{
  "abstract": "We study orthosymplectically invariant supersphere integration at the exceptional superdimensions $M=-2u$, where the harmonic Fischer structure becomes nonsemisimple and the Pizzetti pairing degenerates. For the meromorphically continued homogeneous inverse kernels we obtain the generating function $$ \\mathscr G_μ(ρ;x,y) =\\frac{Γ(μ/2)}{2π^{μ/2}} \\bigl(1+ρ\\{x,y\\}+ρ^2x^2y^2\\bigr)^{-μ/2}. $$ At $μ=-2u$, its Laurent expansion has a polynomial residue and a logarithmic finite part. We prove that these coefficients recover the complete degreewise duality structure on a fixed superspace with nonzero bosonic dimension. In degrees $k\\le u$, the residue inverts a canonical renormalized pairing on $\\mathcal P_k$. In the collision range $u<k\\le2u$, the ordinary pairing has radical $(x^2)^{k-u}\\mathcal P_{2u-k}$; the finite part reproduces the quotient, while the residue reproduces the radical after transport from the reflected degree. For $k\\ge2u+1$, the finite part is the ordinary inverse kernel. We also establish the nondegenerate head--socle pairing on the generalized harmonic modules. As an application, we derive covariant right--left radial $q$-monogenic zonal symbols and identify precise degree-one obstructions to transferring scalar Pizzetti reproduction through a one-sided $q$-Fischer projection.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.21241",
  "author_total": 4,
  "authors": [
    {
      "name": "Juan Bory-Reyes"
    },
    {
      "name": "Baruch Schneider"
    },
    {
      "name": "Diana Schneiderova"
    },
    {
      "name": "Yifan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c4zm6aydsxdrk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.21241",
  "fulltext_url": null,
  "id": "c4zm6aydsxdrk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.21241",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7170859768",
  "pdf_url": "https://arxiv.org/pdf/2607.21241",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c4zm6aydsxdrk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2607.21241",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.21241"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Exceptional supersphere integration and logarithmic Pizzetti kernels",
  "title_text": "Exceptional supersphere integration and logarithmic Pizzetti kernels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/c4zm6aydsxdrk",
  "venue": "arXiv",
  "versions": []
}