The Algebraic Fate of Time: Cosmological Horizons, Observer Algebras, and the Dark-Energy Equation of State

Recent work has shown that gravitationally dressing quantum-field observables to an observer converts type III₁ algebras into type II: the de Sitter static patch yields type II₁ with finite maximal entropy (Chandrasekaran–Longo–Penington–Witten), while inflationary quasi-de Sitter space yields type II∞ (Seo). This note asks what happens when the accelerating phase is not eternal. We classify the asymptotic "ledger" structure of flat FRW cosmologies as a function of the late-time dark-energy equation of state w∞, using three elementary diagnostics: total proper time to the future, existence of a cosmological event horizon, and the asymptotic behaviour of its area. Four phases result: terminal (w∞ < −1: Big Rip; the asymptotic classification is undefined), closed ledger (w∞ = −1: finite horizon entropy; the type II₁ anchor), open ledger (−1 < w∞ < −1/3: divergent horizon area; the type II∞ regime), and no ledger (w∞ ≥ −1/3: no event horizon; the undressed type III₁ situation). Under literal CPL extrapolation the closed-ledger phase is the single point (w₀,wₐ)=(−1,0). If the DESI DR2 preference for evolving dark energy (w₀ > −1, wₐ < 0) holds, the asymptotic observer algebra of our universe is not the type II₁ of eternal de Sitter space: no event horizon survives, the finite holographic ledger of ΛCDM never forms, and the algebra remains ordinary type III₁ (geometry: theorem grade; algebra assignment: explicit conjecture, anchored at both static endpoints). Cosmic acceleration then ends at a ≈ 1.4 (~5 Gyr) for the best-fit parameters. Every claim is tiered (theorem / conjecture / interpretation) in an appendix. All numerical results are reproducible from the included Python scripts.