{
  "abstract": "We give a self-contained proof, in the bounded variation setting, of the Hambly--Lyons uniqueness theorem, which states that (total) signature identifies the path up to tree-like equivalences. The argument is organized around two key geometric observations. First, tree-like paths have trivial signature because factorization over a loop in a tree $τ:[0,1]\\to T$ is preserved under signature lifts, which follows from an elementary property of planar curves. Second, a path with trivial total signature contains a nontrivial subpath with trivial total signature (the sub-interval lemma). This is proven by applying a winding-number argument to a two-dimensional projection of the signature lift. Collapsing all trivial-signature sub-intervals then defines a compact metric tree $T$ through which the original path factors by virtue of the sub-interval lemma.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12283",
  "author_total": 3,
  "authors": [
    {
      "name": "Walter Schachermayer"
    },
    {
      "name": "Josef Teichmann"
    },
    {
      "name": "Valentin Tissot-Daguette"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5xwajv24w4x2b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12283",
  "fulltext_url": null,
  "id": "5xwajv24w4x2b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12283",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7212586024",
  "pdf_url": "https://arxiv.org/pdf/2609.12283",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5xwajv24w4x2b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.12283",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12283"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "An Elementary Proof of the Hambly-Lyons Uniqueness Theorem",
  "title_text": "An Elementary Proof of the Hambly-Lyons Uniqueness Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/5xwajv24w4x2b",
  "venue": "arXiv",
  "versions": []
}