We prove a sensitivity-to-communication lifting theorem for arbitrary gadgets. Given functions $f: \{0,1\}^n\to \{0,1\}$ and $g : \mathcal{X} \times \mathcal{Y}\to \{0,1\}$, denote $f\circ g(x,y) := f(g(x_1,y_1),\ldots,g(x_n,y_n))$. We show that for any $f$ with sensitivity $s$ and any $g$,
\[D(f\circ g) \geq s\cdot \bigg(\frac{\Omega(D(g))}{\log rk(g)} - \log rk(g)\bigg),\]
where ... more >>>