Description This preprint derives a closed-form galactic rotation-curve term from the Aether Physics Model (APM) galactic Aether Lagrangian by treating the scalar field $\Phi$ as a length density (unbound-string density) with dimension $[\Phi]=\mathrm{mass}/\mathrm{length}$. In the Newtonian weak-field limit, the scalar sector contributes an effective gravitating mass density through the standard energy-to-mass bridge\[\rho_\Phi = \frac{u_\Phi}{c^2}, \qquadu_\Phi = \frac{1}{2}(\nabla \Phi)^2 + \frac{1}{2}m_\Phi^2 \Phi^2.\]Within the steady-state, approximately spherical-halo limit, requiring an asymptotically flat circular speed selects a minimal sourced profile with\[\frac{d\Phi}{dr} = \frac{\Phi_0}{r+r_c},\]which yields a cored logarithmic field\[\Phi(r) = \Phi_0 \ln\!\left(1+\frac{r}{r_c}\right) + \Phi(0),\]an effective density scaling $\rho_\Phi \propto (r+r_c)^{-2}$, and an analytic halo rotation term\[v_\Phi^2(r)= v_\infty^2\left[1+\frac{r_c}{r}-\frac{r_c^2}{r(r+r_c)}-\frac{2r_c}{r}\ln\!\left(1+\frac{r}{r_c}\right)\right],\qquadv_\infty^2=\frac{2\pi G\Phi_0^2}{c^2}.\]To test this steady-state prediction on a comparatively regular external spiral, the paper ingests SPARC rotation-curve decomposition data for NGC~3198 and constructs the predicted total curve from the published SPARC baryonic components plus the APM scalar-halo term. The same term is then applied to a Milky Way worked example, where residuals are interpreted in the context of known non-equilibrium structure (bar/spiral non-axisymmetry, warp/flare, accretion debris, and tracer-dependent systematics), emphasizing that the analytic reduction is explicitly steady-state and weak-field. The paper also records falsifiable extensions that follow from the Lagrangian structure: (i) a Yukawa-scale outer cutoff when $m_\Phi \neq 0$, and (ii) potential electromagnetic/polarization signatures in environments where the APM electromagnetic couplings become non-negligible. Finally, it states a torsion-sourcing extension mechanism derived from the bipolar-trumpet coil toy model for black-hole torsion filaments and polar jets, introduced as a future-testable correction channel in torsion-active galactic-center environments without altering the present equilibrium derivation. What is included Newtonian-limit derivation of an analytic scalar-halo rotation term from the APM galactic Aether Lagrangian. Closed-form mapping between the sourced scalar profile, implied source function $S(r)=\nabla^2\Phi$, enclosed scalar mass $M_\Phi(r)$, and the halo rotation term $v_\Phi(r)$. SPARC-based worked example for NGC~3198 using published baryonic decomposition velocities. Milky Way contrast-case demonstration emphasizing limitations of steady-state assumptions in a dynamically complex system. Explicit statements of falsifiable extensions (nonzero $m_\Phi$, electromagnetic/polarization channels, and torsion-sourced corrections). Notes on units Computations in the worked examples are performed in standard astrophysical units (kpc, km/s, $M_\odot$) for rotation-curve modeling. Where relevant to the APM ledger, the paper also highlights the Gaussian-cgs bookkeeping identity $c^2 = A_u \,\mathrm{curl}_{\mathrm{ref}}$ used to clarify the mapping between scalar energy density and gravitating mass density.