Topological Rigidity and the Mathematical H-Theorem for the Ricci Flow

This version presents a refined and fully structured formulation of the Mathematical H-Theorem at the level of Perelman’s $\mathcal W$-entropy. Compared to Version 1, the exposition is significantly clarified, the Lyapunov interpretation is made explicit, and the obstruction to global entropy control is formulated in a canonical and minimal form. The main result establishes a rigid link between the existence of a global Lyapunov functional for the Ricci flow and the topological simplicity of the underlying 3-manifold. No new technical estimates are introduced in this version; the contribution is conceptual and structural, aiming to isolate a minimal topological obstruction and to prepare the ground for a more general Topological H-Theorem.