The Impossibility of Consistent Scalar Aggregation: Why Multidimensional Poverty Indices Must Fail Distributional Sensitivity

Multidimensional ordinal data are frequently summarized by a single scalar index. Twodesiderata are commonly demanded: robust monotonicity under joint dominance improvements,and distributional sensitivity among the deprived. A canonical robust dominance order on a finiteordinal product grid is strict lower orthant dominance. A canonical distributional sensitivityprinciple is a spread axiom that penalizes concentration in the distribution of deprivation severityamong the poor.A known defect of global spread axioms on ordinal scales is Pareto perversity: they canpenalize perfect concentration at the best end of the ordinal support. A standard repair in povertymeasurement is censoring: apply spread sensitivity only within an identified poor subpopulation.This paper formalizes that joint-censoring repair and proves that the scalar aggregation deadlockpersists. Fix any nontrivial poverty identification rule and consider the conditional distributionamong the identified poor. Fix any finite severity scale on the poor set with at least two levels,and fix any order-sensitive severity concentration preorder that treats sufficiently endpoint-peakedseverity distributions as strictly maximally concentrated. If the best corner outcome is classifiedas non-poor, then for every strictly positive distribution P there exists a strictly positive Q suchthatP ≺lower orthant Q ≺joint censored P.Consequently, no scalar index can be strictly increasing under both relations.The proof isolates a structural orthogonality: mixing a distribution with sufficiently largemass at a single non-poor dominance anchor improves all dominance thresholds while leaving theconditional distribution among the poor invariant. This orthogonality yields mixed cycles and ageneral impossibility theorem for scalar indices under joint-censored distributional sensitivity.This version adds five further results in full detail. First, the mixed-cycle impossibilityextends beyond strict positivity: the set of distributions that lie on strict mixed cycles containsan open and dense subset of the full probability simplex, and every boundary distributionadmits an ε-mixed-cycle arbitrarily nearby. Second, the results are stated as a sharp designfrontier: strict anchored dominance and within-poor focus cannot be jointly satisfied by anyscalar index whenever strict anchored dominance is nontrivial. Third, a substantive logical defectis repaired: weak within-poor monotonicity is not automatically satisfied by dominance-basedscalars and remains inconsistent with strict dominance under mixed cycles. Fourth, the argumentis generalized beyond the best corner of a product grid: the mixed-cycle mechanism holds onany finite partially ordered outcome space equipped with any dominance order generated bydown-sets that share a common non-poor dominance anchor, and it admits a measurable-spaceformulation for anchored dominance families under a uniform slack condition.Fifth, and strategically, the paper is extended from a pure no-go statement to a completeclassification theorem and a representation-dimension theory. Under anchored robust dominance,strict scalarizability together with conditional-invariant within-poor sensitivity holds if andonly if the within-poor strict relation is empty (no nontrivial strict within-poor sensitivity).When scalarizability fails, the minimal dimension required for a coordinatewise multi-indexrepresentation is characterized and, in the canonical two-axiom case, equals 2.