A Hidden One-Dimensional Invariant in Late-Time Cosmology

We show that late-time cosmology is not freely parametrizable but constrained to a one-dimensional trajectory in phase space. Under minimal renormalization-group (RG) assumptions and consistency with the contracted Bianchi identity, the coupled evolution of the expansion rate and the vacuum sector reduces to a single degree of freedom. As a consequence, the background dynamics admits a unique first integral: H Ω_Λ1/θ = const., which defines a global invariant of cosmological evolution. This relation is not an additional assumption, but a structural consequence of RG scaling of the vacuum sector, Λ(H) ∝ H2−θ, combined with gravitational consistency conditions. It implies that cosmic expansion and vacuum dynamics are not independent processes, but projections of a single renormalization-group trajectory. The invariant provides a direct null test: any statistically significant deviation from constancy rules out the framework. At the perturbative level, the same mechanism enforces dynamical closure, linking fluctuations of Λ and G to perturbations of the RG scale and eliminating additional phenomenological freedom. A key implication is that a single parameter θ simultaneously governs both the expansion history and the growth of cosmic structure, leading to correlated and testable deviations from ΛCDM, which is recovered as the infrared limit θ = 2. These results suggest that the cosmological constant is not a fundamental parameter, but an emergent quantity selected by the renormalization-group structure of gravity.