Put phase info in the mesh structure....
In[]:=
GraphicsColumn[ArrayPlot[{#},ColorRules->{0->White,1->Lighter[Orange],2->Darker[Orange]},ImageSize->{Automatic,20},Mesh->True]&/@​​Map[First,ResourceFunction["BlockCellularAutomaton"][{{​​{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},​​{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},​​{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},2},​​{CenterArray[{1,1},8],0},10]]]
Out[]=
In[]:=
GraphicsColumn[ArrayPlot[{#},ColorRules->{0->White,1->Lighter[Orange],2->Darker[Orange]},ImageSize->{Automatic,20},Mesh->True]&/@​​Map[First,ResourceFunction["BlockCellularAutomaton"][{{​​{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},​​{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},​​{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},2},​​{CenterArray[#,8],0},10]]]&/@Tuples[{1,2},2]
Out[]=

,
,
,

GraphicsGrid[Partition[ArrayPlot[​​Map[First,ResourceFunction["BlockCellularAutomaton"][{{​​{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},​​{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},​​{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},2},​​{CenterArray[#,20],0},40]],ColorRules->{0->White,1->Lighter[Orange],2->Darker[Orange]}]&/@Take[Tuples[{1,2},8],128],UpTo[24]]]
In[]:=
ResourceUpdate["BlockCellularAutomaton"]
Out[]=
ResourceObject
Name: BlockCellularAutomaton
»
Type: Function
Description: Evolve a block cellular automaton

In[]:=
ArrayPlot[First/@#,ImageSize->60]&/@cyclePaths
Out[]=

,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,

In[]:=
ListLinePlot[Length/@(Counts/@Transpose[Map[Sign[First[#]]&,​​ResourceFunction["BlockCellularAutomaton"][{{​​{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},​​{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},​​{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},2},​​{CenterArray[#,12],0},30]]&/@Tuples[{1,2},6]]),​​PlotRange->All,Frame->True,Filling->Axis,AspectRatio->1/3,​​PlotStyle->$SecondLawColors["Blues",1],​​FillingStyle->$SecondLawColors["Blues",3]]
Out[]=
In[]:=
With[{​​val=6,​​length=12,​​op=ResourceFunction["BlockCellularAutomaton"][{{​​{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},​​{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},​​{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},2}]},​​evols=ResourceFunction["BlockCellularAutomaton"][{{​​{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},​​{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},​​{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},2},{CenterArray[#,12],0},13]&/@Tuples[{1,2},val];​​]
In[]:=
uniquePathGraphs=PathGraph[#,DirectedEdges->True​​]&/@evols;
In[]:=
cgPaths=VertexReplace[#,x_:>Sign[First[x]]]&/@uniquePathGraphs
Out[]=

,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,

In[]:=
g1=Apply[GraphUnion,cgPaths]
Out[]=