I would verbally reward (sincere praise) any still moments, even as short as 2 seconds in the beginning. I wouldn't try it when he's having a difficult time, but more when he is relaxed and happy and still. The more you call attention to the desired behavior, the more he will internalize it as part of his idea of 'who he is.'
I would also try to learn deep breathing techniques and then try to share them if they are something genuine that you can see a use for. It's possible that his gymnastic coach might be recruited to teach deep breathing as part of a 'pre-competition' psych up. Sort of
'The Art of Learning: A Journey in the Pursuit of Excellence' by Josh Waitzkin
A visual you can try is to imagine a diamond in the center of the earth and the pressure it takes to change a lump of coal into a diamond. It may be that when he feels the most urge to move, that he could be encouraged that 'an idea a precious as a diamond is trying to be born - lets sit together and breath all that twitch-desire into the diamond and see what kind of a diamond is being formed'
I have no idea if that would work, but maybe it would, or maybe it will inspire you to think of something similar.
I love for kids to have a flexible range of alternatives for their behavior, if possible, but if not possible right at this age, then there is the alternative of working on keeping yourself as relaxed as possible while he fidgets so that your quiet loving presence can eventually provide leadership. Obviously if it makes you uncomfortable, then he'll be more discomforted, and twitch more and cycle round and round.
My hunch is that if he looks focused to the teachers at 7, when there suspicion is the highest, that you won't have that issue, but of course, it may be crop up.
You may also find that in addition to wearing him out physically with the gymnastics, you need to give him some 'afterschooling' at his actual challenge level to wear him out mentally (if he isn't getting sufficient mental stimulation during the day.)
Love and More Love,
Grinity
