An "opinion reproduction number" for infodemics in a bounded-confidence content-spreading process on networks

The paper’s setup and generating-function analysis are correct: s(x0,c) is defined via the integral over ϕ(x) (with the uniform-on-(0,1) specialization giving the standard piecewise-linear form) and the early-stage reproduction number is R = s(x0,c) g1'(1) = s(x0,c) g0''(1)/g0'(1); for Poisson degree this reduces to R = s(x0,c) λ, exactly as stated in the paper’s Eq. (10) . However, after substituting the uniform ϕ(x) form of s(x0,c) (their Eq. (4)) into R = s λ and setting R = 1, the paper prints the critical c* as reciprocals, c* ∈ {1/(λ + x0), 1/(2λ), 1/(λ − 1 + x0)} with the usual regime conditions . This is algebraically inconsistent with R = s λ and s(x0,c) from Eq. (4): solving λ s(x0,c) = 1 yields the linear-in-x0 solution c*(x0,λ) = 1/λ − x0 (left-boundary regime), c* = 1/(2λ) (interior), and c* = 1/λ − 1 + x0 (right-boundary). The candidate solution derives exactly this piecewise-linear threshold using the same PGF framework, so the model’s solution is correct while the paper’s printed c* expression is wrong.