{
  "abstract": "Long contexts are central to modern transformer systems, but most expressivity results choose a different network for each fixed sequence length. We study whether one masked transformer can approximate causal token-to-token maps uniformly over sequences of arbitrary length sampling a fixed normalized horizon. To relate sampling resolutions, we model tokens by $α$-Hölder sequences or, more generally, a common modulus of continuity. Our notion of continuity across resolutions characterizes the causal families admitting uniform approximation on these compact input classes by a single transformer with length-independent parameters. The result extends to the infinite-length mean-field limit, where tokens form continuous curves and masked attention becomes a causal time integral. For bounded regression with target maps satisfying a $β$-smooth stability condition defined using regular test functions, quantitative approximation yields a generalization bound: exact empirical risk minimization over suitably sized bounded-weight transformers gives root mean-square prediction error $O((\\log\\log N/\\log N)^{β/(d+2)})$ from $N$ iid labeled sequences. The bound holds at fixed confidence on the same sampling distribution, with $d$ the token dimension and no maximum-length factor. Finally, experiments on physical time series support the Hölder-regular token model at observed scales, with dataset-dependent fitted exponents, whereas text input embeddings provide a contrasting case. Native and dense sampling, shuffled controls, and refinement checks delimit this empirical regularity regime.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35055",
  "author_total": 3,
  "authors": [
    {
      "name": "Takashi Furuya"
    },
    {
      "name": "Maarten V. de Hoop"
    },
    {
      "name": "Gabriel Peyré"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/311235c08zj46.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35055",
  "fulltext_url": null,
  "id": "311235c08zj46",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35055",
  "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",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7214806507",
  "pdf_url": "https://arxiv.org/pdf/2609.35055",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/311235c08zj46.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35055",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35055"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Universality and Generalization of Causal Transformers Across Context Lengths",
  "title_text": "Universality and Generalization of Causal Transformers Across Context Lengths",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/311235c08zj46",
  "venue": "arXiv",
  "versions": []
}