{
  "abstract": "Let $p$ be a prime and let $G$ be the diagonal of $N/Q$ in $m$ variables over the $p$-adic integers, with $Q(0)$ a unit. We prove that the $p$-kernel of the coefficients of $G$ modulo $p^α$ has size $K_α(G)\\le p^{(|J|+1)α-1}+(p^α-1)/(p-1)$, where $J$ is an explicit finite support set determined by the Newton polytopes of $N$ and $Q$, and $|J|\\le\\prod_i(h_i+1)$ for $h_i=\\max(°_{z_i}N,°_{z_i}Q)$. Thus the minimal automaton has $p^{O(α)}$ states for fixed input and prime. The general upper bounds of Rowland and Yassawi have exponents that are cubic in $α$ for Furstenberg series and of order $α^{m+1}$ for diagonals in $m$ variables. The proof combines a $δ$-decomposition with synchronous Cartier sections and a count of compatible histories; it requires no smoothness or ordinarity assumption. We give explicit transitions and derive bounds for Furstenberg series. For the Furstenberg series $F$ of $bx^2y^2+(1+ax+dx^2)y+cx^2$ with odd $a,b,c,d\\in\\mathbb{Z}_2$, we prove $K_α(F)\\ge9\\cdot8^α/(64α-32)$ and $\\lim_{α\\to\\infty}α^{-1}\\log_2K_α(F)=3$. This exponent result rests on primitive reachability, parameter-independent normalized window frames, and an overlap estimate certified in a finite quotient of a rational Cartier lattice. An appendix also proves that the zero orbit in the Rowland-Yassawi automaton for $x^2y^2+(1+x+x^2)y+x^2$ has exactly $2^{α+1}+α+1$ states, resolving their orbit-size conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01481",
  "author_total": 1,
  "authors": [
    {
      "name": "Alex Shvets"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eqfwq51sc2gcw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01481",
  "fulltext_url": null,
  "id": "eqfwq51sc2gcw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01481",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01481",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eqfwq51sc2gcw.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01481",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01481"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Linear-exponent bounds for the automatic complexity of rational diagonals modulo prime powers",
  "title_text": "Linear-exponent bounds for the automatic complexity of rational diagonals modulo prime powers",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/eqfwq51sc2gcw",
  "venue": "arXiv",
  "versions": []
}