They currently go with whatever the person looks like. I'm not saying it's right, but it does exist. Before the crackdown on DEI, Gates Scholarship would've rejected me for being white, even if I had some non-white ancestor.
I get that. My comments are strictly within the context of the OP about what would happen if you just lie. If someone rejected you for claiming `asian`, for example, and you have an immigrant great-grandmother from <asian-country>, you would be able to challenge it.
I'm saying they can not prove one is lying because there is no way to be formally classified. I pointed out one of many low hanging fruit ways an argument can be made from the links the other posted included. Being a more creative type, and having money with time to spare, one could find a way to argue whatever race they wanted. Even people with stricter bounds for themself are likely to find a trail of DNA and/or lineage for many of the cherry-picked classes on those forms.(especially in North America)