\begin{abstract}
We prove that the critical capacity of the Hopfield model at zero temperature lies in $[0.138198129,\,0.13820106]$. The best rigorous lower bound was $0.113$, and our lower end exceeds the published one-step and two-step replica symmetry breaking values. The capacity is the supremum of the loads at which the free energy at fixed overlap with one pattern has a well. At every positive temperature, we give a variational formula for this free energy and show that binary patterns and Gaussian rows have the same wells, which characterizes the capacity. The proof builds on the Hamilton--Jacobi solution of the model by Ali, Boursier, Chen, Ho, Issa and Mourrat, and on a strict form of their Jensen inequality, uniform in the number of patterns, which we prove.
\end{abstract}