The article is complete at 2,532 words. Here's what it covers:
- Opens with the A/B testing motivation and why the baseline H log(1/delta) is insufficient
- Explains gap entropy through the geometric shell partitioning and the optimization argument that produces the entropy term
- States both main theorems: instance-wise tight bounds and the uniform algorithm with additive overhead
- Details the lower bound proof: symmetrization, change-of-measure with tied-optimal reference distributions, the counting argument forcing spread across scales, and Gibbs inequality
- Explains the instance-wise upper bound: mixture test statistics across gap scales, trial repetition with decreasing error levels, and the auxiliary confidence-interval procedure for almost-sure stopping
- Describes the uniform algorithm: geometric budget stages, the four subroutines (median estimation, mean estimation, fraction testing, elimination), and the comparison-process error analysis
- Covers error control: hard vs easy arm splitting, the product guarantee, and the exponential-moment bound
- Compares to Track-and-Stop, lil'UCB, Exponential-Gap Elimination, and prior Chen-Li results
- Notes the LLM usage in paper production as a point of interest
All section headings are paper-specific. The article uses `IMPORTANT: no` since this is a theoretical result significant within the bandit theory community but not a widely adopted technique or major capability jump.