We present a mathematical framework for quantifying energy efficiency in intelligent systems by linking energy consumption to information-processing capacity. We introduce a watts-per-intelligence metric that integrates algorithmic thermodynamic principles of Landauer with computational models of machine intelligence. By formalising the irreversible energy costs of computation, we derive rigorous lower bounds on energy usage of algorithmic intelligent systems and their adaptability. We introduce theorems that constrain the trade-offs between intelligence output and energy expenditure. Our results contribute to design principles for energy-efficient intelligent systems.