 No-scale identity from degree-one homogeneity

Let $K=-3\log \Gamma$ where $\Gamma >0$ is twice differentiable and homogeneous of degree one. If the Kähler Hessian is invertible, the Euler theorem for homogeneous functions identities give ${\tau }_i{K}_{ij}=3{\Gamma }_j/\Gamma$, then $K_{ij}^{-1}{\Gamma }_j/\Gamma ={\tau }_i/3$ and ${\Gamma }_i{K}_{ij}^{-1}{\Gamma }_j/{\Gamma }^2=1/3$. Multiplication by nine gives the no-scale identity. A positive degree-one $\Gamma$ need not induce an invertible or positive Kähler metric: $\Gamma ={\tau }_1+{\tau }_2$ gives a rank-one metric for $K$. The Hessian matrix $({\Gamma }_{ij})$ itself always has the radial null vector $\tau$, whereas the Hessian matrix of $K$ can be invertible. In a larger theory the relevant full inverse metric, or a decoupled block, must obey the identity.