We develop Bellman-sufficient information complexity, a formal representation-level framework for sequential decision making. The primitive benchmark is a fixed-truth environment space $Ω$ with unrestricted nonanticipating algorithms. The intrinsic object is a Bellman-sufficient state representation, serving as an interactive notion of sufficient statistics, together with an information index $Y=χ(Ω)$, often the optimal decision or value object rather than the full environment. On the upper-bound side, learning is organized as a dynamic program on the sufficient state, equipped with a logarithmic information potential for the index. On the lower-bound side, a Bellman-Fano certificate uses the same state representation and information index, but propagates separate Bellman recursions for information gain and ghost mass. The central matching statement is therefore a conditional Bellman information-risk sandwich: when the log-penalized Bellman upper value and the ghost-quantile lower certificate close at the same radius, they certify the same complexity scale. Popular algorithms then appear as tractable certificates or relaxations of this common log-potential Bellman program, rather than as separate notions of information complexity.

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