The W. Thurston Algorithm for Real Quadratic Rational Maps

The paper (Bonifant–Milnor–Sutherland, 2020) explicitly proves: (i) uniqueness/rigidity of the conjugacy class from combinatorics and (ii) global convergence of the real pull-back T on X_n/G for all minimal combinatorics except the unique exceptional 6-tuple ((1,3,4,3,1,0)) and its orientation-reversal (Theorem 5.1), with a precise description of the exceptional dynamics (Proposition 5.2). These statements and proofs are present in the uploaded text (see Theorem 4.1 and the construction of T ; Theorem 5.1 and Proposition 5.2, including the explicit two-cycle mechanism v ↦ (v+1)/(v−1) and T∘T fixed-line description ). The paper also isolates the Euclidean-orbifold cases and shows that, among minimal non-polynomial combinatorics, only the 6-tuple produces the anomalous behavior (Lemma 5.6 ). The candidate model’s solution reaches the same conclusions via a different (Teichmüller-theoretic) route: it identifies T as the real restriction of Thurston’s pullback σ_f, invokes Douady–Hubbard contraction for hyperbolic orbifolds to get uniqueness and convergence, and singles out the same exceptional 6-tuple. However, the model’s proof is incomplete: it omits the crucial explicit verification of the exceptional case and overstates the reduction “global attraction on X_n/G fails only if σ_f is not strictly contracting,” overlooking that there are other χ=0 (Euclidean) minimal combinatorics where strict Teichmüller contraction fails but T still globally attracts in the real slice (cf. Lemma 5.6 and Theorem 5.1 ). Net: the paper’s argument is correct and complete for its claims; the model’s outline is essentially correct in conclusion but missing key technical steps.