CATEGORICAL DYNAMICAL SYSTEMS ARISING FROM SIGN-STABLE MUTATION LOOPS

The paper proves that for a sign-stable mutation loop φ (with Conjecture 3.4 assumed), the categorical entropy satisfies h_T(F_φ|D_fd)=log λ_φ for all T and h_0(F_φ|per)=log λ_φ (Theorem 1.1; proved as Theorems 4.7 and 4.11), by carefully using Keller–Yang equivalences, tilts of hearts, mass-growth/entropy inequalities, and spectral radii on K_0; crucially, it shows F_φ(A(t)) is a tilt of A(t), not A(t) itself, and F_φ(Γ) lies in add Γ ⋆ add Γ[1], not purely degree 0. The model’s solution incorrectly assumes t-exactness on the canonical heart and preservation of projectives, which the paper explicitly avoids; the paper instead controls cohomology in degrees 0 and −1 and uses norm/complexity bounds to get T-independence and equality with log λ_φ. Hence the model’s argument is flawed though it reaches the correct numerical conclusion proved by the paper (Theorem 1.1; Lemma 3.19; Theorems 4.5–4.7, 4.11).