Predictive Dynamics: State–Space Flow as a Necessary Structure of Predictive Theories

This work establishes a general, meta-theoretical result concerning the logical structure of prediction. It proves that any theory capable of generating predictions must admit a deterministic evolution on a state space, representable as a flow generated by ordinary differential equations. The result is independent of physical interpretation, empirical content, or representational framework. No new physical model or dynamics is proposed. Instead, the paper identifies a necessary and sufficient structural condition for predictability itself. Classical mechanics, quantum formalisms, probabilistic descriptions, and algorithmic models are shown to be representational encodings of this invariant underlying structure. Apparent indeterminism arises from projection onto incomplete state spaces rather than from fundamentally non-deterministic dynamics. The result has the logical status of a characterization or no-go theorem, comparable in scope to foundational results such as the Church–Turing thesis in computation. It applies to predictive theories in general, not only to physical theories, and fixes deterministic state–space flow as the unique architecture compatible with prediction. The work is purely theoretical and foundational, addressing structural constraints on predictability.