{
  "abstract": "We prove the conjecture of Huang, Jiang, and Oblomkov (HJO) giving a geometric extension of the Rogers--Ramanujan and Andrews--Gordon identities for every torus-knot singularity $X^a=Y^b$ with coprime $1<a<b.$ For a prime power $q$, let $\\mathcal{NC}_n^{a,b}(\\mathbb F_q)$ denote the set of pairs of commuting nilpotent $n\\times n$ matrices $(A,B)$ over $\\mathbb F_q$ satisfying $A^a=B^b$. We establish the threefold equality between their normalized counts, the HJO $q$-series $Z_{a,b}$, and the explicit infinite product $P_{a,b}$: \\[ \\underbrace{\\vphantom{\\Bigg|} \\prod_{m\\geq1}(1-q^{-m}) \\Biggl(\\sum_{n=0}^{\\infty} \\frac{\\lvert\\mathcal{NC}_n^{a,b}(\\mathbb F_q)\\rvert} {\\lvert\\operatorname{GL}_n(\\mathbb F_q)\\rvert}\\Biggr) }_{\\text{point count}} = \\underbrace{\\vphantom{\\Bigg|}Z_{a,b}(q^{-1}) }_{\\text{\\(q\\)-series}} = \\underbrace{\\vphantom{\\Bigg|}P_{a,b}(q^{-1}) }_{\\text{infinite product}}. \\] Our main result is a stronger finite identity: the rank $N$ HJO sum equals $(q;q)_N$ times the generating function for balanced cylindric partitions with entries bounded by $N$. Taking $N\\to\\infty$ yields the HJO conjecture. The proof combines the compositional rational shuffle theorem of Bergeron--Garsia--Leven--Xin and Mellit with a multiplicativity theorem for slope operators and a determinantal model for bounded cylindric partitions, linked by a common $q$-difference equation. The finite identity and the HJO conjecture have been formalized unconditionally in Lean by AxiomProver.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20567",
  "author_total": 3,
  "authors": [
    {
      "name": "Yifeng Huang"
    },
    {
      "name": "Kenny Lau"
    },
    {
      "name": "Ken Ono"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2frx8b95229j0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20567",
  "fulltext_url": null,
  "id": "2frx8b95229j0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20567",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213588808",
  "pdf_url": "https://arxiv.org/pdf/2609.20567",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2frx8b95229j0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2609.20567",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20567"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Rogers--Ramanujan identities from the geometry of $X^a=Y^b$",
  "title_text": "Rogers--Ramanujan identities from the geometry of X^a=Y^b",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/2frx8b95229j0",
  "venue": "arXiv",
  "versions": []
}