Inversion of adjunction for $F$-signature

Let $(R,\mathrm{\Delta }+D)$ be a log $\mathbb{Q}$-Gorenstein pair where $R$ is a Noetherian, $F$-finite, normal, local domain of characteristic $p>0$, $\mathrm{\Delta }$ is an effective $\mathbb{Q}$-divisor and $D$ is an integral $\mathbb{Q}$-Cartier divisor. We show that the left derivative of the $F$-signature function $s(R,\mathrm{\Delta }+tD)$ at $t=1$ is equal to $-s({\mathcal{O}}_D,{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_D(\mathrm{\Delta }))$. This equality is interpreted as a quantitative form of inversion of adjunction for strong $F$-regularity. As an immediate corollary, we obtain the inequality $s(R,\Delta) \geq s($\mathcal{O}$_D, $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}$_D(\Delta))$. We also discuss the implications of our result for the conjectured connection between the $F$-signature and the normalized volume.