Inclusion constants for free spectrahedra

One dichotomic and one arbitrary measurement

The following script allows to compute the bound for k=3 displayed in Table 2 from the paper arXiv:2512.17706

using NCTSSOS
using DynamicPolynomials
using COSMO
settings = cosmo_para()
settings.eps_abs = 1e-7 # absolute residual tolerance
settings.eps_rel = 1e-7 # relative residual tolerance
settings.max_iter = 1e7 # maximum number of iterations

@ncpolyvar x[1:3] b[1:3]
obj = x[1]*b[1] + x[2]*b[2] + x[3]*b[3]
ineq = [1+1.5*b[2], 1+1.5*b[3], 1-1.5*(b[2]+b[3]), 1-(x[1]+4/3*x[2]-2/3*x[3]), 1-(-x[1]+4/3*x[2]-2/3*x[3]), 1-(x[1]-2/3*x[2]+4/3*x[3]), 1-(-x[1]-2/3*x[2]+4/3*x[3]), 1-(x[1]-2/3*x[2]-2/3*x[3]), 1-(-x[1]-2/3*x[2]-2/3*x[3])]
eq = [b[1]^2-1]
eq2 = [(b[2]+2/3)*(b[2]-4/3), (b[3]+2/3)*(b[3]-4/3), b[2]*b[3]-b[3]*b[2]]
pop = [-obj; ineq; eq; eq2]
opt,data = ncpop(pop, [x;b], 4, numeq=4, partition=3, TS=false, obj="eigen")
1/opt # 0.683012