you can prove some things do or don't exist...

Yes, specific claims. Non-specific claims, like this one, can't. It could be that we just aren't aware of an older record, but that can't be proven. It's also really difficult to even prove it's the oldest that we know of. That'd take at least many hours of research, and that's if you're lucky and find a counter-example early. Most likely it'd be days/months.

But the burden of proof works the same with God as with other claims.

Yes. The burden of proof is the same. That's not what I'm saying. I'm saying that you can take their word for it or not. Asking them to prove it is insane. This is a casual internet forum. No one is spending the time or effort to get anywhere near a proof for you. If you don't want to believe it then fine. Just don't. If you want to prove it yourself, you're welcome to do so, and I'm sure posting it would be appreciated. Asking others to is needlessly aggressive for a casual internet trivia post.

Proof, in this case, would be somehow showing that something doesn’t exist. You can provide evidence, but not proof. You can provide proof against this, by just providing an older record though.

it seemed like you just said you can't show proof, then demonstrated how you can provide proof? Maybe I miss your point here.

I demonstrated how you can prove it wrong, not right. There's no way to know what isn't known. Proving it true is effectively impossible. Proving that it's the oldest that we know of is still incredibly challenging.