Specific words have specific meanings, and the usual overlap in meanings doesn’t exist.
Sometimes, words have means that you wouldn’t expect in english.
A theorem, for instance, is an idea about something works, a theory, but specifically one for which a proof exists.
If a proof doesn’t exist, it is a conjecture.
There is one use which appears to be more colloquial, the sort used in classes, but not in text books.
Fact. A fact is something which you are asked to believe, but with no proof.
What is actually asked for, when given a fact, is faith.