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

In a recent breakthrough, Chen, Hirahara and Ren prove that 𝖲_2𝖤/_1 ⊄𝖲𝖨𝖹𝖤[2^n/n] by giving a single-valued 𝖥𝖲_2𝖯 algorithm for the Range Avoidance Problem (𝖠𝗏𝗈𝗂𝖽) that works for infinitely many input size n. Building on their work, we present a simple single-valued 𝖥𝖲_2𝖯 algorithm for 𝖠𝗏𝗈𝗂𝖽 that works for all input size n. As a result, we obtain the circuit lower bound 𝖲_2𝖤⊄i.o.-𝖲𝖨𝖹𝖤[2^n/n] and many other corollaries: 1. Almost-everywhere near-maximum circuit lower bound for Σ_2𝖤∩Π_2𝖤 and 𝖹𝖯𝖤^𝖭𝖯. 2. Pseudodeterministic 𝖥𝖹𝖯𝖯^𝖭𝖯 constructions for: Ramsey graphs, rigid matrices, pseudorandom generators, two-source extractors, linear codes, hard truth tables, and K^poly-random strings.