A two-layer model for coevolving opinion dynamics and collective decision-making in complex social systems

The paper’s Theorem 3 shows that with one stubborn innovator, fully rational decisions (β=∞), and α < d* − 2, if λ < λ* := 1/2 − (2+α)/(4 d* − 4 − 2α), then no nonstubborn adoption occurs and 〈x(t)〉 = −1 + 2/n for all t; the proof proceeds by bounding payoffs for the first potential adopter, with neighbor vs non-neighbor cases, culminating in the same λ* threshold . The candidate solution derives the same condition by computing the exact best-response gap Δπ_i from the paper’s payoff (7a)–(7b) and then applying the same “earliest adoption” logic; it matches the theorem’s statement and logic, differing only in algebraic presentation. Definitions and assumptions (payoff (5), stubborn modeling, 〈x〉) are consistent with the paper .