Arithmetic Rigidity and the Structural Necessity of the Riemann Hypothesis via the Standing–Sitting Band Framework

We develop a fully deterministic arithmetic framework establishing the Riemann Hypothesisas a rigidity consequence of integer structure. Working entirely within Peano arithmetic, weintroduce the Standing–Sitting Band Framework (SSBF), which formalizes forward–causal primepropagation and irreversible composite elimination across logarithmic scales.We prove that arithmetic thinning induced by divisibility propagates globally and cannotbe compensated by prime generation alone. Any global encoding of arithmetic must thereforesatisfy a necessary admissibility criterion, from which a rigidity theorem follows: there exists aunique admissible scaling exponent, namely Re(s) = 12 .Classical analytic structures—Euler products, functional equations, and zero distributions—emerge as secondary encodings of this rigidity rather than as foundational inputs. The RiemannHypothesis is thus established as a structural necessity of arithmetic itself.