A Chiral Adelic Dirac Operator and the Spectral Realization of the Riemann Zeros

This paper develops a chiral adelic operator framework in which the functional–equation symmetry of global $L$–functions is realized directly in the spectrum of a Dirac–type Hamiltonian. Working on the idèle class space, we place a real–place Floquet Hamiltonian into an off–diagonal chiral form to obtain a global adelic Dirac operator with an exact involutive symmetry implemented by real reflection and idelic inversion. Arithmetic information is incorporated through a prime–indexed mass deformation built from spherical Hecke operators; when the coefficient functions are even, the perturbed operator preserves the chiral symmetry and produces isolated $\pm$–paired eigenvalues inside the spectral gaps of the Floquet background. These eigenvalues appear as jump discontinuities of the Dirac spectral shift function, while a separated adelic trace formula expresses the trace as a product of a Floquet orbital factor and a prime–indexed Euler–factor–type term whose logarithmic derivatives yield a prime–orbit expansion reminiscent of the explicit formula. This structure motivates a Dirac reinterpretation of the Hilbert–Pólya idea, identifying the nontrivial zeros of $\zeta (s)$ not with the raw spectrum of a single operator but with the spectral–shift discontinuities of a chiral adelic Dirac system under controlled prime–indexed deformations, with finite–prime truncations providing computable models that converge distributionally and enable numerical exploration of arithmetic spectral flow.