Rational maps with constant Thurston pullback mapping

The paper’s main theorem explicitly states and proves that every non‑trivial regular or mixing CTP polynomial satisfies McMullen’s condition (Theorem 2.1) . The proof proceeds by: (i) using the topological characterization of CTP maps (Theorem A) to deduce that f(E) is a single point and #(A\E) ≤ 2 (Lemma 3.4) ; (ii) treating the Belyi case (#V_f = 3) via monodromy/branched-tree arguments to obtain Stab(a) = Stab∗(E) (Lemma 5.4), and then applying the “power factor” lemma to conclude McMullen’s condition (Lemma 5.5) ; and (iii) reducing the general case to the Belyi case by a pinching construction that produces a non‑trivial CTP Belyi polynomial (Lemma 6.1) and a monodromy injection preserving stabilizers (Lemma 6.2), together with an alternate pinching argument yielding Stab(a) = Stab∗(E) for the original map (Lemma 6.3) . In contrast, the model’s argument hinges on an unsubstantiated differential rank bound that would force |F| = |f(E) ∪ V_f| ≤ 3 and on a claim that degree‑2 constant pullbacks are impossible. The former contradicts the paper’s framework (f may have many critical values even when it factors through a Belyi map via McMullen’s condition), and the latter is false in light of McMullen’s construction, which gives degree‑2 polynomial examples by composing g with z^2 and appropriate markings (see the McMullen example and its use throughout the paper) . Consequently, the model’s proof is unsound, while the paper’s proof is coherent and complete.