Constraint-Induced Coupling and the Failure of MaxEnt–Lorenz Alignment: A General Diagnostic, Unimodal Case Studies, and an HEA Index

Maximum entropy (MaxEnt) is often informally interpreted as selecting the “most heteroge-neous” distribution compatible with constraints. For nonnegative variables with a fixed mean,heterogeneity/inequality is canonically formalized by the Lorenz (majorization) order. Recentwork shows a sharp incompatibility within the fixed-mean log-normal family: distributions form astrict Lorenz chain in the dispersion parameter, yet Shannon and standard generalized entropiesadmit interior maximizers rather than selecting Lorenz-extremal endpoints. We strengthenthat mechanism into a general diagnostic based on constraint-induced coupling between “shape”and “scale” coordinates on the constraint manifold. We then exhibit the same pathology intwo additional unimodal families under fixed mean: Gamma and Weibull. In both families, theLorenz order is a strict chain in a single shape parameter, while Shannon differential entropy hasa unique interior maximizer at the exponential boundary point (Gamma shape k = 1; Weibullshape a = 1). Finally, we propose a quantitative heterogeneity–entropy alignment (HEA) indexwhich measures the misalignment between Lorenz-extremality and MaxEnt selection, and weoutline an empirical protocol for testing MaxEnt–Lorenz alignment in fitted families.