DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra
Abstract
We present the discrete--complex complement quotient (DCCQ) framework through the ordered Bernoulli-word kernel f(p,n,k)=pk(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=z̅ 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+u2 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 γk, the induced level has the exact decomposition Lk=1/4+γk2=Nk+δk, δk=B̃1 (γk2+3/4), where Nk=⌊ Lk+1/2⌋ and B̃1(x)={x}−1/2. The integer--phase pair (Nk,e2πiδk) 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 ℜ s>1. The framework organizes exact coordinate identities, classical zeta connections, and numerical diagnostics; it establishes neither conditional equidistribution nor a zero-location theorem.
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BibTeX
@misc{yılmaz2026dccq,
title = {{DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra}},
author = {Y. Kenan Yılmaz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.03801},
eprint = {2609.03801},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03801},
}
RIS
TY - UNPB
TI - DCCQ: From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra
AU - Y. Kenan Yılmaz
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.03801
UR - https://arxiv.org/abs/2609.03801
AB - 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=z̅ 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=1/4+γ_k^2=N_k+δ_k, δ_k=B̃_1 (γ_k^2+3/4), where N_k=⌊ L_k+1/2⌋ and B̃_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 ℜ s>1. The framework organizes exact coordinate identities, classical zeta connections, and numerical diagnostics; it establishes neither conditional equidistribution nor a zero-location theorem.
ER -