An Extended Continuum Coordinate Framework Based on Uniform Fundamental Elements — Resolving Multiple Geometric Paradoxes Caused by Unconstrained Infinite Scaling

The standard real continuum model in modern analysis defines spatial points as structureless primitive abstract elements and separates set cardinality from Lebesgue measure into two independent axiomatic systems. Although this approach avoids formal contradictions within classical calculus, it leads to a fundamental geometric paradox: regions with distinct lengths, areas, or volumes can establish diffeomorphic bijections, while conventional geometry lacks an intrinsic geometric mechanism to account for essential scale differences. This paper constructs an extended continuous coordinate framework by introducing uniformly sized, indivisible fundamental spatial units. The proposed system fully retains all classical geometric equations, differential rules and integral algorithms, producing numerical results consistent with conventional theories. The traditional structureless continuum merely acts as a limiting approximation when the fundamental unit scale approaches zero. Conversely, the standard continuum cannot accommodate spatial structures with inherent unit granularity. Within this new framework, coordinate diffeomorphisms only correspond to superficial representative points and fail to reflect the underlying aggregate of fundamental units. The Jacobian determinant naturally characterizes the scaling ratio of total fundamental units under coordinate transformation, unifying topological mapping and metric scaling. 本文构建带有均匀不可再分基础单元的连续坐标空间模型,区分微分同胚表层代表元映射与底层空间单元集合体量差异,建立拓扑容量约束原理。该理论补齐经典连续统底层公理缺陷,证明无限缩放无法创造新增空间单元。这套几何框架一方面为紧复流形上霍奇循环与代数循环的对应关系提供全新几何范式,给出霍奇猜想根源性几何解法;同时消解二维挂谷–贝西科维奇等一系列依托无约束无穷假设形成的经典几何悖论。