{
  "abstract": "We present the discrete--complex complement quotient (DCCQ) framework through the ordered Bernoulli-word kernel $f(p,n,k)=p^k(1-p)^{n-k}$ and its inverse-integer level sets. The binary level $2^{-n}$ selects $p=1/2$ as the unique real split-independent anchor. Imposing the additional continuation rule $1-z=\\overline z$ gives the conjugation-symmetric line $z=1/2+iu$. On the central normalized branch, the inverse-level split $κ=k/n$ also has real part $1/2$. The quadratic coordinate $Q(z)=1/4+u^2$ identifies complementary conjugate points, has minimum $1/4$, and has an explicit two-sign inverse; restricting its values to integers defines an arithmetic quantization. For supplied positive critical-line zero ordinates $γ_k$, the induced level has the exact decomposition \\[ L_k=\\frac14+γ_k^2=N_k+δ_k,\\qquad δ_k=\\widetilde B_1\\!\\left(γ_k^2+\\frac34\\right), \\] where $N_k=\\lfloor L_k+1/2\\rfloor$ and $\\widetilde B_1(x)=\\{x\\}-1/2$. The integer--phase pair $(N_k,e^{2πiδ_k})$ retains the level and the positive ordinate. Prime-exponent coordinates describe the factorization of the integer component, while conditional Fourier moments formulate an open question about the residual phases. The finite numerical diagnostics are descriptive. A distinct complex exponent $s$ gives an inverse-level representation of the supplied term $m^{-s}$; direct substitution cancels the auxiliary coordinates. These terms assemble into the classical Dirichlet series and Euler product for $\\Re s>1$. The framework organizes exact coordinate identities, classical zeta connections, and numerical diagnostics; it establishes neither conditional equidistribution nor a zero-location theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03801",
  "author_total": 1,
  "authors": [
    {
      "name": "Y. Kenan Yılmaz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/35fsm1pw1cpr5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03801",
  "fulltext_url": null,
  "id": "35fsm1pw1cpr5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03801",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7208773465",
  "pdf_url": "https://arxiv.org/pdf/2609.03801",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/35fsm1pw1cpr5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.03801",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03801"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra",
  "title_text": "DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/35fsm1pw1cpr5",
  "venue": "arXiv",
  "versions": []
}