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In the negligible bead-mass limit the bead contribution to $\mathcal{Q}$ vanishes. Since the particle starts at rest, $\mathcal{Q}=0$, and hence $r^2\dot{\theta }+b\dot{z}=0$. Assume $b\ne 0$ and a regular massless limit. Integrating along the trajectory giveswhere $C$ is the closed projection onto the plane perpendicular to the coil axis. The given final $r$ and $\theta$ ensure that this projection closes. Green's theorem identifiesThus $A$ is the signed area enclosed by the transverse projection, positive for counterclockwise traversal viewed from positive $z$. Self-intersecting projections contribute their winding-weighted areas; it is neither an unsigned area nor an area of a surface in three dimensions. For $b>0$, a positive signed area gives negative axial displacement. This is signed-area displacement under a helical constraint: returning to the same projected coordinates can leave a nonzero axial displacement.

The plotted loop illustrates the geometric integral only; it is not asserted to be a particular spring trajectory.

If zero conserved helical momentum reduces to $r^2\dot{\theta }+b\dot{z}=0$, a closed transverse projection $C$ has axial displacement $\Delta z=-b^{-1}{\oint }_C(xdy-ydx)=-2A/b$. Here $b\ne 0$ and $A$ is the signed area, with winding multiplicity for self-intersections. This is a path-dependent displacement even when the transverse coordinates return to their initial values. It follows from the momentum constraint and Green's theorem; it is not a prediction that every chosen closed curve is dynamically realizable.