{
  "abstract": "Let \\[ M_{k,\\ell}=[z^k-w^\\ell]\\subset H^2(\\mathbb D^2), \\qquad k,\\ell\\in\\mathbb N,\\quad k\\ne\\ell . \\] We study the large-index asymptotics of Yang's numerical invariants and the boundary structure of their generating function. Starting from the exact staircase formula obtained in our preceding work, we prove that \\[ Σ_j(M_{k,\\ell}) = \\frac{C_{k,\\ell}}{j} + O(j^{-2}), \\] where \\[ C_{k,\\ell} = \\int_0^\\infty \\frac{x^2} {(x+1/k)^2(x+1/\\ell)^2}\\,dx . \\] The first strict descent together with $C_{k,\\ell}$ recovers the unordered pair $\\{k,\\ell\\}$, yielding an asymptotic rigidity principle. At the next order we obtain a periodic correction \\[ Σ_j(M_{k,\\ell}) = \\frac{C_{k,\\ell}}{j} + \\frac{Ψ_{k,\\ell}(j)}{j^2} + O(j^{-3}), \\] whose least period is \\(\\operatorname{lcm}(k,\\ell)\\), and we determine the leading amplitudes of the strict drops. For Yang's generating function \\[ \\mathcal P_{k,\\ell}(t) = \\sum_{j=0}^{\\infty}Σ_j(M_{k,\\ell})t^j, \\] we prove that it has radius of convergence one and a logarithmic singularity at $t=1$; hence it is never a polynomial for $k\\ne\\ell$, giving a negative answer to Yang's polynomiality question within this quasi-homogeneous family. The periodic higher-order corrections generate root-of-unity polylogarithmic boundary modes. The leading logarithmic coefficient together with the second-order boundary support determines the unordered pair $\\{k,\\ell\\}$, while successive renormalized boundary limits recover the Fourier coefficients of every finite-order periodic asymptotic term.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36516",
  "author_total": 3,
  "authors": [
    {
      "name": "Yin Liu"
    },
    {
      "name": "Yufeng Lu"
    },
    {
      "name": "Yixin Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8q25rsj11h7v8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36516",
  "fulltext_url": null,
  "id": "8q25rsj11h7v8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36516",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7214894751",
  "pdf_url": "https://arxiv.org/pdf/2609.36516",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8q25rsj11h7v8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36516",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36516"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Asymptotic Rigidity and Boundary Structure of Yang's Numerical Invariants for the Bidisk Submodules $[z^k-w^\\ell]$",
  "title_text": "Asymptotic Rigidity and Boundary Structure of Yang's Numerical Invariants for the Bidisk Submodules [z^k−w^ℓ]",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/8q25rsj11h7v8",
  "venue": "arXiv",
  "versions": []
}