The Impossibility of Consistent Scalar Aggregation of Multidimensional Ordinal Distributions under Robust Dominance and Joint-Censored Distributional Sensitivity

Multidimensional ordinal data are frequently summarized by a single scalar index. Two desiderata are commonly demanded: robust monotonicity under joint dominance improvements, and distributional sensitivity among the deprived. A canonicalrobust dominance order on a finite ordinal product grid is strict lower orthant dominance. A canonical distributionalsensitivity principle is a spread axiom that penalizes concentration in the distribution of deprivation severity among the poor. A known defect of global spread axioms on ordinal scales is Pareto perversity: they can penalize perfect concentration atthe best end of the ordinal support. A standard repair in poverty measurement is censoring: apply spread sensitivity onlywithin an identified poor subpopulation. This paper formalizes that joint-censoring repair and proves that the scalaraggregation deadlock persists. Fix any nontrivial poverty identification rule and consider the conditional distribution amongthe identified poor. Fix any finite severity scale on the poor set with at least two levels, and fix any order-sensitive severityconcentration preorder that treats sufficiently endpoint-peaked severity distributions as strictly maximally concentrated. If thebest corner outcome is classified as non-poor, then for every strictly positive distribution P there exists a strictly positiveQ such that _ Q _ P. , no scalar index can be strictly increasing under both relations. The proof isolates a structural orthogonality: mixing a distribution with sufficiently large mass at a single non-poor dominanceanchor improves all dominance thresholds while leaving the conditional distribution among the poor invariant. This orthogonalityyields mixed cycles and a general impossibility theorem for scalar indices under joint-censored distributional sensitivity. This version adds five further results in full detail. First, the mixed-cycle impossibility extends beyond strict positivity: the setof distributions that lie on strict mixed cycles contains an open and dense subset of the full probability simplex, and everyboundary distribution admits an -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 any scalar index whenever strictanchored dominance is nontrivial. Third, a substantive logical defect is repaired: weak within-poor monotonicity is notautomatically satisfied by dominance-based scalars and remains inconsistent with strictdominance under mixed cycles. Fourth, the argument is generalized beyond the best corner of a product grid: the mixed-cyclemechanism holds on any finite partially ordered outcome space equipped with any dominance order generated by down-sets thatshare a common non-poor dominance anchor, and it admits a measurable-space formulation for anchored dominance families undera uniform slack condition. Fifth, and strategically, the paper is extended from a pure no-go statement to a complete classification theorem and arepresentation-dimension theory. Under anchored robust dominance, strict scalarizability together with conditional-invariantwithin-poor sensitivity holds if and only 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-index representation is characterizedand, in the canonical two-axiom case, equals 2. Sixth, the fixed-incidence variant (cycles that preserve the poverty headcount) is upgraded from a ``sometimes'' statement to a complete frontier: fixed-incidencemixed cycles through every P exist if and only if an explicit feasibility--richness condition holds for the induced within-poor relation relative to theidentification rule. Seventh, the qualitative impossibility is quantified via a normalized minimax axiom-violation functional for approximate scalarizations, withsharp lower bounds forced by mixed 2-cycles. Eighth, a canonical anchored-core decomposition isolates the anchored component of a general threshold-dominanceorder as the minimal structural witness of the dimension obstruction.