Maximum number of pairwise equidistant lines in $\mathbb{R}^3$

It is proved that no eight affine lines in $\mathbb{R}^3$ have a common positive pairwise distance. Together with the known construction of seven such lines by Bozóki, Lee and Rónyai, this establishes the exact maximum of seven. The upper bound combines a five-line obstruction from hyperbola secants with exact verification over 135 uniform rank-three oriented-matroid types. It settles the full stated maximum-cardinality question; it does not classify all seven-line configurations or address cylinders of unequal radii.

It is proved that no eight affine lines in $\mathbb{R}^3$ have a common positive pairwise distance. Together with the known construction of seven such lines by Bozóki, Lee and Rónyai, this establishes the exact maximum of seven. The upper bound combines a five-line obstruction from hyperbola secants with exact verification over 135 uniform rank-three oriented-matroid types. It settles the full stated maximum-cardinality question; it does not classify all seven-line configurations or address cylinders of unequal radii.