Heluo Geometry - An Introduction From Discrete Sampling to Universal Isomorphism

This paper is the Introductory Volume of the Heluo Geometry system. Heluo Geometry is a new geometric paradigm whose core thesis is: discrete precedes continuous, sampling precedes manifold, structure precedes metric. The system consists of a series of independent papers: the Mathematical Foundations volume establishes a rigorous axiomatic reconstruction from three axioms to the uniqueness of the measurement function and the three-level quantization of Sheng-Ke curvature; the Scientific Applications volume demonstrates the cross-scale projections of this geometric structure in physics, chemistry, biology, cosmology, and aerospace engineering. This Introduction addresses three fundamental questions: What problem do we face? What have we discovered? What does this mean? Core Discovery: From three axioms, discrete sampling geometry necessarily yields the measurement function f(k) = cos(5πk/6) and the three-level quantization of Sheng-Ke curvature (Extremal / Transition / Saddle). This mathematical structure reappears in the discrete scale invariance (DSI) of condensed matter physics and the gauge group dimension decomposition (8+3+1=12) of particle physics, constituting cross-scale isomorphic evidence. Core Conclusion: Mathematics is not a human invention. Mathematics is the underlying code of the universe. Physics is the projection of mathematics at macroscopic scales. The universe is the instantiation of mathematics at all scales from micro to macro.