Lemma: Coordinate envelope of the cell

For every finite $u,h$,

In particular, $F_{\theta }(u,h)\in [-1,1{]}^H$ whenever $h\in [-1,1{]}^H$, independently of input magnitude.

Proof. Each update-gate coordinate lies in $(0,1)$, and each candidate coordinate lies in $(-1,1)$. Equation (eq:cell) is a coordinatewise convex combination of $n_i$ and $h_i$.