The differential of self-consistent transfer operators and the local convergence to equilibrium of mean field strongly coupled dynamical systems

The paper’s Proposition 5 and its proof are coherent and correctly handle the key technical obstacle (only weak-topology differentiability) via an inductive decomposition and a sequential Lasota–Yorke lemma, yielding a block contraction and hence LOSC. The candidate solution reaches the same high-level conclusion but incorrectly claims that Hypothesis (1) (∂Lδ,h: Vw→Bss is bounded) follows from assumption (4). In the paper, (4) only yields a bound Vw→Bs (not to Bss), and Hypothesis (1) is an extra assumption. The model also asserts chain-rule style differentiability of F^N at h in the strong topology and a local s-Lipschitz control that are not justified under the stated assumptions. Hence the paper is correct; the model’s solution contains a critical misstep.