N[qFromd[DimensionOfGenerator[Translate[badSeeds[[4,1,1]],0]]]]

Plot[Evaluate[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],0,d,q,1]],{q,1,2.4},PlotRange->All]

N[qFromd[DimensionOfGenerator[Translate[badSeeds[[4,1,1]],2]]]]

Plot[Evaluate[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]],{q,1,1.7},PlotRange->All]

N[qFromd[DimensionOfGenerator[Translate[badSeeds[[4,1,1]],4]]]]

Plot[Evaluate[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],4,d,q,1]],{q,1,1.7},PlotRange->All]

PositiveOmegaInequality[badSeeds[[4,1]],badSeeds[[4,2]],n,d,q,1]

For Flying Spaghetti Monster, show FSMinequality[1] >0 for n ≥ 4 and q ≥ 1.600.

FSMinequality[1] = 1-2 (q^2)-(q^4)+3 (q^8)+2 (q^10)+(q^12)-4 (q^(4+2 n))-12 (q^(6+2 n))-9 (q^(8+2 n))+4 (q^(12+2 n))+12 (q^(14+2 n))+18 (q^(16+2 n))+4 (q^(18+2 n))-4 (q^(20+2 n))-9 (q^(24+2 n))-12 (q^(26+2 n))-4 (q^(28+2 n))+8 (q^(14+4 n))+24 (q^(16+4 n))+18 (q^(18+4 n))-2 (q^(20+4 n))-18 (q^(24+4 n))-36 (q^(26+4 n))-18 (q^(28+4 n))-2 (q^(32+4 n))+18 (q^(34+4 n))+24 (q^(36+4 n))+8 (q^(38+4 n))-4 (q^(24+6 n))-12 (q^(26+6 n))-9 (q^(28+6 n))-4 (q^(32+6 n))+4 (q^(34+6 n))+18 (q^(36+6 n))+12 (q^(38+6 n))+4 (q^(40+6 n))-9 (q^(44+6 n))-12 (q^(46+6 n))-4 (q^(48+6 n))+(q^(40+8 n))+2 (q^(42+8 n))+3 (q^(44+8 n))-(q^(48+8 n))-2 (q^(50+8 n))+(q^(52+8 n));

Factor[FSMinequality[1]]

If q is more than 1 the first two factors are positive.

FSMinequality[1,1]=(-1+(q^2)+(q^4)+(q^6)+2 (q^(2+n))+3 (q^(4+n))-3 (q^(12+n))-2 (q^(14+n))-(q^(10+2 n))-(q^(12+2 n))-(q^(14+2 n))+(q^(16+2 n)));

FSMinequality[1,2]=(-1+(q^2)+(q^4)+(q^6)-2 (q^(2+n))-3 (q^(4+n))+3 (q^(12+n))+2 (q^(14+n))-(q^(10+2 n))-(q^(12+2 n))-(q^(14+2 n))+(q^(16+2 n)));

If q>1, -1+(q^2)+(q^4)+(q^6)+2 (q^(2+n))+3 (q^(4+n))>0.  So we want to show

-3 (q^(12+n))-2 (q^(14+n))-(q^(10+2 n))-(q^(12+2 n))-(q^(14+2 n))+(q^(16+2 n)) >0;

Since n ≥ 4, q≥0, we find

-3 (q^(12+n))-2 (q^(14+n))-(q^(10+2 n))-(q^(12+2 n))-(q^(14+2 n))+(q^(16+2 n))≥ 
-3 (q^(8+2n))-2 (q^(10+2n))-(q^(10+2 n))-(q^(12+2 n))-(q^(14+2 n))+(q^(16+2 n))== (q^(8+2 n)) (-3-3 (q^2)-(q^4)-(q^6)+(q^8))

So we really want to show

(-3-3 (q^2)-(q^4)-(q^6)+(q^8))≥0

NSolve[-3-3 (q^2)-(q^4)-(q^6)+(q^8)==0]

Since the largest root is less than 1.6, we have f[4,1] is positive for q>1.6.

For FSMinequality[1,2]:

FSMinequality[1,2]-FSMinequality[1,1]

is evidently positive if q≥1.

