= Solution
In a planar cross-section write the <magnetic field> as
$$
\mathbf B=(\psi_y,-\psi_x,0)=\nabla\psi\times\mathbf e_z.
$$
The <Cartesian magnetic flux function> $\psi$ labels the <magnetic field lines>. Ideal induction, with a suitable additive gauge, gives
$$
\partial_t\psi+\mathbf v\cdot\nabla\psi=0.
$$
Thus each flux contour is material. <Incompressibility> also preserves the area inside any material contour. Neither a <magnetic island>'s flux distribution nor the connections of a <separatrix> can be freely altered during relaxation.
A configuration with two or more <magnetic islands> separated by an X-type saddle illustrates the mechanism. As the <magnetic islands> reshape to lower <magnetic energy>, the arms of the <separatrix> can be pressed together. <Magnetic reconnection> would change the topology, so it is excluded in a <perfect conductor>. The X-type structure can instead flatten into an extended interface, with oppositely directed tangential fields on its two sides. The transition shrinks and the <electric current density> concentrates into a <current sheet>. Such collapse in two-dimensional relaxation is documented in section 4 of https://www.damtp.cam.ac.uk/user/hkm2/PDFs/Moffatt_1998_InternationalPress_Tdof_465.pdf[Moffatt's primary discussion]. It is a possible asymptotic outcome, not a finite-time singularity assertion or a property of every planar field.
The local equations show precisely how the interface can support equilibrium. For the planar field,
$$
j_z=-\frac1{\mu_0}\Delta\psi,\qquad \mathbf j\times\mathbf B=-\frac{\Delta\psi}{\mu_0}\nabla\psi.
$$
On a regular connected flux region, <magnetostatic equilibrium> therefore implies $p=P(\psi)$ and
$$
\Delta\psi=-\mu_0P'(\psi).
$$
Different disconnected flux regions can have different functions $P$. Their limiting fields need not join with continuous first derivatives of $\psi$ across a common <separatrix>; continuity of total <pressure> allows the tangential-field jump.
For an explicit local sheet, take
$$
\mathbf B_\epsilon=B_s\tanh(y/\epsilon)\mathbf e_x,\qquad p_\epsilon=p_T-\frac{B_s^2}{2\mu_0}\tanh^2(y/\epsilon).
$$
These satisfy <magnetostatic equilibrium> exactly, with
$$
j_z=-\frac{B_s}{\mu_0\epsilon}\operatorname{sech}^2(y/\epsilon)\ \longrightarrow\ -\frac{2B_s}{\mu_0}\delta(y).
$$
The finite limiting surface current agrees with the field-jump formula. This is a local equilibrium demonstration of the sheet balance, not a claim that this particular family is the full ideal relaxation trajectory from the given initial data.
Strictly planar fields of the form above have zero <magnetic helicity>: choose $\mathbf A=\psi\mathbf e_z$, giving $\mathbf A\cdot\mathbf B=0$. Thus this strictly planar example interprets the last part as a separate illustration of topology-constrained relaxation. It still has an energy obstruction: if $\psi=0$ on the cross-sectional boundary, the advected integral $\int\psi^2\,dx\,dy$ is constant and the <Poincaré inequality> gives $M\geq\lambda_1\int\psi^2\,dx\,dy/(2\mu_0)$ per unit length. If the earlier nonzero-helicity condition is retained instead, a possible extension is a field depending on only two coordinates but permitting an axial component, with periodic boundary conditions in the invariant direction. Write
$$
\mathbf B=(\psi_y,-\psi_x,G(\psi)).
$$
The axial force balance makes the axial component a function of $\psi$ on each regular connected region; the remaining <Cartesian magnetostatic flux-function equilibrium> is
$$
\Delta\psi+GG'+\mu_0P'=0.
$$
In that periodic extension, with consistent flux and gauge conventions, and $\psi=0$ on the cross-sectional boundary, integration by parts gives helicity per unit length $H_M/L_z=2\int\psi B_z,dx\,dy$. It can be nonzero. The same separatrix-collapse mechanism still concentrates current in a sheet. \b[Three-dimensional linkage is therefore unnecessary for sheet formation; planar frozen flux-contour topology already supplies the constraint.]
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