Deterministic Band Geometry, Arithmetic Rigidity, and the Structural Necessity of the Riemann Hypothesis-New Paradigm plus Classical proof for comparison

We investigate whether the Riemann Hypothesis arises as a structural necessity of arithmeticitself rather than as a contingent analytic conjecture. Working entirely within the standarduniverse of arithmetic generated by the Peano axioms and unique factorization, we introduce theStanding–Sitting Band Framework (SSBF), which formalizes forward–causal prime propagationand irreversible composite elimination across logarithmic scales.Sequential integer generation induces an intrinsic logarithmic scale geometry in which arithmeticstructure propagates strictly forward. Within this geometry, unresolved arithmetic structureundergoes deterministic global thinning that cannot be arrested by any local or banddependentmechanism without violating divisibility order. Consequently, any global arithmeticencoding intended to represent arithmetic coherently across all scales must incorporate a compensatingmechanism that remains nondegenerate under scale refinement.We formalize this requirement as the Band–Amplitude Admissibility Principle (BAAP), anadmissibility criterion governing global arithmetic encodings rather than an additional arithmeticaxiom. We prove that BAAP forces a unique admissible scaling exponent, namelyℜ(s) = 12 ; all other exponents lead to exponential collapse or divergence of global arithmeticamplitude and are therefore inadmissible within any coherent encoding.Classical analytic structures—including the Euler product, functional equation, explicit formula,and zero distributions—emerge as secondary representations of this arithmetic rigidityrather than as foundational inputs. Within this framework, the nontrivial zeros of the Riemannzeta function appear as equilibrium labels of admissible global encodings, and the critical line issingled out as the unique locus compatible with scale-coherent arithmetic structure.All claims inthis work are structural rather than proof-theoretic: the results do not assert formal derivabilitywithin Peano arithmetic, but establish rigidity constraints on any arithmetic universe governedby Peano-style sequential construction and forward–causal divisibility. After Appendices and Ai Prompts a Classical proof is shown to validate earlier method.