Paper 3 - Spectral Closure of the SM-Island: Compactness, Palais–Smale Condition, and Uniform Spectral Gap

Paper 3 of the Abebe Carrier Framework under Master Map II. Establishes analytic closure on the Sobolev completion of the reduced configuration space: proves coercivity of the Carrier action, Palais–Smale condition, precompactness of energy sublevel sets, and a uniform spectral gap (α > 0) for the Hessian at all non-SM critical points. This implies exponential convergence of the φ-flow to the SM-Island, completing the attractor result of Paper 2. Part of the complete series (Papers 0–5) with Global Closure Appendices A–G. (Keywords: Carrier Framework, Master Map II, spectral gap, Palais–Smale, SM-Island, analytic closure)