For the translation by 0 case, we have omega = 1,-1, i. we calculate the possible q values, then run tests to get rid of them.

possibleValueOfqForFSM[0,1]=DeleteCases[Solve[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],0,d,q,1]-2==0],{___,q->x_/;Im[x]≠0∨!(x>1),___}]

N[(q+q^(-1))^2/.possibleValueOfqForFSM[0,1]]

possibleValueOfqForFSM[0,-1]=DeleteCases[Solve[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],0,d,q,1]+2==0],{___,q->x_/;Im[x]≠0∨!(x>1),___}]

possibleValueOfqForFSM[0,-1]=DeleteCases[Solve[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],0,d,q,1]+2==0],{___,q->x_/;Im[x]≠0∨!(x>1),___}][[1]]

N[(q+q^(-1))^2/.possibleValueOfqForFSM[0,-1]]

possibleValueOfqForFSM[0,i]=DeleteCases[Solve[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],0,d,q,1]==0],{___,q->x_/;Im[x]≠0∨!(x>1),___}]

N[(q+q^(-1))^2/.possibleValueOfqForFSM[0,i]]

We can rule out two cases as the index obtained is too large. For the other case, we do a cyclotomic test.

CyclotomicQ[z_,n_:200]:=Module[{x},FusionAtlas`CyclotomicTest`Private`PolyCyclotomicTest[MinimalPolynomial[z,x],n,x]]

CyclotomicQ[(q+q^(-1))^2/.possibleValueOfqForFSM[0,-1],200]

N[SolveRelativeDimensionsByDepthEquations[badSeeds[[4,1]],badSeeds[[4,2]],0,d,q,1]/.possibleValueOfqForFSM[0,-1]]

Negative dimensions of bimodules cannot happen. This rules out FSM for translation n=0.
Next we translate FSM by 2.

possibleValueOfqForFSM[2,1]=DeleteCases[Solve[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]-2==0],{___,q->x_/;Im[x]≠0∨!(x>1),___}]

N[(q+q^(-1))^2/.possibleValueOfqForFSM[2,1]]

CyclotomicQ[(q+q^(-1))^2/.possibleValueOfqForFSM[2,1],200]

N[SolveRelativeDimensionsByDepthEquations[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]/.possibleValueOfqForFSM[2,-1]]

Negative bimodule dimensions rule this out.

possibleValueOfqForFSM[2,-1]=DeleteCases[Solve[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]+2==0],{___,q->x_/;Im[x]≠0∨!(x>1),___}]

possibleValueOfqForFSM[2,-1]=DeleteCases[Solve[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]+2==0],{___,q->x_/;Im[x]≠0∨!(x>1),___}][[1]]

N[(q+q^(-1))^2/.possibleValueOfqForFSM[2,-1]]

CyclotomicQ[(q+q^(-1))^2/.possibleValueOfqForFSM[2,-1],200]

possibleValueOfqForFSM[2,(1+i(3)^(1/2))/2]=DeleteCases[Solve[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]-1==0],{___,q->x_/;Im[x]≠0∨!(x>1),___}]

N[(q+q^(-1))^2/.possibleValueOfqForFSM[2,(1+i(3)^(1/2))/2]]

CyclotomicQ[(q+q^(-1))^2/.possibleValueOfqForFSM[2,(1+i(3)^(1/2))/2],200]

N[SolveRelativeDimensionsByDepthEquations[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]/.possibleValueOfqForFSM[2,(1+i(3)^(1/2))/2]]

AlgebraicIntegerQ /@ (Last/@ Flatten[SolveRelativeDimensionsByDepthEquations[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]/.possibleValueOfqForFSM[2,(1+i(3)^(1/2))/2]])

Every bimodule dimension must be an algebraic integer. This rules out this case.

possibleValueOfqForFSM[2,(-1+i(3)^(1/2))/2]=DeleteCases[Solve[OmegaPlusOmegaInverse[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]+1==0],{___,q->x_/;Im[x]≠0∨!(x>1),___}]

N[(q+q^(-1))^2/.possibleValueOfqForFSM[2,(-1+i(3)^(1/2))/2]]

CyclotomicQ[(q+q^(-1))^2/.possibleValueOfqForFSM[2,(-1+i(3)^(1/2))/2],200]

N[SolveRelativeDimensionsByDepthEquations[badSeeds[[4,1]],badSeeds[[4,2]],2,d,q,1]/.possibleValueOfqForFSM[2,(-1+i(3)^(1/2))/2]]

Negative Bimodule dimensions rule this out.