PROJECTION METHODS FOR NEURAL FIELD EQUATIONS

The candidate solution reproduces the paper’s Theorem 4.2 essentially step-for-step: (i) it derives the projection error bound ||u − P_n u|| ≤ α_n via the e_n-variation-of-constants estimate, matching equation (4.2) with the same α_n, and using the same boundedness of the Nemytskii operator F (cf. Lemma 2.5) and the definition of N in (2.10) ; (ii) it establishes the two-sided estimate (4.3) using the identity relating u − u_n and u − P_n u and Grönwall’s inequality, with β_n = T||P_n W|| ||f'||_∞ as in (4.5) ; and (iii) under ||P_n W|| ≤ p, it concludes equivalence of convergence and equality of rates, as stated in Theorem 4.2 and its discussion . Minor differences: the model’s necessity proof briefly invokes choosing ξ to create a constant solution (unnecessary for the theorem with fixed ξ), and its handling of the L^2(Ω) case notes a normalization factor that the paper abstracts away; neither affects correctness. Overall, the proofs are materially the same.