Finite-Horizon Structures III: Canonical Measure Induced by the Homogeneous Invariant

This work establishes the measure-theoretic component of the finite-horizon program.Starting from a single homogeneous invariant expressing the structural balance between magnitude, coherence, and evaluation horizon, we show that any positive measurable representative of this invariant canonically induces a Radon measure on a structured space (Y-object). The construction is purely structural: the invariant is not interpreted as a probability density, but as a finite-horizon structural density encoding the local weight of energetic persistence across an evaluation horizon. The resulting measure is unique up to equivalence, independent of the chosen reference measure, and transforms under admissible structural morphisms by homogeneous rescaling. Together with the companion axiomatic and differential formulations of the finite-horizon framework, this article demonstrates that a single primitive invariant simultaneously generates categorical structure, geometric structure, and measure-theoretic structure. Measure and geometry thus appear as dual manifestations of the same finite-horizon quantity, completing the structural triad of the program. The framework is minimal, model-independent, and does not rely on probabilistic, dynamical, or metric assumptions. It provides a foundational basis for studying finite-time persistence, structural balance, and scale-invariant organization across mathematical physics and related fields.