Methods for improving knowledge use in large language models typically fall into two regimes. Non-parametric retrieval offers flexible access to external knowledge, but adds retrieval latency, context overhead, and only shallow integration with the backbone. Parametric adaptation is efficient at inference time, but entangles knowledge with model weights and can be hard to update, audit, or transfer. Engram-style hashed memory occupies a middle regime: it stores learned information in an external, addressable table, yet consumes that table through a small learned reader. This raises a basic question: when such a memory is moved across backbones, what matters more, the frozen memory itself or the target-side reader? We study this question through cross-model frozen-memory extraction, in which a memory trained on a source model is frozen and attached to a different target model, with only a lightweight reader trained. Ablations show that learned memory content and correct addressing both matter, while the amount of usable signal depends on how well the reader is aligned to the target model. In downstream question answering tasks, a dual-layer, four-branch reader nearly closes the gap between same-model and cross-model reuse, achieving an average score of 38.8 under our controlled evaluation protocol. Moreover, when the provider reader is directly compatible with the target interface, the frozen artifact can provide substantial utility without target-side training, while optional reader adaptation yields further improvement. These results suggest that Engram can serve as a reusable external knowledge artifact, provided that the target has access to a compatible reader interface; target-side adaptation can further improve alignment when direct reader reuse is insufficient.
Multi-path reasoning methods such as self-consistency (SC) sample $K$ reasoning paths and choose the most frequent answer. However, their gains quickly plateau as $K$ increases, and locating this saturation point currently requires fitting a scaling model to accuracies measured at several values of $K$ or stopping online per query. We formalize multi-path LLM reasoning as a diversity combining problem from wireless communications: each path is a noisy channel observation, and the pairwise correlation of path correctness caps the design-effect effective sample size of the vote at a finite ceiling. Generalized least squares (GLS) analysis shows that, for paths with equal gains, the optimal linear combiner of latent embeddings is uniform exactly when every row of the path covariance has the same sum, as under exchangeability, supporting majority vote as the natural default in standard SC while leaving room for weighting or pruning under heterogeneous prompt-template branches. Across 5 models and 12 benchmarks, prompt-template diversity reduces path correlation in $55$ of $60$ model-benchmark cells, with a mean relative reduction of $16\%$ and reductions in every task domain. We derive an Adaptive-K rule that uses a four-path pilot to select $K^*$, retaining $96$--$103\%$ of MV@$K{=}32$ accuracy across Math, QA, and NLU.