Opinion Dynamics with Set-Based Confidence: Convergence Criteria and Periodic Solutions

The paper’s Theorem 1 (A–D) is stated and proved correctly using a recurrent-averaging-inequalities lemma with type-symmetric (cut-balanced) weights and positive diagonals, handling both the no-stubborn and homogeneous-stubborn cases via a reduction argument. The candidate solution proves essentially the same statements, using a Laplacian argument for (A) and a cut-balance consensus theorem for (B–D). For Vs = ∅ the model’s proof aligns well with the paper. For Vs ≠ ∅, the model’s proof sketch implicitly assumes symmetric support of A(t), which is false when stubborn rows are present; however, the result itself is still correct and can be repaired exactly as in the paper by applying the inequality-based reduction to the non-stubborn subdynamics. Hence both are correct; the proofs differ, and the model’s proof needs a minor fix in the stubborn case.