Asymptotic Rigidity of the Raychaudhuri Equation and the Finiteness of Entropic Janus Boundaries

Abstract: This paper establishes a geometric rigidity theorem for the expansion \theta of a hypersurface-orthogonal timelike geodesic congruence in the presence of a positive cosmological constant \Lambda > 0. Using a Riccati-comparison framework and Barbălat's Lemma, we prove that if the matter-energy defect term is L^1-integrable and uniformly continuous, the expansion \theta must converge to the de Sitter attractor \sqrt{3\Lambda}. Summary of Contribution: The primary contribution of this work is the identification of a \Lambda-Obstruction to infinite-loop cosmologies. While various "Janus Point" or "Cyclic" models propose an infinite sequence of entropy-reversing boundaries, this theorem demonstrates that such configurations are mathematically inconsistent with late-time vacuum dominance in a single, smooth, future-complete Lorentzian manifold. Key Findings: • Rigidity Theorem: Proves that vacuum energy (\Lambda) acts as a global attractor that "cleans" the expansion rate, forcing it toward a positive constant. • Finiteness of Resets: Demonstrates that the expansion \theta can only exhibit a finite number of sign changes (reversals) before the de Sitter state becomes permanent. • Cosmological Constraint: Provides a formal "No-Go" result for infinite sequential "bounces" or Janus resets in a universe dominated by Dark Energy.