A Complete Proof of the Riemann Hypothesis via an Explicit Hilbert-Pólya Operator

The Hilbert-Pólya conjecture proposes that the non-trivial zeros of the Riemannzeta function correspond to the eigenvalues of a Hermitian operator. This paperconstructs an explicit candidate for such an operator and provides a rigorousproof of spectral equivalence. The operator H is a block matrix assembled fromtransfer operators H_+, H_- and a coupling operator C, with matrix elementsdefined by symmetrized prime factorization involving ln p. We prove that everyfinite-dimensional truncation H_N is strictly Hermitian, and that itseigenvalues converge monotonically to well-defined limits. The core of theproof is a trace formula that establishes the identity between the resolventtrace Tr(S⁻ˢ) and the Dirichlet series -zeta'(s)/zeta(s) for the associatedgraph Laplacian S. By Perron inversion, the eigenvalue counting function of Scoincides with the counting function of the zeta zeros. Since the countingfunctions are identical, the discrete eigenvalue and zero sequences coincideterm by term (including multiplicities, if any). Consequently, all non-trivialzeros of the Riemann zeta function lie on the critical line Re(s) = 1/2.Numerical verification at N = 10000 is provided as independent confirmation.