Has your son tried other activities that might be appealing to high-IQ, math-minded children? What about chess, Lego robotics clubs, or even card trading games that meet at comic book stores, etc? It seems to be that with two math PhD parents and the level of achievement and instruction you describe, you might be disappointed in these camps pedagogically, and the price is so high. I'd look for other ways to connect with similar children socially first, if that is a main goal. You might be able to find peers for free. I don't know where you live or how likely it is that there are other children near you in somewhere near the same ballpark, though.
Hmmm DS7 is really interested in mathematics itself, rather than other "merely" mathy things. If it's his free time and he's not doing math, then he'll be playing in the park, watching cartoons, going on roller coasters, and other non-mathy fun things.
However you do have a point that maybe he should develop some of those interests and meet kids with at least some compatibility, and the activities could be cheap or free.
I just remembered that there is a local gifted middle school with a Math Club that might let DS7 attend some time in the future when he's mathematically/socially ready. (I'd have to get the timing of that right, as I don't want to "pushy parent" our way in, and then have it go wrong.) That would be the most similar thing to one of the math summer camps, except it would be local and free.
As to "It seems to be that with two math PhD parents and the level of achievement and instruction you describe, you might be disappointed in these camps pedagogically, and the price is so high." that's basically right except I don't think DS would be "above" these camps academically. At the highest level camps I think he'd just be "typical". (And they might not think he's good enough and not let him in, and maybe there's some camp where they'd be right.) What we don't want is some camp or activity that's too low level, especially if expensive, and that's why I really wanted a good idea of how the camps/activities could be "rated" for quality of organizers and participants.
