# Boolean threshold functions, neuron capacity, and memory retrieval

arXiv: https://arxiv.org/abs/2609.29756 · HTML: https://mlsanity.com/paper/2609.29756/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 36 references that are in the map.
- First posted on arXiv 24 Sep 2026.

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

- [Hopfield Networks as Models of Emergent Function in Biology](https://mlsanity.com/paper/2506.13076/) (2025)
- [Chamber geometry and specification numbers of Boolean threshold functions](https://mlsanity.com/paper/2606.29477/) (2026)

## Abstract (arXiv)

How much information can a single neuron remember? How many memories can neural networks retrieve without creating false memories? These questions are related to a basic question: how many Boolean threshold functions f(x)=operatornamesgn(a0+langle a,xrangle), x∈-1,1n, are there? In this paper, we show that the number Tn of distinct Boolean threshold functions is [ Tn=2binom2n-1nbigl(1+O(n-99)bigr). ] Equivalently, the capacity of a single threshold neuron is n2-log2(n!)+1+O(n-99) bits, improving the O(n) error term in the result of Kahn--Koml'os--Szemer'edi to O(n-99). To prove this, we show that, for 1≤ r≤ n-1, and v1,ldots,vr are chosen at random from -1,1n, [ mathbb P!left langle v1,ldots,vrranglecap-1,1n =± v1,ldots,± vr right =1-O(n-99). ] In the context of the Kanter--Sompolinsky Hamiltonian for memory retrieval, this identifies r=n-1 as a sharp threshold, at which, for almost every collection of r memories, the only ground states are these memories and their negatives, confirming a weaker form of the Kalai--Linial--Odlyzko conjecture. It also settles a recent open problem posed by M. Anthony on the specification number of Boolean threshold functions. In addition, we show that, for every 1≤ r≤ n-1, [ mathbb Pv1,ldots,vr are linearly dependent =2binom r2,2-n+O!left(2-ne-cnright), ] confirming a conjecture of Kahn--Koml'os--Szemer'edi.