Global Non-Integrability of Entropic Orientation

We prove a geometric obstruction to the existence of a global time function in Lorentzian spacetimes where the thermodynamic arrow of time reverses infinitely often.Specifically, we show that in a connected, time-orientable Lorentzian manifold, an infinite sequence of separating spacelike hypersurfaces across which entropy monotonicity alternates (an “Infinite Janus” configuration) precludes the existence of any global continuous scalar function whose gradient is everywhere timelike and aligned with entropy increase.The obstruction arises from the disconnected topology of the timelike cone bundle: a continuous timelike vector field cannot transition between future and past light cones without vanishing or becoming null. This result demonstrates that while local thermodynamic arrows may be well-defined everywhere, their integration into a single global time coordinate is mathematically impossible under infinite orientation reversals.This work provides a rigorous geometric proof based on causal structure and constructs an explicit minimal model on ℝ × S³ to demonstrate the obstruction via a direct metric inequality.