Fourteen lonely runners

Abstract

We prove the Lonely Runner Conjecture for fourteen runners by a computer-assisted extension of the finite-checking framework of Sungkawichai and Trakulthongchai. With one runner stationary, their thirteen-runner result supplies the induction input, and their reduction leaves finitely many modular calculations indexed by primes. We certify 111 such prime gates with $\sum_p \log p>681.5292$, exceeding the required threshold $\log B_{13}<670.3498$ by more than $11.17$. For each gate, an exhaustive generator constructs the level-one improper family, a sequence of exact binary lift filters eliminates all but two multiplicative orbits, and an exact branch-and-bound computation treats each remaining fiber of $7^{13}$ lifts at the mixed level $14$. Every no-witness completion remaining at that level has all coordinates divisible by $7$ and is therefore proper by the gcd clause in the framework definition. The same two persistent orbits occur at every closed gate; this is an empirical universality finding, not a theorem beyond the verified gate set. Per-gate certificates and a separate audit of all 111 closed gates support the computation. We also report every gate at which the chosen pipeline failed to close.