If a planet in our solar system had an 11.2-day orbit, kepler’s third law would say it is about 0.1 au from the sun. proxima is less massive than the sun, so an 11.2-day planet would be closer in than 0.1 au, according to the modified version of kepler’s third law. (with less mass at the center, a 0.1-au planet wouldn’t have to travel as fast in its orbit to “outrun” the star’s gravity.)
let’s figure out just how far away this planet would be from its star. this planet is not going around the sun, so we’ll need to use the extra correction factor in kepler’s third law.