 Spin-degeneracy shift of a charge step

Near the first addition threshold of a repulsive single-orbital defect, the empty and singly occupied grand weights are 1 and $2e^{\beta (\mu -ϵ)}$. Equal weights occur at $\mu =ϵ-k_{B}T\ln 2$. Near the second threshold, the single/double aggregate weights are in ratio $2:e^{\beta (\mu -ϵ-U)}$, giving half-step position $ϵ+U+k_{B}T\ln 2$. These are entropic finite-temperature shifts, which vanish in energy units as $T\to 0$. They must still be retained when evaluating statistics exactly at an unshifted degeneracy point.