信息基数：集合论、分形几何与解析数论的统一框架
Informational Cardinality: A Unifying Framework for Set Theory, Fractal Geometry, and Analytic Number Theory
AI总结：
本文提出信息基数作为统一框架，通过引入本质素数集P_ess并计算其Hausdorff维数，揭示了素数驱动分形与经典Cantor集在几何复杂度上的本质差异，并建立了与黎曼zeta函数零点分布的联系。
摘要：
本文研究了一类由算术序列控制构造的确定性分形。我们引入了本质素数集P_ess，它是使用模4素数序列构造的Cantor集变体。我们计算了其Hausdorff维数dim_H(P_ess)，并分析了其几何复杂度。与经典的三分Cantor集C_{1/3}相比，我们证明虽然两个集合都是不可数的且具有相同的基数，但它们不同的分形维数（dim_H(C_{1/3})与P_ess的计算维数）反映了它们在几何复杂度上的根本差异。此外，我们提出了该素数驱动分形的密度与黎曼zeta函数零点分布之间的潜在联系，并通过构造分形零点集Z_F将其形式化。该框架为解析数论提供了新的几何视角，展示了素数的精细结构如何被编码在确定性分形几何中。
英文摘要：
This paper investigates a class of deterministic fractals whose construction is governed by arithmetic sequences. We introduce the essential fractal prime set P_{ess} , a variant of the Cantor set constructed using the sequence of prime numbers modulo 4. We compute its Hausdorff dimension, \dim_H(P_{ess}) , and analyze its geometric complexity. In contrast to the classical middle-third Cantor set C_{1/3} , we demonstrate that while both sets are uncountable and share the same cardinality, their differing fractal dimensions (dim_H(C_{1/3}) versus the computed dimension of P_{ess}) reflect a fundamental difference in their geometric complexity. Furthermore, we propose a potential connection between the density of this prime-driven fractal and the distribution of zeros of the Riemann zeta function, formalized through the construction of a fractal zero set Z_F . This framework provides a novel geometric perspective on analytic number theory, illustrating how the fine-scale structure of primes can be encoded in deterministic fractal geometries.
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