arXiv:2609.18992v2 Announce Type: replace-cross Abstract: Witsenhausen's counterexample is a canonical problem in decentralized control: a two-stage linear-quadratic-Gaussian team problem in which nonlinear policies strictly outperform every affine one. This paper develops a variational and computational framework for the scalar counterexample based on optimal transport. Building on the transport formulation of Wu and Verd\'u, we absorb the first controller into a transport map and recast the problem as $J^\star=\inf_Q\{k^2W_2^2(P,Q)+\mathrm{mmse}(Q)\}$, which trades a quadratic Wasserstein transport cost against a minimum mean-square estimation error. The optimal first controller is the monotone rearrangement that pushes the prior $P$ to a minimizer $Q^\star$. We establish the existence and absolute continuity of minimizers, derive an Euler--Lagrange optimality condition, and obtain a semi-closed-form Gaussian benchmark with an explicit second-variation stability threshold. We then show that restricting to finitely supported laws yields an MMSE-regularized optimal quantizer: its optimal values decrease to $J^\star$, its weak limit points are minimizers of the continuous problem, and its stationarity conditions pair centroid levels with power cells, recovering Lloyd--Max quantization as control becomes expensive. This structure leads to an estimation-aware Lloyd algorithm with signalling-based initialization. As $k\to\infty$, the optimal controllers converge to the classical affine pair; as $k\to0$, $J^\star(k)=\sigma^2k^2+o(k^2)$ and the optimal laws collapse onto $\delta_0$, so signalling constellations are a moderate-$k$ phenomenon. Numerical experiments corroborate the theory, and the framework extends to the vector counterexample, where the finite-level program becomes an MMSE-regularized vector quantizer.