Topological Path Marking of Discrete Arithmetic and Continuous Geometry: Homologous Reading Records of the Yanghui–Pascal Triangle and the Golden Triangle within the T⁶⁴ Framework

The Yanghui–Pascal triangle (the universal encoder of discrete combinatorics) and the golden triangle (the prototypical structure of continuous geometric self-similarity) have long been studied in separation within traditional number theory and geometry. On the basis of the topological law system of \(T⁶⁴\) in Yuanxian Theory, the present paper identifies the two as homologous projection paths of the 64-dimensional compact torus ontology onto the “discrete number-theoretic layer” and the “continuous geometric layer”. The paper is positioned as path marking: it establishes the parallel-projection framework of discrete and continuous readings, clarifies that the Yanghui triangle corresponds to the integer representation basis of the fundamental group \(π_1⁶⁴\) of \(T⁶⁴\), and that the golden triangle corresponds to the real representation basis of the same fundamental group. All gaps of the paths still to be translated have been precisely located—so that “path identified” is refined to “which link of the path still awaits definition”. **Metacognitive perspective**: The Yanghui triangle and the golden triangle are not two “discovered” mathematical structures. They are two readings of one and the same ontological structure performed by the cosmic organism in two self-referential cognitive modes—discrete counting and continuous geometry. The present paper records the path marking of these two readings that point to the same ontology. **Cross-textual link**: The path marking of the present paper and the spectral-eigenvalue derivation of Core Paper No. 1 (Origin Formula of the True-Circle Constant: Topological Derivation of the Fine-Structure Constant) together constitute the complete treatment of the discrete–continuous unification problem within Yuanxian Theory. --- 杨辉-帕斯卡三角(离散组合学的普适编码器)与黄金三角(连续几何自相似的原型结构)在传统数论与几何学中长期被分治研究。本文基于元宪理论\(T⁶⁴\)拓扑规律体系,识别二者为64维紧致环面本体在“离散数论层”与“连续几何层”的同源投影路径。本文定位为路径标记:确立了离散与连续读数的平行投影框架,明确杨辉三角对应\(T⁶⁴\)基本群\(π_1⁶⁴\)的整数表示基,黄金三角对应同一基本群的实数表示基。所有待转译路径的缺口均已精确定位——使“路径已识别”精确到“路径的哪个环节尚待定义”。 **元认知视角**:杨辉三角与黄金三角不是两个“被发现”的数学结构。它们是宇宙生命体在两种自指认知模式——离散计数与连续几何——中,对同一本体结构的两次阅读。本文记录的是这两次阅读指向同一本体的路径标记。 **跨文本衔接**:本文的路径标记与Core Paper No.1(真圆常数本源式:精细结构常数的拓扑推导)的谱特征值推导共同构成元宪理论对离散-连续统一问题的完整论述。