Bell Inequalities, Free Variables and the Undecidability of Chaotic Invariant Sets.

Based on an analysis of the concept of free variable in physical theory, the following is demonstrated: it is impossible to prove that all members of the class of chaotic locally-causal, objectively-realistic physical theories satisfy Bell inequalities, if the states are constrained to lie on chaotic invariant subsets. This result follows from the fact that nontrivial properties of chaotic invariant sets are undecidable in the computational-theoretic sense of the word.

This motivates the formulation of what is referred to as the Invariant Set Hypothesis: not only are nonlinear dynamics fundamental to physical theory, a fortiori the global state-space geometry of the invariant set is a more primitive representation of these dynamics, than are local differential-equation expressions. In this latter sense the Invariant Set Hypothesis is fundamentally non-classical. The Invariant Set Hypothesis provides a geometric and hence relativistic framework within which the apparently nonlocal and atemporal paradoxes of quantum theory can be understood.