= Solution
The central limitation is that a <Hartree-Fock electron gas> includes exchange but omits the correlation <energy> that is not representable by one determinant. Opposite-spin <electrons> can lower their repulsion by avoiding one another in a correlated unpolarized state. Hartree-Fock misses that avoidance while retaining the exchange advantage of aligned spins, so it need not place a magnetic instability at a reliable interaction strength. The relevant <energy> differences between competing <spin> states can be much smaller than their total energies. Correlation calculations of the <electron> gas explicitly address this competition.<Ground State of the Electron Gas by a Stochastic Method by D. M. Ceperley and B. J. Alder>[Author manuscript, physical PDF pages 4–8: competing phases, correlated trial functions and <energy> differences]{ref}
There are further model restrictions. A constant contact coupling discards the <momentum> dependence of the <Coulomb interaction> and the resulting nontrivial screening. Real materials have a <lattice>, band structure and material-dependent effective interactions. Spatial magnetic textures or other competing states are excluded by a comparison restricted to uniform Fermi spheres. Even before those changes, part c's two endpoints do not find the continuous polarization onset.
The symbol $r_s$ also needs care. Here it is an <energy> ratio proportional to $\gamma n^{1/3}$. The usual <Dimensionless Wigner-Seitz radius> of a three-dimensional Coulomb <electron> gas is $r_s^{\rm WS}=[3/(4\pi n)]^{1/3}/a_B$, proportional to $n^{-1/3}$. They are not the same <density> parameter. A quoted experimental value of the latter cannot be compared numerically to $1.175$ without a model-specific mapping. \b[The contact interaction, missing correlations, restricted variational class and parameter convention prevent the number $1.175$ from being a quantitative experimental prediction.] The paper supplies no particular experimental threshold, so none is invented here.
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