Shadowing property and transitivity of a set-valued map and its inverse limit

What the math gives to ML

The paper establishes a structural equivalence: for a surjective upper-semicontinuous set-valued map, shadowing of the map is equivalent to shadowing of the shift on its generalized inverse-limit space, and likewise for the inverse relation. This suggests representing a neural state transition as a set-valued relation rather than a single deterministic update, with an inverse-limit-style sequence consistency constraint enforcing valid forward and backward latent trajectories. The transferable mechanism is a robustness diagnostic and training objective for long-horizon models: small local transition defects should remain close to an exact globally consistent trajectory, while loss of shadowing should mark a sharp breakdown in rollout reliability.