Graph-Parameterized Capacity for Latent Multi-Hop Recall: Usage, Compression, and Composition

When can knowledge stored in model parameters replace bounded knowledge-graph traversal? We price latent multi-hop storage by graph structure rather than raw fact count. We define coverage-conditioned support usage $H$, macro-decomposition credit $S$, and realizable width $W_{\text{D}}$, and give a geometric construction with dimension quadratic in $W_{\text{D}}$ and logarithmic in code count and inverse slack. For witness-executing position-factorized memories, we prove a restricted entry-count lower bound matching the $H-S$ charge on branch-functional, pair-balanced closures under stated selector and representation assumptions. For modular memories, we prove an expected confabulation-plus-abstention floor on unique-witness crossings for weight-local, frame-nonaligning composition with a latent crossing interface, under receiving-cap overlap and path-separation assumptions. Empirically, 14 theory-matched families share a coarse rung index on an $H{\log }_2|V|$-scaled ladder, while realized prefactors remain family-dependent; the distractor control does not identify the ${\log }_2|V|$ factor. Support diversity best predicts thresholds. Macro credit reduces explicit parameters under slot-elastic allocation, while trained macro gains mainly reflect shorter live retrieval depth. A receiving-gate audit isolates frame mismatch. Pretrained LoRA thresholds remain bottom-censored, while CoT reaches the measured floor at $k=4$ on 9/9 chain and 8/9 segmented cell-seeds.