On the Unprovability of Circuit Size Bounds in Intuitionistic ${\mathsf{S}}_2^1$

We show that there is a constant $k$ such that Buss’s intuitionistic theory ${\mathsf{I}\mathsf{S}}_2^1$ does not prove that SAT requires co-nondeterministic circuits of size at least $n^k$. To our knowledge, this is the first unconditional unprovability result in bounded arithmetic in the context of worst-case fixed-polynomial size circuit lower bounds. We complement this result by showing that the upper bound $\mathsf{N}\mathsf{P}\subseteq \mathsf{c}\mathsf{o}\mathsf{N}\mathsf{S}\mathsf{I}\mathsf{Z}\mathsf{E}[n^k]$ is unprovable in ${\mathsf{I}\mathsf{S}}_2^1$. In order to establish our main result, we obtain new unconditional lower bounds against refuters that might be of independent interest. In particular, we show that there is no efficient refuter for the lower bound $\mathsf{N}\mathsf{P}⊈\mathsf{i}\mathsf{.}\mathsf{o}\mathsf{.}\text{-}\mathsf{c}\mathsf{o}\mathsf{N}\mathsf{P}\mathrm{/}\mathsf{p}\mathsf{o}\mathsf{l}\mathsf{y}$, addressing in part a question raised by Atserias (2006).