A Cohomological Programme for the Lonely Runner Conjecture: Toric Arrangements, a Structural Theorem on the Bad Set, and an Honest Account of Approaches Tried

This paper makes three contributions to the study of the Lonely Runner Conjecture (LRC). First, we prove that for all integers k ≥ 2, all m ≥ 1, and all primes p > mk(k+1), the k-tuple m·(k+1)·(1, 2, ..., k) is bad — established by a five-line pigeonhole argument uniform across all k simultaneously. This identifies the unique persistent bad structure and corrects the cohomological target from the full bad set B_k(p) to the coprime bad variety B_k(p)_coprime, whose emptiness for large p is equivalent to LRC. Second, we identify a framework correction: the LRC arrangement is a toric arrangement — level sets of characters on the k-torus — not an affine hyperplane arrangement, and the correct cohomological tool is the toric Orlik-Solomon theory of De Concini-Procesi and Callegaro-Delucchi. Third, we introduce a spacetime reformulation: LRC is equivalent to the statement that every primitive integer worldline in the k-torus intersects the safe region S_k^k = [1/(k+1), k/(k+1)]^k, a region determined solely by k. Two failed approaches are documented with precise failure reasons: a Fourier-analytic argument blocked by the Tao symmetry obstruction, and a Corvaja-Zannier reformulation whose conditions do not hold on the torus. Seven further mathematical directions explored and abandoned are recorded in Annexure B. The paper was developed with AI assistance, disclosed in Annexure A. LRC remains open.