Poincare' Recurrence as a Mechanism for Generating Quantum-like Ensembles.

We explore Poincar´e recurrence as a mechanism for generating quantum-likestatistics from recurrent classical dynamics. Modeling the universe as a singlerecurrent, ergodic trajectory on a compact phase space, we show that an embedded observer’s ignorance of their location along the recurrent trajectory definesa natural ensemble. Using the Kolmogorov Extension Theorem, we constructa Kolmogorov-consistent, normalizable measure on the induced path space. Wefurther show that recurrent loop topology fixes the complex phase weighting ofhistories, reconstructing the Feynman path integral. The Born rule emerges fromthe same consistency requirements.