Unlikely intersection problems for restricted lifts of p-th power

The paper’s Theorem 2 states that for a restricted lift of p-th power F on A^N over K=C_p and a subvariety V with F̄^l: V̄→V̄ finite surjective, if V∩Per(F) is Zariski dense, then V is periodic (indeed V∩Per(F^l)=V∩Per(F) is periodic) . The proof relies on: (i) a bijection between periodic points upstairs and downstairs via reduction (Lemma 33) ; (ii) passing to perfectoid/tilted spaces and an approximation lemma; and (iii) Xie’s Lemma (Lemma 35), which in turn uses the fact that periodic points for finite surjective endomorphisms over algebraically closed fields of characteristic p are Zariski dense (Theorem 36) . The key construction shows that the F^l-orbit of every periodic point lying in V remains in V, yielding a dense F^l-invariant subset O⊆V and hence F^l(V)⊆V; Noetherianity then gives periodicity (see the inequalities and conclusion O⊂V culminating at (4)) . By contrast, the model’s Step 2 invokes a false topological claim: over an uncountable field, a countable union of proper Zariski closed subsets of an irreducible variety cannot be Zariski dense; this is wrong (e.g., countably many distinct lines in A^2 can be Zariski dense). The model also reverses a crucial inclusion in Step 3, asserting F^{lm}(V)⊂V without establishing it, and it dismisses the needed finiteness/surjectivity hypothesis on F̄^l that the paper uses to import density of periodic points from characteristic p via Xie’s lemma . Hence the paper’s argument is essentially correct (modulo minor typographical gaps), while the model’s proof is flawed.