Deterministic Band Geometry, Arithmetic Rigidity, and the Structural Necessity of the Critical Line-Rigidity

We examine whether the critical line ℜ(s) = 12 arises as a necessary structural constrainton coherent global encodings of arithmetic, rather than as a contingent analytic phenomenon.Working entirely within the standard arithmetic universe generated by Peano-style sequentialconstruction, unique factorization, and irreversible divisibility, we formalize deterministic primepropagation across logarithmic scales via the Standing–Sitting Band Framework (SSBF).SSBF enforces unavoidable global thinning of unresolved arithmetic structure across exponentiallywidening bands. Any global representation intended to remain meaningful underunbounded scale refinement must therefore satisfy a nondegeneracy requirement, formalizedas the Band–Amplitude Admissibility Principle (BAAP). BAAP admits a unique scale-neutralregime; all other scaling behaviors induce exponential collapse or divergence and hence fail toencode arithmetic coherently.We further show that SSBF and BAAP jointly supply all structural ingredients required fora full resolution of the Riemann Hypothesis: deterministic prime propagation, irreversible scalecausality, a global admissibility criterion, and a uniquely neutral scaling regime. Classical analyticobjects arise only as encodings of this arithmetic rigidity. Within this setting, the Riemannzeta function emerges as a canonical multiplicative encoding compatible with independent primeaction and global scale coherence, with its nontrivial zeros labeling neutral oscillatory modes.Any putative zero off the critical line corresponds to a globally inadmissible scaling mode andis therefore excluded by BAAP.No claim is made of a completed formal derivation within a specific axiomatic presentation.Rather, we establish proof-sufficiency: any coherent analytic realization faithful to arithmeticstructure is forced into the critical-line regime, of which classical zeta theory provides a distinguishedrealization.