Finite-Time Persistence and the Canonical Scalar Generator

This article develops a conditional scalar reduction theorem for finite-time persistence assessments under explicit admissibility assumptions. Starting from a finite-horizon persistence functional, it shows that, under compositional regularity, continuity, locality, and separability assumptions, the persistence score can be reduced up to monotone reparameterization to a weighted temporal aggregation. This representation yields a causal kernel form and reveals two structurally distinct temporal scales: a maintenance timescale induced by the evaluation architecture and an intrinsic timescale carried by the regime itself. Their comparison defines a canonical dimensionless temporal coherence ratio. The article then addresses the scalar closure problem. Assuming that finite-time viability is represented at the scalar level through effective maintained content, temporal coherence, and finite evaluation horizon, and imposing a homogeneous single-index reduction principle, it is shown that every admissible scalar characterization of finite-time viability reduces, up to monotone reparameterization, to a function of the scalar quantity Y=EC/T. The result is formulated explicitly as a conditional scalar closure result. It does not claim that Y follows from the weakest persistence axioms alone, nor that it constitutes a universal dynamical law. Rather, within the stated homogeneous single-index framework, Y is identified as the unique canonical scalar generator of finite-time viability up to monotone reparameterization. The article is therefore intended as a scalar-level contribution within the broader finite-horizon structural program, while remaining compatible with further projective, differential, measurable, and local structural refinements. This article is part of the Ranesis framework, developed by Alexandre Ramakers.