The Model Operator: Formal Foundations for Model Governance

Abstract

No governance framework for quantitative models provides a formal language for stating what a model is, what it means for two models to compose, or how risk propagates through a pipeline. SR 26-2 (the April 2026 successor to SR 11-7), PRA SS1/23, and the EU AI Act remain entirely qualitative: principles, checklists, and expert judgment.

This paper provides that language. We define a model as a versioned, immutable, typed operator M^v : X × Ω → Y, where Ω is a runtime environment decomposed into components with distinct governance cadences. Five independent axioms make models the morphisms of a symmetric monoidal category. From them we derive an algebra of model operations, a calculus of sensitivities that propagate through composed systems via a chain rule, and a risk functional grounded in Wald's statistical decision theory. The central result for practitioners is that every governance question (reproducibility, equivalence, risk) reduces to the same triple (M^v, x, Ω), and that version transitions are themselves measurable, high-risk operators whose magnitude can trigger revalidation automatically. Worked examples in the appendices demonstrate the formalism on concrete systems, including a diagnosis of the 2012 JPMorgan London Whale losses as a cascade of axiom violations.