{
  "abstract": "We give a systematic creative-microscoping treatment of truncated basic hypergeometric sums attached to three classical third-order mock theta denominator patterns, represented by $ϕ(q)$, $ω(q)$, and $ν(q)$. The main conceptual point is that these denominator patterns naturally admit exact finite theta evaluations at $a=q^{\\pm n}$, microscopic $q$-supercongruences, and cyclotomic/$p$-adic consequences within a single framework. For \\[ S(q,a):=\\sum_{n\\ge 0}\\frac{q^{2n+1}(aq,q/a;q^2)_n}{(-q^2;q^2)_n}, \\] we prove, for odd $n$, a finite theta evaluation at $a=q^{\\pm n}$, equivalently a microscopic $q$-supercongruence. We prove an analogous finite theta evaluation for the $ν$-type denominator \\[ N_m(q,a):=\\sum_{j=0}^{m} q^{2j+1}\\frac{(aq,q/a;q^2)_j}{(-q;q^2)_{j+1}}. \\] We then introduce the one-parameter denominator family \\[ \\T(q;b):=\\sum_{n\\ge 0} q^{2n}\\frac{(q;q^2)_n^2}{(q^2;q^2)_n(b;q^2)_n}, \\] whose specializations include the $ω$-type shifted denominator ($b=q^3$) and the two-color family ($b=q^{2k}$). For its truncated $a$-extension we obtain an exact evaluation at $a=q^{\\pm n}$, deduce microscopic congruences modulo $(a-q^n)(1-aq^n)$, and derive cyclotomic consequences, including zero congruences in the $q^{2k}$-subfamily and nontrivial $p$-adic versions for the cases $b=q$ and $b=q^3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05961",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohamed El Bachraoui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f66ncc6ecsset.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05961",
  "fulltext_url": null,
  "id": "f66ncc6ecsset",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05961",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7211988979",
  "pdf_url": "https://arxiv.org/pdf/2609.05961",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f66ncc6ecsset.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05961",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05961"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A $q$-microscope for mock-theta denominator patterns",
  "title_text": "A q-microscope for mock-theta denominator patterns",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/f66ncc6ecsset",
  "venue": "arXiv",
  "versions": []
}