{
  "abstract": "Starting from the Riemann--$Ξ$ specialization of Weil's explicit formula, we construct finite real-symmetric Prime--Weil matrices $(\\mathbf S)$ from pole, archimedean, and finite prime-power data. This construction is a finite-dimensional arithmetic model within the Hilbert--Pólya program, which seeks a self-adjoint spectral realization of the nontrivial zeta-zero parameters. Their off-diagonal entries form a Loewner-type divided-difference matrix with a rank-two displacement identity. We formulate the spectral quotient as a Hermitian definite generalized eigenproblem on the fixed zero-mean contrast space. This realization is invariant under positive affine rescaling, avoids ground-vector normalization and an ill-conditioned oblique projector, and preserves the finite quotient spectrum. For a dimension-matched zero-side matrix built from $N$ distinct positive ordinates $γ_k$, rational interpolation gives the exact contrast-pencil spectrum $\\{\\pmγ_1,\\ldots,\\pmγ_N\\}$ and positive-parity square spectrum $\\{γ_1^2,\\ldots,γ_N^2\\}$. Since the ordinates are inputs, this is a reconstruction theorem. Assuming RH, the same interpolation vector proves $λ_{\\min}(\\mathbf S)\\to0$. At $N=L=13$, Lemke's ground-state quotient and the contrast pencil agree numerically and reproduce the first three zeta ordinates to the reported precision. The remaining problem is a relative prime-to-zero perturbation theorem with uniform control of the compressed metric. No proof of RH is claimed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04908",
  "author_total": 1,
  "authors": [
    {
      "name": "Yaoming Shi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/evgabra0bdfay.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04908",
  "fulltext_url": null,
  "id": "evgabra0bdfay",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04908",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7211894525",
  "pdf_url": "https://arxiv.org/pdf/2609.04908",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/evgabra0bdfay.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04908",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04908"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Construction of Finite Hilbert--Pólya Matrices from Weil's Explicit Formula",
  "title_text": "Construction of Finite Hilbert--Pólya Matrices from Weil's Explicit Formula",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/evgabra0bdfay",
  "venue": "arXiv",
  "versions": []
}