Uniform entropy and energy bounds for fully non-linear equations

Energy bounds which are uniform in the background metric are obtained from upper bounds for entropy-like quantities. The argument is based on auxiliary Monge-Amp\`ere equations involving sublevel sets, and bypasses the Alexandrov-Bakelman-Pucci maximum principle. In particular, it implies uniform $L^\infty$ bounds for systems coupling a fully non-linear equation to its linearization, generalizing the cscK equation.