{
  "abstract": "Place one grain at every nonsink vertex of the wired $n\\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\\ge1$, \\[ ν_2(L(n))= \\begin{cases} 2,&n=1,\\\\ 1,&n\\ge2\\text{ even},\\\\ ν_2(n+1)+2,&n\\ge3\\text{ odd}. \\end{cases} \\] For even squares, this follows from the domino--sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over $\\mathbb F_2$, giving the exact valuation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10625",
  "author_total": 9,
  "authors": [
    {
      "name": "Turgay Akyar"
    },
    {
      "name": "Artem Beliakov"
    },
    {
      "name": "Konstantin Delchev"
    },
    {
      "name": "Nikita Kalinin"
    },
    {
      "name": "Ernesto Lupercio"
    },
    {
      "name": "Higinio Serrano"
    },
    {
      "name": "Mikhail Shkolnikov"
    },
    {
      "name": "Daniel Tabares"
    },
    {
      "name": "Nikolai Terekhov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0anbafnsjmcr9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10625",
  "fulltext_url": null,
  "id": "0anbafnsjmcr9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10625",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212322713",
  "pdf_url": "https://arxiv.org/pdf/2609.10625",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0anbafnsjmcr9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10625",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10625"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square",
  "title_text": "The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/0anbafnsjmcr9",
  "venue": "arXiv",
  "versions": []
}