Criticism of Reinin Dichotomies

I don't know about the mathematical basis of Reinin dichotomies. The dichotomies are always presented as given and I've never heard of a source about their mathematical basis, which is omitted in all presentations I've seen. Explanations useful for me would involve comments about how did they end up with a blank dichotomy and so on, but this part appears to be unavailable in material written for the general public.

As for your question, I don't understand everything. A dichotomous relationship between what and what? To guess, maybe the relationship is between properties. To "prove" such a relationship one would need to inspect the validity of its mathematical foundation. To "verify" it would mean resorting to empiria to check whether the relationship is also sound and not merely valid.

I don't use bimodal distribution or regression line, but a quick search suggests these operations to generate output that includes irrational numbers. As such, these operations cannot be used to evaluate the validity of Reinin dichotomies. But they could be used to compare actual and expected empirical evidence of dichotomies once someone has collected enough empirical data.

Since Reinin dichotomies have only been used in conjunction with Model A there would be a reason to expect easy success in combining them with Model M. But I'm not very keen about studying these dichotomies because the already published material is methodologically sloppy. For example, at a glance, this list[1] could have been formulated like a truth table so that the top right corner of the table would include the values:
00
01
10
11

Instead, the table is formatted like this:
00
11
10
01

Which could be correct but there's no explanation, which makes me feel this isn't serious research because why would anyone do that?The article goes on to define dichotomies as combinations of each other so that:

"X*Y = (X & Y) or (~X & ~Y) where ~X denotes the opposite of trait X"

I feel uncomfortable reading a source that states the above instead of:

"X ⇔ Y :⇔ (X ∧ Y) ⊻ (¬X ∧ ¬Y)"

This makes me feel like I don't belong to the target audience. Logic can be difficult but it sure is more difficult when one doesn't make use of conventions that are already known to work. Furthermore:

"The relation * can be seen to be associative, meaning (X*Y)*Z = X*(Y*Z) is always true."

The author of this sentence is too afraid of making mistakes. The relation * should be assumed to be associative in order to check whether it really is that. Nobody wants check that if the author acts afraid.

"Proof:
The relation may be restated more concisely as follows:
X*Y = (X == Y)
where == denotes logical equality - parentheses added for clarity."

Okay, so == denotes what I'd call equivalence and denote by ⇔. What does = denote, then?

In any case, a logical consequence of the author's declaration is:

"X*X = (X == X)"

I can't think of any way the author would've meant to state that, because it means either that every trait is the same trait or that this model cannot be used for differentiating traits. Furthermore, his "proof" isn't really a proof. It's just a declaration.

It's a declaration according to which every trait that is (equivalent with) itself also has the relation * to itself. The author can't have possibly meant that because his intention was to define traits in terms of each other.

It would probably be faster to make new dichotomies than repair the existing ones. Anyway I can't expect to learn much from sources that indicate unfamiliarity with logic. I'd use Transactional Analysis in an attempt to define traits p and q so that:

p 1111 q means p and q are the same trait
p 1001 q means p and q are unrelated traits
p 0110 q means p and q are related traits
p 0000 q means p and q are opposite traits

Then I'd attempt to define the dichotomy between obviously opposite traits, such as Aristocratic and Democratic, as D(p,q) so that:

p 1001 r,
q 0110 r,
p 0110 s,
q 1001 s.

In which r and s are other traits.

[1] http://www.wikisocion.net/en/index.php?title=Reinin_dichotomies