ON DYNAMICAL CANCELLATION

The paper proves the Preimages Question affirmatively for étale maps (Theorem 1.2(a)) and, with the additional assumption that f|Y is flat, establishes a uniform bound over all extensions of bounded degree (Theorem 1.2(b)) using a reduction-mod-p counting method and an induction on the dimension of Y, with Lemma 2.2 as a key input . By contrast, the candidate solution incorrectly deduces a uniform bound on the first hitting time from a purported uniform-modulus version of Dynamical Mordell–Lang (DML): even if one has a common difference M for all arithmetic progressions describing N(x), this does not bound the minimal offset, so the step “min N(x) ≤ M−1” is invalid. Moreover, the paper’s part (b) explicitly requires flatness of f|Y, which the candidate dismisses; the proof shows flatness is used crucially in the local-field lifting argument (Proposition 2.5), so the model’s claim is unsupported .