Long-context models repeatedly solve closely related retrieval problems. Attention or global retrieval can retain large stores, but each forward pass again searches for what matters. Recurrent compression avoids this search only by deciding earlier what information to preserve. Neither mechanism directly exploits that useful memories at neighbouring retrieval steps may substantially overlap. We propose an alternative in which retrieval is a persistent local traversal through an address space. We separate a memory's content from its address. Utility is defined by the information revealed by retrieving the content, while geometry determines only how cheaply that information can be reached. This leads to a cost-regularised indexing objective that preserves information value inside local neighbourhoods while charging for every memory made locally accessible. Its marginal-value form gives both attractive and exclusion pressures. Thus, complementary memories should be co-retrievable when the additional value exceeds retrieval cost, whereas redundant memories should not occupy the same retrieval set solely because they are individually useful. As a proof of principle, we test only the simplest fixed-geometry instance on MiniGrid Memory. A persistent local traversal head reaches $0.95$ mean return after $0.59$M training steps with a five-token context, compared with $0.94$M for Gaussian absolute addressing and more than $2$M for categorical absolute addressing. The experiment does not test learned geometry or repeated retrieval. Instead, it motivates the hypothesis that stateful retrieval may be easier to learn than stateless. To test the full proposed method further experimentation is necessary.

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