D-brane supergravity solution ($d{s}_s^2=H^{-1/2}d{x}_{\parallel }^2+H^{1/2}d{x}_{\perp }^2$)

For the asymptotically flat extremal D$p$ solution with $p<7$, $H=1+Q/r^{7-p}$ is harmonic away from the source and $e^{\Phi }=g_s{H}^{(3-p)/4}$. The electric potential can be chosen as $C_{0\cdots p}=g_s^{-1}(H^{-1}-1)$; magnetic descriptions use the dual flux, and a D3-brane has self-dual five-form flux. An equally oriented probe has DBI density $-{\mu }_p{g}_s^{-1}{H}^{-1}$ and Wess-Zumino density $+{\mu }_p{g}_s^{-1}(H^{-1}-1)$, whose sum is position independent. This proves the corresponding parallel D-brane no-force identity in the classical background. The supergravity approximation requires controlled curvature and coupling; it does not resolve every brane core.