Is price discovery lossless?

What exactly is implied by the claim that prices incorporate all information? How strong a claim is strong efficiency? More specifically, are price signals a form of lossless compression? In other words, is there a way to derive its input data from its output data? If so, I’m waiting to see a step-by-step method for doing so. If not, I’m anticipating possible excuses. Perhaps prices incorporate all information which is relevant to allocation. My own speculation is that strong efficiency might be a property of a perfectly transparent economy—but would also be redundant information under such conditions. More important to me than whether prices incorporate all information is whether there are more direct ways to uncloak information about economic production and distribution, and of course strategy. I can accept that this is an asymptotic goal, like the frictionless plane or perpetual motion. I have a harder time accepting that a free market (or even a freed market) would be the best of all possible worlds. I have a really hard time being expected to take it on faith that prices are all the information I need in order to make fully-informed economic decisions. If that is the implication of the theory, then it is entirely fitting that market advocates are called “market fundamentalists,” and it shouldn’t be intended as a compliment.