Prime-Composites →RH (GL1): A Conditional Operator Criterion

We present a conditional GL(1) operator criterion within the cpw/WEF framework. Under explicit standing assumptions (window regularity, thin-band Lipschitz control, identical 1f/2f windowing, phase averaging), a principal-term matching hypothesis on the composite explicit formula, and a scale-complete admissible W-regime, we obtain: (T1) quantitative 2f-dominance with an explicit off-diagonal budget, (T3) band normalization and contraction for the antilinear transfer T= JMβW + RW under the admissible calibration |βW|+ rW ≤1, (T2) a Weil-compatible comparison between composite-side and zero-side test functionals with explicit remainder, and (T4) uniqueness of the spectral measure from the scaled Paley–Wiener test family. Together these imply µC = µζ for GL(1) only (zeta and primitive Dirichlet L-functions). The residual lemma T5 is supplementary quantitative input: it is not part of the minimal uniqueness chain, but provides a uniform line-energy control that is useful in the wider GL1 suite. No GL(2) inputs (modular forms, elliptic curves) or BSD hypotheses are used.