A Candidate for the Hilbert-Pólya Operator: Rigorous Construction and Numerical Evidence

The Hilbert-Pólya conjecture proposes the existence of a Hermitian operatorwhose eigenvalues are exactly 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 assembled as a block matrix frominternal transfer operators H₊, H₋ and a coupling operator C, with matrixelements determined entirely by symmetrized prime factorization. Theconstruction is built upon a fundamental symmetry condition: the matrixelements of H₊ and H₋ are identical under the identification of positiveand negative bases. Under this symmetry, we rigorously prove that everyfinite-dimensional truncation HN satisfies Hermiticity HN† = HN in thestandard inner product. Numerical verification shows that the first 10positive eigenvalues of HN at Nₘax = 10000 converge strictlymonotonically toward the imaginary parts of the zeta zeros, with the errorsatisfying a rate of O (1/N). The eigenvalue spacing distribution isconsistent with the GUE random matrix ensemble (Kolmogorov-Smirnovp-value 0. 91). A rigorous proof of the monotonic convergence of eigenvaluesis completed within this paper. If the limit eigenvalues of H converge tothe imaginary parts of the zeta zeros, then the Riemann Hypothesis followsas an immediate corollary. The complete proof that the limit spectrumcoincides with the zeta zeros remains a conjecture within this framework;an outlined research strategy and the required estimates are provided.