We propose a method which we call "Isotropic Entanglement" (IE), that takes inspiration from Free Probability Theory and other ideas in Random Matrix Theory to predict the eigenvalue distribution of quantum many body (spin) systems with generic interactions. At the heart of this method is a sliding scale, "the Slider", which interpolates the quantum problem between two extrema by matching fourth moments. The first extreme treats the non-commuting terms in the Hamiltonian classically and the second treats them isotropically. By isotropic we mean that the non-commuting subsets have eigenvectors that are in generic positions. We prove The Matching Three Moments and Slider theorems and further prove that the interpolation of the quantum problem in terms of the classical and isotropic is universal, i.e., independent of the choice of local terms. Examples show that IE provides an accurate picture of the spectra well beyond what one expects from the first four moments alone.
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