{
  "schema_version": 1,
  "problem_number": "OWR-13497-006",
  "title": "All Incorrect Equilibria of the Planar Four-Agent Formation Law Are Unstable",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Four agents in the plane move by the standard distance-based gradient law ṗᵢ = Σ_{j≠i} e_ij (p_j − pᵢ), e_ij = ‖pᵢ − p_j‖² − d̄_ij², where all six desired distances come from one planar target. In an Oberwolfach report of 2015, B. D. O. Anderson asked whether, as for rectangular targets, all incorrect equilibria are saddle points when the target is a general quadrilateral or a triangle with an interior agent. We show that the answer is yes for every planar target: at every incorrect equilibrium the Hessian of V = ¼ Σ_{i<j} e_ij² has a negative eigenvalue; all incorrect equilibria except the collocated one also have a positive eigenvalue, and the collocated one is a local maximum modulo translations. In the language of multidimensional scaling, the first statement says that the complete-graph s-stress of four points in the plane has no spurious second-order critical points, for every ground truth. At an incorrect equilibrium spanning the plane, e_ij = κ λᵢ λ_j with κ > 0, where λ is the affine dependency of the agents, and the smallest Hessian eigenvalue is at most −κ‖λ‖²/2 ≤ −√(8V/3); both constants are sharp. The collinear case is known. For the spanning case we combine the classical self-stress of four planar points, a Schur-complement argument as in recent work of Criscitiello on the s-stress, and an inequality between two 2×2 matrices depending only on λ, proved by explicit identities in the elementary symmetric functions of λ. Consequently almost every initial condition converges to a formation congruent to the target, while collinear initial conditions show that literal global convergence fails. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.OC",
    "math.DS"
  ],
  "keywords": [
    "formation control",
    "distance-based control",
    "K4 formation",
    "incorrect equilibria",
    "strict saddle",
    "almost global convergence",
    "gradient flow",
    "s-stress",
    "Euclidean distance geometry",
    "Oberwolfach Reports",
    "UnsolvedMath",
    "OWR-13497-006",
    "math.OC",
    "eess.SY",
    "math.DS",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-01",
  "publication_date": "2026-10-01",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-01",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-13497-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-13497-006/paper.pdf?v=cb7e5d983fce",
  "doi": "10.5281/zenodo.23072208",
  "zenodo_record_url": "https://zenodo.org/records/23072208",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Anderson's question (OWR 12/2015, pp. 647–649) affirmatively for every planar target: every incorrect equilibrium of the standard four-agent distance-based law is unstable, so almost every initial condition converges to the target shape (this also answers OWR-13497-001, -004 and -005 in the almost-global sense; literal global convergence fails for collinear starts). Other graphs, other laws and more agents are not treated.",
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    "source.zip": {
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    "verification_report.md": {
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  "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."
}