Emergent Physical Time in Information-Dynamic Theory: Structural Inertia, Spectral Relaxation, and a Variational Derivation

Abstract Information-Dynamic Theory (IDT) describes the evolution of system regimes on a statistical manifold equipped with the Fisher–Rao metric G, governed by the canonical gradient flow d/d = -G^-1, where = - Z is the information potential and is an internal pre-geometric evolution parameter. A central open problem of the framework is the relationship between this internal parameter and observable physical time. In this work we show that observable time emerges as an inertial reparametrisation of the internal evolution parameter. The effective time scaling is determined by the slowest relaxation mode of the information-geometric dynamics and is given by dt = m () \, d, m () = 1_{ (G^-1H) }, where H = ² is the Hessian of the information potential and _ is the smallest generalized eigenvalue of the pair (H, G). Using a variational principle formulated on the relaxation spectrum of the system, we derive this mobility coefficient from the axioms of IDT through a Kullback–Leibler spectral optimisation problem. This establishes the structural inertia m () as the unique scaling compatible with the spectral structure of the information-geometric flow. The resulting framework predicts a direct connection between curvature of the information landscape and observable dynamical slowdown. As an empirical consistency check, we analyse neural network training dynamics, where the loss landscape forms a statistical manifold. Numerical experiments on multilayer perceptrons trained on MNIST demonstrate a strong correlation between structural inertia and critical slowing down during optimisation (Spearman ρ = 0. 91, p < 0. 001). These results suggest that observable temporal behaviour in complex adaptive systems may emerge from the spectral structure of information-geometric dynamics. Keywords: information geometry, Fisher–Rao metric, structural inertia, emergent time, relaxation spectrum, KL variational principle, critical slowing down.