Each capture paper that we see refers to a certain US Army Circular. This is one of the first circulars, numbered 217 from June 1944. Its worth a read.
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That was a little too broad, so they honed it up a little bit and released circular 353.
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Notice, nothing prohibiting the bring back of machine guns, and that became a problem so they issues circular 155. This created the standardized capture paper that we see the majority of. They all reference circular 155. If you look at the top of the third page of circular 155, you can see it bans auto weapons from being sent home. Apparently there became issues with the need for a form 6 to bring in an MG to the USA caused logistical issues.
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Source: Mark Castel of the P38 forum did a presentation on capture papers in about 2016 at the SOS P38 collectors meeting. I saved these circulars for my records but he was the one that pulled them originally. He has a registered papered MP40.