Finite and Infinite Degree Thurston Maps with Extra Marked Points

The paper’s Main Theorem A is clearly stated and proved: if Sf ⊂ B and f:(S^2,B) is realized, then f:(S^2,A) is realized iff there is no degenerate Levy multicurve consisting of curves non‑essential in S^2\B (Main Theorem A; proof in §3.6) . The argument handles finite degree, the (2,2,2,2) case, and infinite‑degree settings uniformly via pullback dynamics on Teichmüller space and a holomorphic disk factor, together with Proposition 3.9 (any Levy cycle under these hypotheses is degenerate and non‑essential in S^2\B) and the standard obstruction implication (Levy cycle ⇒ obstructed) . By contrast, the candidate’s (⇒) direction relies on rational‑map Fatou theory (e.g., no wandering domains; classification of periodic Fatou components), which does not extend in general to the transcendental/infinite‑degree case treated by the paper. The (⇐) direction is topological but contains a fatal setup error (choosing disks Ua ⊂ S^2\B for every a ∈ A, impossible when a ∈ B) and an unjustified “Alexander‑trick” step to enforce the global conjugacy. Hence the model’s proof is incomplete/incorrect in the generality of the paper.