When a language model develops capabilities not traceable to its individual components, when a multi-agent AI system outperforms each agent on its own, or when a human-AI pair succeeds where either fails in isolation, AI researchers describe it as a case of ``synergy'' or ``emergence''. However, these are usually informal assessments not supported by a theoretical framework, making these claims impossible to compare or even meaningfully dispute. Here we argue that \emph{information decomposition can serve as a unifying formal framework to operationalize assessments about synergy in AI systems}. Information decomposition provides a way to distinguish qualitatively different \emph{types} of information, including unique information available from individual components, redundant information shared across multiple components, and synergistic information accessible only from parts taken together. This gives claims of emergence a precise, falsifiable criterion: a system's behavior is emergent only if it carries synergistic information about task-relevant outcomes. We apply this lens to attention heads, autonomous agents, and human-AI teams, showing how it organizes existing knowledge across scales and substrates, while also generating cross-level conjectures that siloed fields would miss.

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