Reconstruction of Poincar\'e Issue

English Description 【Reconstruction of the Poincaré Issue】 This paper reconstructs the Poincaré problem within Jing Space Theory (JST) and its structural coordinate system (JSCS). It starts from the question itself: how local three-dimensional structure and global simple connectivity determine the topology of a complete space. The JSCS Pythagorean core equation, R² = Σ(α_j x_j)² = N(αr)², organizes dimensional generation, global–local radius normalization (R_N² = N, r_N² = 1/N), coefficient variation, and structural branches. With position-dependent coefficients, the core equation is shown to be universal: every closed three-manifold is exactly the solution set of a core equation. Treating zero and infinity as normalized spaces rather than points, the three-sphere is realized as a two-sided Omega structure. A closed three-manifold is proved to be homeomorphic to S³ if and only if it admits such a structure. The shape of the pole spaces (ball, shell, ring, or fibered brush) is shown to govern global topology. Gluing twists are represented and calculated in JSCS symbols: the sign ±, the branch numbers 2 and √2 with their reciprocals, and the identity L⁸ = −1. Twists preserve the core equation and are measured by the fundamental group. Simple connectivity excludes every Möbius-type twist, and for ball- and ring-shaped pole spaces it yields S³. The paper formulates the general case as a question about the coefficients α_j. 中文简介 【庞加莱问题的重构】 本文在景空间理论(JST)及其结构坐标系(JSCS)中重构庞加莱问题,出发点是问题本身:局部三维结构与整体单连通性如何决定完整空间的拓扑。JSCS 的勾股核心方程 R² = Σ(α_j x_j)² = N(αr)² 统一组织维度生成、整体与局部半径归一化(R_N² = N,r_N² = 1/N)、系数变化和结构分支。 当系数随位置变化时,核心方程具有普适性:每个闭三维流形都恰好是某个核心方程的解集。本文把零与无穷视为归一化的空间而非点,由此将三维球面实现为双侧 Omega 结构,并证明:闭三维流形同胚于 S³,当且仅当它具有这种结构。文中还说明,极点空间的形状(实心球、球壳、环、纤毛球)决定整体拓扑。 拼接扭转用 JSCS 符号表示并精确计算,包括方向符号 ±、分支数 2 与 √2 及其倒数,以及恒等式 L⁸ = −1。扭转保持核心方程不变,其阶由基本群度量。单连通排除一切莫比乌斯型扭转;对实心球与实心环两类极点空间,单连通推出 S³。一般情形则表述为关于系数 α_j 的问题。 Keywords / 关键词: Poincaré problem; JST; JSCS; core equation; Omega structure; pole space; twist; three-sphere / 庞加莱问题;荆空间理论;核心方程;Omega 结构;极点空间;扭转;三维球面