Abstract: This paper establishes a geometric rigidity theorem for the expansion of a hypersurface-orthogonal timelike geodesic congruence in the presence of a positive cosmological constant > 0. Using a Riccati-comparison framework, we identify a minimal condition of eventual -domination, proving that if the matter-energy defect q () is eventually dominated by the vacuum energy, the expansion must converge to the de Sitter attractor 3. Key Contributions: • The -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 can exhibit only a finite number of sign changes (Janus points) before the de Sitter state becomes permanent. • Physical Bridge: We demonstrate that L¹-integrability and uniform continuity of matter fields (standard in -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.
Cameron William Brogan-Higgins (Sun,) studied this question.