This preprint develops a spectral framework for the study of the non-trivial zeros of the Riemann zeta function through the Weil explicit formula, residual energy decompositions, dyadic localization techniques, and operator-theoretic methods. The analysis combines Fourier localization, arithmetic resonance structures, Hilbert-space energy methods, quasi-orthogonal dyadic assemblies, and spectral amplification mechanisms. Particular emphasis is placed on the interaction between microscopic logarithmic oscillations and large-scale spectral stability. Under explicitly stated assumptions, the manuscript investigates how the suppression of coherent arithmetic resonances may be related to the confinement of non-trivial zeros to the critical line. The resulting framework is presented as a conditional spectral rigidity program rather than as an unconditional proof of the Riemann Hypothesis. The work also explores connections between residual spectral collapse, coercive operator estimates, microscopic hyper-congestion phenomena, and entire-function rigidity within a unified analytical setting. Generative AI tools were used for language refinement, formatting, transcription, and organizational assistance. The author assumes full responsibility for all mathematical statements, arguments, conjectures, interpretations, and conclusions contained in this manuscript. This document is released as a research preprint for public dissemination, mathematical scrutiny, and future development.
Joisy Marrugo (Tue,) studied this question.