On the Coupling of Well Posed Differential Models — Detailed Version

The paper’s Theorem 2.6 constructs the coupled local flow F(τ,t0,(u,w)) = (Pw(t0+τ,t0)u, Pu(t0+τ,t0)w), verifies the Theorem 2.4 hypotheses with L = e^{(Cu+Cw)T} and an O(τ)-modulus ω, and concludes existence/uniqueness of a global process along with the invariance of domains. The candidate solution follows the same blueprint: it (i) proves F is a local flow with the same Lipschitz structure, (ii) checks the Euler–polygonal Lipschitz bound with L = e^{(Cu+Cw)T}, (iii) establishes the commutator/defect estimate as O(τ^2), and (iv) derives invariance of domains. Minor constant bookkeeping differs: the paper states ω(τ) = Ct·Cu·τ in the theorem line, while the detailed proof naturally yields Ct·Cw·τ; the candidate obtains 2Ct·Cw·τ and then informally suggests replacing Cu by max{Cu,Cw} to recover Ct·Cu·τ, but drops the factor 2. These are harmless quantitative tweaks that do not affect the validity of the hypotheses of Theorem 2.4 or the conclusion. Overall, both are correct and essentially the same proof, with only minor normalization/notation differences .