Probability-Time as an Informational Ordering Parameter: From Irreversibility to Information Geometry

Probability-time is introduced as a geometric parameter that measures how far a system has progressed along its irreversible evolution in state space, distinct from how long it has evolved in chronological time. It is defined by integrating information-geometric length along entropy-producing trajectories, and is constructed to increase monotonically under dissipative dynamics, remain constant for purely reversible motion, and be invariant under reparametrizations of ordinary time. This framework unifies ideas from thermodynamics, decoherence in open quantum systems, and information geometry by combining three ingredients—a space of states, a metric of distinguishability, and an entropy production functional—into a single parameter that makes irreversible progression explicit rather than implicit, suggesting a refined picture of the arrow of time as ordered motion in an information-geometric manifold. This work does not add new dynamics, instead elevates an implicit geometric quantity to the status of an ordering parameter for irreversible evolution.