# Beyond Fixed Points: Superpolynomial Capacity of Asymmetric Hopfield Networks

arXiv: https://arxiv.org/abs/2605.24611 · HTML: https://mlsanity.com/paper/2605.24611/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 14 references that are in the map.
- First posted on arXiv 23 May 2026.

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

- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [Long Sequence Hopfield Memory](https://mlsanity.com/paper/2306.04532/) (2023)
- [Stabilizing Consensus with Many Opinions](https://mlsanity.com/paper/1508.06782/) (2015)
- [On the Hierarchy of Distributed Majority Protocols](https://mlsanity.com/paper/2205.08203/) (2022)

## Abstract (arXiv)

Classical Hopfield networks are limited to static patterns due to symmetric weights, whereas asymmetric networks can encode temporal sequences via limit-cycle attractors. Achieving high-capacity storage of long sequences in classical synchronous asymmetric networks, however, has remained a challenge. We present a simple and robust construction within the classical asymmetric Hopfield model with binary neurons and synchronous updates, that allows n neurons to support exp!big(Omega(n/(log n)2)big) distinct limit-cycle attractors, each with period exp!big(Omega(√ n/log n)big) and robust to random noise with flip probability up to frac12-o(1), yielding superpolynomial capacity in both the number and length of stored sequences. This is the first demonstration of such capacity for asymmetric Hopfield networks, which we obtain by combining results from combinatorics, number theory and the analysis of opinion dynamics. Our findings show that synchronous asymmetric Hopfield networks possess a sequence-memory capacity which is larger and more robust than previously recognized, demonstrating that, in both biological and artificial neural systems, robust sequence representation can be achieved through coarse architectural motifs rather than complex nonlinearities.