$\partial\mathbb{B}$ nets are differentiable nets that are hard-equivalent to non-differentiable, boolean-valued functions. $\partial\mathbb{B}$ nets can therefore learn discrete functions by gradient descent. Ensuring hard-equivalence requires defining new kinds of activation functions and network layers. `Margin packing' is a potentially general technique for constructing differentiable functions that are hard-equivalent yet gradient-rich. An advantage of $\partial\mathbb{B}$ nets is that `hardening' to 1-bit weights has provably identical accuracy. At inference time $\partial\mathbb{B}$ nets are highly compact and potentially cheap to evaluate. Preliminary experiments demonstrate that $\partial\mathbb{B}$ nets are competitive with existing classification approaches.