Many recent long-context and agentic systems address context-length limitations by adding hierarchical memory: they extract atomic units from raw data, build multi-level representatives by grouping and compression, and traverse this structure to retrieve content under a token budget. Despite recurring implementations, there is no shared formalism for comparing design choices. We propose a unifying theory in terms of three operators. Extraction ($α$) maps raw data to atomic information units; coarsening ($C = (π, ρ)$) partitions units and assigns a representative to each group; and traversal ($τ$) selects which units to include in context given a query and budget. We identify a self-sufficiency spectrum for the representative function $ρ$ and show how it constrains viable retrieval strategies (a coarsening-traversal coupling). Finally, we instantiate the decomposition on eleven existing systems spanning document hierarchies, conversational memory, and agent execution traces, showcasing its generality.

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