The Information Potential as a Lyapunov Function for the Information-Dynamic Theory Flow: A Variational Derivation of the Arrow of Time

We prove a one-line H-theorem for the canonical Information-DynamicTheory (IDT) flow. The information potential Φ(ω) = −ln Z(ω) is astrict Lyapunov function along the natural-gradient dynamics:dΦ/dτ = −‖∇Φ‖²G⁻¹ ≤ 0, with equality only at stationary points.The result is unconditional — no statistical, ergodic, ormolecular-chaos assumption is required; positive-definiteness of theFisher metric suffices. A stronger result establishes monotonedissipation of the gradient norm in convex regions. We further showthat the structural inertia M, previously conjectured to be theLyapunov object, is in fact NOT universally monotone; instead itplays the role of a spectral distance to the C46 trivialisationlocus. The ontological "loss of alternatives" principle is realisedformally as the monotone growth of the smallest GEP eigenvalueλ_min. A foundational separation from probability-based informationgeometry is established: any normalisation of the admissibilityweight forces Φ ≡ 0 and trivialises the dynamics. The workcomplements the IDT programme (Math Foundations, P3, P4, P7,T0-chain, IDT-AP) and constitutes its third flagship, onthermodynamic foundations.