{
  "abstract": "Varkonyi and Domokos proved that homogeneous convex bodies with any prescribed numbers $S\\ge1$ of stable and $U\\ge1$ of unstable static equilibria exist, the case $S=U=1$ being the mono-monostatic Gomboc. We give a complete constructive answer for the simplest nondegenerate shapes: a homogeneous body whose boundary is the unit sphere perturbed radially by a single real spherical harmonic $Y_\\ell^m$ of degree $\\ell\\ge2$ and order $1\\le m\\le\\ell$ has exactly $S=U=m(\\ell-m+1)$ stable and unstable equilibria, for every amplitude in the convex range when $m\\ge2$, and for small amplitude when $m=1$. The reduction to the critical points of the perturbing harmonic is an identity when $m\\ge2$: the symmetry of a single tesseral harmonic pins the centroid at the origin exactly, so the centroid-to-surface distance is a strictly increasing function of the harmonic. We count the critical points of $Y_\\ell^m$ from its separable structure and verify the Poincare-Hopf balance in index form. Three consequences follow. (i) A single harmonic populates only the diagonal $S=U$ of the equilibrium-class lattice; no single harmonic of degree $\\ge2$ is mono-monostatic. (ii) Any centrally symmetric (even-degree) perturbation has even $S$ and $U$ and cannot be mono-monostatic, a parity obstruction. (iii) Mono-monostaticity is therefore intrinsically multi-harmonic. A predict-then-confirm numerical study on eight single-harmonic bodies matches the formula exactly. For the multi-harmonic regime we give a certified equilibrium counter (interval arithmetic with Krawczyk uniqueness and Hessian classification) that, given the centroid, provably neither under- nor over-counts. It certifies that a member of a known analytic family, at an amplitude of our own choosing inside the strictly convex sub-regime, is mono-monostatic. For that body the centroid is exactly the origin, so its certificate is unconditional.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.11213",
  "author_total": 1,
  "authors": [
    {
      "name": "Vincent Wesley Couey"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7q39vp9hwe6k4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.11213",
  "fulltext_url": null,
  "id": "7q39vp9hwe6k4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.11213",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7202375654",
  "pdf_url": "https://arxiv.org/pdf/2608.11213",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7q39vp9hwe6k4.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2608.11213",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.11213"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Static Equilibria of Perturbed Spheres: A Single-Harmonic Class Map, a Parity Obstruction, and a Certified Counter for the Mono-Monostatic Regime",
  "title_text": "Static Equilibria of Perturbed Spheres: A Single-Harmonic Class Map, a Parity Obstruction, and a Certified Counter for the Mono-Monostatic Regime",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/7q39vp9hwe6k4",
  "venue": "arXiv",
  "versions": []
}