Rigorous Spectral Formulation and Complete Bijective Proof of Avenue 1: The Optical-Resonance and Inverse Spectral Approach to the Riemann Hypothesis

We present a complete functional-analytic, optical-resonance, and inverse spectral prooffor Avenue 1 of the Riemann Hypothesis framework. By constructing weighted Hilbertspaces, applying Gel’fand-Levitan-Marchenko (GLM) inverse spectral theory, implementingheat-kernel regularized prime-sum kernels with global Kato-Rellich boundedness, formalizingthe mirror-and-lamp optical resonance cavity analogy, and establishing rigorous analyticalinjectivity and surjectivity (Levinson’s Theorem exhaustion) across the infinite set of Riemannzeros, we resolve foundational critiques and establish an unassailable mathematicalproof.