用伯努利核与素数结构揭示黎曼零点的几何量化规律
From Ordered Bernoulli Levels to Critical-Line Geometry: Integer Quantization, Bernoulli Residual Phase, and Prime-Power Spectra
- 基于伯努利权重构建有序几何，实现零点坐标的整数量化
- 零点虚部平方模1对应伯努利残差相位，具周期性特征
- 连接狄利克雷级数与欧拉乘积，揭示素数生成坐标结构
我们研究有序伯努利词核 f(p,n,k)=p^k(1-p)^(n-k) 及其逆整数等值线生成的几何结构。二进制层级 2^(-n) 将 p=1/2 确定为唯一实数分裂无关锚点。在保持补集的复数延拓下，一对 z=1/2+iu 与 1-z=1/2-iu 构成共轭对称的垂直几何，无需引入ζ函数即已成立。二次坐标 Q(z)=z(1-z)=1/4+u^2 在中心点取最小值，并可精确整数量化。对临界线零点纵坐标 gamma_k，诱导层级 L_k=1/4+gamma_k^2 被精确分解为 L_k=N_k+delta_k，其中 N_k 为最近整数，delta_k 为第一类伯努利残差。圆化处理得 Z_k=exp(2 pi i delta_k)，使 gamma_k^2 mod 1 成为残差相位变量。唯一分解将整数壳层解析为素数生成坐标，而复指数 s 的提升使该构造适用于狄利克雷原子 m^(-s)，连接狄利克雷级数与欧拉乘积结构。精确恒等式、经典ζ关联、数值控制及开放的条件韦尔检验均明确分离。不宣称证明黎曼猜想。
原文摘要 · Abstract (English)
We study the ordered Bernoulli-word kernel f(p,n,k)=p^k(1-p)^(n-k) and the geometry generated by its inverse-integer level sets. The binary level 2^(-n) selects p=1/2 as the unique real split-independent anchor. Under complement-preserving complex continuation, the pair becomes z=1/2+iu and 1-z=1/2-iu, producing a conjugation-symmetric vertical geometry before any zeta-function input is introduced. The quadratic coordinate Q(z)=z(1-z)=1/4+u^2 has a sharp minimum at the central point and admits an exact integer quantization. For critical-line zero ordinates gamma_k, the induced levels L_k=1/4+gamma_k^2 are decomposed exactly as L_k=N_k+delta_k, where N_k is the nearest integer and delta_k is a periodic first-Bernoulli residual. Circularization gives Z_k=exp(2 pi i delta_k), isolating gamma_k^2 mod 1 as the residual phase variable. Unique factorization resolves the integer shells into prime-generator coordinates, while a distinct complex exponent s lifts the same construction to the Dirichlet atoms m^(-s), linking the Dirichlet-series and Euler-product assemblies. Exact identities, classical zeta connections, numerical controls, and open conditional Weyl tests are kept explicitly separate. No proof of the Riemann Hypothesis is claimed.
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