Exact independent arithmetic check (Python fractions; no numerical rounding)
At (x,y,z)=(1,1,1/2), t=z/(x^(1/3)y^(2/3))=1/2.
For R=-log y-log(1+t+t^2):
t_x=-1/6, t_xx=2/9, q=1+t+t^2=7/4, q_t=2, q_tt=2.
R_xx=(q_t^2*t_x^2/q^2)-(q_tt*t_x^2+q_t*t_xx)/q=-110/441 <0.
Therefore R is not convex on int K. This only rejects the naive additive-convex-remainder route; it does not classify all possible decompositions.
