Variety of physical measures in partially hyperbolic systems with multi 1-D centers

The paper explicitly proves Theorem A (T^5, two 1D centers, all Gibbs u-states hyperbolic; for C1+ there is a unique physical measure with χ(Ec1)>0, χ(Ec2)<0 and full-volume basin) and Theorem B (on T^{2m+1}, all Gibbs u-states hyperbolic; for C1+ exactly m physical measures with distinct unstable indices and full-volume basins) using a concrete perturbation of skew products whose base is Morse–Smale on S^1 and fiber is linear Anosov on a torus, together with a DA-type surgery to alter unstable indices; these statements and the proof roadmap are stated verbatim in the abstract and main theorems and carried out in Sections 4–6 . Key steps include establishing hyperbolicity of all Gibbs u-states in a C^1-open neighborhood (Theorem 4.4(3)) and then invoking [24, Theorem B] to obtain finiteness and full-volume basins for C1+ dynamics (Theorem 4.4(5)) . By contrast, the candidate solution incorrectly attributes a different construction to Mi–Zhang (triangular, fiber-bunched skew products over a linear Anosov base with invariant graphs) and replaces the paper’s reliance on [24] by an unrelated PVE hypothesis from Gan–Li–Viana–Yang; neither the “triangular fiber-bunched invariant graph” mechanism nor the PVE route appears in the paper, whose base dynamics is Morse–Smale on S^1 and uses DA surgery rather than normally hyperbolic invariant graphs . The candidate also misstates the unstable indices in the general case (they claim indices dim Eu + j, j=1,…,m, which contradicts the paper’s explicit m=2 case having indices 1 and 2) . Hence, while the model’s final conclusions match the paper’s statements, its proposed proof path and several technical claims are not supported by the paper, making the model solution incorrect relative to the audited document.