 Admissible values for a sandwich extension

Suppose $S$ satisfies the mixed domination on a subspace $Y$, and $z\notin Y$. Apply that domination to $y+y^{\prime }$ and $x+x^{\prime }$, split the $p$ argument as $(x+y+z)+(x^{\prime }+y^{\prime }-z)$, and use superadditivity of $q$. This proves every lower candidate in the displayed interval is at most every upper candidate. The zero-vector candidates bound the lower supremum below by $-p(-z)$ and the upper infimum above by $p(z)$, so both endpoints are finite. Choosing any intervening value $a$ permits $U(y+tz)=S(y)+ta$. For $t>0$, scale the upper comparison by $t$; for $t<0$, scale the lower comparison by $-t$. Both give the required mixed domination, proving the interval characterizes the permitted extension values.