What It Means for AI to Understand

This paper explores a fundamental question in artificial intelligence: what does it actually mean for a machine to understand something rather than simply memorize patterns? It introduces a formal framework for distinguishing genuine comprehension from statistical recall using transformation invariance and generalization tests. The paper argues that true understanding is revealed when an AI system can preserve performance across meaningful changes in context, representation, and structure. It further proposes mathematical criteria for evaluating understanding through equivariance and robustness. Beyond intelligence itself, the framework is extended to AI safety and value alignment, showing that capability alone is insufficient without stable alignment to human objectives. The result is a unified theory connecting machine understanding, generalization, and beneficial AI behavior.