The No¯esis Operator: A Hilbert–Schmidt Framework for the Riemann Hypothesis

This paper presents a formal Hilbert–Schmidt operator framework that realises the Hilbert–Pólya conjecture for the Riemann Hypothesis within a rigorously defined, self-adjoint class.Building upon the author’s earlier work, From Fascia to Fields: A Recursive Operator Model Bridging Soma–Noēsis Mechanics and the Riemann Hypothesis (Zenodo DOI: 10.5281/zenodo.16754802), this study translates the previously established spectral correspondence into a precise operator-theoretic formulation. The Noēsis operator $\widehat{H}_N = -i,d/dx + K$, where $K$ is a Hilbert–Schmidt kernel satisfying self-adjointness and compactness, exhibits a meromorphic resolvent (via the analytic Fredholm theorem) and a trace-regularised spectral zeta function whose zeros align with the critical line.Finite-rank Galerkin truncations reproduce Gaussian Unitary Ensemble (GUE) statistics consistent with the Montgomery–Odlyzko law, providing a concrete spectral model consistent with the Hilbert–Pólya criteria. This paper formalises the operator logic underpinning recursive coherence dynamics, completing the theoretical bridge between Soma–Noēsis Mechanics and the analytic structure of $\zeta(s)$.