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At the lowest temperatures they are expected to pair, so the statement must stop above that scale.","posed_since":"","precise":"Spin-$1/2$ fermions in $R^{3}$ with dispersion $k^{2}/(2m)$ and weak short-range repulsion $\\lambda v$. Prove that for T above the pairing scale ($T \\ge \\exp(-c/\\lambda^{2}) E_F$, say) the momentum distribution has a jump of size $Z = 1 - \\operatorname{O}(\\lambda^{2})$ smeared only on scale T, and $\\operatorname{Im} \\Sigma(\\omega, k_F) \\sim \\omega^{2} + (\\pi T)^{2}$. Answer: a proof with explicit constants.","problem_ref":null,"references":"","settled_by":"A convergent multiscale (renormalization-group) proof of Fermi-liquid bounds in 3D down to an exponentially small temperature.","status_note":"In $2\\mathrm{D}$, Disertori and Rivasseau (2000) controlled jellium above an exponentially small temperature, and Feldman, Knorrer and Trubowitz (2003-2004) proved a $T = 0$ Fermi liquid for Fermi surfaces without inversion symmetry.","title":"Fermi-liquid behavior of weakly interacting fermions above the pairing 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