{
  "abstract": "We associate to finite block size data \\[ \\bm n=(n_d)_{d\\ge1},\\qquad n_1>0, \\] a positive definite cyclotomic form and a Householder reflection representation. Writing \\[ N=\\operatorname{lcm}\\{d:n_d\\ne0\\},\\qquad \\wt=\\sum_d d^2n_d, \\] the form on the cyclic group $C_N$ is \\[ B_{\\bm n}(g,h)=\\frac1{\\wt}\\sum_{(g^{-1}h)^d=1}d^2n_d. \\] Its labelled Fourier spectrum reconstructs the complete multiplicity sequence by Möbius inversion. If the $n_d$ are the Wedderburn multiplicities of a split semisimple algebra $S=\\prod_d M_d(k)^{n_d}$, the divisibility widths are dimensions of characteristic sums of Wedderburn blocks, and the form together with $\\dim_kS$ recovers the isomorphism type of $S$. The main theorem is an arithmetic rigidity result. For $N\\ge2$, the reflection group is finite if and only if it preserves a full rank lattice, equivalently \\[ \\supp(\\bm n)=\\{1,N\\},\\qquad n_1=N^2n_N. \\] In this case $B_{\\bm n}=\\frac12(I+J)$ and the group is $W(A_N)\\cong S_{N+1}$. In semisimple algebra terms, finiteness is equivalent to \\[ S\\cong k^{N^2m}\\times M_N(k)^m \\] for some $m\\ge1$. For $N=2$, every nonfinite case is a dense copy of $D_\\infty$ in the corresponding orthogonal group. For contraction algebras of threefold flops, Toda's width formulas identify the total scale and primitive Fourier mode with $\\dim A_{\\con}$ and $\\dim A_{\\con}^{\\ab}$. This yields infinitude for all irreducible flops of length at least three and, in length two, \\[ Γ_f\\text{ finite}\\quad\\Longleftrightarrow\\quad \\dim A_{\\con}=2\\dim A_{\\con}^{\\ab}. \\] The final part gives a complementary BPS/Riemann--Hilbert realization of the same reflection matrices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00263",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiaobin Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fjj2x1n9n63fv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00263",
  "fulltext_url": null,
  "id": "fjj2x1n9n63fv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00263",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7215933598",
  "pdf_url": "https://arxiv.org/pdf/2610.00263",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fjj2x1n9n63fv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.00263",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00263"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Contraction Algebras, Cyclotomic Forms, and Reflection Rigidity",
  "title_text": "Contraction Algebras, Cyclotomic Forms, and Reflection Rigidity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/fjj2x1n9n63fv",
  "venue": "arXiv",
  "versions": []
}