Synchronization and averaging in partially hyperbolic systems with fast and slow variables

The paper establishes (i) a unique invariant manifold W via a Banach fixed-point argument on the sup norm and then quantifies anisotropic Hölder regularity along v±, (ii) sharp L1/L2 deviation bounds and the maximal-probability estimate m0{sup_n (1−ργ′)^{-n}|(S^n)_ϕ − (Φ_{nρ}+W∘A0^n)|>δ} ≤ C10 ρ/δ^3 (Theorem 7), and (iii) convergence in probability of the continuous-time interpolant with exponential weight (Theorem 8). These are proved through a careful translated/auxiliary-map construction and correlation inequalities for A0, not via a Poisson/cohomological equation. In contrast, the candidate solution hinges on solving (I − U)H = f̃ with U(h)=h∘A0 and then telescoping; this is not valid in general (additive Livšic-type obstructions for Anosov maps), and the claimed uniform resolvent bound on functions in the proposed anisotropic Hölder scale is unjustified. It also incorrectly concludes ‖W‖∞ = O(ρ), while the paper shows only probabilistic smallness of W (e.g., ⟨W⟩, ⟨W^2⟩ = O(ρ)) and, in fact, α− = O(ρ) with ‖W‖∞ = O(1) in general. The paper’s arguments are internally consistent and complete; the model’s crucial step (Poisson equation) fails under the stated hypotheses, so the model’s Phase B is incorrect. Key places in the paper: Hypotheses 1–3 and setup (2.9)–(2.10) , construction of W and regularity via (3.2)–(3.3) , anisotropic exponents α± in the scaling regime (Cor. 2.14) , oscillation bounds for W (Theorem 4) and Chebyshev consequence (2.25) , square-mean comparison (Theorem 6) , and the maximal-probability estimate (Theorem 7) with its proof (Lemma 5.1 and (5.1)–(5.3)) , as well as convergence in weighted sup-norm (Theorem 8 and its conclusion) .