Most apparent facts are irremediably vague. For instance: "The chair in this room is red" may appear at first glance to be a statement that is either true or false. But in fact the meanings of the words "chair", "room", and "red" are all matters of opinion. (This becomes clear especially if one considers borderline cases.)
This is a gross confusion between facts and the statements about them that we make using any natural language. Apparent facts are of course not irremediably vague precisely because they are apparent. Certain facts may be vague, in particular if you are drunk for example, but in this case the fact is not really apparent to you. So, assuming an apparent fact, then by definition it is certainly not vague.
Here you are confusing facts and statement about facts. Your claim can only make sense if it is that statements of fact are irremediably vague. However, once this is clarified, even this falls appart. Of course different people will give different meaning to the word "chair", but the speaker does not care about that for he or she has his or her own meaning in mind, and so the statement "I am sitting in my usual chair right now", is not only clear, it is also true. The wording of my statement will not allow you to decide that it is true, and you may want to say that "chair" and "usual chair" are vague notions, well, not to me, and I decide what I mean by the statement. You don't understand it? Tough.
So here, you are not only confusing facts for statements of facts, but also confusing the truth that many statements seem inevitably somewhat vague for the audience, with the false idea that statements are vague in themselves. No, once you take into account what words mean for the speaker, the statements made by this person will usually be cristal clear. Only your ignorance of what the speaker means by the words he or she used makes the statement vague.
This is in fact demonstrated by your inclusion of mathematics in what you tautologically call "exact facts" (facts are by definition exacts). Mathematics is done by real people, using the same logic as every other human being in good health, and, crucially, using the language of mathematics, language which is... a language, and mathematics is nothing if not all the statements made in the language of mathematics. This demonstrates that statements of fact are not irremediably vague. This, of course, extend to physics and some "hard" sciences.
Truth can be readily understood in all its very few dimensions by looking at logic. If the premises are true and the reasoning is valid, then the conclusion is true. This applies to everything we can think rationally about, even if only in principle. If we cannot think rationally about a subject, even in principle, then there is no truth to be had and there is nothing to say about the subject except perhaps meaningless noises and grunts.
