If all the memories contained within a Boltzmann brain were hypothetical, so would be the physical laws that enable its very existence; therefore, a Boltzmann brain wouldn't be able to explain itself?
This sounds very similar to the skeptic's "Brain in a Vat" (BIV) thought experiment, which challenges our notion that we can know anything outside of our own minds.
If we replace "How do I know I'm not a Brain-in-a-Vat?" with "How do I know I'm not a Boltzmann Brain" I think the same arguments apply almost effortlessly.
The linked article is pretty long, but in summary most anti-skeptical arguments have failed in that they are found to be question-begging. However, there is one argument that has been shown to defeat this style of radical skepticism, but requires use of the Closure Principle (CL) and what is called Putnam's Causal Constraint (CC):
Given such an hypothesis as that of the brain-in-a-vat, the skeptic can go on to argue that there are many commonsense beliefs that we think we know, but that we do not in fact know. One common such argument is based on the widely endorsed closure principle that knowledge is closed under known entailments:
(CL) For all persons S and propositions p and q, if S knows that p, and S knows that p entails q, then S knows that q.
Now pick any proposition about the external world that you think you know to be true but that is inconsistent with your being a brain in a vat, say, the proposition that you have a body. Then the skeptic can argue as follows:
(1) You know that the proposition that you have a body entails that you are not a brain in a vat. [Premise]
(2) If you know that you have a body, then you know that you are not a brain in a vat. [By (1) and (CL)]
(3) But you don’t know that you are not a brain in a vat. [Premise]
(4) You don’t know that you have a body. [By (2) and (3)]
Premise (3) seems justified by the fact that you have the same conscious experiences whether you are a normal human in a normal physical world or a brain in a vat. For again, the evidence you have for each alternative is exactly the same. So you don’t know that you’re not a brain in a vat.
For in order for an utterance of term "X" to refer to actual X, there must be some causal connection between uses of "X" and real X.
Argument for why we know we are not Boltzmann Brains
Putting these together, we can argue for anti-skepticism about our being a BIV/Boltzmann Brain as follows (again, summarized from the article for completeness):
(1) You know that the proposition that you have a body entails that you are not a Boltzmann Brain. [Premise(1) of (SA) ]
(2) You have the concept of a Boltzmann Brain. [By (1)]
(3) But no Boltzmann Brain has the concept of a Boltzmann Brain. [By the causal constraint.]
(4) You are not a Boltzmann Brain. [By (2), (3)]
This argument works because if (1) from SA is true, then we conclude we are not a Boltzmann Brain (and because we deduced this we know this). If we deny this, we cannot conclude we don't know we don't have a body.
(Source: Stanford Encyclopedia of Philosophy, Skepticism and Content Externalism)
You cannot know that you are not a Boltzmann Brain
Whatever physical laws exist, they exist independent of minds which observe them. It may be the case that such a brain does, by some very improbably chance, happen to hold correct beliefs about the physical laws. Of course, there would be many more such brains which hold incorrect beliefs about the physical laws. The brain itself could have no way to differentiate between whether or not it were such a brain that does hold correct beliefs about physical laws and such a brain that holds incorrect beliefs about the physical laws.
Likewise, a Boltzmann brain may or may not be able to explain itself. In a set of infinite possibilities with random brains popping into various universes in various states of completeness, there could still be some brains which happen to pop into existence which hold exactly the correct understanding of what they are and how they came to be. It would, of course, still be held by chance, and there would be many more probable brains which belief they hold the correct explanations of themselves but are wrong.
You could start by affirming some other contrary fact and working back from there, but this is essentially begging the question because if you are in fact a Boltzmann Brain, you would hold those beliefs incorrectly. You don't know that you have a body, and as this shows, even your belief that you have a body may not be arrived at causally.
It is not reasonable to believe that you are a Boltzmann Brain
So, can we know that we are not a Boltzmann Brain? No, there is no way that we can objectively know that we are not a Boltzmann Brain, and there is absolutely no way that we can distinguish from our experiences or thoughts any difference between a "real" world and a Boltzmann Brain world. However, if we change the scope of the question a little, then we can say some useful things about the situation. For instance, you can ask whether it is reasonable to believe that you are a Boltzmann Brain. This qualifiers will reject such a belief.
According to the laws of reason, if a belief we hold is to be reasonable, then there must be some causal relationship between the thing we believe to be true and our belief that the thing is true. It is not enough that the belief is true. For instance, if I have a winning lottery number, that can be true without it being reasonable to believe it to be true because there may be no causal link between the two. In the case of the Boltzmann Brain, beliefs have been formed by chance. Even if you happened to be a Boltzmann Brain which were entirely correct about its own existence, you could not justify that belief as reasonable.
Where this all gets particularly tricky, though, is not in whether you should simply not believe that you are a spontaneous brain. The purpose of Boltzmann's thought experiment was to show that if the universe were formed in such a way that there were more brains which only thought that they could properly reason about their own existence than actual brains which really could, then in such a scenario, one would have to admit that even if one did happen to be a real brain with a real ability to reason they would have to reject such beliefs as being reasonable because there would still be no clear causal link. The thought experiment wasn't to prove such brains existed, but to show that certain theories of the nature of reality were not compatible with reason to prompt searching for better explanations.
There are some strong parallels between this and Alvin Plantinga's Evolutionary argument against naturalism. Consider that in Evolution Theory, the process which creates our minds also produced thousands of minds which do not have the ability to accurately reason. So, if we attempt to use reason to analyze our core beliefs and assumptions about the world, we may find ourselves having some controversial realizations.