{
  "abstract": "Let $X$ be a three-dimensional real Banach space and let $f:S_X\\to S_Y$ be a surjective isometry. We study the propagation of an affine formula for $f$ on a relatively open part of $S_X$. A finite family of antipodal distance coordinates propagates a linear anchor except at three explicitly described degeneracies: a facet, an open face-star, or a family of chord cones with a common cylindrical kernel. We prove that, if the norm has no fixed-direction cylindrical open cone, every affine open anchor is linear and global. This yields a source-geometric criterion for the Mazur--Ulam property and covers, among other non-strictly-convex examples, the Euclidean double cone. We also give a segment-saturation criterion which proves the property for every prism $Z\\oplus_\\infty\\mathbb R$, where $Z$ is an arbitrary real Banach plane. Finally, in the remaining cylindrical regime, we prove that a chord-saturated same-kernel network cannot be confined to one proper projective quotient arc. We then show that this multi-arc difficulty and the transverse-ruled part of producing an initial affine anchor reach the same final obstruction: upgrading norm calibration on an ambient open cone, together with calibration on a few spherical segments, to pointwise calibration of the sphere map on a two-dimensional open patch.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25065",
  "author_total": 1,
  "authors": [
    {
      "name": "Yicen Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/db3d652ps8zeg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25065",
  "fulltext_url": null,
  "id": "db3d652ps8zeg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25065",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7214089978",
  "pdf_url": "https://arxiv.org/pdf/2609.25065",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/db3d652ps8zeg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25065",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25065"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Affine Anchors and Cylinder Obstructions in the Three-Dimensional Tingley Problem",
  "title_text": "Affine Anchors and Cylinder Obstructions in the Three-Dimensional Tingley Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/db3d652ps8zeg",
  "venue": "arXiv",
  "versions": []
}