Treewidth of the $n\times n$ toroidal grid

In this paper, we show that the treewidth of the $n\times n$ toroidal grid is $2n-1$ for all $n\ge 5$. This closes the gap between the previously known upper bound of $2n-1$ (Ellis and Warren, DAM 2008) and the lower bound of $2n-2$ (Kiyomi, Okamoto, and Otachi, DAM 2016). To establish the matching lower bound, we construct a bramble of maximum order by utilizing maximum components obtained after removing $2n-1$ vertices. Our construction relies on the vertex-isoperimetric properties of the infinite grid to establish tight lower bounds on neighborhood sizes, combined with a careful analysis of balls of radius $n\mathrm{/}2-1$ and their boundaries to overcome structural obstructions when $n$ is even.