MultiwaySystem[{"A""AA","B""AB"},{"ABA"},3,"StatesGraph"]

In[]:=

Out[]=

MultiwaySystem[{"A""AA","B""AB"},{"ABA"},5,"StatesGraph"]

In[]:=

Out[]=

FindPath[%,"ABA","AAABA"]

In[]:=

{{ABA,AABA,AAABA}}

Out[]=

In a hypergraph system, the only way to generate distinct objects is to have more elements

JG claim: it’s not enough to just add relations; you have to add elements

Pretty soon you’re only adding multiplicity to relations [and that’s trivial when it comes to applying our rules]

Pretty soon you’re only adding multiplicity to relations [and that’s trivial when it comes to applying our rules]

#### The combinators case

The combinators case

One intermediate case is labeled trees, rather than graphs [combinators]

Combinators:

s[s[s]][s][s[s][k]][k]

In[]:=

s[s[s]][s][s[s][k]][k]

Out[]=

f[x]{rootf,rootx}

f_[x_]Module[{r},{rf,rx}]

Map[f,s[s[s]][s][s[s][k]][k],-1,HeadsTrue]

In[]:=

f[f[f[f[s][f[f[s][f[s]]]]][f[s]]][f[f[f[s][f[s]]][f[k]]]]][f[k]]

Out[]=

Map[Replace[#,f_[x_]Module[{r},{rf,rx}]]&,s[s[s]][s][s[s][k]][k],-1,HeadsTrue]

In[]:=

{r$715413{r$715410{r$715409s,r$715409{r$715408s,r$715408s}},r$715410s},r$715413{r$715412{r$715411s,r$715411s},r$715412k}}[k]

Out[]=

The analog of the graph is tree rewrites, which normally use labels, but could just use tree structure.......

#### Lambdas

Lambdas

[ trivially the same as the functional case ]

#### Functional form

Functional form

f[g[x,y],x]

If you don’t have intermediate variables, you’ll end up with identical nested copies (cf the extra hyperedges)

### Claim about mass

Claim about mass

The mass is some function of the number of elements

[But if you break the object in 2, how do you up with half the mass?]

[But if you break the object in 2, how do you up with half the mass?]

### Edges slicing a hypersurface

Edges slicing a hypersurface

time vector : lapse function + shift vector (indexes leaves in the foliation)

t

μ

Volume of cone is number of events contained in a cone with a certain length of time vector

(Growth rate [i.e. coefficient of t^2] is being interpreted as curvature)

(Growth rate [i.e. coefficient of t^2] is being interpreted as curvature)

Pick a hyperslice: then there a particular space graph

Now evolve to the next hyperslice. We could count the number of edges in the causal graph getting to the next hyperslice. [Which is comparable to the number of elements changed between hyperslices]

Now evolve to the next hyperslice. We could count the number of edges in the causal graph getting to the next hyperslice. [Which is comparable to the number of elements changed between hyperslices]

C

t

Gets a contribution from whatever put edges in the causal graph at a certain density

C(t)~t^d ( 1 + a t^2 )

Out[]=

How do you add nodes without increasing the dimension?

Action: volume average of subleading terms of

C

t

### Consequences of Ct behavior

Consequences of behavior

C

t

If increasing faster than t^d, effectively speed of light increases

Things get more connected; independent experiments get harder

Things get more connected; independent experiments get harder

If approaches t^d .... but there could still be a “dimension wave”

C

t

If there is a lump of increased dimension, what are its equations of motion?

#### Speed of light change

Speed of light change

Multiplies C(X) by a constant

#### Momentum conservation

Momentum conservation

Continuity equation on nodes of the causal graph

[aggregate has a certain form; therefore individual piece of C(X) have to compensate each other]

[aggregate has a certain form; therefore individual piece of C(X) have to compensate each other]

#### Dimension vs. scale

Dimension vs. scale

At a small scale, dimension is not defined

Looking at small scale, there are plenty of dimension fluctuations

#### Dimensionons

Dimensionons

Can a lump of higher connectivity in the causal graph just dissipate? [Has to follow underlying rule]

#### Thermal conductivity

Thermal conductivity

How does this vary with dimension?

3/2kTKE1/2mv^2