It might not chamber so easily... most folks shooting semi-autos give the cases a full length sizing, just to make chambering easier. That way you don't run into function issues.
I don't think .004" is out of spec, if I'm understanding you correctly. I believe the no-go gauge won't chamber until you get to .006" over SAMMI spec. And that's just the no-go... the field gauge is going to give you a bit more than that, so I believe your chamber is fine.
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