For Scorpion and Decagon : prove inequality[1]≥0 for n≥0 and q≥1.577.

inequality[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[inequality[1]]

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

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

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

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}]

The largest root is less than 1.577. We are through.

For Truncated Scorpion: prove inequality[1]≥0 for n≥2 and q≥1.453.

inequality[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[inequality[1]]

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

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

We will show f[1,1] ≥ 0 when n ≥ 2 for allowed q. It is clear that f[1,2] ≥ f[1,1]. This will prove inequality[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 less than 1.453. We are through.

inequality[4]=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));

TODO : prove inequality[4]≥0 for n≥4 and q≥1.600.

inequality[4]=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[inequality[4]]

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

f[4,1]=(-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)));

f[4,2]=(-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 f[2]:

f[2]-f[1]

is evidently positive if q≥1.