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

This paper introduces generative velocity, a new measure–theoretic invariant defined as the fraction of a measure that remains singular under Radon–Nikodym decomposition. Explicitly, for

μ = fλ + μ⊥,

the invariant

vgen(μ) = μ⊥(X) / μ(X)

quantifies how much of the measure cannot be absorbed into any absolutely continuous representation.

This simple ratio provides a structural diagnostic for analytic and geometric persistence. In particular, we prove that

• Taylor remainder persists ⇔ vgen > 0
• singular Radon–Nikodym component ⇔ vgen > 0
• overlap geometry becomes Gram-type ⇔ vgen > 0

Thus analytic remainder, singular measure, and curvature-type geometric behavior are unified as manifestations of a single invariant.

Generative velocity induces an ordering on contextual configurations and defines a geometric stability frontier called the skyline, separating convex (polytope-type) absorption from curvature-producing (sphere-type) regimes.

As a concrete stress test, the Gauss circle problem is reinterpreted in this framework: the classical lattice discrepancy emerges as the analytic trace of a positive generative velocity generated by boundary curvature.

Because generative velocity is defined entirely within standard measure theory, it provides a new invariant linking Radon–Nikodym decomposition, analytic persistence, and geometric regime transition, with applications ranging from discrepancy theory and fractal measures to overlap-based geometric structures.