An Operator-Theoretic Framework for the Riemann Hypothesis: Finite Certificates, Schur-Complement Structure, and Global Positivity Questions

This research presents an operator-theoretic framework for investigating the Riemann Hypothesis through quadratic forms, Schur-complement reductions, finite-dimensional positivity certificates, and computational analysis. The study examines the relationship between reduced quadratic-form positivity and reference-energy estimates, identifying a precise logical obstruction through an exact rational counterexample. It also documents a reported local interval-arithmetic certificate at Y = 7, finite arithmetic verification tests, and finite-element computations conducted in dimension 144. A central objective is to understand how finite-dimensional positivity results can be extended to rigorous global positivity statements in an infinite-dimensional setting. The framework identifies the mathematical conditions required for this transition, including uniform estimates, rigorous control of approximation errors, operator-domain analysis, and a valid connection to established positivity criteria for the Riemann zeta function. The results and methods are presented as research contributions intended to support further mathematical investigation, independent verification, and the development of new approaches toward the Riemann Hypothesis. The manuscript also identifies specific technical questions that remain open within this research framework, providing a structured foundation for subsequent work.