An analysis of exclusion space and octahedral geometry shows us that probabilistic different versions of octahedrons that do not join the cluster are isolated and hence out of scope. That is, they place no further role in the universe. They are the noise. Only octahedrons that joins are signal and so we concentrate only on these.
In order to join they suffer a constraint. No longer able to take on all 54 possible configurations. The second to join is now constrained to only 18. As we add to the chain each new on has 18 possibilites but the inner ones are constrained to only 6. Finally as we move from a chain to a three way joint the possibilities fall to 2. The two-ness just won’t go away, we are stuck with it as this represents each of two mirror images or isomers. We can break this pattern by forming a cross-chain between two out-reached arms. Such a cross-chain sets in place one of the isomers on each side.
It is interesting to note that in octahedral geometry there are three planes that intersect orthogonally. This means that at a three way joint each new chain path is orthogonal to the others. This gives the overall impression of a set of right-angled square pathways, like a crazy jungle-gym in a kid’s park.
Think ahead now to the event surface at around K. The network is littered with loops and cross connections, probability constraints and exclusion have played their part in the network as it stands at n=K. Where now is the signal.
Let’s take a step back to better understand.
We are here because a2 + b2 = c2 has more than one solution.
There are multiple (a, b) and where there is coincidence in the TPPT we get octahedral definitions with matching edges. I would predict that as n gets larger, and as the width of a TPPT generation gets wider the possibilities for edge matching also increases. Hidden deep in the structure are parcels of certainty, and built on top of that uncertainly again outwards to the horizon.
According to some papers on spin foam they are looking for both a notion of spin and an evolution loop defined space over n to produce cells. The problem I have at this level is that what the hell is spin and why would loop space evolution over n be a source of signal. At some stage I would like to formulate a projection into 3-space without n, but I can’t see spin foam ideas at work here.
So, how do loops evolve into spaces? Unlike the spin foam and loop quantum gravity diagrams exclusion space is square, not loopy, there are no four ways vertices and all three way vertices (nodes) have their arms orthogonally. This means that a chain cannot be turned into a three way at any node because there are no free edges and uncertainly has already reduced leaving no option to rearrange the entire chain. Only where an elbow exists is there a free edge. In general there is no preference for straight chains and elbows are just as likely, but this has yet to be proven. Maybe only a large scale simulation will shed some light on this. Nevertheless, the spin foam evolution idea that seems rooted in the simple line diagrams that appear in the papers don’t seem to map easily to a world more like a plumbing game on your phone.
At least we can assign some metric to the square loops. For example, they have a defined perimeter in terms of a node count and there is a notion of loop area and even loop volume. But it’s not signal.