We have a set of new blocks to play with. Instead of octahedrons with the a’s and b’s of three solutions at the vertices we now have each of the six points representing the six pathways to the three solutions.

At each end of a solution pair we have (a,b) and (b,a). These are unique and as such we can not join these blocks up into solution chains. To do this we need solutions. Let’s instead label each vertex as c instead of (a,b), now we have a c at each end and it destroys the information about there being two pathways. Instead label the vertices as +c and -c. We can have a rule such that the c is negative if a-b is negative. It doesn’t really matter as long as we are consistent.

Now we can start to chain out blocks again and we have re-derived a spin foam like structure again. The correction to the mapping has led to a different way of recognizing the joining of octahedral solution clusters into chains and loops, but they are still there.

It also introduces new limitations on chain formation. For example, same c’s will only be found across generations as they are unique within a generation. (Is this true?)

The next step is to be able to project these dimensionally and achieve the dual outcome of projective instancing at a useful 2D density to support MST.

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