This paper develops a spectral framework representing the non-trivial zeros of theRiemann zeta function as eigenvalues of a self-adjoint operator on L2(R). The constructionintegrates a Hilbert–P´olya type operator with confining potential and arithmeticdelta-array, a fully crystallized trace formula with exact base-prime propagation, anddiscrete ternary lattice dynamics seeded by base primes 2, 3, 5, 7. Recursive eigenvaluegeneration ensures exact arithmetic-spectral correspondence, uniform control acrossthe spectrum, and lattice rigidity. This provides a fully controlled operator-theoreticrealization of the Hilbert–P´olya conjecture and strong evidence for the Riemann Hypothesis