Anchored Mixing and the Impossibility of Exact Scalar Aggregation under Multidimensional Ordinal Dominance and Joint-Censored Within-Poor Sensitivity

This paper isolates the minimal structural obstruction behind the scalar aggregation problem for multidimensional ordinal welfare distributions and extends the core impossibility theorem in six directions. The setting is a finite ordinal product grid equipped with lower orthant dominance as the distributional welfare order and a joint-censored within-poor sensitivity principle that evaluates only the conditional distribution inside a fixed identified poor set. The core mechanism is anchored mixing: mixing any interior distribution with a point mass at the best corner of the product grid strictly improves every proper lower orthant threshold because the best corner lies in every threshold, while leaving the conditional distribution among the poor completely unchanged because the best corner is non-poor by assumption. This orthogonality between the dominance improvement and the within-poor conditional distribution forces strict mixed 2-cycles. A universal mixed-cycle theorem is proved: under downward cofinality of the induced within-poor strict relation on the conditional poor simplex, every interior distribution belongs to such a cycle. Under the weaker condition that the induced relation is merely nonempty, one mixed cycle exists, and this single cycle is sufficient to block every strictly monotone scalar index simultaneously. Both results are then extended from lower orthant dominance to every nontrivial anchored threshold family on an arbitrary finite set, covering any dominance order in which all thresholds share a common non-poor anchor element. The six additional results are as follows. First, the abstract hypotheses are verified completely on the explicit grid with outcome space equal to the product of three income levels and two health levels under a union identification rule and a three-level additive severity map, proving both nonemptiness and downward cofinality of the induced within-poor strict relation generated by a local concentration preorder. Second, it is proved that no continuous scalar on the full probability simplex can have a restriction to the interior that is strictly increasing under both anchored dominance and within-poor improvement, and every boundary distribution is shown to be approximable by interior mixed cycles. Third, quantitative ell-1 bounds are derived for the anchored-mixing construction, including an exact headcount-preserving realization formula, an explicit upper bound for the cycle amplitude, and a complementary positive-separation lower bound showing that canonical mixed cycles need not be small even when the worsening conditional distribution is close to the baseline. Fourth, a representation-dimension framework is developed: on finite domains the minimum number of coordinates needed to faithfully represent a collection of strict orders equals the minimum size of an acyclic partition of those orders, and in the ordinal product-grid model with local concentration severity comparisons the pair of lower orthant dominance and within-poor improvement has exact representation dimension 2, with explicit scalar representatives given for each coordinate. Fifth, a general endpoint-domination theorem is proved for local concentration on the interior of the probability simplex over L severity levels for every L at least 2: every interior severity profile is strictly dominated by an endpoint-peaked profile, and therefore every surjective L-level severity map induces a downward-cofinal induced within-poor strict relation. Sixth, the fixed-incidence problem is analyzed: a counterexample shows that fixed-incidence universality can fail even under downward cofinality of the within-poor relation, a sufficient theorem identifies non-poor coverage and a feasibility-richness condition as jointly sufficient for fixed-incidence cycles, and the benchmark grid is solved directly showing that every interior baseline lies on a fixed-incidence mixed cycle despite the single non-poor state violating non-poor coverage.