A Preliminary Study of the P vs. NP Problem

A Preliminary Study of the P vs. NP Problem This paper investigates the P versus NP problem through Jing Space Theory (JST) and the Jing Space Coordinate System (JSCS), introducing a unified algebraic framework for multidimensional structural space, computational counting, and local–global relationships. The study is based on the JSCS Core Equation, [R^2=\sum_{j=0}^{N-1(\alpha_jx_j)^2=N(\alpha r)^2,] Within this framework, dimension is interpreted as a structural counting level, while computational time is associated with accumulated structural displacement. The paper develops a universal counting recursion, examines the relationship between polynomial and exponential counting branches, and derives linear arithmetic-operation bounds for normalized recursive structural updates. The analysis further introduces structural normalization, local–global computational correspondence, and zero–infinity singularity transformations to investigate the relationship between solving and verification. Particular attention is given to the conditions under which normalized structural counting can be associated with deterministic polynomial-time computation. The paper proposes a structural interpretation of the P versus NP problem, identifying global and local computation as related representations governed by the same Core Equation. It also examines the additional algorithmic conditions required to connect this structural correspondence with conventional computational complexity classes. This research presents JST/JSCS as a proposed holographic, non-collapse mathematical model of multidimensional space and computational structure, offering an alternative perspective on recursive counting, computational complexity, and the foundations of mathematical logic. Keywords: P versus NP; Jing Space Theory; Jing Space Coordinate System; JSCS Core Equation; computational complexity; recursive counting; polynomial time; structural normalization; singularity transformation; multidimensional space; topology law. 《P 与 NP 问题初探》 本文基于 Jing Space Theory(JST,荆氏空间理论)与 Jing Space Coordinate System(JSCS,荆氏空间坐标系统),从多维空间的统一代数结构出发,研究 P 与 NP 问题、计算计数递归及局部与全局计算之间的关系。 研究以 JSCS 核心公式为基础:[R^2=\sum_{j=0}^{N-1(\alpha_jx_j)^2=N(\alpha r)^2,]。 在这一理论框架中,维度被定义为结构计数层级,计算时间与结构位移的累积相联系。论文建立了普遍计数递归规律,分析多项式计数与指数计数的不同结构分支,并推导归一化递归结构更新的线性算术运算上界。 论文进一步结合普遍归一化法则、局部—全局计算对应关系以及零与无穷大的奇点转换,研究求解与验证之间的结构联系,并讨论归一化结构计数对应确定性多项式时间计算所需的条件。 本文提出 P 与 NP 问题的一种结构性解释:全局计算与局部计算是受同一 JSCS 核心公式约束的两种相关结构表示。同时,论文讨论了将这种结构对应关系与传统计算复杂度类别联系起来所需的算法条件。 本研究将 JST/JSCS 作为一种全息、无塌陷的多维空间数学映射模型,为递归计数、计算复杂度以及数学逻辑基础提供一种不同于传统计算方法的研究视角。 关键词: P 与 NP 问题;荆氏空间理论;荆氏空间坐标系统;JSCS 核心公式;计算复杂度;递归计数;多项式时间;结构归一化;奇点转换;多维空间多维空间拓扑定律。