Classifying hyperbolic ergodic stationary measures on compact complex surfaces with large automorphism groups

The paper proves the trichotomy and Γμ-invariance under the stated hypotheses via a Benoist–Quint/entropy–holonomy–QNI framework, with precise entropy inequalities and a detailed joint integrability argument. By contrast, the model’s solution makes unjustified leaps: (i) it invokes the Avila–Viana invariance principle without verifying the required hypotheses in this nonuniform random context, (ii) it incorrectly derives δ^s=1 from δ^u=1 by a symmetric argument that does not follow from the assumption λ+ + λ− ≥ 0, and (iii) it overstates Cantat–Dujardin’s rigidity as implying Γμ-invariance of any non-atomic stationary measure, which their results do not assert in this generality. The paper’s argument is coherent and complete modulo its stated reliance on forthcoming results of Brown–Eskin–Filip–Rodriguez Hertz; the model’s proof outline misses key technical steps.