Abstract
By mapping the motion of multiple runners to rotating vectors on a unit circle, this paper invokes Harmonic Analysis and Topological Ergodicity to demonstrate that 'loneliness' is a structural requirement of the system rather than a stochastic occurrence. We develop a novel Fourier-based framework to resolve the Lonely Runner Conjecture specifically for the n = 11 case, providing a rigorous analytical proof for this high-dimensional boundary.
