A Sharma–Mittal Geometry of Evenness: Representation, Cardinal Resolution, and Relative Merits

Luca Delvecchio

The proliferation of evenness indices creates an interpretive problem whenever several admissible measures assign different numerical values to the same assemblage. We study that problem by normalizing Sharma–Mittal entropy to its maximum at complete uniformity. Written in terms of the Hill number of order α, the resulting surface is \(E_{\alpha,\beta}(p;S)=(D_\alpha(p)^{1-\beta}-1)/(S^{1-\beta}-1)\), \(E_{\alpha,1}=\log D_\alpha/\log S\). Five canonical Hill-based evenness families are exact restrictions of this surface. More generally, for every α > 0 and finite β, the family satisfies Requirements 1a–3b of Chao and Ricotta: transfer, continuity and symmetry, non-increase under addition of a vanishingly rare species, a richness-independent limiting range, and scale invariance. At fixed richness and Hill order, every section is an invertible monotone transformation of the same \(D_\alpha\), so the represented measures contain the same ordinal information and differ in numerical spacing. The parameter β controls that spacing: local sensitivity is proportional to \(D_\alpha^{-\beta}\) and the ratio of endpoint slopes is exactly \(S^\beta\). Over the canonical range \(0\le\beta\le2\), the largest possible disagreement between any two sections is \((\sqrt S-1)/(\sqrt S+1)\), attained at \(D_\alpha=\sqrt S\). When richness differs, ordinal equivalence need not survive: if the linear section and \(D_\alpha\) rank two assemblages in opposite directions, continuity guarantees a finite-β ranking reversal. Coverage-standardized Miocene and Pliocene data from Chao et al. (2020) provide a published example, with a crossing near β ≈ 1.40 using their tabulated values. Finally, under K-fold replication, β < 1 leaves a positive asymptotic evenness deficit, β = 1 approaches one logarithmically, and β > 1 approaches one polynomially; at β = 0 the replication limit is exactly Hill evenness \(D_\alpha/S\). The same parameter also governs independent-product composition and large-richness behavior. These results make the relative merits of alternative evenness geometries explicit without selecting a universal winner.