We prove a mathematically rigorous theorem that asserts, under certain carefully stated hypotheses, the validity of the Goertzel and Otsuka conclusions that, in a thermal nuclear reactor that has a minimum critical mass, the fuel must be distributed so that the product of the thermal neutron flux and the adjoint thermal neutron flux is a constant in the core and does not exceed that constant in the reflector. We also furnish some examples that illustrate the necessity of imposing some mathematical hypotheses to obtain the desired conclusions.