Global Non-Integrability of Entropic Orientation and Lambda-Driven Obstructions

Global Non-Integrability of Entropic Orientation and Lambda-Driven ObstructionsThis work develops a geometric obstruction to defining a single global “entropic time” in Lorentzian spacetimes when the arrow of time reverses infinitely often. On a connected, time-orientable spacetime, we introduce an entropic orientation field tied to the monotonicity of a generalized entropy and prove a global non-integrability theorem: if a timelike curve intersects an infinite sequence of “Janus” hypersurfaces across which the entropic orientation flips, then no continuous scalar time function can have an everywhere timelike gradient aligned with the entropic arrow on all components of the complement. An explicit minimal model on \mathbb{R}\times S^3 with alternating entropic orientation illustrates the obstruction concretely.We then connect this to \Lambda>0 cosmology. Using the Raychaudhuri equation, standard energy conditions, and cosmic no-hair–type behavior, we show that in \Lambda-dominated, future-complete spacetimes the generalized entropy becomes eventually monotone along future-directed timelike curves. As a consequence, only finitely many entropy-orientation reversals are allowed: infinite “Janus reset” structures are dynamically incompatible with such \Lambda>0 GR solutions.Finally, we outline a “Raychaudhuri-defect” programme. Assuming only an integral decay condition on the Raychaudhuri defect term along a geodesic congruence, we obtain eventual positivity of the expansion and a rigidity alternative: either the spacetime asymptotically expands in a way that fixes the entropic arrow, or completeness/energy/decay assumptions must fail. This casts classical GR with positive cosmological constant as imposing sharp structural constraints on endlessly reversing arrows of time.