{
  "abstract": "Most of the traditional third-order mock theta functions occur naturally as specialized values of partition-rank generating functions. The same finite rank kernels also occur in the ordinary and odd unimodal-rank series studied previously in the literature. We organize a parallel collection of corner series by retaining these kernels and replacing their quadratic core weights by linear peak weights. Their root-of-unity specializations are governed by partial and false theta identities recorded in Ramanujan's Lost Notebook; in one exceptional case the corner companion is itself a mock theta function. The classical rank series and Ramanujan identities are used as inputs. Our new results include exact two- and three-color partition identities with sparse theta discrepancies and an oriented three-dimensional corner with a distinct tower. In both color cases, positive decompositions separate the colored companion into copies of the relevant odd-corner series and a corresponding decorated even-corner series. Thus both sparse discrepancies become signed differences between natural corner families. Burson's boxed modular involution provides a common rank-refined model for the two corrections, whose surviving fixed points produce the exceptional quadratic sequences. For the side rank of the three-dimensional corners we obtain an exact formula at $z=-1$, a strict parity bias, quartic and cubic root-of-unity rank-difference identities, and exact equidistribution modulo $3$ on the progression $6n+5$. The cubic identity also gives exact congruence formulas on the progressions $6n+2$ and $6n+3$. We further classify the associated binary triangular support modulo every integer and derive general congruence families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25380",
  "author_total": 2,
  "authors": [
    {
      "name": "George E. Andrews"
    },
    {
      "name": "Mohamed El Bachraoui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d8ykv8kx11ew7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25380",
  "fulltext_url": null,
  "id": "d8ykv8kx11ew7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25380",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7223021547",
  "pdf_url": "https://arxiv.org/pdf/2610.25380",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d8ykv8kx11ew7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25380",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25380"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Corners of Ramanujan: Partial- and False-Theta Companions to Third-Order Mock Theta Functions",
  "title_text": "The Corners of Ramanujan: Partial- and False-Theta Companions to Third-Order Mock Theta Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/d8ykv8kx11ew7",
  "venue": "arXiv",
  "versions": []
}