Extended spectral breakdown theory

This work places the Spectral Riemann Conjecture at the heart of a unifying framework for the stability and continuity of polynomial operators. We formalize the SRC as a spectral principle linking the distribution of zeros and their persistence to the dynamics of renormalization and decoherence operators on spaces stratified by the discriminant. The central notion of spectral rupture describes how a spectrum can undergo non‑conservative transitions under perturbation, revealing precise mechanisms of continuity failure near multiple roots. This perspective refines the Hilbert–Pólya hypothesis by proposing structural criteria under which local continuity is restored and ruptures become computable. We prove local continuity of the renormalization operator in neighborhoods of polynomials with multiple roots, adopting a topology compatible with the discriminant’s stratification and the stability of multiplicities. In parallel, we define a decoherence operator on compact spaces and establish its local continuity, providing a tool to model and measure spectral fragility. These results support an operator‑theoretic formulation of the Spectral Riemann Conjecture: the structure of zeros can be understood through the regularity of a spectral flow, and its breakdown through carefully controlled rupture singularities. Beyond the technical proofs, the SRC serves as a guiding thread for reinterpreting arithmetic phenomena through operator dynamics, suggesting bridges between spectral analysis, algebraic geometry, catastrophe theory, and the philosophy of mathematics. This program aims to structure the study of spectral transitions and to unify conjectures within a rigorous, publication‑ready f