Limit theorems for toral partially hyperbolic endomorphisms

The paper proves, for SVPH endomorphisms F on T^2, a CLT with Berry–Esseen bound and an LLT (plus a quantitative LLT for intervals) using the spectral (Nagaev–Guivarc’h/Keller–Liverani) perturbation method on anisotropic Sobolev-type spaces B_s, exactly the approach outlined by the model. The main statements (Theorem A: CLT + Berry–Esseen; Theorem B: LLT and interval LLT with the boundary Gaussian factor and exponent 1/2−1/δ) match the model’s conclusions, including the mixture weights c_k=m(ρ_k)µ_k(f_m), Green–Kubo variances σ_k^2, the non-coboundary/non-lattice hypotheses (B1)/(B2), and the claim that σ_k depend only on (F,τ) and not on the initial density. See the stated results and proof strategy across Sections 2–5 and Appendix A, e.g., the result statements for CLT/Berry–Esseen and LLT with weights and variances, the DFLY-based quasi-compactness and the twisted-operator spectral gap away from 0, and the abstract perturbative expansion yielding Gaussian limits and remainders. Minor differences are purely notational (e.g., a sign convention around λ'' at 0 and whether to write t-dependent projectors), not substantive.