DEGREE GAPS FOR MULTIPLIERS AND THE DYNAMICAL ANDRÉ-OORT CONJECTURE

The paper’s theorem explicitly states the dichotomy: if a family f over X has infinitely many PCF specializations, then for any periodic point P either the multiplier vanishes identically or deg(λf(P)) ≥ hcrit(f) (see the Theorem in the introduction) . The proof uses a combination of local lemmas (including Lemma 4) and the Favre–Gauthier reduction to one infinite critical orbit; in the proof one shows that for a certain set S of places, gcrit,v(f)=0, and then Lemma 4 yields deg(λf(P)) ≥ hcrit(f) . In contrast, the candidate’s Step 2 asserts a general placewise lower bound log^+ |(f^n)'(P)|_v ≥ gcrit,v(f) for any polynomial, which is not in the paper; the only general placewise bound proved is the upper bound log^+ |λf(P)|_v ≤ (d−1) gcrit,v(f) (Proposition 3) . Moreover, the paper shows that without a PCF hypothesis the ratio deg(λf(P))/hcrit(f) can take any rational value in [0,d−1], including values strictly less than 1 (Remark 5) , directly contradicting the candidate’s PCF-free inequality. Hence the model’s proof relies on a false local inequality and drops an essential hypothesis.