{
  "abstract": "Alternating reciprocal transformations of polynomial divisibility systems couple an orbit of integer points to a rational dynamical system on polynomial coefficients. For fixed positive degrees $n,m$, we prove universal four-step return to rational gauge classes. Weighted primitive representatives preserve integral continuation through nondegenerate jumps using only exact division and coefficient gcds; the stronger requirement of monic integral representatives is governed by endpoint $p$-adic valuation defects. After one integral block, constant-term valuations evolve by an explicit unimodular matrix whose trace depends only on $nm$. This gives finite-order behavior for $nm<4$, parabolic dilations for $nm=4$, and hyperbolic dynamics for $nm>4$. In the hyperbolic case, invariant thresholds, irrational eigencones, and discrete valley formulas give finite criteria for forward and bi-infinite integral coefficient orbits, although failure can be delayed arbitrarily long. Every positive-constant canonical monic coefficient chain admits an explicit positive rational four-phase gauge. At $nm=4$, geometric gauges reduce nondegenerate compatible rational point orbits to two-periodic affine recurrences, hence to sequences satisfying linear recurrences with constant coefficients. Polynomial identities of degree at most $h$ on each invariant conic admit certificates from $2h+1$ initial phase states, while integral unimodular recurrence models yield periodic modular obstructions. For $nm>4$, eventually positive normalized point orbits tending to infinity have hyperbolic logarithmic growth and cannot satisfy such linear recurrences. Finally, separated auxiliary-relation transport reduces rational rank to at most one after two alternating jumps; quadratic specialization gives exact local saturation criteria and persistence results across periodic, parabolic, and hyperbolic orbits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20176",
  "author_total": 2,
  "authors": [
    {
      "name": "Max A. Alekseyev"
    },
    {
      "name": "Dmitry I. Khomovsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3qsg2yn06xkfw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20176",
  "fulltext_url": null,
  "id": "3qsg2yn06xkfw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20176",
  "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.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7222322526",
  "pdf_url": "https://arxiv.org/pdf/2610.20176",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3qsg2yn06xkfw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.20176",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20176"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Reciprocal Dynamics in Polynomial Congruence Jumping: Integral Orbits and Valuation Cones",
  "title_text": "Reciprocal Dynamics in Polynomial Congruence Jumping: Integral Orbits and Valuation Cones",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/3qsg2yn06xkfw",
  "venue": "arXiv",
  "versions": []
}