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In[]:=
MultiwaySystem[{"A""AA","B""AB"},{"ABA"},3,"StatesGraph"]
Out[]=
In[]:=
MultiwaySystem[{"A""AA","B""AB"},{"ABA"},5,"StatesGraph"]
Out[]=
In[]:=
FindPath[%,"ABA","AAABA"]
Out[]=
{{ABA,AABA,AAABA}}
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]

The combinators case

One intermediate case is labeled trees, rather than graphs [combinators]
Combinators:
In[]:=
s[s[s]][s][s[s][k]][k]
Out[]=
s[s[s]][s][s[s][k]][k]
f[x]{rootf,rootx}
f_[x_]Module[{r},{rf,rx}]
In[]:=
Map[f,s[s[s]][s][s[s][k]][k],-1,HeadsTrue]
Out[]=
f[f[f[f[s][f[f[s][f[s]]]]][f[s]]][f[f[f[s][f[s]]][f[k]]]]][f[k]]
In[]:=
Map[Replace[#,f_[x_]Module[{r},{rf,rx}]]&,s[s[s]][s][s[s][k]][k],-1,HeadsTrue]
Out[]=
{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]
The analog of the graph is tree rewrites, which normally use labels, but could just use tree structure.......

Lambdas

[ trivially the same as the functional case ]

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

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?]

Edges slicing a hypersurface

time vector : lapse function + shift vector
t
μ
(indexes leaves in the foliation)
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)
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]
C
t
(X)
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
C
t
behavior

If increasing faster than t^d, effectively speed of light increases
Things get more connected; independent experiments get harder
If
C
t
approaches t^d .... but there could still be a “dimension wave”
If there is a lump of increased dimension, what are its equations of motion?

Speed of light change

Multiplies C(X) by a constant

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]

Dimension vs. scale

At a small scale, dimension is not defined
Looking at small scale, there are plenty of dimension fluctuations

Dimensionons

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

Thermal conductivity

How does this vary with dimension?
3/2kTKE1/2mv^2
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