Matings of Cubic Polynomials with a Fixed Critical Point. Part II: Sufficiency of the Limb Condition

The paper proves that if two limbs L1 and L2 satisfy the limb condition Θ(L1) = −Θ(L2), then any postcritically finite Fa ∈ L1 and Fb ∈ L2 are not mateable, by showing the α-cycles lie in a common periodic ray class that must contain a closed loop, which (via Proposition 2.17) yields a non-removable Levy cycle and thus an obstruction . The candidate solution constructs the same loop explicitly from the rays with angles in Θ(L1) and −Θ(L1), notes periodicity under angle tripling (consistent with m3-invariance of Θ(L) in the paper ), and invokes the standard result that a periodic ray class with a loop yields a Levy cycle. Aside from a minor nuance (the paper emphasizes non-removability via separation and Proposition 2.17, whereas the model appeals directly to the ‘periodic loop implies Levy cycle’ fact), the reasoning, mechanism, and conclusion agree.