Geometric Factual Recall in Transformers

Large language models (LLMs) store factual knowledge in their parameters. While recent work has shown that this knowledge resides in MLP layers, existing constructive and mechanistic interpretability models of fact-storage in LLMs fail to explain the surprising empirical phenomenon that they store facts at an information-theoretically optimal rate. In this work, we develop a theoretical account of this phenomenon. We develop the first Transformer-compatible fact-storing MLP closed-form construction that satisfies the following three properties empirically observed in LLMs: it (i) attains optimal fact storage scaling, (ii) handles arbitrary input/output geometries, and (iii) works inside Transformers. Key to our work is to analyze the decoding margin of MLPs, whereas prior work only studies MLP fact storage. Under isotropic embeddings, our construction achieves information-theoretically optimal storage capacity scaling and requires $10$-$104\times$ fewer parameters at matched fact count than prior constructions. For arbitrary key and value embeddings, we show that our construction attains the same storage capacity scaling, up to penalization factors depending on the embedding geometries. Moreover, we demonstrate that our constructed MLPs can be used within Transformer blocks for factual recall tasks at optimal capacity scaling, requiring $15$-$63\times$ fewer parameters at matched fact count than prior constructions. Finally, as a proof-of-concept, we show that fact-storing MLPs enable modular fact editing by swapping a Transformer's MLP with a new one.

In-context learning \ -- performing tasks based on examples given in the prompt \ -- is an important capability that has emerged in large language models and has received significant attention in both theory and practice. Existing theoretical work often focuses on settings where the learning uses information purely from the prompt. However, many practical instances of in-context learning require the model to retrieve factual knowledge stored in the model's parameters, with the context serving to identify which knowledge is relevant. In this work, we study how in-context learning leverages factual knowledge recall. We formalize this behavior by introducing the \emph{in-context factual recall (IC-recall)} task, where a transformer is provided a context of (subject, answer) pairs generated from a hidden relation, along with a query subject, and must both infer this hidden relation and retrieve the corresponding answer. Factual knowledge is modeled by the transformer having access to a simple pre-constructed MLP associative memory storing (subject, relation, answer) triplets. We analyze the supervised fine-tuning dynamics of a one-layer transformer on IC-recall data and prove that the model successfully performs IC-recall by converging to a particular pairwise attention pattern. This fine-tuning stage requires a very small number of samples \ -- only polylogarithmic in the number of stored knowledge triplets. Experiments verify our theoretical predictions and show that the pairwise attention pattern emerges even when the MLP layer is pretrained instead of constructed.

Language models draw on two knowledge sources: facts baked into weights (parametric memory, PM) and information in context (working memory, WM). We study two mechanistically distinct failure modes--conflict, when PM and WM disagree and interfere; and hallucination, when the queried fact was never learned. Both produce confident output regardless, making output-based monitoring blind by design. We show both failures share a unified geometric account. In the hidden-state space of autoregressive generation, learned facts form attractor basins. Conflict is basin competition: WM disrupts convergence to the correct basin without raising output entropy. Hallucination is basin absence: the hidden state drifts freely when no memorized basin exists. The frozen LM head, designed for next-token prediction, cannot distinguish these cases and fires confidently either way. We verify this account in a controlled synthetic task-entity identifiers mapped to unique codes with PM installed via LoRA adapters--where ground truth is exact and component roles can be causally isolated through targeted adapter placement. Geometric margin--the hidden state's distance to the nearest memorized basin--reads this geometry directly and separates correct recall from hallucination far more cleanly than output entropy, with zero false refusals where entropy-based detection cannot avoid rejecting the vast majority of correct outputs. The separation holds on natural-language factual queries from the pretrained model with no adaptation, confirming attractor geometry is structural rather than a fine-tuning artifact. The fraction of confident hallucinations follows a scaling law $C = \exp(-c/\barΔ)$, growing with scale even as overall error rates fall. Hidden states reliably encode epistemic state; the frozen output head systematically erases it--and this erasure worsens with scale.