Symmetric Exponential Time Requires Near-Maximum Circuit Size: Simplified, Truly Uniform

In a recent breakthrough, Chen, Hirahara and Ren prove that ${\mathsf{S}}_{\mathsf{2}}\mathsf{E}{\mathrm{/}}_1\subset ̸\mathsf{S}\mathsf{I}\mathsf{Z}\mathsf{E}[2^n\mathrm{/}n]$ by giving a single-valued $\mathsf{F}{\mathsf{S}}_{\mathsf{2}}\mathsf{P}$ algorithm for the Range Avoidance Problem ($\mathsf{A}\mathsf{v}\mathsf{o}\mathsf{i}\mathsf{d}$) that works for infinitely many input size $n$. Building on their work, we present a simple single-valued $\mathsf{F}{\mathsf{S}}_{\mathsf{2}}\mathsf{P}$ algorithm for $\mathsf{A}\mathsf{v}\mathsf{o}\mathsf{i}\mathsf{d}$ that works for all input size $n$. As a result, we obtain the circuit lower bound ${\mathsf{S}}_{\mathsf{2}}\mathsf{E}\subset ̸i\mathrm{.}o\mathrm{.}$-$\mathsf{S}\mathsf{I}\mathsf{Z}\mathsf{E}[2^n\mathrm{/}n]$ and many other corollaries: 1. Almost-everywhere near-maximum circuit lower bound for ${\mathsf{\Sigma }}_{\mathsf{2}}\mathsf{E}\cap {\mathsf{\Pi }}_{\mathsf{2}}\mathsf{E}$ and ${\mathsf{Z}\mathsf{P}\mathsf{E}}^{\mathsf{N}\mathsf{P}}$. 2. Pseudodeterministic ${\mathsf{F}\mathsf{Z}\mathsf{P}\mathsf{P}}^{\mathsf{N}\mathsf{P}}$ constructions for: Ramsey graphs, rigid matrices, pseudorandom generators, two-source extractors, linear codes, hard truth tables, and $K^{poly}$-random strings.