A Mathematical Theory of Agency and Intelligence

TL;DR

This paper introduces bipredictability, a fundamental measure of shared information in interactions, distinguishing agency from intelligence, and demonstrates its bounds across physical, AI, and biological systems, proposing a feedback architecture for adaptive AI.

Contribution

It defines bipredictability as a fundamental measure of shared information in interactions, deriving bounds for classical and quantum systems, and links agency to the capacity to act on predictions, separate from intelligence.

Findings

Bipredictability P is intrinsic and bounded between 0.5 and 1 in quantum systems.

Current AI systems exhibit agency but lack the full spectrum of intelligence.

A feedback architecture monitoring P enables adaptive and resilient AI.

Abstract

To operate reliably under changing conditions, complex systems require feedback on how effectively they use resources, not just whether objectives are met. Current AI systems process vast information to produce sophisticated predictions, yet predictions can appear successful while the underlying interaction with the environment degrades. What is missing is a principled measure of how much of the total information a system deploys is actually shared between its observations, actions, and outcomes. We prove this shared fraction, which we term bipredictability, P, is intrinsic to any interaction, derivable from first principles, and strictly bounded: P can reach unity in quantum systems, P equal to, or smaller than 0.5 in classical systems, and lower once agency (action selection) is introduced. We confirm these bounds in a physical system (double pendulum), reinforcement learning agents, and multi turn LLM conversations. These results distinguish agency from intelligence: agency is the capacity to act on predictions, whereas intelligence additionally requires learning from interaction, self-monitoring of its learning effectiveness, and adapting the scope of observations, actions, and outcomes to restore effective learning. By this definition, current AI systems achieve agency but not intelligence. Inspired by thalamocortical regulation in biological systems, we demonstrate a feedback architecture that monitors P in real time, establishing a prerequisite for adaptive, resilient AI.