Cubic Polynomials with a 2-Cycle of Siegel Disks

The paper’s Main Theorem explicitly establishes (a) Hausdorff‑continuous dependence of the two Siegel boundaries on the parameter, (b) that Γθ is a Jordan curve separating 0 from ∞, and (c), (d) that Γ0θ and Γ1θ are Jordan arcs lying in the respective complementary components of Σθ \ Γθ containing 0 and 1. Part (a) is proved via Proposition 2.8, building on Zhang’s uniform a priori bounds; parts (b)–(d) are proved using the conformal angle A(α), local injectivity and a rigidity argument, culminating in a global Jordan curve/arc structure for the loci (Γθ, Γ0θ, Γ1θ) . By contrast, the candidate solution makes two incorrect global claims: (i) it asserts that the ‘double‑on‑one‑boundary’ loci are arcs issuing from the punctures and meeting the Jordan curve Γθ; however, the paper proves Γ1θ is a Jordan arc connecting the two finite double‑critical parameters α4 and α5 (and, by symmetry, Γ0θ connects α̃4 and α̃5), with these arcs lying in complementary components and not meeting Γθ ; (ii) it posits a global quasiconformal surgery parametrization by a circle of ‘twist’ angles without supplying the necessary construction or rigidity. While several other steps of the model align with the paper (normalization of fα and Σθ; use of bounded‑type a priori bounds; continuity of boundaries) , the global topological misstatements about Γ0θ and Γ1θ make the model’s conclusion incorrect.