We prove that the bounded arithmetic theory $S^1_2$ is consistent with EXP $\not\subseteq$ P/poly. More generally, we show that certain separations of $V^1_2$ from a theory $T$ imply the consistency of $T$ with EXP $\not\subseteq$ P/poly. For $T=S^1_2$, Takeuti (1988) established such a separation using a variant of Gödel's consistency statement. Analogous results hold for PSPACE $\not\subseteq$ P/poly but the required separations of theories are yet unknown. Finally, we give magnification results for the hardness of proving almost-everywhere versions of these lower bounds.

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