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 $\eta$ can be completed by sparse target-cell residuals into an exact-marginal value-certified sampler with scalar certificate $W_1(\mu,\nu)\leq U\leq W_1(\mu,\nu)+2\Delta$, where $\Delta$ is the public target-partition diameter. The certificate accuracy is controlled by $\Delta$ alone. The field error $\eta$ controls residual communication under a cell-margin condition; without a margin, $\eta$ 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 $\Omega(\varepsilon^{-2d/(d+4)})$ 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.