Infinite Arrows of Time - from Repeated Janus Boundaries (Theorem)

This work develops a formal framework in which thermodynamic arrows of time emerge relative to an infinite set of entropy-minimising hypersurfaces embedded within a time-reversal symmetric dynamical structure. Extending symmetric Markovian reduced dynamics in open quantum systems, the traditional single distinguished low-entropy boundary (commonly associated with the Past Hypothesis) is generalised to a countably infinite set of “Janus” boundaries. Under assumptions of microscopic time-reversal symmetry, finite bath memory, and repeated factorisation conditions, we prove that each entropy-minimising hypersurface induces two outward entropy-increasing directions within the domain of validity of a local Markov approximation. We further demonstrate that, when multiple such boundaries exist with overlapping validity regions, a single globally monotonic entropy functional may be obstructed. Thermodynamic orientation then becomes patchwise rather than globally integrable. The framework is examined within a Friedmann–Robertson–Walker cosmological setting, where geometric time evolution (via the scale factor a(t)) need not coincide with thermodynamic re-anchoring. This decouples entropy increase from monotonic cosmic expansion and weakens the uniqueness of a singular cosmological Past Hypothesis. Philosophically, the proposal reframes the arrow of time as boundary-induced rather than fundamentally encoded in dynamical law. If physically realised, a universe may possess a consistent geometric temporal structure while lacking a single absolute thermodynamic direction, instead containing multiple locally coherent entropy gradients. The manuscript includes formal theorems on local entropy monotonicity, a global non-integrability argument, cosmological embedding analysis, and discussion of the conditions under which the conjecture remains physically nontrivial. The central open problem remains the identification of a dynamical mechanism capable of generating repeated decorrelation hypersurfaces without contradicting empirical entropy monotonicity in observable cosmology.