JORDAN MATING IS ALWAYS POSSIBLE FOR POLYNOMIALS

The paper defines Rd0, the Jordan gluing via Böttcher coordinates with the rotation e^{2πik/(d0−1)} (its formula (1.1)), and the piecewise map F; it states that in the Jordan case F is a branched covering of the sphere and proves the Main Theorem: if f,g ∈ Rd0 and at least one is a polynomial, then F has no Thurston obstruction and so is equivalent to a rational map of degree D = d1 + d2 − d0; in particular, Jordan mating is always possible for polynomials. The proof constructs a monotone quantity on non‑peripheral curves, reduces any obstruction to a Levy cycle, and then deforms it to a Levy cycle for g, contradicting that g (being rational) has no obstruction. All of these elements appear explicitly in the paper, including the degree formula and the “no obstruction ⇒ realizable” step (see the definition and main theorem; the gluing (1.1) and construction of F; the identification F = f outside T and F = g inside T; the canonical obstruction argument and the final deformation to a Levy cycle of g) . The candidate solution follows the same structure (gluing, degree count, unobstructedness via P/R–segment monotonicity, and Thurston realization), so the central argument aligns with the paper. Two caveats: (i) the candidate attributes the final contradiction to producing a Levy cycle for the polynomial f, whereas the paper produces it for g; this is a minor misattribution that does not affect the unobstructedness conclusion since rational maps have no Thurston obstruction. (ii) The candidate’s claim that changing k yields conjugate gluings is not stated or justified in the paper. Apart from these, the solution is essentially the same proof.