Stochastic Causality Constraints: The Information-Theoretic Impossibility of Closed Timelike Curves via Geometric Entropy

General Relativity (GR) allows for exact solutions containing Closed Timelike Curves (CTCs), such as the Gödel universe or the interior of rotating Kerr black holes, theoretically permitting time travel to the past. Hawking's Chronology Protection Conjecture suggests that quantum effects must prevent their formation, but a rigorous mechanism remains elusive. Unified Field Dynamics (UFD) provides a thermodynamic proof of chronology protection based on the stochastic nature of the vacuum.

We demonstrate that the existence of a CTC requires the trajectory of the universe q(t) to form a closed loop in the configuration space Q. However, the evolution of q is governed by the TECD SDE: dq = X_H dt - grad J dt + Sigma dW. The presence of the irreducible stochastic term Sigma dW (Geometric Noise) ensures that the evolution is a Markov process with strictly positive entropy production rate dS/dt > 0 (as derived in UFD-GRA-04). For a system to revisit a past microstate, it must spontaneously decrease its geometric entropy to exactly match the prior value. We calculate the probability of such a recurrence for a macroscopic system and show it is effectively zero (P ~ exp(-S_univ)). Consequently, while the conservative drift X_H might theoretically support loops (unitary cycles), the stochastic agitation Sigma destroys any phase coherence required to close the loop. Time travel is impossible not because geometry forbids it, but because the vacuum noise erases the road back.