Families10 = {{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1}}],DualData[{1},{1},{1,2},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{2,1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}}],DualData[{1},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1},{0,1}}],DualData[{1},{1},{1,2},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1},{0,1}}],DualData[{1},{1},{1,2},{1,2,3}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1},{0,1}},{{0,1}}],DualData[{1},{1},{1,2},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1},{0,1}},{{0,0,1}}],DualData[{1},{1},{1,2},{1,2,3}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1}},{{1}},{{1}}],DualData[{1},{1},{1,2},{1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,1}}],DualData[{1},{1},{1,2},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,1},{0,1}},{{0,1}},{{1}}],DualData[{1},{1},{1,2},{1,2},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1},{0,1}},{{0,0,1}},{{1}}],DualData[{1},{1},{1,2},{1,2,3},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0}},{{1}}],DualData[{1},{1},{1,2},{2,1,3},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1}}],DualData[{1},{1},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{0,0,1},{1,0,0}},{{1,1}}],DualData[{1},{1},{1,2},{2,1,3},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}},{{1,0}}],DualData[{1},{1},{1,2},{1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{0,0,1},{1,0,0}},{{1,1},{1,0}}],DualData[{1},{1},{1,2},{2,1,3},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}},{{1,0},{1,0}}],DualData[{1},{1},{1,2},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1}},{{1,1}}],DualData[{1},{1},{1,2},{2,1,3},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}},{{0,1}}],DualData[{1},{1},{1,2},{1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1}},{{1,1},{0,1}}],DualData[{1},{1},{1,2},{2,1,3},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}},{{0,1},{0,1}}],DualData[{1},{1},{1,2},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1}},{{1,0},{1,0},{0,1}}],DualData[{1},{1},{1,2},{2,1,3},{1,3,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}}],DualData[{1},{1},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1},{0,0,1}},{{1,0,0},{1,0,0},{1,0,0},{0,1,0},{0,0,1}}],DualData[{1},{1},{1,2},{2,1,3},{1,4,5,2,3}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1},{1}}],DualData[{1},{1},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1},{0,1}},{{1,0,0,0},{1,0,0,0}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{2,1,3,4},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1},{0,1}},{{0,1},{1,0}}],DualData[{1},{1},{1,2},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,0,1,0},{0,1,0,0}},{{1,1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}}],DualData[{1},{1},{1,2},{1,2},{3,4,1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,0,1,0},{0,1,0,0}},{{1,0},{1,0},{0,1},{0,1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1,2,3,4}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,1}}],DualData[{1},{1},{1,2},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,0},{1,0},{0,1},{0,1}}],DualData[{1},{1},{1,2},{1,2},{3,4,1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{1,0},{0,1},{0,1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1,2,3,4}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,1}}],DualData[{1},{1},{1,2},{1,2},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,0,1,0},{0,1,0,0}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1,2,3,4}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{1,0},{0,1}},{{1,1}},{{1}},{{1}}],DualData[{1},{1},{1,2},{1,2},{1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0}},{{1}}],DualData[{1},{1},{1,2},{2,1,3},{1,3,2,4},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}},{{0,1}},{{1}}],DualData[{1},{1},{1,2},{1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,0,1},{0,0,0,1}},{{1,0,0},{1,0,0},{1,0,0},{0,1,0},{0,0,1}}],DualData[{1},{1},{1,2},{2,1,3},{1,3,2,4},{1,4,5,2,3}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}},{{0,1}},{{1},{1},{1}}],DualData[{1},{1},{1,2},{1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1}},{{1,0},{1,0},{0,1},{0,1},{0,1}},{{0,1,0,0,0},{0,1,0,0,0}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{2,1,3},{1,3,2,4,5},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}},{{0,1},{0,1}},{{0,1},{1,0}}],DualData[{1},{1},{1,2},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,1}},{{1}},{{1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1},{1}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,1}}],DualData[{1},{1},{1,2},{1,2},{3,4,1,2}]]}}

Map[DisplayBigraph,Families10,{2}]

badSeeds={{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1}},{{1,1},{0,1}},{{0,1}}],DualData[{1},{1},{1,2},{2,1,3},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}},{{0,1},{0,1}},{{0,1}}],DualData[{1},{1},{1,2},{1},{1,2}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,0,1}}],DualData[{1},{1},{1,2},{2,1,3},{1,3,2,4}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}},{{0,1}},{{1},{1}}],DualData[{1},{1},{1,2},{1},{1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0}},{{1,0},{0,1}},{{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}},{{1,0},{0,1}},{{0,1},{1,0}}],DualData[{1},{1},{1,2},{1,2},{2,1}]]},{BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{1,0},{0,1},{0,1}},{{0,1,0,0},{0,0,1,0},{0,0,0,1}},{{1,0,0},{1,0,0},{0,1,0},{0,0,1}},{{1,0,0,0},{0,0,1,0},{0,0,0,1}}],DualData[{1},{1},{1,2},{1,3,2,4},{1,4,3,2}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{1,0},{0,1}},{{0,1},{1,0},{0,1}},{{1,0,0},{0,1,0}},{{0,1},{1,0},{0,1}}],DualData[{1},{1},{1,2},{1,2},{2,1}]]}}

