This is the most important principle of Protagonist metaphysics. In short, everything about our consciousness is actually in our souls and not in our brains. Your thoughts, emotions, and other mental faculties have nothing to do with your brain and things in your brain can't directly alter them. The only thing the brain does (besides the obvious body maintainence tasks) is allow us to control our bodies and forward perceptions from sensory organs to our minds. There are several reasons why you should believe me.
The first is that the existence of free will is irreconcilable with the idea of a physical mind. In a physical system, every event is caused by the previous event, every chemical reaction by the previous reaction. While some modern scientists might argue that physical reality is actually probabilistic rather than deterministic, that doesn't help - randomness isn't any closer to free will than determinism. But the whole idea of free will is that nothing causes it. It could have been different without any of its precendents being different, yet the person has control of it. There's an irreducible concept of "choice" here that clearly cannot be a product of either determined causation or randomness. Therefore your free will has to be coming from somewhere outside of physical space entirely. This is what I call a "soul". Additionally, it should be pretty undeniable that the mental faculties I listed above couldn't possibly exist without each other, so if one of them is only explicable as an immaterial entity separate from your brain then the others must be in your soul too and not in your brain.
The second reason you should believe me is what I call noncontingent morality. If you're a good person, you believe that morality is the same for everyone and that there couldn't (for example) be an alien race for whom it truly is right to steal and murder. If your conscience (or reason if you claim that's the source of morality) is in your brain, then its teaching could have been different because your brain could have been different. That means there could be a sapient being for whom full and honest loyalty to conscience/reason even with all the correct information would lead to a different morality than what you believe is correct. Obviously we can't have that in a sensible moral system. Copy-paste the inseparability argument.
Third, for the same reason you should believe that your own reasoning and self-awareness is reliable. If your thoughts can be directly altered from your brain, then you can't know that an evil scientist hasn't implanted you with a chip that prevents you from ever reasoning validly and from remembering the event. If you've read the lesson on the Principle of Contingent Uncertainty, you already know that you can't technically 'prove' anything about the world around you, although there are still reasons to act like it's real. Well saying that your brain houses or at least controls your thoughts is basically making the same thing true about your mind.
The fourth reason you should believe me, is because you already do and you can't help it. I have another article posted that contains the result of hundreds of hours of philosophy about what primary emotions can exist in the person's mind and exactly what causes each of them. I decided that we feel awe when we see an impressive accomplishment, frustration when we find out we don't understand something we thought we did, fear when we expect something bad to happen, anger when someone does something we don't like (simplified), etc. And that all makes sense, right? But if we accept that our emotions are just chemical reactions in the brain then all of that becomes meaningless because what really fuels those emotions is just some chemical being released, not anything that philosophy can figure out except by coincidence. Is that something you believe? That what fuels emotions is a matter of science and not philosophy?
In fact, I go so far as to say that these immaterial souls we have are the only things that exist - the physical universe does not except insofar as sentient minds perceive it. I can't really explain it, but in my teenage years I just sort of had this epiphany while sitting alone on a pier at night, and ever since then I haven't been able to see it as not self-evident that if a physical object can never be perceived, it by definition doesn't exist. (The reason souls aren't subject to this is because we are conscious of ourselves.)