{
  "abstract": "We study row-softmax self-attention with independent Gaussian query and key weights in the proportional regime, at fixed inverse temperature. Hayase, Collins, and Karakida proved Gaussian equivalence for the empirical squared singular-value distribution after removal of the Perron direction. A global law alone does not exclude finitely many nonleading outliers. We prove that no such outliers persist: the rescaled squared singular value $\\ell s_k(A)^2$ converges in probability to the upper edge of their bulk law for every fixed $k\\ge 2$. In fact, this convergence is uniform over any deterministic sublinear number of leading non-Perron indices. The proof uses an exact decomposition of the softmax normalization, conditions on the key matrix, identifies the conditional covariance exactly with a diagonally conjugated inner-product kernel, linearizes that kernel in operator norm, and applies the outside-support local law of Fan, Ma, Paquette, and Wang. A separate stability argument identifies the finite conditional deformed Marchenko-Pastur edge with the limiting bulk edge. We also derive a scalar formula for that edge throughout the proportional regime and recover the explicit square-model formula of Hayase, Collins, and Karakida, including its physical branch.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23949",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexander Jerschow"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5pz5aenxbjnk5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23949",
  "fulltext_url": null,
  "id": "5pz5aenxbjnk5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23949",
  "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",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7214018826",
  "pdf_url": "https://arxiv.org/pdf/2609.23949",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5pz5aenxbjnk5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.23949",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23949"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Bulk-edge sticking beyond the Perron mode in Gaussian softmax attention",
  "title_text": "Bulk-edge sticking beyond the Perron mode in Gaussian softmax attention",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/5pz5aenxbjnk5",
  "venue": "arXiv",
  "versions": []
}