James, you are probably both artist and scientist. As far as I can figure out, your binary sequences times those fourfold permutations will have a pleasant representation in Clifford algebra. I used to design the dihedral groups, that were once investigated by Claude Lévi-Strauss in CA, and from this I got the 'reorientation groups' that led me, by the time, to the standard model of particle physics. That is to say, physics has its own structuralism, or put differently, the morphogenetic structures of mind, - and paleo society,- can and should be represented in such a convienient way, as you proposed now.

I looked up dihedral group in Wikipedia, including D2 Klein-four-group.   I have attached a 1 page introduction to the formula. For the moment I view the transform sequence as (- -, - - --> - +, ++). This I have interpreted as the basic theatrical (Aristotle's poetics) move from active negation (the actor who performs negative harmful actions) to passive (the passive recipient or victim) to two alternative reflexive moments) in which both the positive and negative polarity of life-energy is held/embraced in reflection (Jung's coincidentia oppositorum; Homeric Shield of Achilles, Chinese yin-yang). The imaginative energy-releasing leap is to the alternative inverses, the transcendences (eight in my version of the rCF). I think this does not fit the Klein group matrix, which if I read it right is (1 0, 0 -1, -1 0, 0 1)? The Weil-Levi-Strauss initial matrices prior to the doubly reflexive inverse (+1 +1) would seem to be, in the dihedral matrix way of indicating valence  (-1 -1, -1 -1, +1 -1, +1 -1). I would naively guess that this may not even be a mathematical formula?  On the other hand, I tentatively see that I might rewrite the active, passive and two reflexive valences as (0 -1, -1 0; 0 1, 1 0). This is actually perhaps more appropriate to psychotherapy and Buddhist meditation, as it has switches the negative (-1) into zero (0) and zero into the positive (+1).  This way the negative actions that cause suffering are no longer held to be negative but rather zero-charged; this means there are harmful things people do but they are not some sort of absolute evil.  In healing process they turn into zeros. Further, the Fx valenced (0) and (b) valenced (0) switch into positives, Fx valenced  (+1) and Fy valenced (+1) [or (a) valenced +1). This would add more nuance to the rCF.  I have to ponder this revision.  Even with such a revision, is the entire Weil-Levi-Strauss transform a Klein group or higher power dihedral group, or none of these?

Very good, wonderful, clever, James, go to the second page of ‘contributions’ on my page, and take a look into the paper ‘Bauhaus – Isometry and Fields’. You identify it immediately, as you see Jung’s Mandala shining through, on the third page or so you find a very basic and important explanation of the dihedral non-commutative group D_2d, also written D_4 (lowered index 4). I can see all the Clifford algebra and iterant representations of your 8-sequences. Basically we have here quaternions (and a two components of Klein 4 groups having positive and negative sides) as I have discussed in the beginning of the ‘Four Forms’ paper.

I will now read your paper attentively, and then show you how it goes.

I have so much to learn about algebra.  I will read these articles.  With respect to basic symbols of space-time orientation and iteration in evolution of graphic marking (palaeoart, proto-art, proto-language) I have hypothesized on the basis of my published article providing inventory of evidence for Oldowan, that the use of fourfolds and polygons arises in the Oldowan 1.8 million years ago. 

https://www.researchgate.net/publication/276033517_Palaeoart_at_Two_Million_Years_Ago_A_Review_of_the_Evidence_2014

Based on our discussions for this project I would say that at 1.8 million years ago the graphic markings precede differentiation between 'art', 'religion/spirituality', 'techne' and 'numerosity'.  Oldowan paleoart is all of these in process of marking stone. 

I have attached 3 examples of Oldowan graphic expression. (1) FxJJ1 #302, from Koobi Fora 1.8 MYA. I saw it in the small Koobi Fora museum when I did field school there.  I give my hypothetical analysis of the  phememes that may have been spoken for each flake removal. (2) MNK-Main, Developed Oldowan B, photo Mary Leakey, Olduvai Gorge. My eye says this is a hexagon, could be natural hexagonal shape, but llooks to me like one or more possible intentional large flake removals  to make the hexagon shape. (3) the Oldowan pecked cobble, Olduvai Gorge, 1.8 MYA, photo Mary Leakey.  I have annotated the image with my hypothesis for numerosity (2 sequential sets of indentations, 2 by 4; possible spoken phemenes). Also circumferential circle by repeated pecking rotation and abrasion on upper part of stone.  Early ethnomathematics of Homo habilis or rudolfensis?

Article Palaeoart at Two Million Years Ago? A Review of the Evidence (2014)

“I have so much to learn about algebra.” That is not necessary. You got a credit, but ..

James, you go too fast. I have just printed out your first reference being “Harrod rCF introduction 2017.docx”, thank you, very good. Look, Lévi-Strauss attempted to formalize these matters with somewhat cumbersome mathematics. I know how these things can be answered properly. But you must decide, if you want to cooperate with me or not. In this case you must wait for my answer before going on. I would also peg you to make it clear what you refer to by quoting “Basic Intelligence Processing v4 2016” in your question starting up this thread. Obviously you mean my paper published in the Journal for Space Philosophy in honor of Joel Isaacson.

Your new additional link to your article “Palaeoart at Two Million Years Ago? A Review of the Evidence (2014)” is wonderful. I agree with your hint, as you can see from my own lectures. But you must decide what you want to do now. You have enough time with your algebra. These problems I can solve for you, in advance if you want that. But it takes a little time, symmetry and honesty where exchange and credit for authorship are concerned. I have a few people to care for, at least more than two. So I need a little time. At present I am writing a paper having a title which I would not even make public in the Research Gate, at present. Fortunately there are titles and ides no one can just copy and paste. So there is a proper chance for real cooperation where everyone is able to know what it is that he or she has created. – For example I have learned a lot from Lévi-Strauss with whom I even exchanged a few letters, and from lectures of dancer and actress Adeline Schebesch with whom I exchanged about this topic. Yes, you are right, there is this early mathematics. But we must go slowly and there must be discipline in our exchange. Otherwise we challenge such an explosion as in most other threads that have fascinating topics. Let us ponder over appropriate procedure instead of rushing in. Look, I have waited for decennia, though my habilitation treatise was exactly about this theme.

@Harrod, to give you a first and definite answer. For me it is immediately clear that your 8-fold vectors which look somewhat unitary, are definitely multivectors from the real quadratic Clifford algebra CL(3,3), as you can see from the analysis of iterant representations of CA in my last book (Decay of Motion - The Anti-physics of space-time; Nova 2014, page 248. But there is still some more humble, more beautiful rep, better adapted to paleolithic thought! Meanwhile best,

Bernd