{
  "abstract": "We view a $k\\times n$ Latin rectangle as $n$ points in the grid $[n]^k$. Each symbol gives one point, whose coordinates are the columns it occupies. The rectangle is balanced when these points are pairwise equidistant in the $\\ell_1$ metric. Its imbalance measures the distance from this ideal. We study the minimum imbalance from three sides. Constructively, we use a simulated annealing that moves symbols within rows and only penalises column clashes. Against the reference table of Voboril et al. (2025), it improves 33 of the 51 non-square values for $8\\le n\\le 13$, by up to 39.4%, and matches the rest. A cheap invariant, the row group, certifies that none of the 33 new rectangles is isotopic to rows of a group table, so the earlier column-swapping local search cannot reach any of them. Exactly, a computer-assisted branch and bound proves that the minimum imbalance of a $3\\times 9$ rectangle is $56$. Díaz et al. found this value, but its optimality was open. A sharper pruning bound, derived from the identity $I=2\\sum(d-t)^+$, reduces the search tree by a factor of $2.4$ to $3.3$ on the instances with $k\\ge3$ that we solve. Asymptotically, the minimum imbalance is $Θ(n^3)$ for every fixed $k$. A pigeonhole argument gives a lower constant of order $k\\,4^{-k}$, and random rectangles give an upper constant of order $\\sqrt k$. For $k=2$, the limit of $\\min I/n^3$ exists, a permuton attains it, and it lies between $4/375$ and $0.1196$. Perfect balance is an $\\ell_1$-equilateral problem, so it forces $k=Ω(n/\\log n)$ rows. For $2\\times n$ rectangles, single two-minute annealing runs improve five of the seven six-hour MIP upper bounds for $14\\le n\\le 20$. Finally, the Lean 4 kernel checks every rectangle reported in this paper against a 74-line specification that imports only Lean's core library. We also prove the pigeonhole lower bound in Lean with Mathlib.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06003",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhengyu Li"
    },
    {
      "name": "Vijay Ganesh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/37r7k1rzn2574.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06003",
  "fulltext_url": null,
  "id": "37r7k1rzn2574",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06003",
  "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",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06003",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/37r7k1rzn2574.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06003",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06003"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "How Unbalanced Must a Latin Rectangle Be?",
  "title_text": "How Unbalanced Must a Latin Rectangle Be?",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/37r7k1rzn2574",
  "venue": "arXiv",
  "versions": []
}