Opinion response functions are key to understanding tipping of social conventions

The paper’s Appendix C (Theorem C.1) proves the fixed‑point structure of Ψ_CM by showing F(CM,r)=Ψ_CM(r)−r is convex in r on [0,1/2], strictly increasing in CM, and then continuing the two simple roots via the implicit function theorem up to a fold CM*=sup{CM where the branch exists}, where ∂rF=0; it also proves monotonicity of r_− and r_+ and their coalescence at r* (and absence of interior fixed points for CM>CM*) . The candidate solution uses a different but compatible route: it introduces H_CM=Ψ_CM−r, leverages symmetry/convexity of Φ on [0,1/2] to show H_CM is strictly convex there, defines g(CM)=min_{r∈[0,1/2]}H_CM(r) to locate a unique threshold C*_M with g(C*_M)=0, and then applies the implicit function theorem to get the two C^1 branches with opposite monotonicity. Both arguments rely on the same structural facts about Φ (definitions, symmetry, sigmoid/convexity) established earlier in the paper . The model also adds a standard saddle‑node square‑root expansion not stated in the paper but valid under the same nondegeneracy conditions. Overall, the conclusions and claims coincide; the proofs are different in presentation (minimization threshold versus branch continuation) but consistent.