How to split a tera-polynomial

The paper’s Theorem 2 states and proves the complete factorization q_{ℓ,n}(z)=∏_{k|n}(h_k(z)^{η_ℓ(k)} ∏_{j=2}^{ℓ} m_{j,k}(z)) with η_ℓ(k)=⌊(ℓ−1)/k⌋+2, and shows that all Misiurewicz factors are simple; it supports this with an inductive recurrence q_{ℓ,n}=q_{ℓ−1,n}(q_{ℓ−1,n}+2p_{ℓ−1}) and degree counts, together with s_{ℓ,n}=q_{ℓ,n}/q_{ℓ−1,n} having simple roots . The candidate solution proves the same statement via a clean telescoping identity q_{ℓ,n}=p_n ∏_{t=0}^{ℓ−1}(p_{t+n}+p_t) and a direct multiplicity count at centers, plus transversality for preperiodic parameters (in line with HT15). Thus both are correct; the proofs differ in method.