MMP for generalized foliated threefolds of rank one

The uploaded paper proves the rank‑one generalized foliated MMP on Q‑factorial klt threefolds: if D=KF+B+MX is pseudo‑effective the (D)-MMP terminates with a minimal model, and if not it ends with a Mori fiber space (Theorem 1.1) . The proof hinges on a new cone theorem for generalized foliated threefolds (Theorem 4.8) and, crucially, on the positivity of the moduli trace along invariant curves (Proposition 4.2), which reduces D‑negative rays to KF‑negative ones so that contractions and flips exist by Cascini–Spicer; termination then follows from their rank‑one threefold termination theorem (Proposition 5.1 and its proof; proof of Theorem 1.1) . By contrast, the candidate solution appeals to a base‑point‑free theorem as an input to produce contractions and also to generalized pairs’ contraction theorems, neither of which directly applies to foliated canonical divisors KF. In the paper, base‑point‑free results are proved as an application after establishing the MMP (Theorem 1.3), not assumed at the outset . The candidate also omits the necessary preliminary reduction to a model with simple foliation singularities that the paper uses when invoking CS20 to run steps and preserve lc (see Proposition 5.1 and Proposition 5.3) . Hence, while the high‑level conclusion matches the paper, key steps in the candidate’s argument rely on inputs that are not valid in this foliated setting or are used circularly.