Well, actually you need to prove you are a citizen if you fit the profile of an immigrant within 100 miles of the border. And if you don't do so to their satisfaction the border patrol will detain you indefinitely. Yes, in America. https://www.dallasnews.com/news/immigration/2019/07/22/dalla...

If you state you are a US citizen, the onus is on them to prove you aren’t.

There is no law saying that Us citizens have to carry identification pricing they are citizens.

And yes, if they have reasonable suspicion that you’re lying, they can detain you.

Constitutional rights are as it relates to the fourth amendment are a bit fuzzy within 100 miles of the US border. Even as a citizen, you'll need to provide proof or sit in lockup until CBP is satisfied you belong in the country. The ability to provide oral attestation of citizenship was deprecated by the justice department many years ago.

https://www.aclu.org/other/constitution-100-mile-border-zone

How do you prove you don't need to provide proof? Or do the police have to prove that you do?

Nope if they think you should have proof and you don't you can be detained and there's no strict limits on how long you can be detained it's essentially till they're satisfied you've proven your citizenship.