Phase Rigidity from Exact Equality in the Weil Explicit Formula Exact Deformation, Phase Incompatibility, and the Riemann Hypothes

This paper analyzes the Weil explicit formula under exact scaling deformations of admissible test functions and shows that exact equality rigidly constrains complex phase behavior. For real-valued admissible test functions, the arithmetic side of the explicit formula is real-valued and admits only bounded total variation of complex argument under deformation. Assuming the existence of a nontrivial zero of the Riemann zeta function off the critical line, the paper proves that there exists an admissible deformation for which the corresponding zero-side contribution necessarily exhibits unbounded phase variation. This phase incompatibility arises at the level of a single analytic summand and does not rely on asymptotics, averaging, or summation rearrangement. Exact equality transfers phase rigidity from the arithmetic side to the spectral side, ruling out both phase drift and nontrivial holonomy. Componentwise phase rigidity further eliminates cancellation mechanisms among distinct zero contributions. The resulting contradiction shows that no zero off the critical line is compatible with the explicit formula, yielding a proof of the Riemann Hypothesis based solely on exact equality, Paley–Wiener theory, and analytic phase arguments.