In that case, there is a very high probability that my brain is a random fluctuation in a maximum entropy universe, rather than a meaningful reflection of reality. Nevertheless, I believe and act as though my memories and perceptions describe the universe around me.

You buy the Boltzmann brain argument? How did you calculate the probabilities? Nobody knows the probability of what seems to be our universe forming, and certainly nobody knows the probability of a Boltzmann brain forming in a universe of unknown size and age. The Boltzmann brain paradox argues that one unknown highly speculative probability is bigger than another unknown highly speculative probability.

If you are a Boltzmann brain, don't worry about it - it's the other case that should be of interest to you.