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 universally incompatible under strict forms of the axioms with scalar aggregation. First, we prove an unavoidable mixed-cycle theorem: for every finite ordinal grid$\{1,\ldots,K\}\times\{1,\ldots,L\}$ with $K,L\ge 2$, we give explicit strictly positive jointdistributions $P,Q$ such that $P\prec_{\mathrm{LO}}Q$ while $Q\prec_M P$. 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 $\R$. 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 also identify universal finite-dimensional order parameters for the axioms (Lorenz signatures formarginals and the principal down-set signature for lower-orthant order) and give a natural two-numberreport $(E,D)$: $E$ is strictly $\prec_M$-monotone and $D$ is strictly $\prec_{\mathrm{LO}}$-monotone,but $(E,D)$ is not a global ranking---it exposes a two-objective conflict. The operational conclusion isto report separate certificates (or signatures) and analyze trade-offs (e.g.\ Pareto frontiers), ratherthan searching for a single scalar that strictly respects both strict axioms.