Rethinking Conventional Inequality Indices: Information Boundaries, Multifractal Spectrum Width, and Spatial Structural Information

Regional inequality is commonly measured with indices (e.g., the Gini and Theil indices) that depend only on the multiset of values and are therefore permutation-invariant: they cannot distinguish configurations differing solely in how the same values are ordered. This paper formalises that limitation as an information boundary and shows that the multifractal spectrum width (Δα) carries qualitatively distinct information. Four propositions are established: (1) with group membership fixed, the Gini, Theil, generalised entropy and Dagum Gini indices are permutation-invariant; (2) Δα is not, and can move in either direction under permutation; (3) Δα still varies appreciably when permutation-invariant indices are held constant, so the two classes are not informationally equivalent; and (4) controlled synthetic fields serve as a proof of concept linking Δα to multiscale heterogeneity. Permutation invariance secures the objectivity and comparability of inequality indices, but it also delineates their information boundary: Δα carries structural information that these indices cannot, in principle, capture, making the two complementary rather than substitutable. A finite-sample baseline and the Multifractal Magnitude Index (MMI) yield an operational diagnostic protocol; because Δα is defined only relative to the ordering rule and the MF-DFA parameterisation, all comparisons are made within a single parameterisation.