The Support of an Optimal Coupling on the Torus is Torus-Cyclically Monotone

In the setting of The Flat Torus: Standing Notation, used with $n=d$ for a natural number $d$ with $1\le d$, and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $\mathcal{P}(\mathbb{T}^{d})$ be as in Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures, let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$, and let $\gamma$ be an optimal coupling of $\mu$ and $\nu$. By Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures the measure $\gamma$ is a Borel measure on the metric space $(\mathbb{R}^{d+d},d_{E})$, so its support $\operatorname{supp}\gamma$ is defined.

1. (Cyclical monotonicity of the support)¶ The set $\operatorname{supp}\gamma$ is torus-cyclically monotone.