Continuous Coordinate Space with Uniform Fundamental Units and Its Topological Capacity Constraint on Complex Manifolds

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 in classical calculus, it leaves a fundamental geometric paradox: regions with different lengths, areas, or volumes can establish diffeomorphic bijections, yet traditional geometry provides no intrinsic geometric mechanism to explain essential scale differences. This paper constructs an extended continuous coordinate framework by introducing uniformly sized, indivisible fundamental spatial units. The proposed system fully preserves all classical geometric equations, differential rules, and integral algorithms, yielding numerical results identical to conventional theories. The traditional structureless continuum serves only as a limiting approximation when the unit scale approaches zero, whereas the standard continuum cannot reciprocally accommodate spatial structures with underlying unit granularity. In the new framework, coordinate diffeomorphisms only correspond superficial representative points and fail to match all underlying fundamental units. The Jacobian determinant naturally characterizes the scaling ratio of total fundamental units during coordinate transformation, unifying topological mapping and metric scaling. This study further establishes a rigorous topological capacity principle: the stable topological complexity of a manifold region is finitely constrained by its total number of internal fundamental units. Infinite scaling operations cannot generate additional spatial elements. Applied to compact complex manifolds, this intrinsic capacity constraint provides a novel geometric paradigm for the correspondence between Hodge cycles and algebraic cycles, offering an independent and innovative approach to the Hodge conjecture. 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,从几何底层解释霍奇猜想具备有限成立边界、并非全域成立,给出该千禧难题唯一根源性几何解法。