sorry for going off-topic (or maybe it's not off topic)
but I have an idea... basically this design should offer more less the same performance as yours
it's pretty self-explenatory....
there is a 3 way spool valve

when it is the 1st positon it directs air into a chamber which is attached to a single air cylinder... pressure extends the air cylinder and it actuates the spool valve
in the second position air is directed from the chamber and aircylinder to the 3rd port (the barrel)
as soon as the air cylinder retracts and the valve switches to the 1st postion the cycle repeats
EDIT
it's not that I suggest him quiting his design or that I am going to try it myself in the near future... let say that posting ideas in his threads dedicated to creating a fullauto valve (yeah there were many of them) is a kind of tradition...
this post is just to show that you don't need a solenoid to use a spool valve for small bore full auto guns... I spent some time playing with 3 way valves and air cylinders and I am sure it will work