On simultaneously preperiodic points for one-parameter families of polynomials in characteristic p

The paper proves the binomial-characterization using local/global canonical heights and a careful valuation analysis of εn together with the special parameter λα = α − f(α). The candidate solution instead relies on a flawed pigeonhole step (fixing one iterate pair (m,n) infinitely often) and an oversimplified Newton–polygon-at-infinity analysis of Δ-polynomials. Over an infinite field, a fixed nonzero polynomial in T cannot vanish at infinitely many λ, so the key common-roots argument collapses. The paper’s statements (Theorems 1.3–1.5) and proof strategy are coherent and documented, while the model’s proof omits essential uniformity and contains incorrect degree/term claims for the “second-leading” T-exponents.