{
  "abstract": "For the intersection of two disks meeting at angle $2α$, let $C(α)$ be the least constant in the associated spectral-set inequality. We give a self-contained M\"obius reduction to the corresponding numerical-range problem on a sector and determine the sharp constant for affine square-zero operators $B=λI+N$, $N^2=0$: $C_{\\mathrm{sq0}}(α)=π\\sinα/(2α)$. A $2\\times2$ matrix attains equality and yields an explicit lens lower-bound certificate. At the right angle, we prove the conjectural $\\sqrt2$ bound in arbitrary dimension for the full palindromic quadratic family, and give exact rational certificates for several larger parameter families, including a complex post-automorphism disk, the complete imaginary diameter, boundary-reaching phase arcs, and symmetric and asymmetric three-node admissible-kernel problems. We also obtain a strict central bound $|w^2|\\leqκ_0<\\sqrt2$, a two-small-zero extension, an exact two-moment criterion for the remaining boundary layer, and a verified angle-dependent envelope. Every computer-assisted assertion has an exact rational verifier. These results are dimension-free but do not determine the unrestricted fixed-lens constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25027",
  "author_total": 1,
  "authors": [
    {
      "name": "Zijian Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1bwbk7sy2scxb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25027",
  "fulltext_url": null,
  "id": "1bwbk7sy2scxb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25027",
  "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.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.25027",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1bwbk7sy2scxb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25027",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25027"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fixed Convex-Lens Spectral Constants: Möbius Reduction, Sharp Model Theorems, and Angle-Dependent Bounds",
  "title_text": "Fixed Convex-Lens Spectral Constants: Möbius Reduction, Sharp Model Theorems, and Angle-Dependent Bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/1bwbk7sy2scxb",
  "venue": "arXiv",
  "versions": []
}