I know you were quick to dismiss inner product spaces, but it might work if we define the inner product to be the "similarity" between the genders. The definition of an inner product space could be met, if addition and scalar multiplication of gender are defined carefully. Addition of genders would represent being both genders at the same time, and scalar representation would reflect the intensity. In this way, addition would be an average with the intensity doubled. So the more intense a gender is, there are more similarities with a similar gender, and more differences with an opposing gender. If the genders are similar, this would be positive, negative if significantly different, and zero in the case of the genders not being opposing nor similar. Since inner product spaces are normed vector spaces, making this a normed vector space would follow the triangle inequality. Although you do make a point about vector spaces being problematic because they use positive and negative numbers, which does in a way reinforce binaries. Although if it was infinite dimensional I don't think this would be too much of a concern. I think it was yewler who suggested using a vector space of real valued functions over some set, this seems like something to consider.
I'm not sure if this helps, this is really just speculation on my part.
I have thought about gender as an N-dimensional space, where N≥0.
This was exactly what I was thinking!
The mental representation is lacking and probably doesnt account for everyones experience (or rather pushes the work of understanding it outside of the representation)
Very much agreed, it is a mental representation after all, but it is useful for contextualizing people's diverse experiences
incoherent rambling inspired by one of DirtOwl's badposts
This bad post by dirt owl got me thinking, oh wait sorry I mean feeling since I am dumb infp tanki with iq of -5, about the way we view gender as a society. Anyways, dirt owl was not the first user to talk about mbti on hexbear. Using the search function of hexbear, this post from four years ago says that it makes no sense that everyone fits into 16 distinct categories. Another comment mentions the anti-worker nature of the personality test.
The comments in the badpost often followed a similar line, with users identifying as personality types that are technically not part of the system. And this is actually valid, since in mbti you are actually the type that you say you are. What these comments show is resistance to a form of categorizing people, but it is even more than resistance, they are not even taking it seriously. The comments are metaphorically sending a giant PPB to the idea of separating people into new categories. And it's pretty obvious how this relates to gender, since both are attempts at limiting us within a binary system. Especially the thinking or feeling part, unfortunately, is connected with people's expectations of gender roles. But just as the badposters are the resistance to this typology, we are the resistance to the idea of gender under the constraints of capitalism. I think that the gender accelerationist manifesto articulates well what I've been thinking of better than I can, even though I didn't read it until today (If you haven't read it yet, perhaps you should...) Some of it talks about how the modern gender binary is a result of colonialism; that sort of relates to this week's megathread topic about gender in different cultures.
So I know that this personality stuff isn't widely used as a class to oppress people, although that is sort of what it is used for when it is used. But the way it creates boxes, it is easy to dismiss. But the typology being wrong doesn't mean that people don't have personalities. It just means that there are more than 16 expressions of personalities, since last I checked, I think that there's at least 16 people alive right now. The more "scientific" way to measure personality is through measuring traits on a multidimensional continuum. There is no need for labels to express your personality. The same goes for gender, the only difference being that gender is oppressive when used as a class. Maybe gender is also something that is multidimensional, and while some people can easily be labeled as a man or woman in the same way that others can be labeled as an extravert or introvert, there is much more variation in reality.
You know how AI language models store words as vectors? In this fashion, an analogy is represented: king is to queen as man is to woman. Here my observation is that there is some way to mathematically represent the idea of royalty, as well as some idea of gender, although most likely in this situation a reactionary version of gender is represented. But here gender is probably stored across multiple dimensions, and the gender binary could be a projection of some underlying phenomenon.
I would like to end this comment discussing the part of the manifesto about gender identity under communism.
Many people fear that, through the abolition of gender, our own gender identities will be taken from us. That, in abolishing gender, we will force you to stop identifying with your gender, however much you might enjoy that identity.
That was me, before yesterday. Now I am fully in support of gender abolitionism.
The end of gender as a system of power is our goal, and the end to gender identities is an eventual result, if it will happen at all, not something of importance or which we should strive toward.
The only way I can describe gender now is like brainwashing. We do what we are told, and we must comply. And almost everyone complies because they're all cis. If institutions didn't enforce this rigid interpretation of gender, life would be so much better for all of us here.
Sorry if this makes no sense or if some of what I am posting is obvious. Feel free to correct anything I said if it is brainwormed.
