The lonely runner conjecture asserts that, for any 
distinct positive integers 
, there is a real
number 
such that
 |
(1)
|
where 
denotes distance to the nearest integer,
not a vector norm. In the running interpretation,
a stationary runner and 
other runners move on a unit-circumference circle, and every runner should at some time be at distance
at least 
from all the others (Hu and Jing 2026).
Hu and Jing (2026) proved that there is an absolute constant 
such that, for every 
and every such speed set,
 |
(2)
|
This is a fixed proportional improvement over 
,
not the full conjectured bound. Hu and Jing (2026) report that ChatGPT suggested
greatest-common-divisor sum estimates and refinements of a cited estimate, and helped
close gaps in their dilation argument. The authors supplied the strategy and framework.
A geometric formulation is the view-obstruction problem for axis-aligned cubes centered at half-integer lattice points. The corresponding problem for Euclidean balls has different obstruction thresholds (Kahlon 2026).
For four distinct positive integer speeds, define the maximum loneliness by
 |
(3)
|
Yi et al. (2026, Theorem 1.2) classify the values strictly below 
as
 |
(4)
|
Thus 
is the sole omitted value from the indicated family. This four-speed formulation
includes a stationary observer, in addition to the four moving runners. The proof
uses an exact search over 800816099 quadruples.
