Transport in Reservoir Computing

The paper’s Theorem 4.16 establishes, for compact X and stationary inputs Θ, (i) existence/uniqueness of the fixed point M_Θ of the sequence-space Foias operator PG together with continuity of the selector S_G(Θ)=M_Θ, and (ii) identification S_G(Θ)=U_g∗Θ under the USP; its proof proceeds via stochastic contractivity, continuity transfer to G, and Banach’s fixed point theorem, plus a shift-equivariance argument T1∘G=G∘T1 for stationarity and the filter identity for (ii) (see Theorem 4.16 and the T1-equivariance steps in equations (50)–(55) in the paper ). The candidate solution proves the same statements but with a different technical route: it uses a product (sum-of-weights) metric on sequence spaces and a direct coupling construction to show that PG(Θ,·) is a contraction when g is stochastically contractive, relies on Banach’s theorem for existence/uniqueness and shift-equivariance for stationarity, proves continuity of Θ↦PG(Θ,μ) by a coupling/Markov-inequality argument plus uniform continuity of g, and identifies S_G via the causal filter under USP. These arguments align with the paper’s structure (stochastic contractivity ⇒ contractive Foias, continuity transfer, fixed points) but differ in technical choices (sum vs. sup weighted metrics; coupling-based continuity vs. the paper’s equicontinuity/Arzelà–Ascoli step underpinning the upper semicontinuity of the contraction modulus and the continuity result in Theorem 4.15 then lifted to G) . Net: conclusions match; proofs are methodologically different but sound.