DYNAMIQUE ANALYTIQUE SUR Z. II : ÉCART UNIFORME ENTRE LATTÈS ET CONJECTURE DE BOGOMOLOV-FU-TSCHINKEL.

The paper proves a uniform bound M for the number of common images of torsion points under distinct standard projections π1, π2 from two elliptic curves, precisely as stated in its Theorem A (Corollary 8.4) , with the framework and definitions (e.g., standard projections) laid out in Section 4.1 and a key uniform lower bound for the mutual energy of the associated Lattès maps (Theorem 8.1) . The method is analytic-dynamical (Berkovich over Z, mutual energy), summarized in the introduction and developed through Sections 5–8 . The candidate model gives a different proof: it pulls back the diagonal to obtain a curve C ⊂ E1×E2 of bounded degree and genus ≥2, then invokes Gao–Ge–Kühne’s uniform Mordell–Lang to bound torsion on C. The paper itself explicitly points out this alternative approach—pullback of the diagonal combined with the uniform Mordell–Lang of Gao–Ge–Kühne—as another route to Theorem A . Minor caveat: the model should note that the uniform Mordell–Lang result is proved over number fields; extension to arbitrary characteristic 0 fields proceeds by specialization (the paper uses such a specialization step to extend its own bound, see the end of Section 8) . Overall, both are correct and the proofs are genuinely different in method.