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Cone-Compatible Monge Geometry for High-Dimensional Ordered Optimal Transport

High-dimensional optimal transport is seldom available in closed form. The one-dimensional case is exceptional because the order of the real line is compatible with convex transport costs, making monotone rearrangement optimal. This paper studies when an analogous Monge structure can be recovered in higher dimensions from a partial order. We introduce a cone-compatible Monge geometry: a closed convex cone (K) induces the order (x\preceq_K y) whenever (y-x\in K), and is compatible with a cost if ordered pairs satisfy a Monge exchange inequality. For squared Mahalanobis costs (c_M(x,y)=(x-y)^\top M(x-y)), we prove a sharp characterization: compatibility holds exactly when (K) is acute under the (M)-inner product, namely (u^\top Mv\ge0) for all (u,v\in K), equivalently (K\subseteq K_M^*). Under this condition, measures supported on cone chains admit a quantile-type closed-form optimal coupling, yielding exact transport under the original ground cost rather than after projection or metric replacement. We distinguish the resulting cone-chain Wasserstein metric on canonically ordered chain distributions from an extended directed cone transport cost on general measures, and develop feasibility, duality, stability, approximation, Gaussian recovery, statistical, and computational results. The theory is complementary to sliced and tree Wasserstein distances: it is not a universal fast surrogate, but a way to obtain interpretable, direction-valid, original-space monotone transport for ordered high-dimensional data.

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