Various gapped maximization problems are known not to be $\mathbf{NP}$-hard under $\mathbf{AC}^0$ reductions, e.g., $\mathsf{Gap}_{1,\epsilon}\text{-}\mathsf{Max3SAT}$ (see, e.g., Proposition 4 of Allender-Ilango-Vafa'21; the analogous result for $\mathsf{MaxClique}$ is attributed to earlier unpublished work by Arora).

The usual argument for showing that, say, $\mathsf{Max\text{-}3SAT}$ requires large $\mathbf{AC}^0$ circuits is to $\mathbf{AC}^0$-reduce from $\mathsf{Parity}$, but for these gapped problems, the $\mathbf{AC}^0$ reduction from $\mathsf{Parity}$ is precisely what fails. My question is, can $\mathbf{AC}^0$ lower bounds for these problems (e.g. for $\mathsf{Gap}_{1,\epsilon}\text{-}\mathsf{Max3SAT}$, to make it concrete) be recovered in some other way?



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