AI 中文总结
该研究分析有序伯努利词核及其逆整数水平集的几何，将临界线零点纵坐标对应的水平分解为整数与伯努利残差，关联狄利克雷级数与欧拉乘积，明确区分相关恒等式、检验等内容，未声称证明黎曼假设。
AI 中文摘要
我们研究有序伯努利词核函数f(p,n,k)=p^k(1-p)^(n-k)及其逆整数水平集生成的几何。二元水平2^(-n)选取p=1/2作为唯一与实分割无关的锚点。在保补复延拓下，该对变为z=1/2+iu和1-z=1/2-iu，在引入任何ζ函数输入前产生共轭对称的垂直几何。二次坐标Q(z)=z(1-z)=1/4+u²在中心点处有尖锐极小值，并允许精确整数量化。对于临界线零点纵坐标γ_k，诱导水平L_k=1/4+γ_k²可精确分解为L_k=N_k+δ_k，其中N_k是最近整数，δ_k是周期一阶伯努利残差。圆化给出Z_k=exp(2πiδ_k)，将γ_k² mod 1分离为残差相位变量。唯一因式分解将整数壳分解为素生成元坐标，而不同的复指数s将同一构造提升到狄利克雷原子m^(-s)，连接狄利克雷级数与欧拉乘积集合。精确恒等式、经典ζ函数关联、数值控制和开放的条件性魏尔检验被明确区分，未声称证明黎曼假设。
英文摘要
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.
Comments18 pages, 2 figures. Substantially revised: clarified inverse-level branch handling, Bernoulli/zeta/zero connections, statistical scope, rounding and phase conventions, and numerical provenance; streamlined figures and exposition. Main propositions and conclusions unchanged
