The information geometry of large language models is shared, learned, and controllable

Post-training can substantially alter language-model behavior, yet aggregate behavior rates do not reveal whether training removes an existing mechanism, creates a new one, or changes how an inherited mechanism is used. We study this question through two mechanistically distinct failures, repetition as a decoding-attractor pathology and sycophancy as a preference-related alignment failure. We introduce behavioral manifold analysis, which isolates behavior-specific geometry by selecting sparse behavior-associated coordinates and lifting them into low-dimensional local charts. We construct these charts in two complementary spaces. ACT captures runtime activation states, while NOC quantifies how strongly the model routes functional information flow through the shared behavior-associated subspace. Across multiple model families, the resulting charts are highly compressed and partially alignable across architectures. Contribution-space charts expose a more architecture-robust shared core, whereas activation-space charts retain stronger family-specific structure. Tracking these charts through controlled post-training reveals a consistent asymmetry. Supervised fine-tuning substantially alters the inherited behavioral geometry, whereas reward optimization changes behavior while largely preserving the underlying chart. This geometric perspective provides a unified framework for understanding the mechanistic distinction between the two objectives. SFT tends to rewrite behavioral geometry, whereas reward optimization primarily reweights it. Code is available at https://github.com/ronglingze/Manifold-Analysis

GPT-style language models are sensitive to single-token changes at generation points where the predicted probability distribution is spread across multiple tokens. Viewing this sensitivity as a geometric property, we derive an $\mathfrak{so}(n)$-valued 1-form that depends only on the geometry of the token embeddings. Despite this purely geometric origin, we show that its curvature is semantically meaningful: On chess reasoning tasks, the curvature couples to the world model of an off-the-shelf instruction-tuned model, with transformations clustering by board region and respecting piece importance. Our findings suggest that token space geometry directly reflects how models internally represent problems.

A defining feature of human intelligence is the ability to adapt to changing environments by inferring latent task structure from sparse observations. Neuroscientific research indicates that this capability relies on the hippocampus constructing abstract representations, expressed as low-dimensional, approximately orthogonal manifolds in neural state space. However, the internal mechanisms of large language models (LLMs) remain largely opaque, making it unclear whether they form comparable abstract representations or instead rely on task-specific statistical regularities when performing comparable reasoning tasks. Here we adapt a contextual reversal-learning paradigm to a text-based setting and compare humans and LLMs at both the Behavioural and representational levels. We report that although LLMs exhibit generalizable reasoning less frequently than humans, when such inference occurs, their internal states exhibit abstract geometric structures that resemble those reported in the hippocampus. Notably, this representational geometry is not uniformly distributed but is organized hierarchically across model depth: whereas lower layers show early, stable encoding of stimulus identity, higher layers form a hippocampal-like functional band enriched for abstract context geometry associated with inference. Furthermore, complementary intervention experiments mechanistically implicate geometry in reasoning: task-sequence language modelling induces geometric disentanglement, whereas geometric regularization of higher layers increases the emergence of generalizable inference. Together, these findings establish abstract representational geometry as a mechanistic principle supporting inference in large language models.