Finite Time Hyperbolic Coordinates

The paper proves, under quasi-hyperbolicity type (I) alone, the exponential convergence bound ‖e(k)−e(i)‖ ≤ Q1 (Γ Γ̃ c/λ)^i (Proposition 3.3), without any extra cone assumption; the proof uses an a-priori telescoping estimate (Lemma 3.1) and the ratio ‖DΦ^j‖·‖DΦ_{ξ_j}‖/‖DΦ^{j+1}‖ ≤ const·(Γ Γ̃/λ)^j, giving the required base (Γ Γ̃ c/λ)^i (see 3.2.1 and (34)–(36) , with the defining inequalities (I) in Definition 2.3 ). The candidate solution incorrectly asserts that the sharper rate (Γ Γ̃ c/λ)^i under (I) requires an additional cone invariance; their argument only derives a weaker rate (Γ̃^2 c)^i absent that extra assumption. For type (II), the model’s bound matches the paper’s result ‖e(k)−e(i)‖ ≤ Q̃1 (c/c̃)^i (Proposition 3.4) . For the derivative bound, the model’s conclusion agrees with Theorem 2.5, which the paper proves by decomposing contributions E(k)_0, Σ E(k)_i, and a third term involving Σ F(k)_i (Proposition 6.1 and Lemmas 6.3–6.7), yielding ‖Dξ0 e(k)‖ ≤ K1‖D^2Φ_{ξ0}(e(1),·)‖ + K2 c ≤ K1√2‖∂_ς DΦ_{ξ0} e(1)‖ + K2 c, uniformly in k .