{
  "abstract": "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 $f(x)=\\operatorname{sgn}(a_0+\\langle a,x\\rangle)$, $x\\in\\{-1,1\\}^n$, are there? In this paper, we show that the number $T_n$ of distinct Boolean threshold functions is \\[ T_n=2\\binom{2^n-1}{n}\\bigl(1+O(n^{-99})\\bigr). \\] Equivalently, the capacity of a single threshold neuron is $n^2-\\log_2(n!)+1+O(n^{-99})$ bits, improving the $O(n)$ error term in the result of Kahn--Komlós--Szemerédi to $O(n^{-99})$. To prove this, we show that, for $1\\le r\\le n-1$, and $v_1,\\ldots,v_r$ are chosen at random from $\\{-1,1\\}^n$, \\[ \\mathbb P\\!\\left\\{ \\langle v_1,\\ldots,v_r\\rangle\\cap\\{-1,1\\}^n =\\{\\pm v_1,\\ldots,\\pm v_r\\} \\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 $1\\le r\\le n-1$, \\[ \\mathbb P\\{v_1,\\ldots,v_r\\text{ are linearly dependent}\\} =2\\binom r2\\,2^{-n}+O\\!\\left(2^{-n}e^{-cn}\\right), \\] confirming a conjecture of Kahn--Komlós--Szemerédi.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29756",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinyuan Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e9hc7cdsgtfhv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29756",
  "fulltext_url": null,
  "id": "e9hc7cdsgtfhv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29756",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "1702|Neural Networks and Applications"
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  "openalex": "W7214358360",
  "pdf_url": "https://arxiv.org/pdf/2609.29756",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e9hc7cdsgtfhv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29756",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29756"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Boolean threshold functions, neuron capacity, and memory retrieval",
  "title_text": "Boolean threshold functions, neuron capacity, and memory retrieval",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:21Z",
  "url": "https://research.draconitic.com/paper/e9hc7cdsgtfhv",
  "venue": "arXiv",
  "versions": []
}