Deterministic Band Geometry, Arithmetic Rigidity, and a Spectral Framework for the Riemann Hypothesis — Bridging Emergent Arithmetic to Classical Formulas

We develop a deterministic arithmetic framework that isolates the critical line ℜ(s) = 12 asthe unique band-stable exponent compatible with sequential integer construction and forwardcausalprime propagation. The analysis is based on a logarithmic band decomposition of thenatural numbers and a Standing–Sitting Band Framework (SSBF) that deterministically resolvescomposites using only previously identified primes, enforcing arithmetic thinning withoutprobabilistic, asymptotic, or global density assumptions.The logarithmic band geometry reflects the same scale structure underlying the Euler product,the explicit formula, and Hilbert–P´olya heuristics, placing the framework on the naturalarithmetic scale at which zeta phenomena arise. Within this setting, arithmetic influence propagatesstrictly forward across bands, and the exponential widening of bands enforces dilution ofunresolved composite structure.To connect arithmetic structure with spectral behavior, we introduce a Band–Amplitude AdmissibilityPrinciple (BAAP), which replaces square-integrability as the mode-selection criterionfor dilation dynamics. Under this principle, a forcing lemma shows that all dilation exponentswith ℜ(σ) ̸= 12 are unstable across bands, while ℜ(σ) = 12 is uniquely compatible with persistentband-level amplitude. The argument relies solely on band geometry and arithmetic thinning anddoes not invoke Mellin transforms, Dirichlet series, analytic continuation, or random models.A canonical dilation generator provides a realization of the admissible spectrum, and theWeil explicit formula appears as a consistency identity expressing arithmetic–spectral balancerather than as an analytic input. Within this framework, the Hilbert–P´olya criterion is satisfiedin the sense of realization: arithmetic rigidity determines the allowable spectral support, whileself-adjointness is compatible but not causative. The results identify the critical line as the onlyexponent consistent with deterministic arithmetic propagation on logarithmic scales, clarifyingthe structural origin of the Riemann Hypothesis without circular dependence between primes,zeros, or operators.