# Exclusion Space and Uncertainty

## More on Uncertainty

Exclusion space is directly implied by the counting numbers, I didn’t have to invent it, I just discover it by following the signal. This is a natural quantum space both discrete (the octahedrons) and uncertain. I’m going to talk more here about uncertainty.

The first octahedron is a pick from 54 isomers, they are all equally uncertain in terms of which takes up the space, and therefore exclude the others. That’s why its exclusion space!

As you introduce more octahedrons, if they don’t connect to a preexisting octahedron then they are isolated and there is no communication possible. When an octahedron appears with the possibility of a shared node or pair (edge) then we get our first real look at uncertainly at work.

There are two possibilities here. 1) like high school probability, the uncertainly of the system increases. One might think if you roll one die you have 6 states, but roll two and the number of possible states increases, OR 2) uncertainly decreases because we already have the pre-instantiation of the first die. The first and second event are not independent. The second octahedron only comes into view WHEN it shares a node or edge, so history is important.

Take the example of a shared edge. The uncertainty of the first octahedron drop to 1 in 18 when an edge is known. Similarly the second octahedron now has only 18 isomers to be uncertainly about and not 54. So did the uncertainly of the system increase or decrease. I put it to you that the uncertainly of a system, unlike probability, is measured at its boundary. When there was one, it was 1 in 54, now with two the boundary has increased in size, but at each new interface the uncertainly has fallen to 1 in 18. If we add a third to the chain the lowering of uncertainly will ripple back into the interior. Once the interior has its connection possibilities exhausted it is out of scope and unobservable.

A bit like a singularity, you only know of it through its event horizon.

## Where from here

There is only one way and that’s up. The problem with exclusion space is finding new signal. This is virgin territory unlike the counting numbers which have been studied for hundreds of years.  The goal is clearly to approach concepts like the quantisation of space, gravity, charge, type and instance. Eventually confirm the dimensionality of spacetime, derive the Standard Model and explain mass and energy as we perceive them.

One of my thoughts was that at this level uncertainly behaves like temperature or entropy, and so in the long run a sum over history is essentially thermodynamic. Simply put, the more you have the less uncertain it all becomes and so entropy decreases. Which is why time appears to run forward, its essentially just thermodynamic memory.

## K

Looking at exclusion space in the low numbers is I suggest very misleading. We will only really get to know its properties once we use some supercomputing to simulate upward of billions of octahedrons. This means pushing n into unheard of regions. The universe as we see it today behaves in the region of K, where K is a counting number defined to be larger than any other counting number you can nominate or become aware of the ability to express or communicate. What happens at K is a mystery, do the number of solutions to a2 + b2 = c2 get more dense, or less. Are there many isolated clusters of connected octahedrons, or just one big one cluster. Do they loop, and is this common enough at scales to suite the spin foam theory.

# A Differentiable Manifold

Take a look at this paragraph from the Wikipedia article on Spin Foam:

“A spin network is defined as a diagram (like the Feynman diagram) that makes a basis of connections between the elements of a differentiable manifold for the Hilbert spaces defined over them. Spin networks provide a representation for computations of amplitudes between two different hypersurfaces of the manifold. Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network. A spin foam may be viewed as a quantum history.”

A differentiable Manifold is just one over which there is local linear structure where calculus can be done. In this case as simple as the ability to measure the lengths of chains and loops by counting. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured.

The exclusion space defined by generated octahedrons fits both these definitions.

I will also refer to “Sum over Surfaces” form of Loop Quantum Gravity (Michael P. Reisenberger, Carlo Rovelli) which canvasses the idea that:

“By giving a graphical representation `a la Feynman of this expansion, we find that the theory can be expressed as a sum over topologically inequivalent (branched, colored) 2d surfaces in 4d.”

