And furthermore.... to answer your question above re: 'surely not', Professor... the subject and predicate are inferred and understood: ('Fine' is) surely not (a sentence).
Try saying 'surely not' to someone out of context and see what reaction you'll get. It lacks meaning without a subject and predicate. Hence, it is not a sentence.

Professor Ballantyne, clearly, my post needs elaboration. My definition of a sentence is not 'because I say so'. Most dictionaries give the same definition. If you disagree with their definitions, then take it up with them.
Oxford Dictionary defines a sentence as 'a set of words that is complete in itself, TYPICALLY containing a SUBJECT AND PREDICATE, conveying a statement, question, exclamation, or command, and consisting of a main clause and sometimes one or more subordinate clauses.'
Webster Dictionary's definition is 'A combination of words which is complete as expressing a thought, and in writing is marked at the close by a period, or full point.'
Therefore, by both definitions, 'fine' and 'sure' are not sentences. Neither are 'not so', 'there', and 'done'. These words do not mean anything if written or uttered by themselves since they do not express a complete thought. However, put them into a proper context, and the thought is complete. They are then sentences. The subject and predicate are inferred or understood (and therefore not necessary).
Example: 'fine' by itself is not a sentence, but an adjective. Put it a context: 'How are you feeling today?' 'Fine' is accepted as a sentence since it means 'I am fine' ('I am' is understood)
Example: 'not so' by itself is not a sentence. Put it in a context: 'A sentence has a subject and a predicate.' 'Not so.' means 'It is not so', so it is acceptable as a sentence. ('it is' is understood)
So your assertion that 'One-word sentences can and do exist.' has to be qualified by saying it depends on the context in which they are used.

No, 'fine' is not a sentence. A sentence needs a subject and predicate. People do not always speak in complete sentences (by definition), but often in single words or phrases, like 'no', 'surely not', 'in the house'... there are endless examples.

Sorry, that should've been 'a' sound in my post.

Most Canadians pronounce 'aunt' like 'ant'....it's in the dictionary. Likewise, we use the same 'ay' sound in Tanya, Sarah, tacos, pajamas (second 'a'). But on American tv shows I hear the 'aw' sound in the same words (Tawnya, Sawrah, tawcos, pajawmas). I think it's a dialectal difference, and it may vary within the US accordingly.