Marginal straight-line launch at thirty degrees ($\Delta {\Phi }_{\mathrm{eff}}/(GM/R_0)=-7(s/R_0{)}^3/16+O(s^4)$)

At exactly thirty degrees the quadratic curvature vanishes, but the specified straight-line point-mass effective potential is downhill for small positive displacement at cubic order. Thus the strict negative-curvature criterion does not exclude every nonlinear one-sided marginal launch. An exactly stationary bead remains at equilibrium until displaced.