Dimension reduction in recurrent networks by canonicalization

The paper proves a canonical realization theorem (Theorem 3.2) for causal, time-invariant filters with the input-forgetting property via Nerode equivalence on semi-infinite pasts, constructing X = ZZ_- / ∼, F([z], a) = [za], and h([z]) = HU(z); it establishes well-definedness, echo state property (at the state level), canonicity (strong reachability + observability), and uniqueness up to system isomorphism (via Proposition 2.4 and Corollary 2.6) . The candidate solution mirrors this construction and all key logical steps. The only notable gap is that it proves output-level uniqueness (ESP at the output) rather than the paper’s stronger state-level ESP used to assert canonicity formally; however, the rest of the argument aligns and can be completed by the paper’s state-level uniqueness argument .