A few facts:

1. Yes, there is a male brain and a female brain. Examples:

-Men have 8-10% larger brains on average -Women have a higher percentage of grey matter -During emotional memory processing, men have greater activation in the right amygdala, while women have greater activation in the left amygdala -Women have greater cortical thickness -Men have a higher serotonin synthesis rate -Women have higher neuronal density in language-processing regions -Women have higher orbitalfrontal to amygdala ratio -During resting states, men show stronger functional connectivity within motor and sensory networks while women show stronger functional connectivity within social and emotional networks -INAH-3 is 2-3x larger in men than women -BNST is 50% larger in men

These are also there before puberty, it’s built in due to prenatal hormones

2. No, the fact the brain is a mosaic does not matter when discussing the fact that there is a male and female brain. People like to talk about the fact that individually these structures are really just group averages and the brain is a mosaic of masculinised and feminised structures, which is true, but when you look at the brain as a whole it really does cleanly fall into male and female brains. We know this because a stanford study trained an AI on resting-state fMRI scans of roughly 1,500 young adults, and it maintained over 90% accuracy in predicting sex across multiple independent datasets from different research centers. Now some may think that 10% inaccuracy means they're not their sex 10% of the time, which is absolutely not true, in reality it just shows the limits of AI and it could have been pushed further with different methods, more training data, less measurement noise, scanner effects etc.

3. These structures are definitely built by prenatal hormones. Not by x or y chromosomes (with some exceptions) or anything else. We know this because when looking at structural MRI, functional MRI and DTI of people with complete androgen insensitivity syndrome (CAIS), their brains are completely female typical in all the areas you’d expect. CAIS is a condition where the androgen receptor has a loss of function mutation and therefore cannot actually read the testosterone in the body, which makes the fetus develop along a female-typical pathway despite having xy chromosomes. The reason x and y chromosomes are talked about in sex is just because the y chromosome contains the SRY gene which starts the production of testosterone and turns the fetus into a male, that’s literally it. And the absence of that would make the fetus just develop along the female-typical pathway. But whichever pathway it develops on is entirely contingent on actual prenatal hormones and sensitivity to the hormones. Those with CAIS exhibit completely female-typical interests, social behaviour, sexual preferences, and a female identity. No, this stuff is not caused by socialisation, all the DSD and endocrinology literature strongly supports that socialisation actually does not play a major role in these sex differences and the vast majority of it is attributable to prenatal sex hormones. Of course this is just an N=1 but take david reimer for example, he was born male but was reassigned female with surgery at 22 months old following a botched circumcision, no one ever told him and everyone raised him as a girl named brenda, his hair was grown out, he was made to wear traditional female clothing, given hormone treatments at puberty etc, and he never knew he was born male until the age of 14 and he transitioned back to male. Long before he ever knew, he rejected his female identity, having male typical play preferences, saying he felt like a boy despite being told otherwise, was attracted to girls, etc.

4. No, unless you have an extremely rare genetic disease, prenatal hormones cannot be changed sufficiently enough to make a sex switch in the brain. There’s a pretty narrow range for variation within sex when it comes to prenatal hormones. Yes, some differences in testosterone or changes in timing of testosterone may change the brain slightly, but overall it’s still fairly sex-typical. The absolute difference in male vs female androgen signalling is massive, especially when talking about the crucial first trimester.

5. No, trans people actually do not sufficiently differ in their brain structure compared to their sex at birth, which is exactly what we would expect given all the other information. Sure, a few papers with a small sample size and ideologically driven funding may show small local changes compared to their assigned sex, but these papers are more focused on finding the differences between their assigned sex rather than looking at the big picture, which would be that the vast majority of them, if not all of them, would still have an overall brain that looks typical for their sex assigned at birth

As you can clearly see, in people without some rare genetic deformity, they simply would not have a sex inverted brain. Whether the validity of trans people is based on them having a sex-inverted brain is another discussion, but the actual claim that they have any major differences neurologically from their assigned sex itself is false.