Where the Gauss--Bonnet coupling enters the Bondi--Sachs hierarchy

We integrate the Bondi–Sachs hierarchy of scalar–Gauss–Bonnet gravity with an arbitrary coupling function $f(\mathrm{\Phi })$ and a radiating massless scalar, and determine the first order in $1\mathrm{/}r$ at which the coupling $\lambda$ becomes visible in each field equation. We obtain the asymptotic falloff of the effective Gauss–Bonnet stress tensor together with its leading coefficients in covariant form on the sphere: every component decays between one and three powers of $1\mathrm{/}r$ faster than the slowest falloff compatible with a smooth null infinity. As a consequence, (i) the Bondi mass-loss formula and the evolution equation of the angular-momentum aspect are those of the minimally coupled Einstein–scalar system, a statement that is exact in $\lambda$ under the assumption of a smooth expansion; (ii) the coupling generates no logarithmic terms through the orders considered, even though the Horndeski representation of the theory involves logarithmic functionals; and (iii) the coupling first enters a dynamical equation in the evolution of the $O(r^{-3})$ coefficient of the angular metric, through the cross term $-2\lambda f^{\mathrm{\prime }}({\text{vp}}_0){\text{vp}}_1{\mathrm{\partial }}_u{N}_{AB}$ between the asymptotic scalar charge aspect and the time derivative of the news. In Newman–Penrose form, the evolution equation of the leading coefficient of ${\mathrm{\Psi }}_0$ acquires the source $-12\lambda f^{\mathrm{\prime }}({\text{vp}}_0){\text{vp}}_1{\bar{\mathrm{\Psi }}}_4^0$, while those of ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ are unchanged. The Gauss–Bonnet invariant decays as $r^{-6}$ and affects the scalar hierarchy only from the evolution of the $r^{-5}$ coefficient onward. For vacuum black-hole spacetimes, where the scalar is sourced by the coupling itself, all these effects are of order ${\lambda }^2$; if $f^{\mathrm{\prime }}({\text{vp}}_0)=0$ they are pushed one further order in $1\mathrm{/}r$.