In[]:=

ResourceFunction["EnumerateSubstitutionSystemRules"][{22},2]

Out[]=

{{AAAA},{AAAB},{AABB},{ABAA},{ABAB},{ABBA}}

In[]:=

ResourceFunction["SubstitutionSystemCausalGraph"][{"AB""BAAAB"},"AB",8]

Out[]=

In[]:=

LayeredGraphPlot[ResourceFunction["SubstitutionSystemCausalGraph"][{"AB""BAAAB"},"AB",8],AspectRatio1/2]

Out[]=

#### This case involves superluminal expansion

This case involves superluminal expansion

In[]:=

ResourceFunction["SubstitutionSystemCausalGraph"][{"BA""AB"},"BBBAAA",8]

Out[]=

In[]:=

TimeConstrained[LayeredGraphPlot[ResourceFunction["SubstitutionSystemCausalGraph"][#,"AABB",15],AspectRatio1/2],3]&/@ResourceFunction["EnumerateSubstitutionSystemRules"][{22},2]

Out[]=

,

,

,

,

,

In[]:=

TimeConstrained[LayeredGraphPlot[ResourceFunction["SubstitutionSystemCausalGraph"][#,"AABB",15],AspectRatio1/2],3]&/@ResourceFunction["EnumerateSubstitutionSystemRules"][{23},2]

Out[]=

,

,

,

,

,

,

,

,

,

#### The mass of a black hole is equal to the number of causal edges that go into the BH ??

The mass of a black hole is equal to the number of causal edges that go into the BH ??

nth cousin for nodes?

Note: the distance function is not symmetric; [[ “removedness is not symmetric in cousin relations” ]]

Lowest layer distance matrix:

#### Claim: for BH horizon, the distance function is asymmetric; for cosmic horizon it’s symmetric

Claim: for BH horizon, the distance function is asymmetric; for cosmic horizon it’s symmetric

#### True existence of an event horizon involves intersection of future propagation of causal edges to infinity, and will in general be undecidable.

True existence of an event horizon involves intersection of future propagation of causal edges to infinity, and will in general be undecidable.

#### But the existence of event horizons is basically a failure of confluence-like merging for the causal graph.

But the existence of event horizons is basically a failure of confluence-like merging for the causal graph.

#### In general, there is a transition graph of what vertices can reach what other vertices.

In general, there is a transition graph of what vertices can reach what other vertices.

What we are currently doing is approximating null infinity by whatever level we’ve currently reached.

Time step evolution:

## Hypergraph case

Hypergraph case