I think you need to go to sleep. You are not even making sense. There are all kinds of correlations between biological factors and social factors. Not every correlation between gender and biology refers to sex. You can study, as some studies have done, relations between say gender and propensity for development of physiological conditions such as bulimia. Some genome-wide association studies (GWAS) have suggested genetic markers associated with gender identity, though no single "gender identity gene" has been identified. Variants in androgen receptor genes, for example, have been explored in relation to transgender identity.
All categories are invented, since they are products of human discourse, not innate.
There are no 'immutable' categories, not even fundamental categories such as a matter. Categories emerge in time and can be displaced, revised, or modified in a variety of ways. We change our categorial frameworks as our theories change, on the basis of research, new methods, experiments, and so on.
Gender is a category, and it is widely used to study phenomena across many sciences, including biology, sociology, psychology, philosophy, history, and so on. That you think you are in a position to say it is a 'fake' category shows an incredible amount of arrogance, and ignorance.
But the reasoning you give is actually even more tendentiously stupid.
The fact that there are always correlations between sex and gender does not mean that the category of gender is redundant or 'fake' (whatever that means).
Trivially, for example, the category "chimpanzee" will always be correlated to atomic physical structure. But it would be retarded to say that this makes "chimpanzee" a category that does not refer to anything that atomic physics does not already tell us. Or else, biology would be redundant. Similarly, every single individual of which gender is predicated is instantiated in a biophysical body, at least for now.
You can trivially name correlations between biology and gender that do not refer to sex:
All gendered individuals are members of the human species.
All gendered individuals have sensory organs.
That all gendered individuals are also individuals with a sex is a different issue, and trivially true.
Gender is not a species of the sex genus. Gender roles are not reducible to sexual developmental categories or their anomalies.
A huge part of the category of gender concerns social roles that are irrespective of sexual development: girls play with dolls, and boys like sports, and women belong in the kitchen are not biological facts, since they refer to the normative structure of social institutions, like toys, sports, and so on. The study of gender is also the study of the institution, functioning and revision of these norms which include, but are not exhausted by, biological considerations.
Social constructs are correlated to biological factors, as I've already mentioned. Hormonal production is one set of factors. Others include anatomical ones. Here is another example from another study:
Findings from neuroimaging studies focusing on brain structure suggest that the brain phenotypes of trans women (MtF) and trans men (FtM) differ in various ways from control men and women with feminine, masculine, demasculinized and defeminized features. The brain phenotypes of people with feelings of gender incongruence may help us to figure out whether sex differentiation of the brain is atypical in these individuals, and shed light on gender identity development. Task-related imaging studies may show whether brain activation and task performance in transgender people is sex-atypical."
If self identifying as whatever counts as insane, which is an empirical question, it does not entail that studying such self-attributions and their relations to a variety of factors is useless. In fact, we do study the physiology and psychology and behavior of ill, deranged, and maladapted people all the time.
I never said that, nor does that follow from anything I said.
I don't believe that is true, nor does anyone that I cited.
You're throwing strawmen because you are desperate.
Right, the entire scientific community are quacks.
But you have not a single study to show that indicates that the distinction between sex and gender is scientifically not credible.
Not one. You can't even write coherent sentences in English, and you think you are somehow smarter than the entire scientific establishment. And even more pathetically, you accuse those who try to share studies and information as "deferring to authority".
Well, if that is the case, I think that's a bit better than to believe that I know better than professionals.
If anyone believes that because someone says they are a cat they are a cat, they are either very naive, or have a very odd conception of identity. But you are talking about a subset of people at a very minoritarian extreme.
I don't think that one is what one says one is simply because one says one is. Also, cat is not a gender. But I'm sure you know this.
The question under what conditions are self-attributions of gender reliable? is not an easy question. I don't think self identification as being of a gender makes one a member of that gender because I think one can be wrong about one's gender, without this entailing that gender is equivalent to sex. I also don't happen to believe gender identities are semantically stable.