We examine whether the critical line ℜ(s) = 12 arises as a structural constraint on coherentglobal encodings of arithmetic rather than as a contingent analytic phenomenon. Working entirelywithin the standard arithmetic universe generated by Peano-style sequential construction,unique factorization, and irreversible divisibility, we formalize deterministic prime propagationacross logarithmic scale using the Standing–Sitting Band Framework (SSBF).We show that SSBF enforces unavoidable global thinning of unresolved arithmetic structureacross exponentially widening bands. Any global representation that remains meaningful underunbounded scale refinement must therefore satisfy a non-degeneracy condition, formalized as theBand–Amplitude Admissibility Principle (BAAP). BAAP admits a unique scale-neutral regime;all other scaling behaviors lead to exponential collapse or divergence and hence fail to encodearithmetic coherently.Classical analytic objects are introduced only as encodings of this arithmetic rigidity. Withinthis setting, the Riemann zeta function appears as a canonical multiplicative encoding compatiblewith independent prime action and scale coherence. Its nontrivial zeros label neutral oscillatorymodes of the encoding. Any putative zero away from the critical line would correspondto a globally inadmissible scaling mode and is therefore excluded by BAAP.The result is structural rather than proof-theoretic: no claim is made of formal derivabilitywithin Peano arithmetic. Instead, we establish a rigidity constraint governing all coherentglobal encodings of arithmetic structure, of which classical zeta theory provides a distinguishedrealization.