Maximum Entropy versus Lorenz-Extremality Log-Normal Families under Fixed Mean and the Limits of "Maximal Heterogeneity"

Maximum entropy (MaxEnt) is often interpreted as selecting a “most heterogeneous”distribution compatible with constraints, while heterogeneity/inequality for nonnegativevariables with common mean is canonically formalized by the Lorenz order (majorization).This paper proves a sharp incompatibility: within the log-normal family under fixed arithmeticmean, distributions form a strict Lorenz chain in the shape parameter, yet Shannondifferential entropy has a unique interior maximizer. We strengthen the result by restoringgeneralized entropies: R´enyi and Tsallis entropies admit closed forms along the sameconstraint manifold and typically yield interior maximizers or boundary divergence dependingon order; Kaniadakis entropy exhibits parameter-dependent interior-versus-boundaryregimes determined by competing L1±κ contributions. Consequently, none of these standardentropy functionals is strictly decreasing along the Lorenz chain, hence none is strictlyLorenz-Schur-concave on the mean-constrained log-normal family. We discuss why practitionersmight expect alignment, why it fails here (constraint-induced coupling of scale andshape), how this contrasts with monotone scale-free families where boundary MaxEnt selectioncan coincide with maximal tail heterogeneity, and what alternative heterogeneitytargets align naturally with Lorenz order.