#### What if number of nodes = mass?

What if number of nodes = mass?

#### Mass density

Mass density

#### Size of particle related to e

Size of particle related to e

#### What if particle is fixed clump of nodes?

What if particle is fixed clump of nodes?

#### Phase space vs. ℏ

Phase space vs. ℏ

#### Position is given by location in hypergraph

Momentum (flux) is number of edges crossing a surface per unit “time”

Position is given by location in hypergraph

Momentum (flux) is number of edges crossing a surface per unit “time”

Momentum (flux) is number of edges crossing a surface per unit “time”

#### Somehow branching sets scale of granularity

Or related to e.g. spin?

Somehow branching sets scale of granularity

Or related to e.g. spin?

Or related to e.g. spin?

#### nodes / volume

events / volume

nodes / volume

events / volume

events / volume

#### Constant of proportionality in Vr related to mass?

Constant of proportionality in related to mass?

V

r

#### Elementary light cone

Is there an elementary energy-momentum cone

Elementary light cone

Is there an elementary energy-momentum cone

Is there an elementary energy-momentum cone

#### Multiway causal graph contains everything

Multiway causal graph contains everything

Multiway has neither cones nor balls, really

#### volume of causal cone

volume of causal cone

(c T)^d = spacetime volume

#### Causal graph as tree = no interaction

Causal graph as tree = no interaction

#### Can we measure the dimension of the multiway graph?

Can we measure the dimension of the multiway graph?

#### Is the multiway graph exponential (as opposed to the causal graph)

Is the multiway graph exponential (as opposed to the causal graph)

Exponent in multiway graph ~ quantum decoherence time

#### Two pieces of spacetime with the same dimension, and curvature... but different constants of proportionality [cf conformal invariance]

Two pieces of spacetime with the same dimension, and curvature... but different constants of proportionality [cf conformal invariance]

#### G determines the actual mass, based on node density

G determines the actual mass, based on node density

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G ~ L^3 m^-1 t^-2

1/G ~ m t^2 L^-3

1/G ~ m t^2 L^-d

1/G ~ m t^2 L^-d

Newton law of gravitation:

E ~ G m^2/L^(d-2)

E ~ G m^2/L^(d-2)

1/r^(d-1) is force

#### [Charge relates distance to gauge degrees of freedom] <events per spatial hypergraph volume??>

[Charge relates distance to gauge degrees of freedom] <events per spatial hypergraph volume??>

How many events can happen in this given hypergraph volume

[Given a particular updating order, (roughly) how many events would occur in a generation in that volume]

[Given a particular updating order, (roughly) how many events would occur in a generation in that volume]

#### Vertex degree is independent degree of freedom from dimension

Vertex degree is independent degree of freedom from dimension

#### Gauge fields: ? isomorphism in hypergraph

In a rewrite, many ways to align the rewrite with the actual graph

But if you do one alignment locally, there is some connection to neighboring updates (?)

Gauge fields: ? isomorphism in hypergraph

In a rewrite, many ways to align the rewrite with the actual graph

But if you do one alignment locally, there is some connection to neighboring updates (?)

In a rewrite, many ways to align the rewrite with the actual graph

But if you do one alignment locally, there is some connection to neighboring updates (?)

#### Imagine you have a grid, and your rewrite is triples of nodes in a line

Imagine you have a grid, and your rewrite is triples of nodes in a line

GridGraph[{10,10}]

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Covariant derivative:

∂ - i (e / ℏ) A

∂ - i (e / ℏ) A

{{1,2},{2,3}}{{1,2},{2,3}}

Causal graph looks at shared edges

HypergraphPlot/@WolframModel[{{1,2},{2,3}}{{1,2},{2,3}},List@@@EdgeList[GridGraph[{5,5}]],3,"StatesList"]

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HypergraphPlot/@WolframModel[{{1,2},{2,3}}{{1,2},{2,3}},List@@@EdgeList[GridGraph[{5,5}]],3,"CausalGraph"]

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HypergraphPlot/@WolframModel[{{1,2},{2,3}}{{1,2},{2,3}},List@@@EdgeList[GridGraph[{5,5}]],3,"CausalGraph","EventOrderingFunction""Random"]

