Finite-time geometric focusing and current-sheet precursors in helical magnetic flux ropes

We study a cylindrical helical magnetic flux rope subject to the requirement that its electromagnetic field retain the same local two-plane structure as it evolves. On the nondegenerate branch considered here, F∧F ≠ 0, equivalently E · B ≠ 0, the Maxwell tensor splits canonically into two decomposable parts spanning mutually orthogonal two-planes; we require the magnetic-type part to keep a fixed radial leg and a helical direction, with a prescribed longitudinal profile. Imposing the source-free Maxwell equations dF = 0 on the total field then yields a convex scalar conservation law for the field-line pitch variable Φ =r\, |Bφ ̂|/Bχ ̂, whose ratio to r is the tangent of the local field-line pitch angle. Smooth compressive pitch profiles therefore steepen, and the radial characteristic map loses invertibility after a finite accumulated drive while the pitch and the field amplitudes can remain finite. Provided an explicitly computable prefactor does not vanish, the Ampère current required to support the rope diverges at that point. We call this endpoint a current-sheet precursor, because the analysis does not show that a self-consistent plasma reaches it, nor which layer resistive, Hall, or kinetic physics would select. The focusing survives the addition of radial magnetic through-flux. An exact homologous family shows that pitch-gradient blow-up is not the essential feature: the characteristic Jacobian vanishes and the rotation rate of the unit helical direction diverges while ∂rΦ → 0. Since E · B ≠ 0, no four-velocity satisfies Fabub = 0 and no frame removes the electric field – an algebraic property that fixes no resistivity, dissipation rate, or reconnection model.