The loops that are generated (at any point in the iteration) are 2D structures with a defined length and area. As the iteration continues two things happen, 1) new octahedrons join the network, and 2) multi-stated octahedrons tend to more limited ranges of states as exclusion bites and limits some octahedral solutions from participating in the network. It can be said then, that there is an sum over surfaces (area) projected into 4D.

There are number of generalizations in the “Sum over Surfaces” approach. There are assumptions for example about the micro structure and nature of loop branching. Never-the-less it is clear that over “time” the change in the 2D surfaces produces the desired 4D structures.

As a final thought, if you were wondering how the area of a loop is calculated, you simply imagine octahedrons unimpeded by exclusion. How many do you need to fill in the loop. You were right, its integral.

# Quantum Uncertainty

As previously described in earlier posts the very nature of exclusion space is that it is populated progressively with octahedrons. Each octahedron can be in any one of 54 states. Most states do not form common edges are so can be ignored as they do not participate in any linked structures. Where there is more than one possible arrangement of states the various states are mutually exclusive. In this way the very nature of the construction and placement of octahedrons in uncertain. This is the underlying nature of all quantum uncertainty.

As these uncertain structures form and reform as the core iterator generates more structure the foam implied in loop quantum gravity exhibits the same uncertainty. Hence the vision of a bubbling or boiling foam where loops come and go from existence.

As the iterator moves on some structures become resolved as options for placement in exclusion space run out. This leaves a core of lower uncertainty and a horizon or higher uncertainly.

# Spin Foam and Quantum Gravity

Further examination of the chains developed in the last post lend me to believe that this theory makes an excellent candidate for Quantum Gravity like Spin Foam.

As Carlo Rovelli says in Loop quantum gravity:

“In gravity the loops themselves are not in space because there is no space. The loops are space because they are the quantum excitations of the gravitational field, which is the physical space. It therefore makes no sense to think of a loop being displaced by a small amount in space. There is only sense in the relative location of a loop with respect to other loops, and the location of a loop with respect to the surrounding space is only determined by the other loops it intersects. A state of space is therefore described by a net of intersecting loops. There is no location of the net, but only location on the net itself; there are no loops on space, only loops on loops.”

I think this sums up what the chains and loops can be used to characterize. Any loose chains are now seen as noise and only looped chains are signal. The geometry of the octahedral unit and its three way connector also lends itself to loops upon loops. In further posts I will explore the spin state and half spin numbering that quantum gravity suggests may be a feature of the spin foam loops.

# Spacetime from causality

I was just posting a comment on the blog of the author of this paper.

I had actually been reading a related paper from the same author, “Spacetime from causality” which I found very depressing at first. Then I realized that everything Wuthrich is complaining about is practically the same things I complain about when it comes to thinking about causal sets.

His main complaint is that you can’t just make this stuff up, apply arbitrary rules to sprinkings etc. and hope to get anywhere. Naturally, I agree because my cause set theory, it we continue to call it that, is a primitive view on the counting numbers. I didn’t have to make it up, if you think about the counting numbers long enough, and I did, then this interpretation is evident. Please note, that although this exclusion space is 3 dimensional it is NOT Euclidean.

• There is no origin
• isolated octahedrons are just that … noise
• chains are the only signal
• common nodes are the only structural mechanism

The problem now, is what do with this structure. Maybe it needs some giant computing resource to map the exclusion space and model the chains. I just don;t have the predictive ability at the moment to peer forward into the first 10,000 octahedrons, find the common edges, and follow the strands. In all probability the early chains are likely to reveal very little and the really interesting work begins somewhere after the first terra octahedrons.

On the grand assumption that the law of large numbers kicks in, and there is not something fundamentally broken with the chain dynamics of spatial exclusion space then the next problem is to show how to derive a mapping to our precious illusion of Minkowski spacetime.

As Wuthrich suggests/implies, a better starting mapping may be to ideas associated with quantum gravity as this is not strictly Minkowskian or spatial in the traditional sense, but you can see it from there on a nice day.

