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 by the true zeta spectrum is equivalent to RH. The framework reveals that classical analytic and spectral methods lack the expressive power to exclude off-critical configurations, because they operate only along metric directions detectable by traditional invariants. In contrast, the admissibility functional detects deformations along coherence directions—global structural variations invisible to Riemannian or isospectral considerations. The paper isolates the final unresolved step of any operator-theoretic proof of RH as a single, well-defined structural statement: establishing that the true zero configuration Γζ\Gamma_\zetaΓζ satisfies A(Γζ)=0\mathcal{A}(\Gamma_\zeta)=0A(Γζ)=0. This reduction provides the necessary and logically prior architecture for any future admissibility-based resolution of the Riemann Hypothesis. A companion work will introduce the global coherence principle required to enforce this admissibility condition. Notes:This release contains the full operator–admissibility formalism, including the construction of TΓT_\GammaTΓ, the defect operators, the admissibility functional A(Γ)\mathcal{A}(\Gamma)A(Γ), and the main rigidity theorem. It establishes the exact logical location of the remaining gap and provides the structural reduction required for any future proof based on admissibility principles.