Spectral Gap and Stability of the Geometric Phase

This preprint develops the protected spectral-gap sector of the Theory of Structural Articulation (TSA). Its aim is to formalise the mechanism by which topologically protected defect sectors retain finite relaxation, finite Green--Kubo response, and stable macroscopic transport after projection from the microscopic overlap structure. The paper introduces the protected-sector decomposition, the charge-resolved state spaces, the restricted Dirichlet forms, and the corresponding generators. It proves the equivalence between protected spectral positivity, Poincare-type coercivity on the zero-mean sector, exponential relaxation, and finiteness of the protected Green--Kubo integral. A constructive Cheeger-type route to a uniform protected gap is also formulated through local move connectivity, core--tail decomposition, and conductance bounds. Within the TSA corpus, this work supplies the spectral-stability component needed by the companion papers on diffusion, time, emergent geometry, and gravitational normalisation. In particular, the protected gap controls the relaxation time of defect sectors and fixes the characteristic correlation scale used in the downstream geometric and response-sector constructions. The results are theorem-grade relative to the stated imported infrastructure of the TSA corpus: subcritical Gibbs/DLR mixing, reversible Dirichlet-form structure, local move completeness, finite-core normalisation, and core--tail conductance estimates. The paper is intended as a self-contained spectral bridge between the microscopic projected overlap graph and the macroscopic transport/geometric sector of TSA.