{
  "abstract": "A magic square exemplifies balance in inequality: its $n^2$ entries $\\{1,\\dots,n^2\\}$ are distinct, yet every row, column, and main diagonal sums to $μ_n=n(n^2+1)/2$. Lifting centred grid cells into $\\mathbb{R}^3$ as radial vectors $\\mathbf{p}_{ij}(λ)=(x_j,y_i,λz_{ij})$ and taking their convex hull---the Magic Gem $\\mathcal{G}_λ(S)$---asks: when does line-sum equality suffice for equilibrium in 3D, and when a nonlinear realization breaks balance, how can an opposing inequality restore it? We test this across all $880$ order-$4$ magic squares in four regimes. For radial spokes from $O$, natural lengths balance identically, while unit spokes break equilibrium on all $832$ non-associative squares in 3D and on all squares in 2D, where the ring-radius ratio $r_2/r_1=\\sqrt{5}$ bounds the resultant below by the Pell convergent $9/4\\approx\\sqrt{5}$. By length--weight duality, balance is restored via tip weights $\\mathbf{w}^*>\\mathbf{0}$ on $S^2$ or power-law spoke scaling ($r^{α^*}$ in 2D and $(\\mathbf{V}_3)_z=0$ in 3D). In 3D rotation, equal point masses make every semi-magic square an axisymmetric top with spherical-top transition $λ_{\\mathrm{iso}}=(n^2+1)^{-1/2}$, whereas entry masses $m_{ij}=s_{ij}$ expose the quadrupole $Q_{xy}=\\sum z_{ij}x_jy_i$, triggering $180^\\circ$ Dzhanibekov flips on $824$ squares. Passing to the uniform solid hull $\\mathcal{G}_0(S)$ shifts the centroid off $O$ on $829$ squares; re-engaging the opposing core $\\mathcal{G}_1(S)$ restores balance via two-density inclusions, nested wireframes, or the $K_{n^2}$ strut network, while all $880$ hulls unfold into non-overlapping nets with $f_0-1$ glue tabs. Finally, $λ=1/5$ stabilizes $74.5\\%$ of facets on a table, and alternating $\\mathcal{G}_λ(S)$ with its mirror twin $\\mathcal{G}_λ(r^2\\overline{S})=-\\mathcal{G}_λ(S)$ locks chiral blocks into a vertical tower.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03991",
  "author_total": 1,
  "authors": [
    {
      "name": "Kyle Elliott Mathewson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7swz14v2w08xx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03991",
  "fulltext_url": null,
  "id": "7swz14v2w08xx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03991",
  "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.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "physics.class-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03991",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7swz14v2w08xx.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:25Z",
      "native_id": "2611.03991",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03991"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.class-ph",
      "subfield": "general"
    }
  ],
  "title": "Balance in Inequality: Testing the Physical Equilibrium, Precession, and Structural Stability of Magic Gems",
  "title_text": "Balance in Inequality: Testing the Physical Equilibrium, Precession, and Structural Stability of Magic Gems",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/7swz14v2w08xx",
  "venue": "arXiv",
  "versions": []
}