In[]:=

ResourceFunction["MultiwayFunctionSystem"][n|->{n+1,2n+1},{1},4,"StatesGraph"]

Out[]=

ResourceFunction["MultiwayFunctionSystem"][n|->{n+1,2n+1},{1},4,"ExpressionEventsGraph"]

In[]:=

NestGraph[n|->{n+1,2n+1},{1},5,VertexLabelsAutomatic]

Out[]=

NestGraph[n|->{n+1,2n+1},{1},5,VertexLabelsAutomatic]

In[]:=

testrel=And@@Flatten[Table[{p[n]<p[n+1],p[n]<p[2n+1]},{n,7}]]

Out[]=

p[1]<p[2]&&p[1]<p[3]&&p[2]<p[3]&&p[2]<p[5]&&p[3]<p[4]&&p[3]<p[7]&&p[4]<p[5]&&p[4]<p[9]&&p[5]<p[6]&&p[5]<p[11]&&p[6]<p[7]&&p[6]<p[13]&&p[7]<p[8]&&p[7]<p[15]

Trying to find sequence for the p[i] that effectively says what “layer” they are in....

In[]:=

<<SetReplace`

In[]:=

Graph[TokenEventGraph[VertexLabels->Placed[Automatic,After]]@GenerateAllHistories[MultisetSubstitutionSystem[{n_,m_}:>{n-m,n+m}],2]@{1,2},AspectRatio1]

Out[]=

Non-deduped....

In[]:=

Graph[TokenEventGraph[VertexLabels->Placed[Automatic,After]]@GenerateAllHistories[MultisetSubstitutionSystem[{n_}:>{n+1,2n+1}],3]@{1},AspectRatio1]

Out[]=

In[]:=

Graph[TokenEventGraph[VertexLabels->Placed[Automatic,After]]@GenerateFullEventSet[MultisetSubstitutionSystem[{n_}:>{n+1,2n+1}],5]@{1},AspectRatio1,GraphLayout"LayeredDigraphEmbedding"]

Out[]=

(For numbers, the whole state is one token...)

Different events convention:

In[]:=

ResourceFunction["MultiwayFunctionSystem"][n|->{n+1,2n+1},{1},4,"EvolutionEventsGraph"]

Out[]=

In the hypergraph case, the expressions/tokens [which are hyperedges] are related because they share atoms of space

In combinators, the whole graph

How does the reconstruction work from the multiway graph to get to the structure of space at individual moments?

## Finding foliations

Finding foliations

Is spacelike separation ever possible?

#### We can’t get a notion of space without nontrivial antichains, and nn+1 defines a number-line-like ordering

We can’t get a notion of space without nontrivial antichains, and nn+1 defines a number-line-like ordering

#### If we have a rule that operates on pairs rather than integers, then we can get an antichain...

If we have a rule that operates on pairs rather than integers, then we can get an antichain...

Compare this to the hypergraph case.... The only difference here is that we’re looking at values inside the hyperedges rather than at matching of atoms of space....

Possible antichain structures: https://oeis.org/A006126/internal

If there are antichains, then there are multiway antichains, AKA a sequence of branchlike hypersurfaces

On a single branch, is there a state with multiple events that can occur in that state: this is the criterion for spacelike separation in a global multiway system

Pure spacelike separation:

Branchlike separated:

#### Deriving a singleway causal graph:

Deriving a singleway causal graph:

Pick a foliation. As soon as there multiple events at a single foliation, the causal graph will be nontrivial.

Do you need branchlike separation to get a nontrivial causal graph?

#### Can you have spacelike antichains even without having a multiway system?

Can you have spacelike antichains even without having a multiway system?

I.e. does the operation of GR require QM?

#### Can there be simultaneity without the possibility of branchlike separation?

Can there be simultaneity without the possibility of branchlike separation?

#### For relativistic transformations, must be able to have different evaluation orders for events.

For relativistic transformations, must be able to have different evaluation orders for events.

#### We could think about a non-overlapping MaxScan like a CA, and that would have nontrivial causal graph .... but as soon as we think about different events separately, we have a multiway graph...

We could think about a non-overlapping MaxScan like a CA, and that would have nontrivial causal graph .... but as soon as we think about different events separately, we have a multiway graph...

### Goal of the observer: sequentialize to make a multicomputational system merely unicomputational...

Goal of the observer: sequentialize to make a multicomputational system merely unicomputational...

#### Observer has to use foliations.

Observer has to use foliations.

(They could make a total ordering and deny the existence of space.... )