{
  "abstract": "Although the Heil--Ramanathan--Topiwala conjecture is false in full generality, its failure makes the classification of positive geometric and arithmetic regimes more urgent. We settle the critical mixed-arithmetic regime for four time-frequency shifts. For \\(z=(x,ω)\\) and \\(w=(y,η)\\), set \\(σ(z,w)=xη-yω\\). Let \\(u,v\\in\\R^2\\) satisfy \\(\\lvertσ(u,v)\\rvert=1\\), write \\(ν=αu+βv\\), and assume that \\(0,u,v,ν\\) are distinct. If \\(\\dim_{\\Q}\\operatorname{span}_{\\Q}\\{1,α,β\\}=2\\) then, for every nonzero \\(f\\in L^2(\\R)\\), the vectors \\(f,π(u)f,π(v)f,π(ν)f\\) are linearly independent. This includes both four-point configurations in Chris Heil's Conjecture~9.2. The proof first converts a putative dependence into a scalar cocycle over an irrational rotation. On a positive-measure family of zero-free fibres, winding and continued-fraction returns force the periodic holonomy to be constant. The return multiplier is Laurent polynomial, however, so its ungauged holonomy is algebraic; the constant-holonomy law simultaneously makes it an irrational character. These conclusions are incompatible. The principal theorem and its complete formal dependency chain have been fully certified end-to-end in Lean~4, including the physical all-nonzero case, the complete three-point HRT theorem, the coefficient-reduction step, and the final four-point conclusion. The accompanying \\href{https://www.dropbox.com/scl/fi/jd9xeqqera147wdc82r8k/3f260460-07d5-45f5-8d1c-ee1d2879c0af-aristotle-32-.tar.gz?rlkey=a24njadfds5ty9ez473lk03lv\\&e=1\\&dl=0} {Lean~4 certification archive} contains reproducible source, pinned build instructions, and transitive axiom audits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27970",
  "author_total": 1,
  "authors": [
    {
      "name": "Vignon Oussa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2qwn8vrvc682x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27970",
  "fulltext_url": null,
  "id": "2qwn8vrvc682x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27970",
  "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.CA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7214171339",
  "pdf_url": "https://arxiv.org/pdf/2609.27970",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2qwn8vrvc682x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27970",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27970"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "The critical mixed-arithmetic four-point HRT theorem",
  "title_text": "The critical mixed-arithmetic four-point HRT theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/2qwn8vrvc682x",
  "venue": "arXiv",
  "versions": []
}