# A Hand-Checkable Proof That Two Hidden ReLU Layers Compute the Maximum of Six Numbers

arXiv: https://arxiv.org/abs/2610.04256 · HTML: https://mlsanity.com/paper/2610.04256/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 5 references that are in the map.
- First posted on arXiv 3 Oct 2026.

## Builds on (most cited first, 5 of 5)

- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Towards Lower Bounds on the Depth of ReLU Neural Networks](https://mlsanity.com/paper/2105.14835/) (2021)
- [Better Neural Network Expressivity: Subdividing the Simplex](https://mlsanity.com/paper/2505.14338/) (2025)
- [Shallower ReLU Network Representations via Exact Linear Algebra](https://mlsanity.com/paper/2607.21651/) (2026)
- [Representing MAX functions using two-hidden-layer ReLU networks](https://mlsanity.com/paper/2608.25221/) (2026)

## Abstract (arXiv)

Exactly computing the maximum function is a standard test case for studying depth in ReLU networks. Two hidden layers are known to suffice for up to twelve inputs through computer-assisted constructions. For six real inputs, we give an explicit hexagon identity whose local structure yields a self-contained analytical proof of this depth bound. The identity was found by computer-assisted search; we prove it through explicit cancellations that can be checked entirely by hand, without executing a verification program. The identity also yields an explicit network with hidden widths 17 and 41, zero biases, and rational weights.