{
  "abstract": "We investigate the deterministic hierarchy of prime distribution biases in arithmetic progressions modulo $N$ using a regularized spectral approach. Classical studies on Chebyshev's bias attribute prime races primarily to the accumulation of prime squares $p^2 \\equiv 1 \\pmod N$, which creates a systematic deficit in quadratic residue classes. However, this classical mechanism fails to explain or distinguish any bias among residue classes sharing identical quadratic residue status (e.g., $3, 5, 7 \\pmod 8$). To overcome the long-standing analytical obstacles of jump discontinuities and non-convergent boundary fluctuations inherent in classical Perron-type step-function truncations, we introduce a smooth $C^\\infty$ Gaussian mollifier into Weil's explicit formula for Dirichlet $L$-functions. By defining the spectrally normalized individual mollified sums $\\widetilde S_T(x, a)$ and adopting the virtual character $χ_{1,a}(x) := \\mathbf{1}_{\\{x \\equiv 1 \\pmod N\\}} - \\mathbf{1}_{\\{x \\equiv a \\pmod N\\}}$, the principal character component $χ_0$ cancels identically since $1 - \\overline{χ_0}(a) = 0$. This automatic algebraic elimination erases both the universal logarithmic growth $\\log x$ and the background noise $\\log L(1, χ^2)$. Under the Deep Riemann Hypothesis (DRH), we uncover a hitherto undetected \\textbf{fine-structure bias} (or \\emph{secondary bias}) strictly governed by the special values $\\log L(1, χ)$. We prove that $\\widetilde S_T(x, χ_{1,a}) := \\widetilde S_T(x, 1) - \\widetilde S_T(x, a) = C_N \\cdot \\log L(1, χ_{1,a}) + \\mathcal{O}((\\log x)/\\sqrt x)$ as $x \\to \\infty$, where $C_N > 0$ depends solely on $N$. Consequently, we establish a deterministic multi-way ranking (such as $7 > 3 > 5 > 1 \\pmod 8$) that completely transcends the classical quadratic residue framework.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.28931",
  "author_total": 1,
  "authors": [
    {
      "name": "Shin-ya Koyama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2n21z30xcf3v4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.28931",
  "fulltext_url": null,
  "id": "2n21z30xcf3v4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.28931",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7172276091",
  "pdf_url": "https://arxiv.org/pdf/2607.28931",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2n21z30xcf3v4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.28931",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.28931"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Fine-Structure Hierarchy of Prime Biases and the Universal Dominance of $-1 \\pmod N$",
  "title_text": "The Fine-Structure Hierarchy of Prime Biases and the Universal Dominance of −1 pmod N",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/2n21z30xcf3v4",
  "venue": "arXiv",
  "versions": []
}