The Mathematics of Model Discipline

This paper develops a general mathematical framework for model discipline. A model is disciplined when it is not evaluated only by internal coherence or in-sample empirical fit, but also by its preservation of structure, causal response, information sets, operational constraints, regime behavior, linguistic and semantic invariants, and external transfer. The framework applies to parametric models, statistical learning systems, generative models, agent-based simulators, language models, image generators, social simulators, macroeconomic models, financial time-series models, and synthetic economies. Leontief’s critique of formalism detached from empirical measurement is used only as historical motivation; the framework itself is stated in model-agnostic terms. Beyond the definitions we prove four structural facts. The observational diagnostics are refinements of a single object, the observable path law; causal and external validity are not identified by observational fit, so they must be certified by intervention or invariance rather than estimated from fit; the discipline vector controls deploymentregret through a path-space optimal-transport bound; and information leakage can only inflate measured performance, which is why information validity acts as a gate rather than as one term among many. The resulting criterion is deliberately diagnostic rather than scalar: it records what has been measured, what has been certified, and what remains assumed. This separation is important across domains: high predictive accuracy can coexist with linguistic shortcut learning, contaminated information, infeasible policies, unstable regimes, invalid counterfactual responses, or brittle transfer. These results organize the framework into validation certificates with a precise account of what each score can and cannot certify. A minimal numerical illustration computes the discipline vector on observationally equivalent bivariate systems; real- data macroeconomic and stock-return stress tests show finite-sample observational scoring with uncertainty bands; and a controlled language shortcut stress test shows a non-economic failure mode in which surface fluency and coverage remain high while negation, semantic-role, and order-sensitive inference fail