{
  "abstract": "Every semi-magic square of order $n$ satisfies the axial moment identities $\\sum_{i,j}s_{ij}j^m=\\sum_{i,j}s_{ij}i^m=M(n)\\sum_jj^m$ for all $m\\ge1$, where $M(n)=n(n^2+1)/2$. For real matrices the converse requires $1\\le m\\le n-1$, whereas for arrangements of $\\{1,\\dots,n^2\\}$ fewer moments suffice once $n\\ge14$. We define the permutation threshold $k^*(n)$ as the least $k$ such that every arrangement satisfying these identities for $1\\le m\\le k$ is semi-magic. In terms of the column-sum deviations $γ_j$, the degree-$k$ conditions state that $\\sum_jγ_jt^j$ is divisible by $(t-1)^{k+1}$, linking $k^*(n)$ to the Prouhet--Tarry--Escott problem. While Schur's bound on real zeros gives $k^*(n)\\le\\sqrt{2(n-1)\\log(n^4/4)}$, we prove the converse realisability bound $k^*(n)\\ge(\\sqrt2-o(1))\\sqrt n$ (and $k^*(n)\\ge(\\sqrt6-o(1))\\sqrt n$ when $4\\mid n$) by perturbing semi-magic squares with complementary row pairs via horizontal swaps. We also show that $k^*(n)=n-1$ for $3\\le n\\le13$, $k^*(n)=n-2$ for $14\\le n\\le17$, and $k^*(n)\\le n-3$ for $n\\ge18$, and determine $k^*(n)$ for $18\\le n\\le40$ with computer assistance. Finally, at the $k=1$ level of the hierarchy, we specialise the Mantel--Hubert quadratic assignment variance formula to obtain exact means, variances, and limit laws under a uniform random arrangement for the full line-sum energy $E_{\\mathrm{full}}$, for the four-term energy $E_4$ combining the first moments and the two diagonals, and for Loly's moment of inertia, and find that $|L_4|=14{,}639{,}440$ arrangements of order $4$ satisfy $E_4=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38386",
  "author_total": 1,
  "authors": [
    {
      "name": "Kyle Elliott Mathewson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fjdwce497w4f4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38386",
  "fulltext_url": null,
  "id": "fjdwce497w4f4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38386",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38386",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fjdwce497w4f4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38386",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38386"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Moments of Magic Squares: Line-Sum Conditions, a Permutation Threshold, and Random Energies",
  "title_text": "Moments of Magic Squares: Line-Sum Conditions, a Permutation Threshold, and Random Energies",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/fjdwce497w4f4",
  "venue": "arXiv",
  "versions": []
}