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}]]}}

Families10[[10]]

Map[DisplayBigraph,Families10][[10]]

Families10[[12]]

Map[DisplayBigraph,Families10][[12]]

Families10[[20]]

Map[DisplayBigraph,Families10][[20]]

OmegaPlusOmegaInverseR[r_,m_,q_] :=Together[Simplify[qi[m,q]qi[m+2,q](r+(r)^(-1)-2)-2]-2]

explodePolynomial[f_]:=Reverse[Sort[{Coefficient[Exponent[#,q],n],Exponent[#,q]/.n->0,#/.q->1}&/@List@@f]]

negativeCoefficients[f_]:=-Plus@@Cases[explodePolynomial[f],{_,_,a_/;a<0}:>a]

MinimalNSoInequalityIsReallyEasy[f_,q0_]:=Module[{e,c},
e=explodePolynomial[f];
c=Ceiling[Log[q0,negativeCoefficients[f]/e[[1,3]]]];
Ceiling[Max[(((e[[1,1]]-#[[1]]))^(-1))(#[[2]]-e[[1,2]]+c)&/@Rest[e]]]
]

RelativeBranchFactorForFamilies[Families10[[1,1]],Families10[[1,2]],n,d,q,0]

RelativeBranchFactorForFamilies[Families10[[2,1]],Families10[[2,2]],n,d,q,0]

RelativeBranchFactor[Families10[[3,1]],Families10[[3,2]],n,d,q,0]

OmegaPlusOmegaInverseR[(((q^2) (-2-(q^2)-(q^4)+(q^(8+2 n))+(q^(10+2 n))+2 (q^(12+2 n))))/((1+(q^4)) (-1+(q^(12+2 n))))),n+3+1,q]

N[qFromd[DimensionOfGenerator[Families10[[3,1,1]]]]]

inequality10Families[3] = -1+4 (q^2)-4 (q^4)+2 (q^6)-4 (q^8)-(q^12)+(q^(8+2 n))+2 (q^(10+2 n))-4 (q^(12+2 n))+6 (q^(14+2 n))+4 (q^(18+2 n))+(q^(20+2 n))+2 (q^(22+2 n))-2 (q^(18+4 n))-(q^(20+4 n))-4 (q^(22+4 n))-6 (q^(26+4 n))+4 (q^(28+4 n))-2 (q^(30+4 n))-(q^(32+4 n))+(q^(28+6 n))+4 (q^(32+6 n))-2 (q^(34+6 n))+4 (q^(36+6 n))-4 (q^(38+6 n))+(q^(40+6 n))

Factor[inequality10Families[3]]

inequality10Familes[3,1]= 1-2 (q^2)-(q^6)-(q^(4+n))-2 (q^(8+n))+(q^(10+n))

For n≥ 2 we have  1-2 (q^2)-(q^6)-(q^(4+n))-2 (q^(8+n))+(q^(10+n))≥ q^(2+n)(-2q^(-2)-(q^2)-(q^2)-2 (q^6)+(q^8))

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

This means that for all n ≥ 2 the factor in question is always positive in the allowed q-range.

RelativeBranchFactorForFamilies[Families10[[4,1]],Families10[[4,2]],n,d,q,0]

N[qFromd[DimensionOfGenerator[Families10[[4,1,1]]]]]

OmegaPlusOmegaInverseR[((1+(q^2)+(q^4)+(q^8)+(q^10)+(q^12))/(1+(q^2)+2 (q^4)+2 (q^6)+2 (q^8)+(q^10)+(q^12))),n+3+1,q]

inequality10Families[4] = 1+4 (q^2)+6 (q^4)+4 (q^6)+(q^8)-4 (q^(2 n))-4 (q^(4+2 n))+4 (q^(6+2 n))-5 (q^(8+2 n))+4 (q^(10+2 n))-11 (q^(12+2 n))-11 (q^(16+2 n))+4 (q^(18+2 n))-5 (q^(20+2 n))+4 (q^(22+2 n))-4 (q^(24+2 n))-4 (q^(28+2 n))+(q^(20+4 n))+4 (q^(22+4 n))+6 (q^(24+4 n))+4 (q^(26+4 n))+(q^(28+4 n))

Factor[inequality10Families[4]]

Both of these quantities are always positive.

