatorwhose eigenvalues coincide with the imaginary parts of the non-trivial zerosof the Riemann zeta function. This paper constructs an explicit candidate forsuch an operator. The operator H is a block matrix assembled from transferoperators H_+, H_- and a coupling operator C, with matrix elements defined bysymmetrized prime factorization. Under a symmetry condition identifying thematrix elements of H_+ and H_- on positive and negative bases, we prove thatevery finite-dimensional truncation H_N is strictly Hermitian in the standardinner product, and that its eigenvalues converge monotonically as N increases.Numerical verification at N_max = 10000 reveals a systematic convergencepattern toward the zeta zeros, with the error approximately halving when Ndoubles, from which predicted values for N_max = 20000 are derived. Theeigenvalue spacing distribution is consistent with the GUE ensemble. Acomplete numerical verification and prediction procedure is given in AppendixB. A research program toward proving spectral equivalence is outlined.