FLOP BETWEEN ALGEBRAICALLY INTEGRABLE FOLIATIONS ON POTENTIALLY KLT VARIETIES

The paper’s Theorem 1.3 exactly matches the solver’s question and is proved with care: after lifting the two (K_F+B)-MMPs to small Q-factorial modifications using a dedicated lifting proposition (Proposition 3.1), the authors show the induced birational map is small (isomorphism in codimension one) and then factor it into flops; crucial steps rely on Lemma 2.9 (crepancy and smallness for two D-MMP endpoints when D is pseudo-effective) and on termination and NQC results for algebraically integrable foliations from the recent literature, all in the relative setting over U . By contrast, the candidate’s solution essentially repeats the classical variety-proof: it asserts smallness via a negativity-lemma computation and then invokes the “standard flops connect minimal models” argument with scaling to conclude. However, the paper explicitly explains why a naive Kawamata-style argument fails for foliations—Bertini-type issues for lc structures, and nontrivial termination of MMP with scaling—requiring generalized foliated quadruples and recent termination/NQC results (LMX et al.) to overcome obstacles (i)–(iii) . The model omits these foliation-specific difficulties and the needed machinery (e.g., NQC, polytopes, generalized pairs), and treats the Q-factorial lifting as routine, whereas the paper proves it carefully via Proposition 3.1 . Therefore, the paper’s argument is correct and complete at the level claimed, while the model’s solution is incomplete and does not justify the key termination and lc-compatibility steps required in the foliated setting.