MINIMAL MODEL PROGRAM FOR ALGEBRAICALLY INTEGRABLE ADJOINT FOLIATED STRUCTURES

The paper establishes the cone theorem, contraction theorem, and flips for algebraically integrable adjoint foliated structures, and deduces the existence of the K(X,F,B,M,t)-MMP (and with scaling) for Q‑factorial klt data (Theorems 1.3, 1.4, 1.5, 1.6), with preservation of key properties along the program (Lemma 8.1, Theorem 8.2) . The candidate solution invokes exactly this standard MMP skeleton: select a K‑negative extremal ray via a supporting function, contract it, flip if necessary, and iterate with scaling of an ample divisor. The only minor mismatch is that the candidate phrases the supporting-ray selection via a K+λA nef-threshold; in the paper this selection is formalized through a slightly different but equivalent device (Proposition 5.6 uses thresholds for D+s(K+A)) or directly by taking a supporting function HR=K+A, plus the contraction/flip package (Theorems 6.3, 6.9, 7.1). These are routine adjustments that do not affect correctness. Hence both are correct and follow substantially the same proof strategy.