### Causal Sets Revisited

I’ve voiced my displeasure at causal set paradigm’s before. The main point at issue being that it was all just made up. A random sprinkling is a really poor way to derive a Minkowski Spacetime. This is one of the reasons I started out to find a better way.

In this vein of thought and with my recent discovery of how to fabricate spatially exclusive octahedrons I have come back around to consider what this mechanism means for causal sets.

To cut a long story short its pretty evident that you can stick octahedrons together. That is, they form chains. Have a look this wiki page again, notice anything strange.

Tree of primitive Pythagorean triples

Take a look a the tree diagram to the right. Note the grand children of (5, 12, 13) these create the concepts that underlie the construction of 54 octahedrons. Now note the grandchildren of (15, 8, 17).

Both sets of octahedrons contain the pair 205 an 305. This means that its possible to construct two octahedrons that SHARE an edge, and remain spatially excluded from each other except the shared nodes.

Project your mind now into the far flung limits of the TPPT. At some point isn’t it possible that a chain of octahedrons is possible. I’m guessing they would all live in the same generation, but that needs some work to prove.

### Whats Legal Octahedral Geometry

I can imaging two octahedrons joined along one edge. To me this is great signal. But, now consider a simple shared node. Does this happen at all? and how many times can it happen and still be in keeping with my claims of spatial exclusion. For example, if three octahedrons shared a node, then by definition they also share edges and so more nodes. The complexity increases quickly. This would be clearer if I could find a proof of a conjecture about the number of solutions to the core identity a2 + b2 = c2

Clearly from our wiki example, multiple solutions are allowed and hence form common edges, but what happens if there are more. Already I’m imagining a whole raft of diagrams to map out all the possibilities engaged in corner, edge and face sharing and octahedral geometry in general. This will be my next research area.

Staying with causal sets, lets assume that we go with an edges only chain paradigm. Each new octahedron can join at only one of three edges opposite the prior joining edge. How can we imagine the ideas around the way the chain may branch. To branch at a right angle to the left or right would involve a triple node inclusion. The only way to branch that preserves the edge only paradigm in the transverse edges.

These are the edges that are opposite to the prior chained edge and extend at a angle from the node at the top to one fr side and from the node at the bottom to the other far node. These two edges share no nodes. Its the only clean way to branch and preserve spatial integrity.

So now you can chain in a choice up to 5 ways, and branch in a choice of 2 ways. This places some severe limits on causal set construction. It also demonstrates that its possible to construct loops, and for branches to merge.

# At the edge of reality and dimensionality

In the previous post, I said that we are taking generation at the limit of the open 4-vertex TPPT graph. What this means is NOT at infinity or anything silly like that, but from the zone where its not possible to tell anymore what generation your at.

Have a read of the following: clifford-algebra-a-visual-introduction

Especially the part about the inverse function. As with Clifford I also share the problem with infinities. In my blogs I certainly do not invoke the need for infinity, its enough to say that the core iterator is there and its still there. Without iteration you don’t move, think or read, so it’s there BUT it’s not an infinity.

Have a read of the following Maths Stack Exchange posting here

It seeks to calculate the fractal dimensionality of the TPPT, and I think it succeeds rather well, apart from the fact I wrote the original post and Ricardo Buring was good enough to work through the proof.

But, you ask where did I get this crazy idea that the fractal scale would be anything like 3(3 – 2√2) in the first place. To understand this read my as yet unpublished paper, as follows:

Octahedral Graph Scaling

The main result it that the fractal scale is the same for both the TPPT (a open 4-vertex graph) and the octahedron (a closed 4-vertex graph). Because they have the same fractal scaling factor the direct implication is that at its limit the TPPT produces a notion of spatial separation, and the things being separated are closed 4-vertex graphs. This is the first instance where we can be at last allowed to view the octahedral graph as a spatial octahedron.

I think in this post I can say that I have a rival derivation of the concept of 3-space.