@Springbird
Thanks for all the advice. However. I am still not getting it - why Masters? If he is on campus, and wants to socialize with anyone, he is free to do it, whether he is frosh or grad student, or even 5-th year grad student in PhD program. There is no need to sit in Masters. He can socialize with anyone, period. No one is going to ask him for an ID to verify he is 16 or 24 (he is 16 now, will be 17 next school year).
Look. He is used to be in the older crowd - it started when he was in Community College at age 11, and explaining Differential Equations to his 22-year old classmates. In fact, if anything, he always was easier socializing with adults than with kids. And yes, he does have interest outside of math.
During the school breaks, like now, - he goes to his job (internship), from 8 to 4pm, sits in the office, interacts with co-workers, with his boss, with his co-authors over e-mail, writes his proofs of theorems and the code to illustrate it - i.e., exists in an adult office environment. He is a regular young scientists, like other some other young guys on the floor, some of whom are 23, but no one else is 16. But no-one, unless specifically told, knows or suspects that he is as young as he is.
As a scientist myself, I really, REALLY do not understand what the fuss is about! It’s a guy, with publications, with research, with respect from his co-authors, etc. Why would his age matter? So far, all adult scientists that I talked with, including professors at Universities (several of them), - all are telling me that all what counts is his professionalism and his math ability, - for both admissions and his functioning as a PhD candidate. It is not undergraduate admissions, where you have to describe what clubs at the University you are going to join to contribute to the culture, or count how many community hours you have served.
The application season is over anyway. He made his choices, he applied everywhere he wanted, and everywhere his science boss (from his still active internship) wanted him to apply. Let’s see how it all pans out. It’s out of our hands now, for the most part.