Field Codes for Distributed Coupling Samplers and Certified Empirical Transport

In this paper, we formulate three communication tasks for empirical optimal transport: distributed coupling sampling, cost-evaluable coupling output, and scalar value-certified sampling. Our main result is a field-code compiler: any communicated transport field approximating an optimal empirical Monge map to error η can be completed by sparse target-cell residuals into an exact-marginal value-certified sampler with scalar certificate W_1(μ,ν)\leq U\leq W_1(μ,ν)+2Δ, where Δ is the public target-partition diameter. The certificate accuracy is controlled by Δ alone. The field error η controls residual communication under a cell-margin condition; without a margin, η alone does not bound residuals. We instantiate the compiler via adaptive local-affine and tensor-product spline codes with d(m+1)^db field bits in the spline case, plus residual lists charged separately. For lower bounds, exact Gap-Hamming embeddings prove certified output is hard, including a smooth cell-packing diffeomorphism family requiring Ω(\varepsilon^{-2d/(d+4)}) communication for any cost-evaluable, cost-certified, or value-certified protocol. The same gadgets admit zero-communication samplers, formally separating the sampler and certificate-bearing output models. These results identify the transport field as the right communicated object whenever a field code is available, primarily as a residual-sparsity tool.