AI 中文总结
该论文将DCCQ框架从二元推广到多项式计数，定义坐标映射并证明其为实解析微分同胚，涵盖临界线至临界带，且不声称证明黎曼猜想。
AI 中文摘要
我们将离散复补商（DCCQ）框架从二元伯努利计数推广到多项式计数组合。对于m+1个类别，m是独立概率自由度的数目。整数计数向量模公共缩放决定了m维概率单纯形的有理点。基于标准单纯形和对数比坐标几何，对于m≥2，我们定义了完整的多项式DCCQ坐标映射，并证明它是实解析微分同胚。先前建立的二元基线m=1给出临界线坐标，而三元情形m=2给出完整的开临界带；更高的多项式模型保留m-2个额外的实对比度。我们还给出了一对多（one-versus-rest）特化、三元坐标的精确整数格实现，以及其对数比的双曲表示。我们不声称任何零点位置定理或黎曼猜想的证明。
英文摘要
We extend the discrete complex complement quotient (DCCQ) framework from binary Bernoulli counts to multinomial count compositions. For m+1 categories, m is the number of independent probability degrees of freedom. Integer count vectors modulo common scaling determine rational points of the m-dimensional probability simplex. Building on standard simplex and log-ratio coordinate geometry, for m >= 2 we define the full multinomial DCCQ coordinate map and show that it is a real-analytic diffeomorphism The previously established binary baseline m=1 gives a critical-line coordinate, while the ternary case m=2 gives the full open critical strip; higher multinomial models retain m-2 additional real contrasts. We also give a one-versus-rest specialization, an exact integer-lattice realization of the ternary coordinate, and a hyperbolic representation of its log-ratio. No zero-location theorem or proof of the Riemann Hypothesis is claimed.
Comments17 pages, 4 figures
