The ATF has the authority to regulate the manufaturing, sale and transportation of firearms. Theirs is a discrectionary power extended by congress through the ability to levy taxes. If you really look at the ATF regulations the locus of authority comes from the power to tax and regulate taxable manufactured goods. I do not know if this power has ever been challenged up to the SUpreme Court. It should be a case for the Supremes because it is a Constitutional issue, a matter of taxes and interstate commerce. But so far, the justices have stayed away from anything that has to do with the Second Amendment.
I agree with Zot. You are asking for more trouble than it may be worth. If you are dead set on doing this, get the proper forms, pay the $200 fee, claim you are making the sub-gun for "experimental, educational and historic" purposes, wait your six months for approvals, then go for it. There are plenty of LEO's and ATF agents who would simply love to do their part in getting you free room and board at the local pen for having an illegal machine gun. Even pro-2nd leo's I know wouldn't think twice about busting someone with an illegal firearm.
In a nutshell: 1) forget it, go semi-auto. -or- 2) buy one and file the proper paperwork. -or- 3) jump through all the hoops, pay the fees and wait for permission and do a "U-build it". The choice is yours.
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You can find the price of freedom, buried in the ground.