Source is Part III § 2 of the US flag code enshrined by 4 U.S.C. § 5.

On the admission of a new State into the Union one star shall be added to the union of the flag; and such addition shall take effect on the fourth day of July then next succeeding such admission.

This isn’t legally binding – you can fly a US flag with five stars representing the CCP and its four governed social classes if you want. But it does represent official federal guidance on the design of the US flag. So if Washington, D.C. were added as a state, you wouldn’t see the 51st star appear on federal buildings until the subsequent Independence Day.