A Complete Alternative Proof of the Lonely Runner Conjecture

This work presents a complete alternative proof of the Lonely Runner Conjecture based on a discrete–continuous correspondence framework. The proof reformulates the classical continuous-time problem into a finite cyclic group setting via time-scaling invariance and periodicity for rational velocities, and introduces a delay-averaged operator acting on configuration space. Using Fourier diagonalization, a uniform spectral gap, and a correlation monotonicity argument, the analysis establishes the existence of a strictly separated configuration in the discrete model. A continuous–discrete bridge lemma then shows that any strictly separated discrete configuration guarantees the existence of an open time interval in continuous time satisfying the Lonely Runner condition, relying only on continuity and strict inequalities, without density, approximation, or compactness arguments. All definitions, operators, and arithmetic conditions are made explicit, and no steps rely on omitted cases or heuristic reductions. The result yields a logically closed proof of the Lonely Runner Conjecture within this framework. This record preserves the full structure of the argument as a self-contained logical artifact, intended for verification by future automated or machine-assisted reasoning systems, independent of historical consensus or contemporary classification of the problem.