I love my trans comrades 
Posting more. If you search "intel arc gpu in xcpng" on duckduckgo, hexbear is one of the first results. Search up "how to set up radial menus in vrchat," and hexbear shows up on page one in google. If we simply make more posts, the website will be visible to more people. The more varied the topics of discussion are, the more likely someone is to stumble upon hexbear. Also, making the site more friendly to new users will go a long way. I can't count the number of times I've accidentally clicked on PPB because it is at the bottom of the page. Maybe that shouldn't be something that happens to people who visit the website for the first time.
I've been seeing a lot of users suggesting exposure to hexbear on other social media. I think that besides search, that is the best way for hexbear to expand, since hexbear is embarrassing for many users. Another thing is that one of the places that gives hexbear the most publicity is lemmy.world. If 
gets users to migrate to lemmy, that means more people finding out about hexbear. And this goes for other websites too, since people leaving xitter will discover the federation as an alternative. Also, these days short form video content is very popular and more engaging for many people than text based content, maybe there could be some federated tik tok alternative or something.
Also the power of the trans userbase on hexbear keeps on growing! Making hexbear a safe space for our trans comrades will help maintain this growth. 
Although, maybe the best way to grow hexbear isn't to increase the number of users, but decrease. Ban everyone but bureaucrat, and we shall continue posting under one name, more powerful than ever. This is how we can make hexbear great again.
Thanks for your reply. You're right, a lot of that "theory" doesn't make any sense at all.
Do you want to be a girl?
Yes. That means I'm trans. But if I'm going to be honest, it all sounds scary. Hopefully I'll be able to overcome these fears.
while I have not said much
Thank you for this! It means a lot to me knowing that there are other people out there with similar experiences.
Hi, I'm new here 
CW: dysphoria, transmedicalism, sad, transphobia
:(
I want to be a woman, but I'm not. It's been like four years of wanting to be a girl, but my egg only really cracked this June. Before my egg cracked, my concept of gender identity was that I wasn't really a boy, and I wanted to be a girl, but obviously I was totally cis because even though I knew about the existence of trans people, I couldn't possibly be one of them, after all, trans people know that they are trans at the age of four and are conviced that they are the other gender. I don't know what I was thinking.
Last year someone in my ap computer science class came out as trans, our teacher told us that the student wanted to tell us something, and people were like "did [deadname] die?" and the teacher said that the student wanted to go by a different name and pronouns. Then someone goes said that he wouldn't use she/her pronouns to refer to someone who doesn't look like a girl. At least some people were respectful, though. The point is, I can't trust the people around me to not be bigoted, although most of the people in that class, have graduated. The trans student was very cool though, she was one of the only students who took the class seriously (I think there might be something in the computers turning people trans!) and was probably a gamer. Too bad she also graduated. I'm not a gamer and I don't have anything interesting about myself. I kind of looked up to her, even though I didn't identify as trans.
My whole life has been a lie of pretending to be someone that I'm not. I'm a minor, and I live with my parents in the wonderful state of Florida, where woke goes to die. I have reason to believe that my parents are transphobic. My family is Catholic. I am not, but they don't know that. I have been pretending for years. I don't say anything about how I feel about my gender.
I am growing more body hair every day. It's awful, and I don't think I've ever had this much. As much as I would like to do something about it, I am scared of what other people would say. I don't even know how to remove it without accidentally hurting myself.
I feel like a robot. I do the same stuff every day, and by that I mean that I don't do anything. I just do my schoolwork, and school gets repetitive. It distracts me from thinking. But guess what, when I do think, I just think about how I want to be a girl, and then I get sad. But I don't actually get sad, since I don't feel my emotions very deeply.
Read at your own risk CW: autogynephilia
I think I might have that? But also maybe not? I mean, it can be arousing to imagine myself as a woman? But also I can think of myself as a woman and not be aroused? Autogynephilia has been one of those things that makes me question if I'm really trans. Most research on trans women does make a distinction between two types of trans people, those who find out early and are attracted to men, and those who find out later and are attracted to women, and sometimes have autogynephilia. Yes, I know that this research is stupid. That doesn't help. I still think about it. Also, I'm not really sure if I meet the criteria for gender dysphoria from a transmed perspective. Wikipedia shows the dsm description, but it says that it has to cause clinically significant distress or an impairment in functioning. The thing is, I do well in school, and to most people, especially adults, I appear well functioning. But I don't know, I don't have many friends and no one that I am close to, but I don't think that gender issues are what is causing this?
If you made it this far, I'm sorry for putting you through all of this. I'm okay, even if it sounds like I'm not.