Sharp cardinality for unique-Bayes full-support counterexamples

Determine the sharp minimum output-space cardinality required for a full-support counterexample to argmax Fisher consistency of the structured max-margin loss among metrics satisfying the common-geodesic condition, under the additional requirements that the Bayes output be unique and the argmax set be disjoint from it.

Background

The paper establishes that four outputs suffice for an unrestricted counterexample, using a unit star and a boundary distribution, while five outputs are necessary and sufficient when the counterexample distribution must have full support. The five-point complete-bipartite example has multiple Bayes outputs, whereas the eight-output Hamming-cube example has a unique Bayes output and an argmax set disjoint from that Bayes output.

The unresolved issue is whether a smaller output space can realize a full-support inconsistency witness satisfying both additional properties simultaneously: a unique target Bayes output and complete separation between the surrogate score maximizers and the Bayes set. Resolving this would refine the paper’s sharp cardinality results.