Scale–Localized Divisibility Rigidity and the Generalized Riemann Hypothesis

We develop a deterministic framework for analyzing weighted prime sums based on a logarithmic decomposition of the natural numbers and a forward divisibility principle. Within this framework, cancellation at fixed scale is formalized through an admissibility condition and shown to be rigid: any persistent deviation from alternating behavior produces scale-level growth that cannot be neutralized locally. We then show that this rigidity propagates across scales, yielding an obstruction to boundedness for prime-indexed sums with incompatible weights. A logarithmic truncation principle is established that connects scale-localized behavior of prime sums to vertical growth properties of Euler products. Applying the framework to Dirichlet characters, we prove that admissibility of the associated prime weights is not an additional assumption, but a necessary consequence of the classical analytic properties of Dirichlet L-functions, including the existence of an Euler product and polynomial growth in vertical strips. Combining admissibility with transported rigidity yields a structural obstruction to analytic vanishing away from the critical line. As a consequence, all nontrivial zeros of Dirichlet L-functions are shown to lie on the critical line. The argument is unconditional and is carried out entirely within standard set-theoretic foundations, using only elementary arithmetic, classical complex analysis, and operator-theoretic rigidity principles.