Uniform Spectral Saturation and Structural Closure of Regulated Operators
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The author presents a structural consolidation of the Additive–Multiplicative (AM) regulated operator framework. Let T ρbe a symmetric densely defined operator subject to uniform spectral saturation. Without invoking Sobolev embeddings, geometric compactness, or boundary coercivity, the author shall prove that,
Closure of the admissible domain under the graph norm, [1][4]
Collapse of deficiency indices under uniform regulation, [1][4][5]
Compactness of the resolvent via sequential precompactness, [1][3][7]
Failure of compactness under non-uniform saturation, [3][8]
Stability under bounded perturbations, [2]
Degeneracy recovery as regulation vanishes. [6][9]
All arguments are operator-theoretic and depend solely on regulated accumulation. [1][4] This establishes a structurally closed framework suitable for spectral applications.