RelativeBranchFactorForFamilies[Families10[[5,1]],Families10[[5,2]],n,d,q,0]

RelativeBranchFactorForFamilies[Families10[[6,1]],Families10[[6,2]],n,d,q,0]

OmegaPlusOmegaInverseR[((1-(q^2)+(q^4)-(q^6)+(q^8))/(1-(q^2)+2 (q^4)-(q^6)+(q^8))),n+3+1,q]

DimensionOfGenerator[Families10[[6,1,1]]]

This means no translates are below index 5.

N[qFromd[DimensionOfGenerator[Families10[[6,1,1]]]]]

NSolve[1-2 (q^2)+4 (q^4)-5 (q^6)+6 (q^8)-5 (q^10)+4 (q^12)-2 (q^14)+(q^16)==0]

MinimalNSoInequalityIsReallyEasy[1-4 (q^(2 n))+16 (q^(2+2 n))-36 (q^(4+2 n))+60 (q^(6+2 n))-81 (q^(8+2 n))+88 (q^(10+2 n))-81 (q^(12+2 n))+60 (q^(14+2 n))-36 (q^(16+2 n))+16 (q^(18+2 n))-4 (q^(20+2 n))+(q^(20+4 n)),1.618]

RelativeBranchFactorForFamilies[Families10[[7,1]],Families10[[7,2]],n,d,q,0]

OmegaPlusOmegaInverseR[((1+(q^2)+(q^4)+(q^6)+(q^8))/((1-q+(q^2)) (1+q+(q^2)) (1+(q^4)))),n+3+1,q]

N[qFromd[DimensionOfGenerator[Families10[[7,1,1]]]]]

MinimalNSoInequalityIsReallyEasy[1-4 (q^(2 n))-4 (q^(4+2 n))+4 (q^(6+2 n))-(q^(8+2 n))+8 (q^(10+2 n))-(q^(12+2 n))+4 (q^(14+2 n))-4 (q^(16+2 n))-4 (q^(20+2 n))+(q^(20+4 n)),1.425]

RelativeBranchFactorForFamilies[Families10[[14,1]],Families10[[14,2]],n,d,q,0]

OmegaPlusOmegaInverseR[((1-(q^4)+(q^8))/((1-(q^2)+(q^4)) (1+(q^2)+(q^4)))),1+3+n,q]

N[qFromd[DimensionOfGenerator[Families10[[14,1,1]]]]]

MinimalNSoInequalityIsReallyEasy[1-(q^(2 n))+2 (q^(2+2 n))-(q^(4+2 n))-2 (q^(8+2 n))+2 (q^(10+2 n))-2 (q^(12+2 n))-(q^(16+2 n))+2 (q^(18+2 n))-(q^(20+2 n))+(q^(20+4 n)),1.594]

RelativeBranchFactorForFamilies[Families10[[15,1]],Families10[[15,2]],n,d,q,0]

RelativeBranchFactorForFamilies[Families10[[16,1]],Families10[[16,2]],n,d,q,0]

RelativeBranchFactorForFamilies[Families10[[17,1]],Families10[[17,2]],n,d,q,0]

RelativeBranchFactorForFamilies[Families10[[18,1]],Families10[[18,2]],n,d,q,0]

RelativeBranchFactorForFamilies[Families10[[19,1]],Families10[[19,2]],n,d,q,0]

RelativeBranchFactorForFamilies[Families10[[23,1]],Families10[[23,2]],n,d,q,0]

KnownObstructions[Families10[[3,1,1]]]

KnownObstructions[Families10[[6,1,1]]]

KnownObstructions[Families10[[6,2,1]]]

KnownObstructions[Translate[Families10[[6,1,1]],2]]

KnownObstructions[Translate[Families10[[6,1,1]],4]]

KnownObstructions[Families10[[7,1,1]]]

KnownObstructions[Translate[Families10[[7,1,1]],2]]

KnownObstructions[Translate[Families10[[7,2,1]],2]]

KnownObstructions[Translate[Families10[[7,1,1]],4]]

KnownObstructions[Families10[[14,1,1]]]

KnownObstructions[Translate[Families10[[14,1,1]],2]]