Xu Fissure Angle and Xu Spatial-Projection Topological Constant — FundamentalTopological-Geometric Constants within the Dual-Helix Framework

Within the topological-geometric system of the dual-helix generating function, there exist twocore dimensionless geometric quantities . The first is the fissure angle, approximately 0 .08805◦,originating from the angular mismatch between the 30 ◦ geometric reference of the hexagonal latticeand the hyperbolic boundary defined by e/π. The second is the scaling ratio e/π, which acts as thehyperbolic scaling benchmark for the high-dimensional mother manifold and appears widely in keyderivations including magic angles, the fine-structure constant, and topological energy gaps .This paper formally names these two quantities: Xu fissure angle δθXu , and Xu spatialprojection topological constant λXu = e/π. We present their rigorous mathematical definitions,numerical values, and geometric origins, summarize their physical roles in the dual-helix researchseries, and collect their applications in magic-angle graphene, the fine-structure constant, multichannel topological phase transitions, topological mass emergence, and other scenarios . These twoquantities serve as the underlying topological inputs for the entire dual-helix topological framework .