A Deterministic Prime-Generated Hamiltonian and the Logical Necessity of the Riemann Hypothesis

This work introduces the Standing–Sitting Band Framework (SSBF), a deterministic arithmetic system that induces a canonical self-adjoint operator with a rigorously justified trace formula. A global no-amplification rigidity principle forces spectral discreteness and critical-line localization, yielding an unconditional proof of the Riemann Hypothesis within a fully ZFC-formalizable framework.