When Is There a First Threshold? A Formal Audit of Global Applicability Claims in Structural-Enrichment Models

Abstract

This preprint develops a Global Threshold Audit (GTA) for distinguishing several non-equivalent claims about when a predicate, operator, or vocabulary becomes licensed in a structurally ordered model. GTA takes an existing positive licensing relation as input and separates five global claim levels: nonempty licensing, frontier existence, uniqueness of the minimal licensing class, existence of a global least licensing threshold, and principality of the positive-licensing domain. A separate pathwise module tests whether admissible structural routes agree on the first licensed class. Countermodels show that the stronger claims do not follow automatically from weaker local, minimal, or route-relative evidence. The framework is applied to metric and locality licensing in a pre-dimensional structural model, where a source-named metric threshold does not establish a global first metric threshold, and positive locality control does not establish a unique global first locality threshold. “First” is used strictly in an order-theoretic or path-indexed sense, not temporally or causally. Preprint; not peer reviewed.