Circuit orderability of cographic matroids of complete and complete bipartite graphs

We classify the circuit-orderable cographic matroids associated with two complete graph families. For every integer n >= 4, M*(K_n) is orderable if and only if n = 4 or n = 6. For all positive integers r,s, M*(Kr,s) is orderable if and only if min(r,s) <= 2. The main argument reconstructs a closed simplicial surface from an arbitrary consistent ordering, then uses a mod-two face equation and a link-degree obstruction. The positive K6 case is from an earlier projective-plane construction. The complete bipartite classification combines known results with a three-neighbor obstruction for K3,s. These classifications concern the two specified graph families, not all cographic matroids. Status: Internally checked preprint; external mathematical review and formal peer review are pending. AI-assisted research and writing: OpenAI Codex assisted with research, verification, and manuscript preparation. Carptopus is the sole author and assumes responsibility for the mathematical claims and final text.