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 HN = -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 (s).
Fiona Mcgeough (Mon,) studied this question.