Kravitz’s denominator conjecture for the Lonely Runner spectrum

Kravitz’s conjecture proposed a specific rational-denominator pattern for values in the Lonely Runner spectrum. It was proved in low dimensions but later disproved in general, including by a six-speed example; the current question concerns the remaining six-speed pattern and its finite exceptions.

Known results

• Kravitz proved the original spectrum conjecture for $n=2,3$ and obtained partial results for $n=4,6$.
• Fan and Sun exhibited counterexamples for $n=4$ and $n=6$, including $\operatorname{ML}(5,6,11,17,23,28)=8/51$ (2023).
• The amended pattern allows $\operatorname{ML}(v_1,\ldots,v_n)=s/(ns+k)$ with $k\leq n$; computations observed $k\in\{1,3\}$ for $n=6$, but the explanation was previously conjectural.

September 3, 2026 six-speed theorem

A new paper determines three infinite two-parameter families, characterizes when $k=3$ occurs, and reports an exhaustive search of more than two billion sextuples with speeds at most $110$ finding no exception. It establishes the odd-denominator pattern apart from finitely many six-speed and five-speed tuples, but does not settle those remaining tuples.

Current status (as of September 2026): The original universal conjecture is false, while the revised six-speed denominator pattern has claimed progress but remains unverified and leaves finitely many tuples open.