Beyond Zeros: The Riemann Hypothesis as an Admissibility Constraint

This work develops a fully explicit operator–admissibility framework in which the Riemann Hypothesis (RH) appears as a rigidity condition rather than an analytic estimate. To every zeta-compatible zero configuration Γ\GammaΓ we associate a compact operator TΓT_\GammaTΓ whose symmetry encodes the horizontal placement of zeros. Two defect operators measure failure of Hermitian symmetry and failure of functional-equation invariance, and their Hilbert–Schmidt combination defines a coherence defect functional A(Γ)\mathcal{A}(\Gamma)A(Γ). We prove that A(Γ)=0\mathcal{A}(\Gamma)=0A(Γ)=0 if and only if all zeros in Γ\GammaΓ lie on the critical line. This identifies a precise admissibility condition whose satisfaction…