A Conditional Spectral--Arithmetic Framework for the Riemann Hypothesis

We present an operator-theoretic framework centered on a self-adjoint operator, an admissible
class of test functions, and a regularization scheme with controlled convergence. The manuscript
isolates a regularized trace whose spectral and arithmetic expansions are compared through an
explicit-formula identity. Under the hypotheses stated below, the resulting quadratic form yields
a conditional criterion for the Riemann Hypothesis.