Irreducibility of periodic curves in cubic polynomial moduli space

The paper proves that, for every p ≥ 1, the curve Sp in the moduli space of cubic polynomials with a marked critical point is irreducible, answering Milnor’s question. It works on the double cover with coordinates (a,v), fa,v(z) = (z − a)^2(z + 2a) + v, noting that polycm3 ≅ C^2/± with the involution (a,v) ↦ (−a,−v), and that Sp upstairs maps as a regular double cover onto Sp for p ≥ 2 . The main result is stated explicitly as Theorem 1: for all p ≥ 1, Sp is irreducible . The proof is dynamical: it decomposes Sp into the connectedness and escape loci, analyzes parameter external and internal rays, and shows connectedness by moving between escape regions until reaching the distinguished one; combined with Milnor’s smoothness of Sp for p ≥ 2, connectedness implies irreducibility . By contrast, the model’s algebraic proof hinges on a specialization step that invokes “Bousch irreducibility” for the unicritical cubic family z^3+v. However, the paper only cites Bousch (and later Buff–Tan) for the quadratic dynatomic curves, not for the cubic unicritical polynomial; the key external claim that Φp(0,v) for z^3+v is irreducible is neither supported nor referenced in the paper’s discussion (which emphasizes the quadratic case) . Without a correct and cited source that the unicritical cubic dynatomic polynomial (for the critical point 0) is irreducible, the model’s argument is unsupported at its critical step. The rest of the model’s pipeline (monicity in v, use of marked dynatomic polynomials, and image of an irreducible set under a finite morphism) is plausible but depends crucially on that specialization claim. Consequently, the paper’s result stands on a complete, independent proof, while the model’s solution is incomplete/wrong due to a missing, nontrivial ingredient.