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I realize I'm butting in on an old debate, but thinking about this caused me to come to conclusions which were interesting enough that I had to write them down somewhere.

I'd argue that rather than thoughts containing extra contents which don't exist in brain states, its more the case that brain states contain extra content which doesn't exist in thoughts. Specifically, I think that "thoughts" are a lossy abstraction that we use to reason about brain states and their resulting behaviors, since we can't directly observe brain states and reasoning about them would be very computationally intensive.

As far as I've seen, you have argued that thoughts "refer" to real things, and that thoughts can be "correct" or "incorrect" in some objective sense. I'll argue against the existence of a singular coherent concept of "referring", and also that thoughts can be useful without needing to be "correct" in some sense which brain states cannot participate in. I'll be assuming that something only exists if we can (at least in theory if not in practice) tie it back to observable behavior.

First, I'll argue that the "refers" relation is a pretty incoherent concept which sometimes happens to work. Let us think of a particular person who has a thought/brain state about a particular tiger in mind/brain. If the person has accurate enough information about the tiger, then they will recognize the tiger on sight, and may behave differently around that tiger than other tigers. I would say in this case that the person's thoughts refer to the tiger. This is the happy case where the "refers" relation is a useful aid to predicting other people's behavior.

Now let us say that the person believes that the tiger ate their mother, and that the tiger has distinctive red stripes. However, let it be the case that the person's mother was eaten by a tiger, but that tiger did not have red stripes. Separately, there does exist a singular tiger in the world which does have red stripes. Which tiger does the thought "a tiger with red stripes ate my mother" refer to?

I think it's obvious that this thought doesn't coherently refer to any tiger. However, that doesn't prevent the thought from affecting the person's behavior. Perhaps the person's next thought is to "take revenge on the tiger that killed my mother". The person then hunts down and kills the tiger with the red stripes. We might be tempted to believe that this thought refers to the mother killing tiger, but the person has acted as though it referred to the red striped tiger. However, it would be difficult to say that the thought refers to the red striped tiger either, since the person might not kill the red striped tiger if they happen to learn said tiger has an alibi. Hopefully this is sufficient to show that the "refers" relationship isn't particularly connected to observable behavior in many cases where it seems like it should be. The connection would exist if everyone had accurate and complete information about everything, but that is certainly not the world we live in.

I can't prove that the world is fully mechanical, but if we assume that it is, then all of the above behavior could in theory be predicted by just knowing the state of the world (including brain states but not thoughts) and stepping a simulation forward. Thus the concept of a brain state is more helpful to predicting their behavior than thoughts with a singular concept of "refers". We might be able to split the concept of "referring" up into other concepts for greater predictive accuracy, but I don't see how this accuracy could ever be greater than just knowing the brain state. Thus if we could directly observe brain states and had unlimited computational power, we probably wouldn't bother with the concept of a "thought".

Now then, on to the subject of correctness. I'd argue that thoughts can be useful without needing a central concept of correctness. The mechanism is the very category theory like concept of considering all things only in terms of how they relate to other things, and then finding other (possibly abstract) objects which have the same set of relationships.

For concreteness, let us say that we have piles of apples and are trying to figure out how many people we can feed. Let us say that today we have two piles each consisting of two apples. Yesterday we had a pile of four apples and could feed two people. The field of appleology is quite new, so we might want to find some abstract objects in the field of math which have the same relationship. Cutting edge appleology research shows that as far as hungry people are concerned, apple piles can be represented with natural numbers, and taking two apple piles and combining them results in a pile equivalent to adding the natural numbers associated with the piles being combined. We are short on time, so rather than combining the piles, we just think about the associated natural numbers (2 and 2), and add them (4) to figure out that we can feed two people today. Thus the equation (2+2=4) was useful because pile 1 combined with pile 2 is related to yesterday pile in the same way that 2 + 2 relates to 4.

Math is "correct" only in so far as it is consistent. That is, if you can arrive at a result using two different methods, you should find that the result is the same regardless of the method chosen. Similarly, reality is always consistent, because assuming that your behavior hasn't affected the situation, (and what is considered the situation doesn't include your brain state) it doesn't matter how or even if you reason about the situation, the situation just is what it is. So the reason math is useful is because you can find abstract objects (like numbers) which relate to each other in the same way as parts of reality (like piles of apples). By choosing a conventional math, we save ourselves the trouble of having to reason about some set of relationships all over again every time that set of relationships occurs. Instead we simply map the objects to objects in the conventional math which are related in the same manner. However, there is no singular "correct" math, as can be shown by the fact that mathematics can be defined in terms of set theory + first order logic, type theory, or category theory. Even an inconsistent math such as set theory before Russell's Paradox can still often produce useful results as long one's line of reasoning doesn't happen to trip on the inconsistency. However, tripping on an inconsistency will produce a set of relationships which cannot exist in the real world, which gives us a reason to think of consistent maths as being "correct". Consistent maths certainly are more useful.

Brain states can also participate in this model of correctness though. Brain states are related to each other, and if these relationships are the same as the relationships between external objects, then the relationships can be used to predict events occurring in the world. One can think of math and logic as mechanisms to form brain states with the consistent relationships needed to accurately model the world. As with math though, even inconsistent relationships can be fine as long as those inconsistencies aren't involved in reasoning about a thing, or predicting a thing isn't the point (take scapegoating for instance).

Sorry for the ramble. I'll summarize:

TL;DR: Thoughts don't contain "refers" and "correctness" relationships in any sense that brain states can't. The concept of "refers" is only usable to predict behavior if people have accurate and complete information about the things they are thinking about. However, brain states predict behavior regardless of how accurate or complete the information the person has is. The concept of "correctness" in math/logic really just means that the relationship between mathematical objects is consistent. We want this because the relationships between parts of reality seem to be consistent, and so if we desire the ability to predict things using abstract objects, the relationships between abstract objects must be consistent as well. However, brain states can also have consistent patterns of relationships, and so can be correct in the same sense.