# When the noncommutative AM-GM inequality holds

arXiv: https://arxiv.org/abs/2610.04874 · HTML: https://mlsanity.com/paper/2610.04874/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 12 references that are in the map.
- First posted on arXiv 4 Oct 2026.

## Builds on (most cited first, 2 of 2)

- [Recht-R'e Noncommutative Arithmetic-Geometric Mean Conjecture is False](https://mlsanity.com/paper/2006.01510/) (2020)
- [A Resolution of the SS--RS--GD Inequalities](https://mlsanity.com/paper/2607.22620/) (2026)

## Abstract (arXiv)

In this note, we prove that the noncommutative AM-GM inequality holds if n≥ 2lceil m/2 rceil2. The motivation comes from counterexamples constructed in [De Sa, Random reshuffling is not always better, NeurIPS2020]. The proof constructs a vertex measure based on the Chebyshev nodes on the Boolean cube to extract the distinct indices. The main difficulty is that the measure is not positive on non-integer nodes. The key technique comes from [Grigoriev, Complexity of Positivstellensatz proofs for the knapsack, Computational Complexity (2001)] and eventually transforms the problem into a quadrature estimate.