In[]:=

cg=ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},5,"LayeredCausalGraph"]

Out[]=

In[]:=

ceg=ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},5,"EventGenerations"]

Out[]=

{1,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}

In[]:=

Position[%,4]//Flatten

Out[]=

{7,8,9,10,11}

In[]:=

Rest[VertexOutComponent[cg,#]]&/@Flatten[Position[ceg,4]]

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{{12,16,13,14},{15,18,22,17},{16,17,18},{19,21},{20,21,22}}

In[]:=

Graph[Flatten[(UndirectedEdge@@@Subsets[#,{2}])&/@%]]

Out[]=

In[]:=

ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},5,"FinalStatePlot"]

Out[]=

In[]:=

Clear[SpatialReconstruction]

This measures distance relative to the past. We could measure distance to the future....

In[]:=

SpatialReconstruction[wmo_,dt_Integer:1]:=Module[{cg=wmo["CausalGraph"],ceg=wmo["EventGenerations"],ev0,ev1,oc},ev0=First/@Position[-(ceg-Max[ceg]),dt];ev1=First/@Position[-(ceg-Max[ceg]),0];oc=Select[Rest[VertexOutComponent[cg,#]],MemberQ[ev1,#]&]&/@ev0;Graph[ResourceFunction["WolframPhysicsProjectStyleData"]["SpatialGraph","Function"][Graph[ev1,Flatten[(UndirectedEdge@@@Subsets[#,{2}])&/@oc]]],VertexStyle->ResourceFunction["WolframPhysicsProjectStyleData"]["CausalGraph","VertexStyle"]]]

In[]:=

SpatialReconstruction[ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},5],1]

Out[]=

In[]:=

SpatialReconstruction[ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},5],2]

Out[]=

[[[ This is like a construction of space by looking overlaps of [open] sets ?? ]]]

In[]:=

SpatialReconstruction[ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},8],1]

Out[]=

In[]:=

SpatialReconstruction[ResourceFunction["WolframModel"][{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}}},{{0,0},{0,0}},8],2]

Out[]=

A given pair of events can appear in multiple

In[]:=

SimpleGraph[%]

Out[]=

What is the effective view of space in terms of a Turing machine?

#### The distance between “points” can vary with time [what counts as the “same point”?]

The distance between “points” can vary with time [what counts as the “same point”?]

#### All sorts of “motion” is possible in a part of the universe that is being updated when others are not

All sorts of “motion” is possible in a part of the universe that is being updated when others are not

But the causal graph says generally this isn’t possible.... [Definitely not possible if there is a causal invariance]

## Light Cones

Light Cones

This the causal graph:

This shows the events from the final layer that have as common ancestors events in the previous layer:

Eventually, everything is connected to the Big Bang:

#### Relation between balls in the spatial graph, and the reconstructed spatial graph....

Relation between balls in the spatial graph, and the reconstructed spatial graph....

Fake/apparent spatial light cone:

#### Relation to light cone balls:

Relation to light cone balls:

Actual light cone:

## Comparison between genuine light cone slice region and the inferred effective spatial region

Comparison between genuine light cone slice region and the inferred effective spatial region

#### We can either compare a bag of events, or a bag of expressions (i.e. hyperedges)

We can either compare a bag of events, or a bag of expressions (i.e. hyperedges)

Starting from a given point.... Which is either an event on a previous step, or some successor of that event in the target slice....

Compare either the light cone of that selected event; or compare the effective disk of the successor of that event.... [e.g. from the middle of the light cone]

### Case 1: “Points” are approximated by hyperedges

Case 1: “Points” are approximated by hyperedges

### Case 2: “Points” are approximated by events

Case 2: “Points” are approximated by events

### Case 3: “Points” are approximated by elements

Case 3: “Points” are approximated by elements

### Case 1: Points are hyperedges

Case 1: Points are hyperedges

Start of light is a hyperedge

This includes the atom number in each expression:

This includes the expression indices:

Pick an expression at a particular step, then compute its out component to get its genuine light cone

Now construct the final version of space based on expression connection by common elements (which is the dual of the usual case):

#### Function to deduce the effective spatial ball from the genuine light cone:

Function to deduce the effective spatial ball from the genuine light cone:

#### Function to construct the effective spatial hypergraph from expressions:

Function to construct the effective spatial hypergraph from expressions:

This is labeling expression indices:

#### This is our deduced structure of space, based on local criss-crossing at a particular slice in the foliation...

This is our deduced structure of space, based on local criss-crossing at a particular slice in the foliation...

#### This is the “genuine light cone” [though it may be centered in the same place]

This is the “genuine light cone” [though it may be centered in the same place]

These are the “genuine successors” of expression 18; and of these could be center of the ball in the effective spatial hypergraph.

#### To decide our value of c, we have to normalize the “average reconstructed distance gone” ... which will effectively tell us some version of the rule size

To decide our value of c, we have to normalize the “average reconstructed distance gone” ... which will effectively tell us some version of the rule size

### Highlighting the hypergraph

Highlighting the hypergraph

## Definitions

Definitions

### Both these functions return expression indices

Both these functions return expression indices

#### Function to deduce the effective spatial ball from the genuine light (causal) cone:

Function to deduce the effective spatial ball from the genuine light (causal) cone:

#### Function to construct balls in the effective expression-deduced spatial hypergraph:

Function to construct balls in the effective expression-deduced spatial hypergraph:

Construct the effective expressions hypergraph:

#### Function to find the center of the effective spatial ball from the genuine light cone:

Function to find the center of the effective spatial ball from the genuine light cone:

### These functions take expression indices and make plots

These functions take expression indices and make plots

Genuine light cone intersecting the reconstructed spatial slice:

Deduced geodesic ball in the reconstructed spatial slice:

Genuine light cone:

Effective reconstructions:

### Workflow: pick an expression at a particular generation ... then see how its genuine light cone expands, and compare it with the sizes of reconstructed spatial balls with certain radii

Workflow: pick an expression at a particular generation ... then see how its genuine light cone expands, and compare it with the sizes of reconstructed spatial balls with certain radii

“subspace navigator” / “subspace demon”