{
  "abstract": "The Connes-van Suijlekom and Connes-Consani-Moscovici truncations of the Weil quadratic form, at a prime cutoff c>1 and frequency band N, produce finite Galerkin matrices whose spectra are the finite-rank window on Weil positivity. We prove two exact finite theorems about this truncation. First, every real even Galerkin coefficient vector v determines, in closed form, a band-limited Guinand-Weil test function g_v whose zero sum over the nontrivial zeros of zeta equals the quadratic value exactly: every value of the truncated form is an exact sum over the zeros. The construction factors through an exact source quotient of dimension 2N+1 and admits a non-collapsing pole-neutral subfamily. Second, beyond the Galerkin band the omitted archimedean tail is a totally positive Cauchy-Stieltjes increment. This yields a two-sided certification rule with an explicit budget B_T ~ (2N+1) rho log(T) / (pi^2 T), where T is the archimedean cutoff and rho = 2 pi / log c: finite-cutoff positivity certifies cutoff-free positivity, a finite-cutoff eigenvalue below -B_T certifies a cutoff-free negative, and a negative eigenvalue in the band [-B_T, 0) certifies nothing. Resolving a spectral scale of 10^-59 at c=100 by brute cutoff would require T of order 10^63; a cutoff-free interval LDL^T factorization resolves it directly. The dictionary is verified over the first 512 zeros of zeta and by three independent computational routes; all scripts and artifacts ship with the paper. The paper makes no Riemann Hypothesis, prime-counting, next-prime, or factoring claim.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Akiva Groskin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/beyfqnh5x6g9e.bib",
  "cited_by": 7,
  "doi": "10.48550/arxiv.2607.02828",
  "fulltext_url": null,
  "id": "beyfqnh5x6g9e",
  "issns": [
    "2331-8422"
  ],
  "landing_url": "https://arxiv.org/abs/2607.02828",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7167606751",
  "pdf_url": "https://arxiv.org/pdf/2607.02828",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-02",
  "publisher": "Cornell University",
  "ris": "https://research.draconitic.com/paper/beyfqnh5x6g9e.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T13:21:00Z",
      "native_id": "W7167606751",
      "source": "openalex",
      "url": "https://arxiv.org/abs/2607.02828"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form",
  "title_text": "A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form",
  "type": "preprint",
  "updated_at": "2026-11-06T13:20:22Z",
  "url": "https://research.draconitic.com/paper/beyfqnh5x6g9e",
  "venue": "arXiv (Cornell University)",
  "versions": []
}