Eddington-regulated galactic gas balance ($M_{s,\max }=\Phi /A$)

For an Eddington-luminous hole with $M_{\mathrm{BH}}=\beta M_s$ and wind mass flux $\lambda L$, define $A=\lambda \beta (4\pi G{m}_{p}c/{\sigma }_T)$. The gas budget ${\dot{M}}_g=\Phi -ϵM_g-A{M}_s$ has positive instantaneous balance only for $M_s<\Phi /A$. A positive constant-gas trajectory is incompatible with ongoing ${\dot{M}}_s=ϵM_g$ unless the secular black-hole coupling is neglected.