Thank you Anton for the excellent post.

As you mentioned there was a difference in the numbers between the pdf and in gigabrain's.

I did some testing for the difference and looks like the reason is the difference in the Wolfram model type.

The example uses the parameter EdgeCountList and the documentation mentions it's "the list of edge counts for each complete generation". And it's the same as edges in the FinalState.

ResourceFunction[ "WolframModel"][{{1, 2, 3}, {2, 4, 5}} -> {{6, 1, 7}, {6, 5, 8}, {7, 5, 3}, {2, 4, 6}}, Automatic, 20, {"EdgeCountList"}]

I used the edge count from the LayeredCausalGraph by calculating the Length of the EdgeList.

And the difference is the LayeredCausalGraph has edges 626 and FinalState has edges 630.

We could continue with this more by email. If you send mail to tuomas@gigabrain.io and we can look this in more detail.

Thank you very much,

Tuomas

Hi Tuomas, appreciate the feedback!

I did test it on gigabrain this past week and it took me some time to figure out how the requests are processed and where they end up showing. I tried it on 2 WMs (24528 and 3975) and it does take a while (days) for higher number of steps, however if it saves it to a cloud it should not matter in the long run. Unfortunately, generations above 35 seems to give an error…

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Code>AccessDenied</Code Message>Access Denied</Message RequestId>JQEG9VE85B4NMT8A</RequestId HostId>kZeWl/i9YBPWU4n0G9GlJ+ODqY83BsuyAxKQZ7F6UHq+kptCjH1yA5RTVeoPjW99m9tsjS0IyuI=</HostId /Error>

Hopefully, this part can be fixed as all the interesting stuff will be happening at later generations. One idea/question I’ve had is the ability to upload computation directly into gigabrain for a particular model. Say I ran a search for singularities for 100 generations (5_100) all day on my computer and it finally gave me the results, perhaps those results can be uploaded directly instead of having to repeat the computation. I can see that some sort of authenticity check might be a good idea where it makes sure that the data matches the model number and a computation type.

As far as a general method of examining these singularities it might end up being a little tricky. Causal invariance that is so often mentioned in this Wolfram Physics Project is basically a claim that any divergence in the causal graph is always temporary. In the view of these singularities, I think the same claim means that none of them will persist for very long. One could come up with a new parameter called “causality” that describes how causally invariant it is at a certain number of generations. A value of 1 would indicate that every generation is causally invariant. Any singularities would then lower that value of 1 depending on how long they persist. That lowered value can be a ratio of Black Hole regions over total number (or rather the inverse of that). So, say a model 24528 was ran for 7 generations and has a total of 18 steps (partial generations). At the 5th generation we have a singularity form at step 7. At the 7th generation we can count total number of steps (18 in our case) and also steps that end up inside that singularity (7). Our “causality” parameter would be (1- 7/18) = 0.61 indicating overall tendency of the causal structure to recombine its steps. If Steven and Jonathan are right, then after large enough number of generations that value will approach 1. Nested singularities present an added difficulty here and perhaps should be treated as extra anti-causal hit to the value of 1. Unfortunately, all of this means that it would require a rigorous analysis of the causal graph. A graph of 100 generations would have to be not only ran as 1 _ 100, 2 _ 100, 3 _ 100….99 _ 100 but also ran at a lower number of steps to determine duration of singularities. That is a lot of computation! It would almost be easier to visually examine it with some sort of an AI algorithm. It would also have been if nice singularity regions could be highlighted even if the singularity itself went away at some later generation.

The most interesting part about all of this is that macroscopic causality appears to emerge from microscopic randomness (randomness in a sense that we cannot pick and choose which update will be applied to our foliation of space). Underlying rule will ultimately determine whether causal structure emerges at all and if it does at what scale (number of generations). Personally, I am hoping for a very tiny but persistent % of singularities at very high generations pointing to some interesting physics.

Interesting experiment. Please develop your ideas and publish some results in a journal of physics. Most of my publications are based on experiments made on a personal computer.