Probability-Time as an Information-Geometric Ordering Parameter for Irreversible Quantum Evolution (Revised)

We propose probability-time (τ) as an information-geometric ordering parameter on the manifold of quantum states undergoing irreversible evolution under completely positive trace-preserving (CPTP) dynamics. Probability-time is defined as the accumulated information-geometric arc length of the entropy-producing component of a trajectory, distinguishing it from chronological time (t), which parameterizes duration rather than irreversible progression. (Revised Version)