I have two different answers to your question: one practical, one more theoretical. On a practical level, what I gain from peak experiences depends on where my attention is. If I'm out and about, or doing something materially, then the main advantage I gain is noticing new aspects of a situation, or seeing the same aspects in a different light; I believe this is a result of greater flexibility in choosing the cognitive map I apply to the territory. These, I suppose, would be the "unknown dots": information that was present in the environment, but which my brain never bothered to record. If I'm sitting around thinking, on the other hand, then I tend to find a lot of unconventional connections. Even here, though, there is new information to be gained; I've learned a lot about my own nervous system simply by careful observation of my internal experience.

On a theoretical level, I have some trouble with the distinction between information and deduction. In the strictest sense, mathematical truths contain zero information, since they are automatically true in every possible world (or insert your own X-Rationalist translation of this claim). Yet we are still surprised to learn, for example, that e^(pi.i)+1=0 - or I know I was surprised, at the very least. I think this is a result of the fact that our evaluation of mathematical claims is based on manipulation of tangible stuff: we can check our memory to determine if we've ever seen a proof before, or we can manipulate symbolic expressions, encoded in our wetware, to attempt to prove or disprove it. Even wood pulp and graphite can be leveraged for this purpose, and so the result of this computation comes in as an observation of an uncertain outcome in the world.

Yet the no-information claim takes on new complications when we realize that our brains use the same basic processes to encode relations between facts, as it does to encode facts. This leads to kind of "flat" ontology, in which we can treat relations as facts and draw relations between them, or even between a relation and a fact. We can even draw relations between internal experience (including mathematical knowledge) and external information. Once we have recognized these connections, natural systems can actually point us toward mathematical truths (for example: we might never have learned about solitons if we had not observed them in nature). The hierarchy of "levels of organization", and the division between information and deduction from information, therefore appear to be constructed post hoc, and I'm not sure if I can cleanly distinguish between unknown dots and unknown relations.