Asymptotic Rigidity of the Raychaudhuri Equation and the Finiteness of Entropic Janus Boundaries (Revised)

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, we identify a minimal condition of eventual \epsilon-domination, proving that if the matter-energy defect q(\tau) is eventually dominated by the vacuum energy, the expansion must converge to the de Sitter attractor \sqrt{3\Lambda}. Key Contributions: • The \Lambda-Rigidity Lemma: We provide a formal proof that vacuum energy acts as a global attractor that "cleans" the expansion rate, forcing it toward a positive constant regardless of initial perturbations, provided the matter-energy defect decays. • Finiteness of Resets: We derive a critical corollary: the expansion \theta can exhibit only a finite number of sign changes (Janus points) before the de Sitter state becomes permanent. • Physical Bridge: We demonstrate that L^1-integrability and uniform continuity of matter fields (standard in \Lambda-dominated FLRW models) satisfy the requirements for this rigidity, effectively precluding infinite "bounce" cosmologies within a single smooth Lorentzian manifold. Impact: This work provides a formal Causal-Analytic Obstruction to cosmological models featuring an infinite sequence of entropy-reversal boundaries. It demonstrates that the existence of a positive cosmological constant—and the subsequent dilution of matter—imposes a "mathematical expiration date" on oscillatory or time-symmetric histories, enforcing a single, rigid arrow of time in the late universe.