Spectral Realization of the Nontrivial Zeros of the Riemann Zeta Function via a Hermitian Operator Framework

Abstract

We present a spectral construction of a Hermitian operator whose spectrum coincides exactly with the imaginary parts of the nontrivial zeros of the Riemann zeta function  The operator, denoted H_∞, is defined on a discrete geometric space modeled by a 20-vertex dodecahedral graph, incorporating a discrete Laplacian, an entropy-based coherence potential, and a prime-indexed infinite-order algebraic term derived from Infinity Algebra. We show that H_∞ is self-adjoint, spectrally complete, and compatible with the analytic continuation and functional symmetry of ζ(s). A spectral determinant constructed from its eigenvalues matches the Hadamard product representation of ζ ¹ + it , and no extraneous roots appear off the critical line. Numerical approximations from a truncated version of the operator validate this correspondence. The construction yields a functional-analytic framework that supports a spectral- theoretic resolution of the Riemann Hypothesis