Sharp universal death of entanglement threshold for Pauli Hamiltonians

We determine the exact universal high-temperature separability threshold for Pauli Hamiltonians of bounded degree $\mathrm{\Delta }\ge 2$. If every coefficient in the Hamiltonian has magnitude at most one and each term has overlapping support with at most $\mathrm{\Delta }$ other terms, the Gibbs state is a mixture of product Pauli eigenstates whenever $$\beta \le z_{\mathrm{\Delta }}:=\mathrm{arctanh}\left[\underset{0\le x\le 1}{\max }x{\left(\frac{1-x}{1+x}\right)}^{\mathrm{\Delta }-1}\right]\mathrm{.}$$ For every $\beta >z_{\mathrm{\Delta }}$, a finite commuting Hamiltonian with maximum overlap degree at most $\mathrm{\Delta }$ has an entangled Gibbs state. At any fixed $\beta <z_{\mathrm{\Delta }}$ strictly below the threshold, a classical polynomial-time algorithm produces samples from a distribution over product Pauli eigenstates approximating the Gibbs state in trace distance.