Two-variable polynomials with dynamical Mahler measure zero

The uploaded paper proves the two directions of Theorem 1.5: under the Dynamical Lehmer conjecture it classifies irreducible P in Z[x,y] with mf(P)=0 as lying in the Medvedev–Scanlon/GNY family, and it shows the converse mf(P)=0 for those building blocks. The converse (our Part (2)) is established via invariance of µf under maps commuting with an iterate, giving mf(f̃^n(x)−L(f̃^m(y)))=mf(x−y)=0 (Lemma 7.1 and Corollary 7.3) . For the forward direction (our Part (1)), the paper uses Weak Dynamical Boyd–Lawton (Prop. 1.3) and a bounded-orders property (BOP) to produce infinitely many preperiodic points on the curve, then applies the Ghioca–Nguyen–Ye characterization of (f,f)-preperiodic curves (Prop. 8.2) to conclude the classification (Theorem 8.1) . By contrast, the candidate’s Part (1) relies on a resultant identity Qℓ(x)=Resy(P(x,y),fℓ(y)) and asserts m_f(Qℓ)/d^ℓ→0. This crucial limit ignores the leading-coefficient term d^ℓ·m_f(A_y) arising from Resy(P,fℓ), so in general m_f(Qℓ)/d^ℓ→m_f(A_y)≠0; consequently the step invoking the Lehmer gap to force m_f(Qℓ)=0 for infinitely many ℓ fails. The argument also implicitly uses positivity m_f(P(·,v))≥0 for v with fℓ(v)=0, which need not hold when coefficients are not integral. Finally, the move from “infinitely many points with one coordinate in Kf” to “infinitely many (f,f)-preperiodic points” is not justified, whereas the paper supplies a complete route through BOP and GNY. The analytic preliminaries (energy identity I(µf)=0 for monic f, Jensen-type formulas) used by the paper and partially echoed in the candidate are correct (see (1.8), Prop. 3.1–3.2) .