  This paper presents both rigorous results and physical theory on the
breakdown of magnetic flux conservation for ideal plasmas, by nonlinear
effects. Our analysis is based upon an effective equation for
magnetohydrodynamic (MHD) modes at length-scales $>\ell,$ with smaller scales
eliminated, as in renormalization-group methodology. We prove that
flux-conservation can be violated for an arbitrarily small length-scale $\ell,$
and in the absence of any non-ideality, but only if singular current sheets and
vortex sheets both exist and intersect in sets of large enough dimension. This
result gives analytical support to and rigorous constraints on theories of fast
turbulent reconnection. Mathematically, our theorem is analogous to Onsager's
result on energy dissipation anomaly in hydrodynamic turbulence. As a physical
phenomenon, the breakdown of magnetic-flux conservation in ideal MHD is similar
to the decay of magnetic flux through a narrow superconducting ring, by
phase-slip of quantized flux lines. The effect should be observable both in
numerical MHD simulations and in laboratory plasma experiments at moderately
high magnetic Reynolds numbers.
