The earliest it could possibly have dissolved would probably have been the 900s; it actually did dissolve for a time, with no one claiming the throne, but Otto the Great revived it in the 960s after he took control over Italy. If Italy somehow remains independent of France and Germany, then it is very possible for the Empire (that is, the Carolingian claim) to never be revived, and hence it would be, in practice, dissolved.
How you do this is a whole different ball of wax, but you could probably work something out...