Order-Cycle Impossibility for Scalar Aggregation; Lower-Orthant Monotonicity versus Order-Sensitive Marginal Deconcentration

Scalar aggregation of multivariate distributions is routinely used to rank social outcomes,risks, or system performance, but “reasonable” monotonicity desiderata can point in incompatibledirections. We study scalar evaluation rules under two strict monotonicity requirements.(LO) Strict lower-orthant monotonicity: if a joint distribution increases the probabilityof every joint low-threshold event (every principal down-set on a finite ordinal grid, or everyorigin-anchored rectangle in a continuous product order), with at least one strict increase, thenany admissible scalar must strictly improve.(M) Strict order-sensitive marginal deconcentration: if in every coordinate the marginaldistribution becomes less locally concentrated along the given order, with at least one strictmarginal improvement, then any admissible scalar must strictly improve. In the finite model,local concentration is measured by maximal probability mass in contiguous blocks of categoriesat each block length. In the continuous model on 0, 1, local concentration is measured by theL´evy concentration function (maximal interval mass) at each interval length; this continuousaxiom presumes a fixed normalized scale on 0, 1 and is not invariant under arbitrary increasingreparametrizations.On every nontrivial finite ordinal product grid in every dimension d ≥ 2, we prove a pointwisemixed-cycle richness theorem: for every strictly positive joint distribution P we explicitlyconstruct Q such thatP ≺LO Q ≺M P.This yields strict scalar impossibility, and we further show that weakening strictness does notrescue meaningful scalarization: strict–weak hybrids are impossible and weak–weak monotonicityforces constancy. To place the marginal axiom in a standard order-theoretic canon, we provethat the classical dispersive order implies our marginal concentration preorder, both on finitechains and (under continuous strictly increasing CDFs) on 0, 1.On 0, 1d we obtain two continuous cycle mechanisms. A boundary mixed-cycle theorem usesthe corner Dirac measure. An atom-free interior mixed-cycle theorem is proved on a uniformlyregular class with explicit marginal density bounds; we establish transport-map regularity(absolute continuity, derivative bounds, and a correct Jacobian formula) for the coordinatewisequantile transports used in the construction. Finally, on bounded-density classes we definea finite-dimensional discretized surrogate of the infinite-dimensional continuous profile (gridevaluations of the joint lower-orthant CDF and the marginal concentration functions) andprove deterministic uniform approximation bounds and dominance certificates with explicittolerances. A general factorization lemma shows that any report monotone in the underlyingtest families factors uniquely through the canonical profile. These cycle mechanisms supply aformal backbone for the index–dashboard split that has intensified in multidimensional povertymeasurement and welfare reporting: headline counting indices are increasingly paired withrobustness checks, indicator dashboards, and within-poor distributional supplements, becausethe underlying desiderata cannot be compressed into a single strict scalar without contradictio