You're asking about CBP form 4457.

"Certificate of Registration for Personal Effects Taken Abroad"

Fill it out prior to leaving the USA, made my re-entry into the USA, with my hunting rifle, quite easy. The rifle was inspected by US agents on my departure and return from Washington State. . The Canadians were actually pleased to see it, and just charged me their $25 entry fee for the rifle.

https://www.cbp.gov/document/forms/form-4457-certificate-registration-personal-effects-taken-abroad

Regards, Guy