{
  "abstract": "Let $A$ and $B$ be polynomials with integer coefficients. We study nonzero integer solutions of $$y\\mid A(x),\\qquad x\\mid B(y),$$ using quotient transformations that may change the polynomial pair. We call this process \\emph{congruence jumping}. We establish a general reciprocal-polynomial rule governing such jumps, without coprimality, monicity, or unit assumptions, and give exact criteria for when the resulting polynomial states can be normalized integrally. Companion-surface identities provide a mechanism for constructing infinite integral quotient chains, including an explicit mixed-degree example. The quadratic case is considerably more rigid. We obtain a quantitative denominator bound for finite chains, classify the exceptional nonconstant one-sided infinite chains with nonintegral conic parameter, and show that changing quadratic states reduce to ordinary Vieta dynamics on a fixed conic. We further construct a genuinely nonunit recurrent four-cycle with infinitely many positive integral points and show that its dynamics admits a uniform Pell-type linearization. A motivating nonunit divisibility system is analyzed through Pell orbits and changing-state ladders. Finally, an independent relation-lattice criterion reduces certain solution sets to a finite divisor search.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00808",
  "author_total": 2,
  "authors": [
    {
      "name": "Max A. Alekseyev"
    },
    {
      "name": "Dmitry I. Khomovsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3nssasmzpm9ny.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00808",
  "fulltext_url": null,
  "id": "3nssasmzpm9ny",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00808",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7216324014",
  "pdf_url": "https://arxiv.org/pdf/2610.00808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3nssasmzpm9ny.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00808"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets",
  "title_text": "Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/3nssasmzpm9ny",
  "venue": "arXiv",
  "versions": []
}