Arithmetic Trace Correspondence under Regulated Spectral Saturation

The author constructs the arithmetic interface layer required to transition from regulated operator theory to a Hilbert–Pólya–compatible framework. Building on uniform spectral saturation and compact resolvent structure established previously; ^[1][5] deriving a regulated trace formula, ^[3][4] compare it to the Riemann explicit formula, and isolate the precise structural conditions under which spectral identity with the nontrivial zeros would follow. The paper does not assume equality with the zeta spectrum; instead it reduces Hilbert–Pólya realization to a single trace correspondence condition.