Yes, he's that smart.

Those are really high scores. You might look into the Davidson Young Scholars Program if you haven't already. Those scores would qualify, and it sounds like your child fits the bill:
http://www.davidsongifted.org/youngscholars/ . It's a nonprofit, and many of our kids are YSers.
There are plenty of people who are making it work for their kids within a public school. Much depends upon the school--even the specific teachers the child gets within the school--and the child. There is no one right answer for all HG+ kids; it's all a case-by-case basis.
We looked at all 3 options you mention: public, private GT and homeschooling. In our case, with a resistant school system, a very small GT school with few kids, and a very active and diverse homeschooling community in our area, homeschooling was the clear winner...for now. Ask me again in 6 months and my story might change though.

It's a reality that pretty much all of us with HG+ kids have to re-examine our options about that often because the ideal situation becomes not nearly so ideal after just a short time.
If you think the school is serious and will work with you, then it's fine to give that a try. If your gut is telling you that they're feeding you a line, then it's never a bad idea to explore other options. You can even do both at the same time.
Homeschooling often scares people who know nothing about it. There are lots of myths out there that just aren't accurate. So if you have questions, there are a number of us here who are homeschooling and are happy to give you more info if you want to ask about it. We've all been in your shoes and we're happy to help.