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### Tensors

Tensors

From the outside, can paint values on tuples of hyperedges

In GR, the tensor is obtained by looking at the structure of the network by “geodesic probing” [network structure tensor]

Another type of tensor would measure things about “orientation” of events / updatings [event probing tensor]

#### Fact about “geodesic probing tensor”

Fact about “geodesic probing tensor”

Divergence of geodesic bundle vanishes in “vacuum” (i.e. uniform node density)

#### In fiber bundle

In fiber bundle

Base space is collection of nodes

Fibers are the collections of edges of each node

Fibers are the collections of edges of each node

#### Lie group for local gauge transformations

Lie group for local gauge transformations

Relates to interchange between hyperedges at a node

Minimally there is a permutation that relates hyperedges

Elements of gauge group map between different updatings, where each update is defined by a subset of hyperedges at a node

Elements of gauge group are transformations between subsets

G: powerset(fibers)->powerset(fibers)

Elements of gauge group are transformations between subsets

G: powerset(fibers)->powerset(fibers)

Permutations[Range[4]]

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For a given rule, what is the actual transformation?

In reality,

G:possible LHS assignments -> possible LHS assignments

G:possible LHS assignments -> possible LHS assignments

{{x,y},{x,z}}XXXXXX

G is the hypergraph automorphism group

GraphAutomorphismGroup[{xy,xz}]

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PermutationGroup[{Cycles[{{2,3}}]}]

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Imagine the automorphism group for a coarse-grained update involving many nodes

Individual updates on larger hypergraphs

Lie group comes from subgroup of group of infinite permutations....

Group can be represented by : a certain set of generators which are permutations....

What are the limit automorphism groups of grids?

Table[GraphAutomorphismGroup[TorusGraph[{n,n}]],{n,4}]

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Table[GraphAutomorphismGroup[TorusGraph[{n,n}]],{n,3,6}]

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Length[First[#]]&/@%

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{3,4,3,3}

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Table[Length[First[GraphAutomorphismGroup[TorusGraph[{n}]]]],{n,3,12}]

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{2,2,2,2,2,2,2,2,2,2}

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GraphAutomorphismGroup[TorusGraph[{10}]]

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PermutationGroup[{Cycles[{{2,10},{3,9},{4,8},{5,7}}],Cycles[{{1,2,3,4,5,6,7,8,9,10}}]}]

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GraphAutomorphismGroup[TorusGraph[{11}]]

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PermutationGroup[{Cycles[{{2,11},{3,10},{4,9},{5,8},{6,7}}],Cycles[{{1,2,3,4,5,6,7,8,9,10,11}}]}]

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CayleyGraph[GraphAutomorphismGroup[TorusGraph[{11}]]]

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Table[Length[First[GraphAutomorphismGroup[TorusGraph[{n,n}]]]],{n,3,12}]

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{3,4,3,3,3,3,3,3,3,3}

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GraphAutomorphismGroup[TorusGraph[{10,4}]]

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CayleyGraph[%]

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Table[Length[First[GraphAutomorphismGroup[TorusGraph[{n,n,n,n}]]]],{n,3,7}]

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{5,8,6,6,6}

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Table[Length[First[GraphAutomorphismGroup[TorusGraph[{n,n,n,n,n}]]]],{n,3,7}]

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{6,10,8,7,7}

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CayleyGraph[GraphAutomorphismGroup[GridGraph[{10,4}]]]

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CayleyGraph[GraphAutomorphismGroup[GridGraph[{10,10}]]]

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CayleyGraph[GraphAutomorphismGroup[GridGraph[{10}]]]

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#### graphical regular representation of the automorphism group

graphical regular representation of the automorphism group

Can be like the graph itself

Claim: nilpotency / non-Abelianness

Claim: nilpotency / non-Abelianness

### Aggregated hypergraph rewrite

Aggregated hypergraph rewrite

Do n rewrites; how can we do that as a single bigger rewrite?

{{1,2},{2,3}}{{3,2},{2,1}}

For a given rewrite, what does it do to a series of hypergraphs?

EnumerateHypergraphs[{{3,2}}]

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What were the effective LHSs of multiple rule applications....

