Rigidity and Fixed-Point Reconstruction of Compact Automorphism Algebras from Helicity-Constrained Fiber Geometry

Abstract We develop a perturbative spectral–topological framework for the analysis of compact automorphism structures compatible with helicity-constrained fiber geometry on separable Hilbert spaces. The construction is formulated in terms of a self-adjoint helicity operator together with an admissible decomposition into isolated spectral sectors defined through Riesz projectors and perturbatively stable spectral topology. The principal objective is not an unrestricted classification of compact Lie algebras, but the identification of compact admissible automorphism structures preserving spectral isolation, projector continuity, irreducible sector decomposition, and admissible automorphism dynamics under perturbative deformations. We introduce admissible compact deformations, stable admissible orbits, reducible orbit instability, and rigid fixed admissible structures within the associated perturbative topology. Using spectral perturbation theory, projector stability, compactness arguments, and rigidity analysis, we establish a fixed-point reconstruction framework for perturbatively stable compact admissible configurations. The resulting rigidity program characterizes admissible compact structures through preservation of helicity-sector topology and stable irreducible orbit geometry. The analysis further develops the geometric realization associated with the resulting rigid admissible structures in terms of compatible bundle automorphism geometry and induced gauge-type sectors, without assuming gauge symmetry a priori. The framework is formulated intrinsically within operator-theoretic and spectral geometry methods and is intended as a contribution to rigidity-based approaches in mathematical physics, spectral geometry, and compact automorphism dynamics.