### What is the meaning of light cones in MM space?

What is the meaning of light cones in MM space?

Things that can be proved equal symbolically

Incompleteness: causally disconnected parts of the MM universe

(If have to consider paths of infinite length, potentially cannot ever reach them with a finite number of added axioms)

#### Accumulation of theorems: relation to causal connections?

Accumulation of theorems: relation to causal connections?

With[{g=ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},{"A"},5,"EvolutionCausalGraph"]},HighlightGraph[g,Subgraph[g,PathGraph[FindShortestPath[g,"A","ABBA"]]]]]

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With[{g=ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},{"A"},5,"EvolutionEventsGraph"]},HighlightGraph[g,Subgraph[g,PathGraph[FindShortestPath[g,"A","ABBA"]]]]]

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LayeredGraphPlot[With[{g=ResourceFunction["MultiwaySystem"][{"A""AB","BB""A"},{"A"},7,"EvolutionCausalGraphStructure"]},HighlightGraph[g,Style[Subgraph[g,PathGraph[FindShortestPath[g,"A","ABAA"]]],Red,Thick]]],AspectRatio1/2]

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LayeredGraphPlot[With[{g=ResourceFunction["MultiwaySystem"][{"B""ABA","AA""B","BB""A"},{"B"},7,"EvolutionCausalGraphStructure"]},HighlightGraph[g,Style[Subgraph[g,PathGraph[FindShortestPath[g,"B","BAB"]]],Red,Thick]]],AspectRatio1/2]

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FindShortestPath[ResourceFunction["MultiwaySystem"][{"B""ABA","AA""B","BB""A"},{"B"},7,"EvolutionCausalGraphStructure"],"B","BAB"]

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{B,{BABA,B,{,}},{AAB,AA,{,AB}},BAB}

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### What is the meaning of dimension/growth rate in MM space?

What is the meaning of dimension/growth rate in MM space?

### Branchial graph

Branchial graph

Known relationships in mathematics at a given time..... [at least for 2-way edges]

### Causal invariance

Causal invariance

In a branch pair .... are the two branches “equal” ?

{xa,xb},{ay,by}

Is the key axiom that then a is equal to b? [Cinv any branch pair is equal .... i.e. that connected parts of the branchial graph are equal]

[[[ Euclid: Things that are equal to the same thing are also equal to one another (the Transitive property of a Euclidean relation) ]]]

## Euclid

Euclid

[[ partly copied from MM...-11.nb ]]

ResourceObject["Theorem Network from Euclid's Elements"]

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ResourceObject

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euclid=ResourceData["Theorem Network from Euclid's Elements"]

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#### Dimensions

Dimensions

#### Statistics etc.

Statistics etc.

#### Dependence on continuity of space....

Dependence on continuity of space....

#### Longest path

Longest path

Platonic solids

### Entity Version

Entity Version