{
  "abstract": "Determining the momentum missing on a Poincaré section of a two-degree-of-freedom Hamiltonian is usually posed as the root-finding problem $H(0,q_2,p_1,p_2)=E$, which degenerates at the boundary of the energetically allowed region. We argue that the degeneration is the visible symptom of a geometric object and reorganize the problem around it. The section is a coisotropic hypersurface, the missing momentum is its characteristic direction, and recovering it is a \\emph{reconstruction} through the associated coisotropic reduction. Its carrier is not the fibrewise least-energy function, which we show is informationally incomplete, but the correspondence $Σ_E^{\\mathrm{sec}}=Σ_E\\cap C$ together with the reducing projection $π_E$, a branched reconstruction over the admissible region whose discriminant is identified with the Hill boundary under a base-side regularity condition. We prove, isolating for each conclusion its minimal hypothesis: that $π_E$ is proper and a covering away from the discriminant (coercivity); that the fibres have exactly two points over the interior (unimodality); that along the discriminant $π_E$ is a fold (fibrewise nondegeneracy and transversality); and that the discriminant coincides with the topological boundary of the admissible region precisely under a regularity of the least-energy function \\emph{on the base}, which no fibrewise hypothesis supplies. Strict convexity of $H$ in the momenta is shown to be a sufficient but non-fundamental bundle of these conditions. The boundary is a discriminant, not a Lagrangian caustic. Only afterwards does the squared fibrewise energy residual $F_b=(h_b-E)^2$ appear, as the analytic resolution of the fold; inside the admissible region it is a double well with two global minimizers, so branch selection is a separate rule.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16715",
  "author_total": 1,
  "authors": [
    {
      "name": "E. Chan-López"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/96zetjf0bgfg1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16715",
  "fulltext_url": null,
  "id": "96zetjf0bgfg1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16715",
  "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.DG"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7213399406",
  "pdf_url": "https://arxiv.org/pdf/2609.16715",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/96zetjf0bgfg1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16715",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16715"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Reconstruction through a coisotropic reduction: the missing momentum on a Poincaré section as a branched covering",
  "title_text": "Reconstruction through a coisotropic reduction: the missing momentum on a Poincaré section as a branched covering",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/96zetjf0bgfg1",
  "venue": "arXiv",
  "versions": []
}