Map[DisplayBigraph,badSeeds,{2}]

Crab = {BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{0,1}},{{1,0},{0,1},{0,1}},{{1,0,0},{0,0,1}}],DualData[{1},{1},{1,2},{2,1,3}]],BigraphWithDuals[GradedBigraph[{{1}},{{1}},{{1}},{{1},{1}},{{1,0},{1,0}},{{0,1}},{{1},{1}}],DualData[{1},{1},{1,2},{1}]]}

Map[DisplayBigraph,Crab]

Crab[[1,1]]

N[qFromd[DimensionOfGenerator[Append[Crab[[1,1]],{{1,0}}]]]]

N[qFromd[DimensionOfGenerator[Append[Crab[[1,1]],{{0,1}}]]]]

N[qFromd[DimensionOfGenerator[Append[Crab[[2,1]],{{1,0}}]]]]

N[qFromd[DimensionOfGenerator[Append[Crab[[2,1]],{{0,1}}]]]]

N[qFromd[DimensionOfGenerator[Append[Append[Crab[[2,1]],{{1,0}}],{{1}}]]],50]

N[qFromd[DimensionOfGenerator[Append[Append[Crab[[2,1]],{{0,1}}],{{1}}]]]]

N[qFromd[DimensionOfGenerator[Crab[[1,1]]]]]

Plot[Evaluate[OmegaPlusOmegaInverse[Crab[[1]],Crab[[2]],0,d,q,1]],{q,1,1.5},PlotRange->All]

N[qFromd[DimensionOfGenerator[Translate[Crab[[1,1]],2]]]]

Plot[Evaluate[OmegaPlusOmegaInverse[Crab[[1]],Crab[[2]],2,d,q,1]],{q,1.3,1.5},PlotRange->All]

N[qFromd[DimensionOfGenerator[Translate[Crab[[1,1]],2]]]]

RelativeBranchFactor[Crab[[1]],Crab[[2]],n,d,q,1]

PositiveOmegaInequality[Crab[[1]],Crab[[2]],n,d,q,1]

For Crab: prove crabInequality[1]>0 for n≥2 and q≥1.467.

crabInequality[1]=-1+2 (q^2)+(q^4)-3 (q^8)-2 (q^10)-(q^12)+(q^(8+2 n))+4 (q^(10+2 n))-(q^(12+2 n))-2 (q^(14+2 n))-(q^(16+2 n))+4 (q^(18+2 n))+5 (q^(20+2 n))+2 (q^(22+2 n))-2 (q^(18+4 n))-5 (q^(20+4 n))-4 (q^(22+4 n))+(q^(24+4 n))+2 (q^(26+4 n))+(q^(28+4 n))-4 (q^(30+4 n))-(q^(32+4 n))+(q^(28+6 n))+2 (q^(30+6 n))+3 (q^(32+6 n))-(q^(36+6 n))-2 (q^(38+6 n))+(q^(40+6 n));

Factor[crabInequality]

crabInequality[1,1]=1-(q^2)-(q^4)-(q^6)-(q^(4+n))-(q^(6+n))-(q^(8+n))+(q^(10+n));

crabInequality[1,2]=-1+(q^2)+(q^4)+(q^6)-(q^(4+n))-(q^(6+n))-(q^(8+n))+(q^(10+n));

We will show crabInequality[1,1] > 0 when n ≥ 2 for allowed q. It is clear that crabInequality[1,2] ≥ crabInequality[1,1]. This will prove crabInequality[1] > 0.

Chop[NSolve[q^(8)-q^6-q^4-q^2 ==0]]

NSolve[{q^(-2)-1-(q^2)-(q^4)-(q^(2+n))-(q^(4+n))-(q^(6+n))+(q^(8+n)) ==0}/.{n->2}]

The largest root is smaller than 1.467. We are through.

N[qFromd[DimensionOfGenerator[badSeeds[[1,1,1]]]]]

N[qFromd[DimensionOfGenerator[badSeeds[[2,1,1]]]]]

For Scorpion and Decagon : prove crabInequality[1]>0 for n≥0 and q≥1.526. We use the same method.

Chop[NSolve[q^(8)-q^6-q^4-q^2 ==0]]

NSolve[{1-(q^2)-(q^4)-(q^6)-(q^(4+n))-(q^(6+n))-(q^(8+n))+(q^(10+n)) ==0}/.{n->0},WorkingPrecision->20]

The largest root is less than 1.526. We are through.

KnownObstructions[Families10[[8,1,1]]]

KnownObstructions[Families10[[9,1,1]]]

KnownObstructions[Families10[[10,1,1]]]

KnownObstructions[Families10[[11,1,1]]]

KnownObstructions[Families10[[12,1,1]]]

KnownObstructions[Families10[[13,1,1]]]

KnownObstructions[Families10[[20,1,1]]]

KnownObstructions[Families10[[21,1,1]]]

KnownObstructions[Families10[[22,1,1]]]

This shows that all extensions of translates of the Decagon and the Scorpion along with all translates of (8)-(13) and (20)-(22) don't work.