I. INTRODUCTION AND MOTIVATION The configuration-interaction shell model is a useful framework for a detailed under- standing of low-energy nuclear structure [1–3]. The many-body basis is a large dimension (10 3 − $^{10}$) set of Slater determinants, which are antisymmeterized products of single-particle states. One must truncate the single-particle states, corresponding to one or a few shells (typically using the harmonic oscillator as an approximation to the mean-field); the many- body basis may be further truncated. For phenomenological calculations one writes the Hamiltonian in terms of single-particle energies and two-body matrix elements, while for ab initio calculations one may extend this to three-body interactions [4]. The two-body matrix elements are the matrix elements of the residual interaction in the lab frame, V$_{JT}$ ( ij, kl ) = 〈 ij ; JT | ˆ V | kl ; JT 〉 (1) where | ij ; JT 〉 is the normalized, antisymmeterized product of particles in orbits labeled by i and j and coupled to good angular momentum J and isospin T . If one starts from a translationally invariant interaction between particles, one can either compute the integral in the lab frame or start in the relative frame and then transform to the lab frame; in either case there are correlations between the matrix elements, although they are not obvious to the casual observer. Often for semi-phenomenological calculations, one starts from a “realistic” interaction, and then adjusts the two-body matrix elements until the rms error on a set of experimentally known energy levels is minimized [1]. In the 1 s -0 d shell, such a semi-phenomenological interaction has been recently derived [5], improving on an earlier interaction[6]. It has been found that the fits are empirically dominated by a few linear combinations of matrix elements. The physical meaning of those dominant combinations is not immediately obvious. One might naively guess the linear correlations are due to an underlying transla- tionally invariant interaction (although a density dependence would destroy this). Somewhat more phenomenologically, it has been argued by appealing to mean-field properties that one can improve fits primarily through adjusting the monopole-monopole part of the interaction, that is, interaction terms that look like n$_{a}$ ( n$_{b}$ − δ$_{ab}$ ), where n$_{a}$ is the number of particles in the a th orbit. This protocol for shifting monopole strengths has been successfully applied to several semi-empirical interactions [7–11] 2