Topological obstructions in probability-localization quotients

Characterize which cycles, homological features, and other topological obstructions are preserved or destroyed by probability-localization quotients and their state-space refinements.

Background

The paper interprets probability localization models as adaptive quotients of an underlying dynamical state space. State-space refinement is intended to split observational fibers until their future observables become sufficiently predictive, but the discussion does not analyze the topological consequences of these quotient and refinement operations.

In particular, the paper identifies the preservation of cycles, homology, and related topological obstructions as unresolved questions requiring work beyond the presented probabilistic, geometric, and predictive-closure framework.