I think you guys are technically correct, but I would rather be safe than sorry.
(I have always heard that you have to carry around a copy of the paperwork. Probably it was more "avoid trouble" advice-at least in this state.)
If I shot at public ranges I would probably carry paperwork with me. Since I only shoot on private land I don't worry about it too much.
I do have copies of everything saved to the cloud and I can access them from my phone or any computer if I absolutely need to.