[pclmulqdq]

It doesn't have to be a document, but I think it does need to be a thing. For example, most courthouses have an eagle feather you can swear oaths on if you want (several native American tribes do this).

I think you can also "affirm" instead of "swearing an oath," and that avoids any need for an object.

[JohnFen]

Why does there need to be a thing?

[pclmulqdq]

I think it's traditionally supposed to be a representation of a deity who is "backing" your oath.