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For an isolated cold initially resting razor-thin disk, the displayed pointwise bound makes the absolute gravity-equivalent magnetic surface density smaller than its actual surface density. The positive-kernel comparison of thin-disk fields then makes the integrand in the horizontal virial balance of a cold magnetized fluid negative above and below the disk. Consequently its radial second moment begins to decrease. This establishes initial contraction in the global virial sense, not inward acceleration of every individual fluid element or unlimited subsequent collapse.

Integrate Poisson equation for Newtonian gravity through a narrow slab around the disk. The horizontal derivative contributions vanish as its thickness tends to zero, leaving the normal-derivative jumpThe even function symmetry of $\Phi$ makes the derivatives opposite, soIn the current-free simply connected upper half-space, Ampère's circuital law gives $\nabla \times \mathbf{B}=0$ and permits a magnetic scalar potential. Rescale it so that $\sqrt{4\pi G/{\mu }_0}\mathbf{B}=-\nabla {\Phi }_M$. The divergence-free condition makes ${\Phi }_M$ satisfy Laplace's equation, withCompare with the gravitational jump condition. Subject to the same isolated-field boundary condition at infinity, ${\Phi }_M$ is the harmonic potential of the effective surface densityThis gravity-equivalent magnetic surface density can have either sign; it is a mathematical representation of the exterior magnetic field, not physical negative mass. An imposed nondecaying field would require additional boundary data and would not be fixed by the disk surface density alone.

The vertical field of a disk has kernel $Gz/(|\mathbf{R}-{\mathbf{R}}^{\prime }{|}^2+z^2{)}^{3/2}>0$. The triangle inequality therefore bounds a signed-surface density field by the field of its absolute surface density. If that surface density is everywhere smaller than a nonnegative comparison surface density, their vertical fields satisfy the displayed strict comparison, provided the integrals exist and the strict bound holds on positive measure. This is useful when comparing gravity-equivalent magnetic surface density with actual disk mass.