Observable Continuity as an Effective Gravitational Principle: Horizon Thermodynamics and the Cosmological Arrow of Time

The physical origin of the cosmological arrow of time remains an open problem despite the thermodynamic interpretation of spacetime horizons and the success of the standard cosmological model. Existing approaches typically attribute irreversibility either to the initial conditionsof the Universe or to the statistical behavior of microscopic degrees of freedom, while a direct effective connection between microscopic trajectory statistics, horizon thermodynamics, and infrared cosmological dynamics has remained elusive. In this work, we propose an effective framework in which observable continuity acts as the statistical organizing principle governing the contribution of microscopic trajectories to macroscopic gravitational dynamics. Observable continuity is described by a trajectory functional, 𝒞[𝛾] = 𝐴[𝛾]𝑃[𝛾]𝑇[𝛾], where admissibility, persistence, and transmission are formulated as local exponential functionals satisfying consistency under trajectory concatenation. This structure uniquely determines a local continuity action, 𝐼𝐴𝑃𝑇 [𝛾] = −ln 𝒞[𝛾], which reweights the microscopic trajectory ensemble and generates an effective bulk action after coarse graining. Within this formulation, variations of the continuity-induced effective action define an effective gravitational source tensor whose projection onto the apparent horizon naturally generates the macroscopic horizon entropy current. As a consequence, the apparenthorizon entropy flow and the observable infrared variable are no longer introduced as independent phenomenological assumptions but emerge from the same effective statistical construction. The standard identity 𝑢 = 𝑑 ln 𝑆𝐴 / 𝑑𝑁 is recovered directly from apparent-horizon thermodynamics, while the continuity-weighted ensemble yields the effective observable relation, 𝑢obs = 𝑢 + 𝑑 ln⟨𝐴𝑃𝑇⟩𝑅𝐴/𝑑𝑁 The resulting framework provides a unified effective description connecting microscopic trajectory statistics, induced gravitational dynamics, horizon thermodynamics, and the cosmological arrow of time while remaining independent of any specific ultraviolet completion. The theory predicts that departures from the standard infrared evolution are controlled by the evolution of the continuity functional and may be tested through late-time cosmological observables, including the expansion history, structure growth, and weak gravitational lensing. The present formulation is proposed as an infrared effective theory. Its principal open problem is the microscopic derivation of the underlying trajectory action from which the effective continuity action, the induced gravitational source, and the horizon entropy current ultimately emerge.