### Grothendieck limit of symmetric groups

Grothendieck limit of symmetric groups

CayleyGraph[SymmetricGroup[4]]

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CayleyGraph[SymmetricGroup[5]]

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Graph3D[%]

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CayleyGraph[SymmetricGroup[6]]

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GraphPlot3D[%]

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GraphNeighborhoodVolumesIndexGraph@UndirectedGraph,{1}

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1{1,4,10,20,36,60,89,110,116,119,120}

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GraphNeighborhoodVolumes[IndexGraph@UndirectedGraph[CayleyGraph[SymmetricGroup[6]]],{1}]

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1{1,4,10,21,41,76,131,212,321,449,575,670,710,716,719,720}

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First[Values[%]]

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{1,4,10,21,41,76,131,212,321,449,575,670,710,716,719,720}

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Ratios[%]//N

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{4.,2.5,2.1,1.95238,1.85366,1.72368,1.61832,1.51415,1.39875,1.28062,1.16522,1.0597,1.00845,1.00419,1.00139}

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ListLinePlot[LogDifferences[%168]]

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First[Values[GraphNeighborhoodVolumes[IndexGraph@UndirectedGraph[CayleyGraph[SymmetricGroup[7]]],{1}]]]

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{1,4,10,22,44,86,159,283,486,789,1214,1773,2451,3164,3910,4521,4876,4998,5026,5036,5039,5040}

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ListLinePlot[LogDifferences[%]]

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Ratios[%171]//N

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{4.,2.5,2.2,2.,1.95455,1.84884,1.77987,1.71731,1.62346,1.53866,1.46046,1.3824,1.2909,1.23578,1.15627,1.07852,1.02502,1.0056,1.00199,1.0006,1.0002}

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ListLinePlot[LogDifferences[2^Range[20]]]

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First[Values[GraphNeighborhoodVolumes[IndexGraph@UndirectedGraph[CayleyGraph[SymmetricGroup[8]]],{1}]]]

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{1,4,10,22,45,89,169,311,558,969,1631,2650,4131,6190,8935,12400,16526,21159,26072,30849,35012,38091,39703,40191,40285,40310,40316,40319,40320}

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ListLinePlot[LogDifferences[%]]

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## Putting units on graphs

Putting units on graphs

#### Spatial graph

Spatial graph

Each edge is an elementary length

Each node has a certain elementary mass

Edges also have a certain energy [relation between length and energy is like cosmological constant]

Edges also have a certain energy [relation between length and energy is like cosmological constant]

#### Causal graph

Causal graph

Each edge is a spacetime interval:

the edge has unit of time

the edge has unit of time

Each node in a causal graph has units of charge ???

#### Multiway graph

Multiway graph

Each edge has units of time

If we embed a branchlike hypersurface then the “branchial graph” edges are in units of energy

If we embed a branchlike hypersurface then the “branchial graph” edges are in units of energy

[Each edge of the branchial graph is a critical pair]

Is the volume element in the multiway graph an elementary action?

Edge has units of energy ???

cf https://en.wikipedia.org/wiki/Energetic_space

MultiwaySystem[{"A""AB","B""A"},"A",4,"CriticalPairs"]

In[]:=

{{AA,ABB},{AAB,ABA},{AAB,ABBB},{ABA,ABBB},{AAA,AABB},{AAA,ABAB},{AABB,ABAB},{AAA,ABBA},{ABAB,ABBA},{AABB,ABBA},{AABB,ABBBB},{ABAB,ABBBB},{ABBA,ABBBB}}

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Table[MultiwaySystem[{"A""AB","B""A"},"A",t,"CriticalPairs"],{t,5}]

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Graph[UndirectedEdge@@@Catenate[%]]

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Graph[UndirectedEdge@@@Catenate[Table[MultiwaySystem[{"A""AB","B""A"},"A",t,"CriticalPairs"],{t,8}]]]

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Graph[UndirectedEdge@@@#]&/@Table[MultiwaySystem[{"A""AB","B""A"},"A",t,"CriticalPairs"],{t,7}]

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We can also compute the branchial graph of the multiway causal network