{
  "abstract": "At a junction, a score field can reveal weighted tangent rays, yet these first-order quantities do not determine how individual branches bend or how their densities change away from the center. Recovering this missing information is necessary for describing local continuation beyond a single point, but finite observations must separate branchwise second-order effects while allowing error in the estimated center. We address this inverse problem using matched score queries at noise scales $σ$ and $λσ$. For a finite union of $C^{2,α}$ half-branches in $\\mathbb{R}^D$, the normalized score has the expansion $F_σ=F_0+σG+O(σ^{1+α})$. Matched subtraction cancels the tangent contribution and exposes $G$, which depends linearly on branchwise curvature and log-density slope. Given tangent directions and weights on distinct rays, $G$ uniquely identifies all $sD$ branch parameters, and $sD$ scalar component observations are necessary. An $O(σ^2)$ center error introduces $D$ translation modes, leading to $(s+1)D$ observations under full-rank calibration, except for a translation-invariant full line. We also establish a perturbation bound and a conditional kernel-density-estimation rate. Experiments reproduce the predicted population and $N^{-1/5}$ trends and remain full rank up to $D=20$ with 16 supplied branches. In end-to-end tests for $D=3$--$5$, a known-count first-order frontend yields full rank in all 135 population systems and a median relative jet error of 0.132. With strong first-order error, matched responses reduce median parameter error by a factor of 49.4 relative to naive tangent subtraction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01319",
  "author_total": 2,
  "authors": [
    {
      "name": "Ziqi Zhao"
    },
    {
      "name": "Qingjian Ni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2c2xswysxdybj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01319",
  "fulltext_url": null,
  "id": "2c2xswysxdybj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01319",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7206217423",
  "pdf_url": "https://arxiv.org/pdf/2609.01319",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2c2xswysxdybj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:10Z",
      "native_id": "2609.01319",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01319"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Matched Queries for Curvature and Density at Branching Junctions",
  "title_text": "Matched Queries for Curvature and Density at Branching Junctions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:10Z",
  "url": "https://research.draconitic.com/paper/2c2xswysxdybj",
  "venue": "arXiv",
  "versions": []
}