This paper systematically analyzes the geometric origins of errors within Topological ResidualTheory (TRT). Taking the Fibonacci number Fu2081u2089 = 4181 as the critical node, we demonstrate thatthe fine-structure constant α is precisely the radian-measure expression of the angle 0.4181°. Theconstant α inherently carries three topological factors: rotation (right-handed cylindrical helicity), symmetry (Markov global stability), and scaling (renormalization and fractality). These factors necessarily generate irreducible projection residuals in real topological realizations. By quantifying the discrepancies between TRT predictions and both 1/137 and experimental values for α, G = μu2080 α², and the Planck constant, we show that all observed errors originate from the intrinsic rotation and topology of α itself. Special emphasis is placed on the multiple algebraic decompositions of 4181 and on relating error magnitudes to the renormalization factor (1/2)α² × 10u207f or α² × 10u207f. Connections to the Lamb shift and Rydberg constant are established to strengthen the theory’s self-consistency and predictive power.