Finite-Horizon Structures VI: Fertility Spectrum and Structural Thresholds in Superlevel Y-Filtrations

This article develops the measure-theoretic realization of projective Y-structure within the finite-horizon framework. Starting from a positive measurable representative of the homogeneous scalar associated with a finite-horizon Y-object, it shows how, relative to a chosen full-support Radon reference measure, this representative induces a corresponding Radon measure whose density is given by the same structural scalar. The construction is explicitly structural: the measurable representative is interpreted as a density of finite-horizon structural weight rather than as a probability density. A central result is that the intrinsically meaningful measurable object is not, in general, a single canonical measure, but a Y-measure class defined up to equivalence of reference measures. This preserves the projective nature of the underlying Y-structure while providing a rigorous measurable realization of the framework. Once a reference measure is fixed, the associated measure is uniquely determined by the measurable representative, and measurable Y-morphisms act on these measure representatives through the same constant rescaling factor that governs the transformation of the homogeneous scalar itself. Together with the companion axiomatic and differential formulations, this work shows that one and the same finite-horizon organization admits categorical, geometric, differential, and measure-theoretic realizations. Within the broader Ranesis program, the article provides the measurable layer of the finite-horizon framework and helps complete the structural articulation between its axiomatic, geometric, and measurable components.