Order-Cycle Impossibility for Scalar Ordinal Aggregation: Marginal Lorenz Monotonicity and Lower-Orthant (Principal Down-Set) Monotonicity Cannot Be Simultaneously Scalarized

Two strict desiderata are natural in multidimensional ordinal aggregation: (i) strict marginal Lorenz(monotone-in-majorization) response to marginal equalization, and (ii) strict principal down-set(monotone-in-lower-orthant) response to strengthening of joint low-ordinal mass. We show these twostrict requirements are generically incompatible with scalar aggregation. On finite domains we also isolate a canonical obstruction certificate: in the directedmultigraph obtained by overlaying the desiderata, strict scalarization is equivalent to the existenceof a potential (a $0$-cochain) whose coboundary is strictly positive on every desideratum edge, andfailure is witnessed by a nonnegative circulation (a $1$-cycle) via a sharp potential--circulationduality alternative. First, we prove an unavoidable mixed-cycle theorem: for every finite ordinal grid\1,…,K\×\1,…,L\ with K,L≥ 2, we give explicit strictly positive jointdistributions $P,Q$ such that PLOQ while QM P. Any scalar strictly monotone inboth relations would increase along this two-step cycle and return to its start, impossible. The samecontradiction extends to any strict dependence relation satisfying a minimal principal down-setmonotonicity axiom. Second, even when mixed cycles are blocked by restricting dependence comparisons to fixed marginals(marginal-invariant dependence), scalarization can still fail for an independent reason: on anuncountable chain of marginal classes totally ordered by strict marginal Lorenz equalization, strictwithin-class dependence monotonicity forces uncountably many pairwise disjoint nonempty open intervalsin . We prove a sharp classification on such chain domains: scalarization is possible if and onlyif only countably many marginal classes are dependence-nontrivial and each such class admits aninternal scalar representation. We then give a general no-go template (mixed-cycle and interval-packing principles), define a canonicaldesiderata profile reporting standard, and prove that Pareto-frontier reporting is the uniqueassumption-minimal axiom-respecting replacement for scalar ranking. We also identify universalfinite-dimensional order parameters for the axioms (Lorenz signatures for marginals and the principaldown-set signature for lower-orthant order) and give a natural two-number report $(E,D)$: E isstrictly M-monotone and D is strictly LO-monotone, exposing a two-objectiveconflict.