Order-Cycle Impossibility for Scalar Ordinal Aggregation: Marginal Lorenz Monotonicity and Concordance Monotonicity Cannot Be Simultaneously Scalarized

Scalar aggregation of multidimensional ordinal data is often required to satisfy two natural monotonicityprinciples: (i) monotonic response to marginal dispersion, formalized via Lorenz or majorization order oneach marginal, and (ii) monotonic response to strengthening dependence, formalized via a concordance ororthant order on the joint distribution. This paper shows that these requirements are jointly incompatible.On a finite 3×3 ordinal grid, we construct an explicit directed cycle that alternates marginalLorenz equalization and dependence strengthening, implying that no scalar index can be strictly monotonewith respect to both principles. The obstruction is structural rather than analytical: it arises fromnon-embeddability of the induced ordinal relations into a one-dimensional scale. As a consequence,scalar ordinal indices necessarily conflate marginal dispersion with cross-dimensional dependence. The result provides afoundational justification for state-vector architectures in ordinal aggregation, such as OrdinalMajorization Entropy (OME) and its multidimensional extension r-OME-IV, which separate marginal anddependence coordinates rather than attempting to collapse them into a single number.