AIM-GEOMETRY-0201 · Singular example and minimality theorem

Expansive Symmetry Breaking at a Five-Vertex Linkage Singularity

Manuscript 11 October 2026 · Online 11 October 2026

math.MGmath.COUnrefereed preprint

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.

Record

Affiliation
Mercury Software GmbH
Result
Singular example and minimality theorem
Categories
math.MG · math.CO
Manuscript
11 October 2026
Online release
11 October 2026
Version
1.0
License
Creative Commons Attribution 4.0 International

Files and verification

The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.

Citation

Alper Ferudun, “Expansive Symmetry Breaking at a Five-Vertex Linkage Singularity,” EulerSolve Research Papers, AIM-GEOMETRY-0201, 2026. https://doi.org/10.5281/zenodo.23294757.

BibTeX
@misc{ferudun2026expansivesingularity,
  author = {Ferudun, Alper},
  title = {Expansive Symmetry Breaking at a Five-Vertex Linkage Singularity},
  year = {2026},
  howpublished = {EulerSolve Research Papers},
  url = {https://eulersolve.org/papers/aim-geometry-0201/},
  doi = {10.5281/zenodo.23294757},
  note = {AIM-GEOMETRY-0201; unrefereed preprint}
}

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