I think Heather Heying (I called her Heyer earlier, sorry) describes sex as the hardware, gender as the software or operating system but that it is in her view tied to biology and I think she distinguishes that from socialized or psychological aspects of it. So, be it instinct or something else, "a rose by any other name" might apply here.

Trans arguments are literally all over the map, and all in bad faith and all committing so many logical fallacies and rhetorical trickery that cover all the bases will be very tough. I do not envy you as the task is daunting.

But, here is one thing we have on our side: at best all they can hope to do is produce a theory predicts things as they are and perhaps some thing yet discovered that explains the world better than our current reality based set of theories and models, though to call definitions of man and woman "theories" shows how far they have to go to create something worth using.

In Geology the "continental drift" theory was laughed at, especially in the US. As the ideas around it became more refined and it became better defined, it was called plate tectonics and predictions were made from it that the older theory, called Geosynclines, fell out of favor. In spite of that there were people who did not see the need for plate tectonics in their work and still used geosyncline ideas long after the theory stopped being useful and was no where near as useful as plate tectonics. There are still uncertainties about some of the mechanisms involved in tectonic processes, but the theory is vastly superior to geosynclines. Another theory that tried to usurp a dominant ideology was Intelligent Design which attempted to compete with Evolutionary Theory (which is wrongly called Darwin's Theory of Evolution by some today but the theory has come very far from Darwin's possible stolen ideas, much further than say the change from Continental Drift to Plate Tectonics). Where Plate Tectonics answered many questions better and provided clarity where there was little to none, Intelligent Design advocates most often relied on two fallacies: they argued from a conclusion and they argued for an irreducible complexity basically claiming there are things we can not know. Ultimately they did not put forth any actual theory that could be compared to evolution, they just tried to nit-pick away at some oddities and really leaned on a lack of understanding as if that proved something.

Gender propagandists do not have anything like a theory that answers any questions, where as gender critical critics of them do in the sense that the rapes of women by trans-identified males in women's prisons was predicted. There are recent crime stats that show trans-identifed males do in fact commit violent crimes at male rates and not female rates, I'm fairly certain this will fall into a prediction made by team GC and a failure by team gender.

As we understand it, a man has the possibility of impregnating a woman to them it is too complicated to know-- irreducible complexity. They may bring up men who do not produce viable sperm, arguing from the margins I guess you'd say, but we're not the ones saying that person is not a man instead we will go to other ways of classifying that person and that person will be classified a man probably very easily. But that man who is not producing viable sperm does not make someone who is producing viable sperm not a man and certainly does not make them a woman.

In the GC Debates QT sub, attempts were made to define woman. Team QT absolutely failed to even bring forth a workable definition. They hid behind a fake kind of speaking in set notation, but it was not workable. They tried to say a woman is a natal woman and anyone who identifies as a woman, and then tried to say that the were not using the word in its own definition. I wish I could say they were more artful than that but they were not artful but awful. They tried to argue that WOMAN is a set with members in that set that include natal woman and anyone identifying as a woman, then they claimed they defined the set and the words in the set were some how different than the name of the set. Maybe not "they" but one guy, I lose track of names on these places. Maybe I'm playing it safe and not mxgenderxng them.

So, if we accept some other definition for woman, how workable is that definition and what does it help us do? They can not give us a definition that does not allow the state of being a woman to be reduced down to a claim made by somenone, and they can not give a reason better than the defintion we use is definite in that it delineates what is and what is not a woman based on physical criteria. Even if the criteria is complicated and multivariate, to cover all the intersex and DSD conditions, and even if there are a very few (1 in 5000 I think is the number) who can not fit the male and female classification, that does not make an adult human female anything other than a woman and it does not make a trans identified person anything more than their physically determinable sex.

They can say that due to those factors and due to the concept of "passing" and anything else they might invoke, that we might not be able to discern a person's sex, and even if we grant them a momentary confusion someone might have over that and let them then imply that if you spent long periods of time with them that one might never know the persons's sex: they are arguing that deception is reality.

Colin Wright has a long twitter thread that I think he turned into an article somewhere that goes into the multivariate aspects of determining sex when dealing with DSDs or intersex conditions. He did it in the context of ripping apart science journals publishing things that fell for this irreducible complexity argument, he has a couple of specific names for the fallacy the use and one might be just The Multivariate Fallcy or the Univariate Fallacy, where since there are multiple factors involved the fact that there is no one factor that determines it every single time means the categories are insufficient. He says it better than I.

But, all of this is arguing intersex and DSD and then they argue from their conclusion and just say "so trans can be a thing". Nah.

I doubt any of this is helpful to you or your purpose, and I don't want to like slow you down or send you off in directions you're not planning on going in. Perhaps grouping their arguments by fallacy would help. There is no study that shows any of this, those all are just arguing from a conclusion, because there simply is no way to say some inner identity is "gendered" in a way that mimics or sexed differences. There is no part of the brain that behaves or reacts in such a way that it means anything about sex or gender that doesn't rely on the biological categories to first be established and then be added to by the trans people they want to add to it, and even if there were some part of the brain where it was a match between trans identified males and actual women, that does not mean these guys are women-- that is arguing from the conclusion.

I guess I'll end with this: I think establishing what we know to be true and unassailable, establishing the firm ground we are on the way Wright talks about defining men and women, and establishing the percent of humans this covers, all but 1 in 5000, would make all their arguments weaker just from the very start. All their arguments can be dealt with from those easy and clear physical facts. You might have to throw away the "one is not born but made" type rhetoric though as that seems to serve them far better than team GC even though she was team GC.

Good luck, I hope some of this helps, but I fear it is just pulling you away from your work on it and I apologize for that. Good luck!