Spectral Admissibility and Logarithmic Rigidity for the Generalized Riemann Hypothesis

We develop a deterministic arithmetic framework establishing the Generalized Riemann Hypothesis(GRH) as a rigidity consequence of forward–causal arithmetic structure rather thanan analytic conjecture. Working entirely within the standard arithmetic universe generated byPeano sequential construction and unique factorization, we extend the Standing–Sitting BandFramework (SSBF) to incorporate Dirichlet characters as deterministic multiplicative weightson logarithmic arithmetic bands.Sequential integer generation induces an intrinsic logarithmic scale geometry in which divisibilitypropagates strictly forward and unresolved arithmetic structure undergoes deterministicglobal thinning. We show that any global arithmetic encoding intended to represent twisted multiplicativestructure coherently across all scales must satisfy a Band–Amplitude AdmissibilityPrinciple (BAAP). This admissibility criterion governs representations of arithmetic structurerather than introducing new axioms.Within this framework, we prove that BAAP admits exactly one stable scaling regime,namely ℜ(s) = 12 , independent of the choice of primitive Dirichlet character. All off–criticalexponents induce exponential collapse or divergence of band amplitudes and are therefore inadmissible.As a consequence, all nontrivial zeros of Dirichlet L–functions are forced onto thecritical line.The results are structural rather than proof–theoretic: we do not assert formal derivabilitywithin Peano arithmetic, but establish rigidity constraints governing any arithmetic universeobeying Peano–style sequential construction and forward–causal divisibility