Exact Algebraic Features of Horizon Thermodynamics in Flat ΛCDM: ℏ-Cancellation and Equipartition Epochs

The flat ΛCDM background contains two exact algebraic features of the late-time horizon thermodynamics that do not appear to have been previously highlighted. First, the product of the Gibbons-Hawking temperature and Bekenstein-Hawking entropy at the cosmological horizon yields a purely classical invariant power floor P_horizon = c⁵/2G: the quantum parameters ℏ and k_B cancel identically, and the result is independent of horizon radius and cosmic epoch. Second, imposing the condition that the volumetric expansion-work rate at the horizon equals this classical floor yields a quadratic in the standard background that factors exactly as (2Ω_m x − Ω_Λ) (Ω_m x − 2Ω_Λ) = 0, where x = (1+z)³. The two roots give crossover epochs z₁ ≈ 0.632 and z₂ ≈ 0.028, at which the deceleration parameter takes the exact values q = 0 and q = −1/2 respectively. Both results follow from established horizon-thermodynamic relations and the standard Friedmann background with no adjustable parameters. The classical power floor bounds the expansion-work rate at all epochs; the de Sitter terminal state is the unique configuration at which this bound is saturated, identifying it as a thermodynamic attractor by constraint rather than by assumption.