  The large-scale structure of the magnetic field in the solar corona provides
the energy to power large-scale solar eruptive events. Our physical
understanding of this structure, and hence our ability to predict these events,
is limited by the type of data currently available. It is shown that the
multifractal spectrum is a powerful tool to study this structure, by providing
a physical connection between the details of photospheric magnetic gradients
and current density at all size scales. This uses concepts associated with
geometric measure theory and the theory of weakly differentiable functions to
compare Amp\`{e}re's law to the wavelet-transform modulus maximum method. The
H\"{o}lder exponent provides a direct measure of the rate of change of current
density across spatial size scales. As this measure is independent of many
features of the data (pixel resolution, data size, data type, presence of
quiet-Sun data), it provides a unique approach to studying magnetic-field
complexity and hence a potentially powerful tool for a statistical prediction
of solar-flare activity. Three specific predictions are provided to test this
theory: the multifractal spectra will not be dependent on the data type or
quality; quiet-Sun gradients will not persist with time; structures with large
current densities at large size scale will be the source of energy storage for
solar eruptive events.

Solar and Stellar Astrophysics