We develop Bellman-sufficient information complexity, a representation-level framework for studying information-theoretic complexity in sequential decision making. The primitive object is an environment space $Ω$ and an admissible algorithm class. The intrinsic object is a Bellman-sufficient state representation together with an information index $Y=χ(Ω)$, often the optimal decision or value object rather than the full environment. This replaces syntactic model realizability with representation-level sufficiency for decision making. On the upper-bound side, learning is organized as a dynamic program on the sufficient state with a logarithmic information potential for the index. In fixed-truth analysis this potential is represented by the coordinate log loss $γ\log(1/q_t(χ(ω^\star)))$; in the indexed Algorithmic Information Ratio (AIR) regret identities it gives rise to the log-posterior telescope, and after Bayesian posterior averaging it corresponds to an entropy term. On the lower side, a Bellman-Fano certificate uses the same state and index to compare the indexed information telescope with the ghost-good mass of low-regret reference trajectories. 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 on the same representation and at the same radius. UCB, E2D/DEC, and AMS/EBO then appear as tractable certificates or relaxations of this same log-potential Bellman program, rather than as separate notions of information complexity.

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