A Mathematical Theory of Agency and Intelligence

Information theory measures prediction as a reduction in uncertainty between interacting systems. In most applications, however, the state spaces over which prediction is made are treated as fixed. This assumption breaks down in open environments, where the scope and distribution of interactions can change over time. Current AI systems show the consequences: they can perform well within a training interface yet degrade when that interface shifts. Here we introduce bi-predictability , P , which measures how much of an interaction’s total uncertainty is converted into shared predictability. Rather than measuring predictive success in isolation, P normalizes it by the full uncertainty available in the observation-action-outcome loop, allowing interaction quality to remain interpretable even when the scope of the interaction changes. We show that P is bounded across regimes. In classical non-agentic systems, P ; in quantum entangled systems, P can in principle reach 1; and in classical agentic systems, responsive action lowers P below the classical ceiling through an informational agency penalty. We test this predicted hierarchy across physical, computational, and linguistic systems. These bounds also provide operational definitions of agency and intelligence. In this framework, agency is the capacity to choose actions that affect outcomes, whereas intelligence additionally requires monitoring whether interaction quality is degrading and adapting when it does. On this view, learning within a fixed interface is not enough for robust intelligence under changing conditions. We operationalize this distinction with the Information Digital Twin (IDT), an auxiliary, thalamus-inspired monitoring architecture that estimates P continuously from the observable interaction stream, without access to model internals. By shifting attention from prediction within fixed state spaces to regulation of interaction structure, this framework offers a mathematical basis for systems that can monitor and stabilize their own coupling to the world.