A CONVERGENCE CRITERION FOR THE UNSTABLE MANIFOLDS OF THE MACKAY APPROXIMATE RENORMALISATION

The paper precisely formulates the unstable-manifold series (Eq. (1.2)) and proves that it converges iff lim_j→∞ log q_{j+1}(k_{n_{-1}}^{-1}) / q_j(k_{n_{-1}}^{-1}) = 0 (Proposition 1.1), via a careful block decomposition, a justified reordering that requires two vanishing conditions (Eq. (1.6)), and identities linking the product ∏(-k_{n_j}^{-1}) to β-terms of the Gauss map together with Stirling bounds for the inner sums (Eqs. (1.7)–(1.9); Lemma 1.3) . The candidate’s solution reproduces parts of the block and Stirling analysis but makes two critical errors: (i) it asserts that log(q_{j+1})/q_j → 0 automatically because q_j grows at least exponentially, which is false and contradicted by the paper’s explicit counterexample (Example 1.4) ; and (ii) it appeals to an alternating-series heuristic to claim “convergence iff block magnitudes → 0,” ignoring the paper’s necessary reordering conditions (Eq. (1.6)) and the need to control both the boundary and block terms separately (Eq. (1.5)) . Hence the model’s argument is incorrect/incomplete, while the paper’s proof is sound.