In[]:=

{ResourceFunction["MultiwayFunctionSystem"][n{2n,n+1},0,7,"StatesGraph",ImageSize{Automatic,350}],ResourceFunction["MultiwayFunctionSystem"][n{2n,n+1},0,10,"StatesGraphStructure",GraphLayout"LayeredDigraphEmbedding",ImageSize{Automatic,350}]}

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,

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ResourceFunction["MultiwayFunctionSystem"][n{2n,n+1},0,4,"EvolutionEventsGraph",GraphLayout->"LayeredDigraphEmbedding","IncludeEventInstances"->True]

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ResourceFunction["MultiwayFunctionSystem"][n{2n,n+1},1,4,"EvolutionEventsGraph",GraphLayout->"LayeredDigraphEmbedding","IncludeEventInstances"->True]

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In[]:=

ResourceFunction["MultiwayFunctionSystem"][n{2n,n+1},1,5,"EvolutionEventsGraph",GraphLayout->"LayeredDigraphEmbedding","IncludeEventInstances"->True]

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In[]:=

ResourceFunction["MultiwayFunctionSystem"][n{2n,n+1},0,4,"EvolutionEventsGraph",GraphLayout->"LayeredDigraphEmbedding"]

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#### Single-Integer Tokens (“monatomic tokens”)

Single-Integer Tokens (“monatomic tokens”)

With a single integer for every state, which is also every token, which is also every atom

#### Diatomic Tokens

Diatomic Tokens

States = tokens, but tokens = (2 atoms)

E.g. apply an integer affine transformation to 2-vector

#### In an integer system, the “atom names” are integers, which “mean something”

In an integer system, the “atom names” are integers, which “mean something”

#### With a tupling (pairing) function, this multi-atom-token system is just like the single-atom token system

With a tupling (pairing) function, this multi-atom-token system is just like the single-atom token system

#### This is a system that maps a single token to multiple tokens

This is a system that maps a single token to multiple tokens

## 22 token case

22 token case

#### Levels of interpretation:

atoms ; tokens ; states ; transversals ; [higher categories]

Levels of interpretation:

atoms ; tokens ; states ; transversals ; [higher categories]

atoms ; tokens ; states ; transversals ; [higher categories]

[ atoms + tokens ~ syntax ; above is semantics ]

## SW partial code

SW partial code

This is a single history.... because Overlaps->False

## 22 on finite alphabet

22 on finite alphabet

We want all cases of this with each of the atoms being from 0 to k-1:

Either a given {a,b} can have a mapping, or it can an “ϵ move”

Represent each pair by FromDigits[Sort[pair], k] ; i.e. numbers from 0 to k^2-1

These are the bijections:

All mappings:

#### [With a finite set, one can explicit enumerate all possible tokens]

[With a finite set, one can explicit enumerate all possible tokens]

#### Kneser graphs: mutually exclusive

Kneser graphs: mutually exclusive

#### Does the TEG ring bells in all possible permutations?

Does the TEG ring bells in all possible permutations?

### What is the relationship between events? [Basically an “event graph”]

What is the relationship between events? [Basically an “event graph”]

In the following, the edges could be labeled by events:

#### The “event-knitting graph” [i.e. how possible events are connected by tokens]

The “event-knitting graph” [i.e. how possible events are connected by tokens]

#### Token-event graph is either a bipartite ordinary graph with explicit nodes for events ... or is a hypergraph with “bipartite hyperedges”

Token-event graph is either a bipartite ordinary graph with explicit nodes for events ... or is a hypergraph with “bipartite hyperedges”

In general, there are many possible data structures which could be the hyperedges in a hypergraph

Wolfram Model: list

unordered hypergraph: set

unordered hypergraph: set

Expressions as hyperedges: atoms (AKA symbols, integers, etc.) are atoms

[with attributes like Orderless as needed]

[with attributes like Orderless as needed]

#### Token-event hypergraph:

Token-event hypergraph:

hyperedge: set set

(I.e. it is a rewrite (i.e. event) from [unordered] multisets of tokens to multisets of tokens)

(I.e. it is a rewrite (i.e. event) from [unordered] multisets of tokens to multisets of tokens)

<Can these multisets of tokens actually be ordered?>

#### Event sets are like the factored instances of the laws of physics (“theoretical-possibility sets”)

Event sets are like the factored instances of the laws of physics (“theoretical-possibility sets”)

cf mapping defined on all of a space, but it might be that much of that space can never be reached

[cf fiber bundle without a global section]

[cf fiber bundle without a global section]

#### Event set is like a group; actual evolution is like a groupoid

Event set is like a group; actual evolution is like a groupoid

### Rules vs. rule schemas

Rules vs. rule schemas

Explicit rules for a finite set of atoms are “rules”

Pattern rules are like rule schemas

Pattern rules are like rule schemas

“Concrete rule” / “Explicit rule” / “Verbatim rule” / ((“Instantiated rule”)) [ rule in which certain specifically named tokens occur ]

Verbatim rule yields a verbatim rewrite AKA verbatim event

### Consider a case with a finite number of possible tokens, and a finite number of possible (concrete) events

Consider a case with a finite number of possible tokens, and a finite number of possible (concrete) events

### Crucial assumption: a token (in multihistory) can be consumed any number of times

Crucial assumption: a token (in multihistory) can be consumed any number of times

### < Ultimate deduplication: ignore what the universe actually does; just consider its rules >

< Ultimate deduplication: ignore what the universe actually does; just consider its rules >

#### Partial deduplication ... in a collection of slices

Partial deduplication ... in a collection of slices

## Fibonacci

Fibonacci

https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/foliations-and-coordinates-on-causal-graphs/#p-255

How do you coordinatize the possible total orders?