Counterexample to majority optimality in NICD with erasures

TL;DR

This paper presents a counterexample to the conjecture that majority functions are optimal in NICD with erasures, discovered using GPT-5 Pro and verified manually, highlighting AI's role in theoretical computer science.

Contribution

It provides the first explicit counterexample for NICD with erasures and demonstrates AI-assisted discovery in theoretical research.

Findings

Counterexample function surpasses majority at p=0.40

Majority is optimal near p=0 for odd n

Manual verification confirms the computational result

Abstract

We asked GPT-5 Pro to look for counterexamples among a public list of open problems (the Simons ``Real Analysis in Computer Science'' collection). After several numerical experiments, it suggested a counterexample for the Non-Interactive Correlation Distillation (NICD) with erasures question: namely, a Boolean function on 5 bits that achieves a strictly larger value of $\mathbb{E}|f(z)|$ than the 5-bit majority function when the erasure parameter is $p=0.40.$ In this very short note we record the finding, state the problem precisely, give the explicit function, and verify the computation step by step by hand so that it can be checked without a computer. In addition, we show that for each fixed odd $n$ the majority is optimal (among unbiased Boolean functions) in a neighborhood of $p=0$. We view this as a little spark of an AI contribution in Theoretical Computer Science: while modern Large Language Models (LLMs) often assist with literature and numerics, here a concrete finite counterexample emerged.