A Candidate for the Hilbert-Pólya Operator: Rigorous Construction of an Explicit Hermitian Operator from Symmetrized Prime Factorization

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 candidatefor such an operator. The operator H is a block matrix assembled fromtransfer operators H₊, H₋ and a coupling operator C, with matrix elementsdefined by symmetrized prime factorization. Under a symmetry conditionidentifying the matrix elements of H₊ and H₋ on positive and negativebases, we prove that every finite-dimensional truncation HN is strictlyHermitian in the standard inner product, and that its eigenvalues convergemonotonically as N increases. Numerical verification at Nₘax = 10000reveals a systematic convergence pattern toward the zeta zeros, with theerror approximately halving when N doubles; predicted values forNₘax = 20000 are also provided. The eigenvalue spacing distribution isconsistent with the GUE ensemble. A complete numerical verification andprediction procedure is given in Appendix B. A research program towardproving spectral equivalence is outlined.