Regression-based projection for learning Mori–Zwanzig operators

The paper formalizes a regression-based projection P for Mori–Zwanzig and derives: (i) the discrete-time GLE and operator definitions Ω(n)=PK[(I−P)K]^n, W_n=[(I−P)K]^{n+1}g, (ii) the GFD Ω(ℓ)=P(W_{ℓ−1}∘F), (iii) a constructive, regression-based extraction of Ω(0),Ω(1),… from a snapshot tensor D via a base regression (x=D(·,·,1)→y=D(·,·,2)) and an inductive residual-target construction, and (iv) linear-case closed-form formulas Ω(0)=C(1)C(0)^{-1} and Ω(n+1)=[C(n+2)−Σ_{ℓ=0}^n Ω(ℓ)C(n−ℓ+1)]C(0)^{-1} . The candidate solution reproduces exactly these steps, including the base and inductive regression targets and the linear closed forms. Minor issues in the paper (arg max vs. arg min; N vs. K indexing) are typographical. Hence both are correct and follow substantially the same proof.