<p>This is true for PHD’s, too. Not set in stone, but the undergrad would have to be absolutely exceptional in order to stay in the same dept for grad school.
However, your son does sound exceptional (not because you’re his mother, but because of his laser focus on a discipline in which he’s excelled since the age of 5.) So, who knows, he may be the exception. And when grad school comes around, he may be more interested in Stanford or Harvard. 
He should email the Head of Dept at UCB to ask an honest opinion on the matter.</p>