@sindri to be honest, it sounds like you are throwing around math terms without fully understanding them. I can see that you understand how to multiply dichotomies but perhaps you aren't too familiar with group theory and formal proof methods?

When it comes to "physical" theories (theories that apply to the real world, that is) the notion of "proof" is a bit hazy -- what is an assumption/axiom and what is derived is usually subject to some choice. Once a consistent model has been presented it's just a matter of working out its structure.

There is some nuance to this, as I think you have commented in your own article, because you can make the model appear unbalanced with out a full explanation. I think you define the supervisions rings as Dih4 if I remember right. This is unnecessary if you account for the rationality dichotomy, but i cannot just define the type dichotomies as z2 sign group and have that point understood.
Maybe it's because abelian groups are easier to understand, or some fascination with Reinin dichotomies, but I am a bit puzzled why people want to somehow "prove" that the socion is actually abelian / a vector space (I've seen this attitude from Russians too). The relationship group is nonabelian no matter how you define it, because the mirror of my benefactor is not the benefactor of my mirror. The rings may derive from some underlying binary reality but the questions of why and how that occurs are still open. Until then, I am inclined to focus more on the relationships than the Reinin dichotomies.