Graph3D[GeneralizedGridGraph[{4"Directed",4"Directed",4"Directed"}]]
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Projective Space

Metric in branch space

Minimum number of updates to derive both points
(i.e. its the common ancestor distance in the tree)
​
Finding that common ancestor distance might be computationally hard
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---> geometric group theory
​
TransitiveClosureGraph[GridGraph[{5,5}]]
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TransitiveReductionGraph
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TransitiveReductionGraph
,VertexLabelsAutomatic
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Table[TransitiveReductionGraph[MultiwaySystem[{"A""AB","B""A"},"A",t,"BranchialGraph"]],{t,2,5}]
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Projective space

Modding out the metric by a constant
Tuples[Range[3],2]
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{{1,1},{1,2},{1,3},{2,1},{2,2},{2,3},{3,1},{3,2},{3,3}}
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If these were critical pairs, this must define a transitive binary relation......
What are the transitive binary relations on n elements?
https://math.stackexchange.com/questions/243773/amount-of-transitive-relations-on-a-finite-set
https://oeis.org/A006905
https://oeis.org/A000798/a000798_12.pdf
[ Quantum states are rays in Hilbert space ]

<Transitivity is discrete rescaling>