If vacuum energy can be negative, why is mass always positive?: Uses of the subdominant trace energy condition

Diverse calculations have shown that a relativistic field confined to a cavity by well defined boundary conditions can have a negative Casimir or vacuum energy. Why then can one not make a finite system with negative mass by confining the field in a some way? We recall, and justify in detail, the not so familiar subdominant trace energy condition for ordinary (baryon-electron nonrelativistic) matter. With its help we show, in two ways, that the mass-energy of the cavity structure necessary to enforce the boundary conditions must exceed the magnitude of the negative vacuum energy, so that all systems of the type envisaged necessarily have positive mass-energy.