{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0201",
  "title": "Expansive Symmetry Breaking at a Five-Vertex Linkage Singularity",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give an explicit planar framework with five distinct vertices spanning the plane and six bars whose locally one-dimensional configuration set has two analytic branches at a reflection-symmetric configuration. Both outgoing branches are expansive and immediately break the reflection. Every nonedge distance strictly increases on an entire interval. The initial bars overlap, but every strictly positive-time configuration in the stated interval is noncrossing. The starting infinitesimal flex space modulo rigid motions has dimension two, not one. Five is the smallest number of vertices for immediate expansive symmetry breaking among connected simple frameworks with an injective full-span initial placement and a one-dimensional configuration germ. In contrast, a finite relabeling symmetry fixes a regular one-dimensional squared-distance curve pointwise whenever a nonconstant expansive path leaves a fixed point. We also record analytic continuation and a fixed-frame formulation. The example applies classical conic four-bar kinematics and distinguishes singular local dimension from a regular one-degree-of-freedom mechanism. It is not a counterexample under the latter hypothesis or an initially noncrossing requirement, and does not unconditionally resolve AIM Problem 14. This AI-assisted manuscript is self-audited and unrefereed; no independent review, formal certification or absolute priority is claimed.",
  "result_type": "COMPLETE_SCOPED_SINGULAR_COUNTEREXAMPLE_AND_MINIMALITY",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "keywords": [
    "expansive motion",
    "framework",
    "linkage",
    "symmetry",
    "singular mechanism",
    "rigidity",
    "four-bar linkage",
    "vertex minimality"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0201/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0201/paper.pdf?v=bf35fede36db",
  "doi": "10.5281/zenodo.23294757",
  "zenodo_record_url": "https://zenodo.org/records/23294757",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete explicit singular expansive example, positive-time noncrossing opening and five-vertex minimality within the stated connected simple injective full-span class. The initial bars overlap and the initial infinitesimal dimension is two. Regular distance curves preserve symmetry. This is not an initially noncrossing or regular-1DOF counterexample and does not unconditionally close the original AIM question. Classical conic-I kinematics is credited. Self-audited and unrefereed; no independent review, formal certification or absolute priority claim.",
  "files": {
    "paper.pdf": {
      "sha256": "bf35fede36db65d74fdc425d877909cc7d4038e3c6d05f2af16424c9ff9fda88"
    },
    "source.zip": {
      "sha256": "da24c821474a8745cf175ebae3489b137fc1a4eeec9262d7d781c02f83750e71"
    },
    "verification_report.md": {
      "sha256": "e0506aae0626f7b82cfe8d0dad81107007ca5aebec150ae185d2cda498fdc812"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}