I think this is relatively simple.
If the question is "here's my theory; is there evidence within the canon material to support it?", I think it is okay.
If instead the question is "here's my theory; what do you think?", it needs to be fixed.
While I agree that a good version of a "pet theory" question should provide examples and citations to justify the theory, I don't think lack of those citations should be grounds for closure. Downvoting, maybe, but not closure (at least, not without other factors in favor of closure).
In The Kingkiller Chronicles, I've heard Auri is really Princess Ariel. Is that true? What do you think?
- Close as primarily opinion based, if the OP can't be persuaded to elaborate. Even if the OP elaborates, edit out the "Is that true? What do you think?" part and replace with "what evidence exists within the books (or in other material from Patrick Rothfuss) that would support or disprove this theory?"
In The Kingkiller Chronicles, I've heard Auri is really Princess Ariel. Is there any evidence for this?
- I'd probably downvote this, but not vote to close.
Note that the distinction here is that the intent of the OP in the second example is clearly looking for evidence. In the first example, even though edits can make it functionally the same as the second example, the intent of the OP was clearly to start a discussion, and so long as they don't explicitly agree to reformat it, it should remain closed.
In The Kingkiller Chronicles, Kvothe says "I have stolen princesses back from sleeping barrow kings".
He also tried to convince Aaron (the blacksmiths apprentice) not to take the Kings coin and if he stayed to listen to the rest of his story would hear the truth about Princess Ariel.
Auri hates personal questions, and is specifically said to have bolted for weeks when Kvothe asked her name. This is consistent with a person who has a hidden identity.
She also acts like a noble at times, is described as always being prim and proper, and is often formal in her form of address. She eats and drinks as might someone raised in court.
She is adamant that she won't wear secondhand clothing, saying it is "filthy".
All of this makes me wonder if perhaps Auri is secretly Princess Ariel. Even the names are pretty similar, and there is a strong theme of similar-sounding names within the series.
Are there any other reasons mentioned either in the books, or directly from Patrick Rothfuss, that would support or refute this theory?
I don't think notability is a requirement. All that does is ensure that really cool theories appear somewhere else first.