{
  "abstract": "Fusing independently normalized attention heads can increase the number of explicit key-value entries, or atoms, needed to preserve their output. We prove a sharp separation for $W_d(q)=\\sum_{j=1}^dσ(q_j)$: its $d$ two-atom heads use $2d$ head-atom terms, whereas its minimum single-head representation has exactly $2^d$ atoms, even with arbitrary fixed replacement keys, scalar values, and positive masses. The same lower bound holds for every fixed uniform error below $1/2$ on $\\mathbb R^d$, and the threshold is sharp. The proof uses exposed-key normal cones; exact equality on any nonempty open query domain inherits the bound. The resulting atom bound for a fixed head count is tight when that count divides $d$. A normalized family retains its exponential exact minimum while approaching a constant on fixed-radius query balls. For pretrained-model caches, a classical affine-rank bound and a certified instance exclude exact vector preservation at a specified atom count. Learned-cache comparisons on two disjoint banks of 64 contexts show budget-dependent reversals and calibration effects.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02943",
  "author_total": 4,
  "authors": [
    {
      "name": "Hiroyasu Usami"
    },
    {
      "name": "Keisuke Hara"
    },
    {
      "name": "Ayato Tsuboi"
    },
    {
      "name": "Naohiko Matsuda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e7t2sc6ykbpmh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02943",
  "fulltext_url": null,
  "id": "e7t2sc6ykbpmh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02943",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NE"
    ],
    [
      "arxiv",
      "stat.ML"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02943",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e7t2sc6ykbpmh.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:21Z",
      "native_id": "2611.02943",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02943"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Exact Fusion of Softmax Attention Heads Can Require Exponentially Many Atoms",
  "title_text": "Exact Fusion of Softmax Attention Heads Can Require Exponentially Many Atoms",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/e7t2sc6ykbpmh",
  "venue": "arXiv",
  "versions": []
}