Perturbative Hairy Black Branes in the $G_2$-Invariant Sector of Dyonic ISO(7) Gauged Supergravity

We construct perturbative black-brane solutions carrying neutral scalar hair in the $G_2$-invariant truncation of four-dimensional dyonic $\mathcal{N}=8$ ISO(7) gauged supergravity, an exact consistent truncation of massive type~IIA supergravity on $S^6$. Expanding around the $G_2$-symmetric AdS${}_4$-Schwarzschild black brane in powers of a dimensionless scalar-charge parameter $\text{eps}$, we show that the perturbation hierarchy reduces, order by order, to a nested sequence of inhomogeneous Legendre equations (demonstrated through second order). The first-order scalar satisfies a Pöschl–Teller equation with exact parameter $\nu (\nu +1)=2\mathrm{/}3$, giving $\nu =-\frac{1}{2}+\frac{\sqrt{33}}{6}$, and the first-order metric is fixed algebraically by the same function, $A^{(1)}=-\frac{\sqrt{7}}{2}{\phi }^{(1)}$, so the entire $\mathcal{O}(\text{eps})$ sector is one Legendre function. An identity $\nu +1={\mathrm{\Delta }}_{+}\mathrm{/}3$, with ${\mathrm{\Delta }}_{+}=(3+\sqrt{33})\mathrm{/}2\approx 4.372$ the conformal dimension of the dual scalar operator, relates the Pöschl-Teller parameter to the AdS/CFT dictionary via the nonlinear Fefferman-Graham map $z\propto z_{\mathrm{F}\mathrm{G}}^3$. The $\mathcal{O}({\text{eps}}^2)$ back-reaction yields explicit corrections to the renormalized free energy and Bekenstein-Hawking entropy density at fixed temperature, consistent with the first law; equivalently, at fixed entropy or energy density it \emph{lowers} the Hawking temperature by $\delta T\mathrm{/}T=\mathcal{O}({\text{eps}}^2)$. For the leading transport coefficients the shear viscosity saturates $\eta \mathrm{/}s=1\mathrm{/}(4\pi )$, while the Eling-Oz horizon formula gives a nonzero bulk viscosity $\zeta \mathrm{/}\eta =(\frac{{\mathrm{\Delta }}_{+}-3}{2}{)}^2{\text{eps}}^2=\frac{3(7-\sqrt{33})}{8}{\text{eps}}^2\approx 0.471{\text{eps}}^2$. With $c_s^2=\frac{1}{2}-\mathcal{O}({\text{eps}}^2)<\frac{1}{2}$, this indicates a softening of the equation of state.