My Share of Rabbit Holes

To anyone who has read my maths adventure this far you will realise that maths is full of twists and turns and it’s fair share of rabbit holes. After some soul searching I have decided to leave this blog intact for posterity with all the rabbit holes in place.

The one conclusion I have come to after all this time is that my original result (the paper on Graph Scaling) still stands out for it’s core meaning. The implication is that the triples inherently define a three dimensional substructure of the counting numbers. That is, three dimensions are directly implied, the rest being noise. These dimensions are pure, they are NOT Minkowski space time, but the underlying spatiality of MST is core and doesn’t need complex and artificial construction (such as with the bubbles in quantum gravity) … it’s just there.

I have still to make any headway in terms of perturbation (nature of randomness) and instancing in this model. My older model seemed to have these features, but it was a rabbit hole!

I feel that the nature of the dimensional mapping is still important, and that modular forms may be in there somewhere giving rise to interference and perturbation. At any rate, the paper needs to be rewritten, i’s a first cut, but the significance of the result is the same.

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It’s a Square World After All

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.

 

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.

If you have an opinion, please leave a comment.

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.

Christian Wuthrich –  The Structure of Causal Sets

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.