I am looking for information on "Pouzet's lemma," which I learned about in slides of Chris Hartman on Sperner's lemma. I have copied the statement of the result from that site. I would love any published references, and in particular a more direct proof (since the proof on that site goes Sperner's lemma $\Rightarrow$ Connector theorem $\Rightarrow$ game of Hex $\Rightarrow$ Pouzet).
Def: In $d$-space, call the unit vectors along the axes $\{u_1, u_2,\dotsc, u_d\}$. A box of the integer lattice points is the set of points $\vec{x} = (x_1, x_2, \dotsc , x_d)$ satisfying $a_i \le x_i \le b_i$, $1 \le i \le d$. We call two points neighbors if their coordinates differ by no more than one in any position.
Pouzet’s Lemma: For any mapping $f$ from the lattice points of a $d$-dimensional box $A$ to the set of unit vectors $\{\pm u_1, \pm u_2,\dotsc, \pm u_d\}$ such that $\vec{x} + f(\vec{x})$ is in $A$ for all $\vec{x}$ in $A$, there exist two neighbors $\vec{x}$ and $\vec{y}$ such that $f(\vec{x}) = −f(\vec{y})$.
