All Incorrect Equilibria of the Planar Four-Agent Formation Law Are Unstable

Four agents in the plane move by the standard distance-based gradient law ṗᵢ = Σ_{j≠i} e_ij (p_j − pᵢ), e_ij = ‖pᵢ − p_j‖² − d̄_ij², where all six desired distances come from one planar target. In an Oberwolfach report of 2015, B. D. O. Anderson asked whether, as for rectangular targets, all incorrect equilibria are saddle points when the target is a general quadrilateral or a triangle with an interior agent. We show that the answer is yes for every planar target: at every incorrect equilibrium the Hessian of V = ¼ Σ_{i<j} e_ij² has a negative eigenvalue; all incorrect equilibria except the collocated one also have a positive eigenvalue, and the collocated one is a local maximum modulo translations. In the language of multidimensional scaling, the first statement says that the complete-graph s-stress of four points in the plane has no spurious second-order critical points, for every ground truth. At an incorrect equilibrium spanning the plane, e_ij = κ λᵢ λ_j with κ > 0, where λ is the affine dependency of the agents, and the smallest Hessian eigenvalue is at most −κ‖λ‖²/2 ≤ −√(8V/3); both constants are sharp. The collinear case is known. For the spanning case we combine the classical self-stress of four planar points, a Schur-complement argument as in recent work of Criscitiello on the s-stress, and an inequality between two 2×2 matrices depending only on λ, proved by explicit identities in the elementary symmetric functions of λ. Consequently almost every initial condition converges to a formation congruent to the target, while collinear initial conditions show that literal global convergence fails. This is an unrefereed note.