Environmental Rigidity and the Decoupling Theorem for Deutsch Time Loops

This preprint investigates environmental effects in the Deutsch model of quantum closed timelike curves (CTCs). In the Deutsch framework, consistency of a time loop is enforced by a fixed-point condition on the quantum state circulating within the loop, leading to effective nonlinear dynamics on interacting systems. Focusing on finite-dimensional Deutsch CTC models with well-defined fixed-point selection rules, we examine whether loop self-consistency alone implies environmental invariance, meaning that degrees of freedom external to the loop remain unchanged after a traversal. Using explicit system–environment couplings, we analyze the reduced environment dynamics induced by a single loop interaction. We show that for generic nontrivial couplings, there exist environment preparations for which a detectable disturbance is induced, even though the loop state satisfies the Deutsch self-consistency condition. Exact environmental invariance arises only in the trivial, factorized coupling limit within the class of models considered. These results distinguish internal paradox resolution on the loop register from the operational behavior of external degrees of freedom, indicating that environmental invariance is not enforced by Deutsch self-consistency alone. The conclusions are independent of the specific fixed-point selection rule and rely only on the structural properties of Deutsch consistency and completely positive dynamics. The analysis provides an operational framework for comparing environmental signatures of different quantum CTC constructions and can be extended to alternative consistency models, including postselected and trace-preserving formulations.