The Geometric Origin of the Disk-Edge Shoulder\\[1ex]\large The Curve Morphology Locked by the Static Weak-Field Boundary-Value Problem, and the Specification of the Dark-Matter Halo

Given the configuration "baryons in a disk, dark matter in a halo", the static weak-field boundary-value problem locks the morphology of the rotation curve beyond the disk edge: the part in excess of Kepler is a positive-term multipole series with radial index $2k+1$ (slowest $r^{-3}$), whose coefficients are uniquely determined by the ratio of two moments of the source; whereas the contribution of a spherically symmetric halo to all $\ell\ge2$ vanishes identically. Consequently, within the domain of validity of the rotation-curve morphology, the inner region of a spherically symmetric halo (or its $\ell=0$ part) is a zero-parameter specification, not an object of fitting. The complete theoretical input of this paper is: the Poisson equation given by the static weak-field limit, the non-negative energy condition, axisymmetry, and equatorial reflection. (The specification of \S{}5.4 additionally requires the observed plateau value $v_{\rm flat}$ as a data input.) Two testable predictions: the amplitude--slope lock of the disk-edge shoulder, $C/W=3.31,\,3.28,\,3.24$ (independent of absolute calibration), and the zero-parameter character of the halo specification. The series agrees with an independent direct integration to $10^{-10}$ (Appendix B), and that direct integration does not pass through the multipole expansion. Of the two predictions, the halo specification is the unconditionally testable one.