Perturbative Double-Scalar Hair on Planar Black Holes in AdS${}_4$ Einstein-Scalar Gravity

We construct a two-scalar hairy planar black hole in four-dimensional Einstein gravity coupled to two scalars whose potential descends from the ABJM consistent truncation of eleven-dimensional supergravity. Starting from the known single-scalar hairy black brane, we switch on the second scalar $\text{phim}$ perturbatively in its amplitude $\text{eps}$ and solve the coupled Einstein-scalar system to order ${\text{eps}}^2$. The linearized $\text{phim}$ profile is an exact hypergeometric function, independent of the $\text{pp}$ hair, whose horizon-regular branch always carries a source, so there is no spontaneous $\text{phim}$ hair. Because the dual operator ${\mathcal{O}}_{-}$ is irrelevant ($\mathrm{\Delta }=4$), the second-order response develops a logarithmically running condensate with a closed-form, scheme-independent slope $\frac{\sqrt{21}}{70}{C}_s^2$. The renormalized free energy is scheme dependent, but the geometric temperature shift $\delta T\mathrm{/}{T}_0\approx -0.345{\text{eps}}^2$ at fixed entropy is not. The full metric and $\text{pp}$ back-reaction is obtained in closed form, diagonalizing into two Pöschl-Teller problems with $\delta \text{pp}=\frac{6}{\sqrt{7}}\delta H$. Three transport coefficients of the dual fluid remain exactly conformal to this order, while the sound speed, like the free energy, is scheme dependent.