WOLFRAM NOTEBOOK

Theoretical Predictions

Reproduce General Relativity

Yes for the structure of the Einstein equations
Matter vs. background of space [ probe: scalar field ]
Not yet: trajectories of test particles [ topological soliton as a faux particle ]

New Phenomena

Early Universe

Dimension change / inflation

In[]:=
ResourceFunction["WolframHausdorffDimension"][RandomGraph[{200,600}],1]
Out[]=
3.28979
In[]:=
Table[ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,600}]]],1],400];
In[]:=
Histogram[%]
Out[]=
In[]:=
ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,600}]]],All,"AllDimensions","VertexMethod"Mean]
Out[]=
3.30127
In[]:=
Histogram[ParallelTable[ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,600}]]],All,"AllDimensions","VertexMethod"Mean],1000]]
Out[]=
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Histogram[ParallelTable[ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,800}]]],All,"AllDimensions","VertexMethod"Mean],1000]]
Out[]=
In[]:=
ParallelTable[ResourceFunction["WolframHausdorffDimension"][First[ConnectedGraphComponents[RandomGraph[{200,800}]]],1],4000];
In[]:=
Histogram[%]
Out[]=
In[]:=
RandomGraph[{200,800}]
In[]:=
init=
;
In[]:=
ResourceFunction["WolframModel"][{{{x,y},{y,z},{z,w},{w,v}}{{y,u},{u,v},{w,x},{x,u}}},List@@@EdgeList[init],5,"StatesPlotsList"]
Out[]=
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In[]:=
ResourceFunction["WolframModel"][{{{x,y},{y,z},{z,w},{w,v}}{{y,u},{u,v},{w,x},{x,u}}},List@@@EdgeList[init],10,"StatesPlotsList"]
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Is the dimension “cool down” continuous or does it have phase changes?

Phase change ⇔ percolation ?

What is the effect on dimension of random reconnection?

Not even local, etc. If it’s completely random, the dimension can increase.

Is dimension cool-down an “approach to equilibrium” that does not show universal behavior?

Relationship to variable speed of light

r^d vs. c t
Projecting a causal cone to an instantaneous spatial structure
Events in the causal cone (c t)^d t
If c ~ t^α, then dimension is d + α

Cosmological constant : events associated with maintenance of space / events associated with matter

Grid-graph isotropy?

While geodesics are preserved by rotations; the whole space also has reflections...
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