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

Frozen-in Fractals All Around: Inferring the Large Scale Effects of Small-Scale Magnetic Structure