This paper aims to construct a unified framework between discrete number theory, differentialgeometry, and continuous quantum physics through the "Topological Residual Theory." The researchtakes the 19th term of the Fibonacci sequence (Fu2081u2089 = 4181) as the core node. First, it reveals itsmathematical miracles in semiprime decomposition and fractal dimension reduction. Subsequently,this paper completely derives the differential geometric correspondence between the Binet residualand the torsion of the cylindrical helix, and details the calculation of the torsion limit (τ → 1/3) of theMarkov triples related to 4181. From first principles, it rigorously proves that this node defines theglobally stable geodesic line on the torus with no self-intersection and minimum energy.Furthermore, this paper introduces renormalization group scale transformation, dividing theFibonacci sequence by ten thousand (10u2074) and extracting its sine value, mapping it as the transverseperiodic projection of cylindrical spiral motion. Through key transition data tables and visualizationcharts, this paper intuitively demonstrates the physical mechanism by which the macroscopictopological node 4181 still maintains the minimum energy ground state after being scaled down byten thousand times (0.4181°). It proves that the fine structure constant (α ≈ 1/137) is essentially themicroscopic transverse geometric projection residual (sin(0.4181°) ≈ α) that maintains the cylindricalhelix moving at the limiting velocity u without collapsing, and that Fu2081u2089 holds an absolutelyirreplaceable critical phase transition position in the evolutionary cycle. Finally, this paper proposesthat the space around objects is structured on the basis of right-handed cylindrical spiral motion,providing solid geometric and dynamic support for the holographic universe principle