This form has the features that we want the N L term of Eq. 37 to have: it decreases with S i and it ranges from 0 to 1/2. Originally, such form has been used as the only term for the original fluctuation function at λ = 1 (Eq. 11). Because of this and its simplicity it is likely that σ c,λ=1 i,N L (r) of Eq. 38 is not optimal when used with the other two terms in Eq. 37. However, instead of fully optimizing MRF, we aim here to better understand how we can further constrain it by using Eq. 37, and understand how each term of the fluctuation function of Eq. 37 can improve different aspects of MRF. To complete the construction, we still need the L term of Eq. 37, σc,λ=1 L (r). Since we want it to primarily depend on the local density, the UEG gives us a natural choice for its construction. For this term, we cannot use σλ=1 (r s ) -σx from the right panel of Fig. since it was computed without the N L term, which we will need to "subtract off" to make Eq. 37 highly accurate for the UEG. This is what we will consider in the next section.