The Exact Comass Criterion for the Tsai--Wang Form

For a map $F:Ω\subset\mathbb{R}^n\to\mathbb{R}^m$, let $Θ(F)$ be the $n$-form on $Ω\times\mathbb{R}^m$ introduced by Tsai and Wang [arXiv:2604.04336]. If $λ_1,\ldots,λ_{r}$ are the nonzero singular values of $d F$ at any fixed point, we prove that $$ Θ(F) \text{ has comass one if and only if }~