Syntropy in complex systems: A complement to Shannon’s Entropy

Using the paper’s stated definitions for H_N and S_N on the two-point simplex, one shows H_N is strictly concave with unique maximum at p=1/2, S_N is strictly convex with unique minimum at p=1/2, so F_N(p)=H_N(p)-S_N(p) attains its global maximum at p=1/2. From this, equilibria exist iff F_N(1/2)≥0. A short calculation gives F_N(1/2)=(ln 2)/(ln N)−β_N(e^{1/2}−1) with β_N=1/[N(e^{1/N}−1)], and F_N(1/2) strictly decreases in N. One checks F_2(1/2)>0, F_3(1/2)>0, but F_N(1/2)<0 for all N≥4, so no equilibrium exists for N≥4. Hence there is no sequence (p_N) for large N and no limiting value near 0.01. The paper’s claim of apparent convergence to p≈0.01 relies on a numerical routine in the annex that actually solves (e^p−1)/[N(e^{1/N}−1)]=H_N(p) on the restricted interval [0.01,0.99], and moreover uses a syntropy expression that omits the (1−p) term and the p-weights stated earlier in the text. This inconsistency, together with the analytic argument, shows the convergence claim is false as stated (and is likely an artifact of the code and bracketing).