{
  "abstract": "This paper proves asymptotically optimal linear degree lower bounds for Nullstellensatz and Polynomial Calculus, yielding a four-way polynomial-versus-exponential proof-size classification for uniform polynomial consequences over fields of positive characteristic. The size bounds use full, explicit coefficient lists, including zeros. Apart from a fixed finite initial range, exactly one of four cases occurs: $\\mathrm T_1$, bounded degree; $\\mathrm T_2$, degree at least linear in the universe size $n$; $\\mathrm T_3$, constant degree on some residue classes and at least linear degree on the others; or $\\mathrm T_4$, infinitely many recurring finite degrees, with bounded-degree subsequences and finite $Ω(n)$ peaks at arbitrarily large sizes. Thus the size patterns are polynomial bounds, exponential lower bounds, a periodic mixture, or polynomial subsequences with finite exponential peaks, respectively. Underivable targets have degree $\\infty$, and both proof systems assign the same type. The result strengthens my 2008--2009 refutation classification and extends it to uniform consequences. The proof uses degree-preserving lifting with onset and period bounds linear in the degree. A common $p$-adic hardness spectrum describes the four types topologically. Distinguishing any two types is undecidable already for refutations. The linear degree bounds, four-way classification and pairwise undecidability results have been formalised and verified in Lean. The Lean formalisation, verification records and an accompanying tutorial are available in the supporting materials (https://doi.org/10.5281/zenodo.23965838).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07280",
  "author_total": 1,
  "authors": [
    {
      "name": "Søren Riis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8nedxrj61vwhh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07280",
  "fulltext_url": null,
  "id": "8nedxrj61vwhh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07280",
  "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.CC"
    ],
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.LO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07280",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8nedxrj61vwhh.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07280",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07280"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Degree profiles of uniform polynomial consequences: Linear lower bounds for Nullstellensatz and Polynomial Calculus",
  "title_text": "Degree profiles of uniform polynomial consequences: Linear lower bounds for Nullstellensatz and Polynomial Calculus",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/8nedxrj61vwhh",
  "venue": "arXiv",
  "versions": []
}