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We investigate the short, medium, and long-range structure of soft diskconfigurations for a wide range of area fractions and simulation protocols byconverting the real-space spectrum of volume fraction fluctuations for windowsof width $L$ to the distance $h(L)$ from the window boundary over whichfluctuations occur. Rapidly quenched unjammed configurations exhibitsize-dependent super-Poissonian long-range features that, surprisingly,approach the totally-random limit even close to jamming. Above and just belowjamming, the spectra exhibit a plateau, $h(L)=h_e$, for $L$ larger thanparticle size and smaller than a cutoff $L_c$ beyond which there are long-rangefluctuations. The value of $h_e$ is independent of protocol and characterizesthe putative hyperuniform limit. This behavior is compared with that forEinstein solids, with and without hyperuniformity-destroying defects. We findthat key structural features of the particle configurations are more evident,as well as easier and more intuitive to quantify, using the real-space spectrumof hyperuniformity lengths rather than the spectral density.

The Spectrum of Structure for Jammed and Unjammed Soft Disks