The Riemann Hypothesis and the Critical Line: A Framework-Internal Characterization by Elimination

Abstract We synthesize results from two companion papers to establish that within the logarithmic-potential operator framework for the Riemann ξ-function, the critical line σ = 1/2 emerges as the unique viable configuration—not by assumption, but by internal consistency requirements. Paper I proves that for all σ ≠ 1/2, the potential Vσ = ∂²[-log|ξ(σ + it)|] fails to be real-valued, obstructing self-adjoint operator construction unconditionally. Paper II establishes that at σ = 1/2, conditional on the Riemann Hypothesis, the framework admits consistent operator realizations with spectral properties equivalent to RH. The synthesis yields a characterization theorem by elimination: σ = 1/2 is the unique fixed point of the functional equation symmetry s ↔ 1−s, and this is precisely what permits real-valued potentials. Scope: We do not prove the Riemann Hypothesis. We prove that within this framework, the critical line is the only place where the framework remains coherent. Keywords: Riemann Hypothesis, critical line, operator framework, structural obstruction, characterization by elimination