In fundamental-measure theories the bulk excess free-energy density of a hard-sphere fluid mixture is assumed to depend on the partial number densities $\left\{{\rho }_i\right\}$ only through the four scaled-particle-theory variables $\left\{{\xi }_{\alpha }\right\}$, i.e., $\Phi \left(\left\{{\rho }_i\right\}\right)\to \Phi \left(\left\{{\xi }_{\alpha }\right\}\right)$. By imposing consistency conditions, it is proven here that such a dependence must necessarily have the form $\Phi \left(\left\{{\xi }_{\alpha }\right\}\right)=-{\xi }_0\ln (1-{\xi }_3)+\Psi \left(y\right){\xi }_1{\xi }_2/(1-{\xi }_3)$, where $y\equiv {\xi }_2^2/12\pi {\xi }_1(1-{\xi }_3)$ is a scaled variable and $\Psi \left(y\right)$ is an arbitrary dimensionless scaling function which can be determined from the free-energy density of the one-component system. Extension to the inhomogeneous case is achieved by standard replacements of the variables $\left\{{\xi }_{\alpha }\right\}$ by the fundamental-measure (scalar, vector, and tensor) weighted densities $\left\{n_{\alpha }\left(\mathbf{r}\right)\right\}$. Comparison with computer simulations shows the superiority of this bulk free energy over the White Bear one.

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