Generative Velocity as a Measure--Theoretic Invariant: Skyline Structure and the Radon--Nikodym Decomposition

This paper introduces a measure–theoretic invariant called generative velocity, defined as the proportion of a measure that remains singular under contextual refinement. Unlike classical velocity, which is defined as the derivative of position within a fixed geometric background, generative velocity arises directly from the Radon–Nikodym decomposition μ = fλ + μ⊥ and quantifies the extent to which structure escapes absorption into an absolutely continuous regime. Within this framework, singular measure acquires a dynamical interpretation: it records the persistence of generative directions that cannot be flattened into convex accumulation. This interpretation provides a unified explanation for phenomena traditionally treated as analytic irregularities, including Taylor remainders, discrepancy terms, and singular measures. We show that generative velocity governs the transition between two structural regimes: a polytope regime, characterized by complete contextual absorption and convex flattening, and a sphere regime, characterized by persistent singular structure and the emergence of Gram overlap geometry. The extremal envelope of admissible configurations under fixed generative velocity defines the skyline, which serves as a stability frontier for contextual transformation. As an accessible stress test, the Gauss circle problem is reinterpreted in this framework. The classical discrepancy between lattice counting and area is shown to correspond structurally to a velocity gap between bulk absorption and boundary curvature, illustrating how analytic remainder functions as a regime indicator. This work establishes generative velocity as a structural invariant linking measure decomposition, analytic persistence, and geometric regime transition, providing a unified foundation for analyzing singular behavior across analytic, geometric, and arithmetic contexts.