The Standard Model of modern physics contains dozens of free parameters (such as the finestructure constant ) that cannot be derived theoretically. Abstract "point particles" and "probability waves" do not exist in the universe. This paper establishes the "Spatial Right-Handed Cylindrical Helical Light-Speed Motion" as the sole first-principle axiom. By combining fluid dynamics vortex theorems with Knot Theory, we prove that electrons, protons, and neutrons are essentially topological vortex knots of spatial fluid. Furthermore, this theory rigorously derives the pure geometric origin of the fine-structure constant and directly introduces a second-order topological residual mechanism governed by the "square law of amplitude." This leads to the deduction of an infinite-order residual geometric cascade enclosed entirely within the axiom, constructing the topological full-order energy level formula for the hydrogen atom. This paper not only conducts a rigorous, zero-fitting algebraic examination of this formula but also provides a pure geometric resolution to the "Proton Radius Puzzle," demonstrating the paradigm-shifting predictive power of topological geometry.