  We consider a gas of bosonic particles confined in a box with Neumann
boundary conditions. We prove Bose-Einstein condensation in the
Gross-Pitaevskii regime, with an optimal bound on the condensate depletion. Our
lower bound for the ground state energy in the box implies (via Neumann
bracketing) a lower bound for the ground state energy of the Bose gas in the
thermodynamic limit.
