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Let $X$ be a building of arbitrary type. A compactification ${𝒞}_{\mathrm{sph}}\left(X\right)$ of the set ${\text{Res}}_{\mathrm{sph}}\left(X\right)$ of spherical residues of $X$ is introduced. We prove that it coincides with the horofunction compactification of ${\text{Res}}_{\mathrm{sph}}\left(X\right)$ endowed with a natural combinatorial distance which we call the . Points of ${𝒞}_{\mathrm{sph}}\left(X\right)$ admit amenable stabilisers in $\text{Aut}\left(X\right)$ and conversely, any amenable subgroup virtually fixes a point in ${𝒞}_{\mathrm{sph}}\left(X\right)$. In addition, it is shown that, provided $\text{Aut}\left(X\right)$ is transitive enough, this compactification also coincides with the group-theoretic compactification...