Here’s the rule.
OBR 6.05 A batter is out when—
(k) In running the last half of the distance from home base to first base, while the ball is being fielded to first base, he runs outside (to the right of) the three-foot line, or inside (to the left of) the foul line, and in the umpire’s judgment in so doing interferes with the fielder taking the throw at first base, in which case the ball is dead; except that he may run outside (to the right of) the three-foot line or inside (to the left of) the foul line to avoid a fielder attempting to field a batted ball;
Rule 6.05(k) Comment: The lines marking the three-foot lane are a part of that lane and a batterrunner is required to have both feet within the three-foot lane or on the lines marking the lane. The batter-runner is permitted to exit the three-foot lane by means of a step, stride, reach or slide in the immediate vicinity of first base for the sole purpose of touching first base.