Categorical Perception in Large Language Model Hidden States: Structural Warping at Digit-Count Boundaries

Large language models (LLMs) generalize smoothly across continuous semantic spaces, yet strict logical reasoning demands the formation of discrete decision boundaries. Prevailing theories relying on linear isometric projections fail to resolve this fundamental tension. In this work, we argue that task context operates as a non-isometric dynamical operator that enforces a necessary "topological distortion." By applying Gram-Schmidt decomposition to residual-stream activations , we reveal a dual-modulation mechanism driving this process: a class-agnostic topological preservation that anchors global structure to prevent semantic collapse, and a specific algebraic divergence that directionally tears apart cross-class concepts to forge logical boundaries. We validate this geometric evolution across a gradient of tasks, from simple mapping to complex primality testing. Crucially, targeted specific vector ablation establishes a strict causal binding between this topology and model function: algebraically erasing the divergence component collapses parity classification accuracy from 100% to chance levels (38.57%). Furthermore, we uncover a three-phase layer-wise geometric dynamic and demonstrate that under social pressure prompts, models fail to generate sufficient divergence. This results in a "manifold entanglement" that geometrically explains sycophancy and hallucination. Ultimately, our findings revise the linear-isometric presumption, demonstrating that the emergence of discrete logic in LLMs is purchased at an irreducible cost of topological deformation.

While large language models (LLMs) are trained purely on textual data, prior work has shown that their internal representations can exhibit rich geometric structure in embedding space. Building on this line of work, we investigate whether such structure is similar to human perceptual organisation across different domains (e.g., color, pitch, emotion, and taste). Specifically, we study the layer-wise emergence of intrinsic geometrical structure corresponding to perceptual modalities within the residual streams of multiple open-weight transformer architectures. Our results reveal three key findings. First, we observe the emergence of layer-wise geometric structure across multiple perceptual domains, despite the absence of any direct perceptual supervision during training. Second, these perceptual domains exhibit distinct emergence profiles, with both geometric structure and its alignment with human baselines following domain- and model-specific trajectories across depth. Third, this emergence follows a consistent representational trajectory: geometry is weak or diffuse in early layers, becomes progressively organised in intermediate layers, and is attenuated in later layers, suggesting that perceptual geometry arises transiently as part of the model's internal transformation pipeline. This provides new insight into how and where human-like perceptual geometry arises in LLMs, offering a principled pathway for mechanistic analysis of internal representations.

Large language models (LLMs) exhibit remarkable flexibility: they can adapt to novel tasks from in-context examples without any parameter updates, a capability known as in-context learning (ICL). Prior work on synthetic tasks has shown that ICL can implement specific algorithms, demonstrating architectural competence, and mechanistic analyses have identified key circuits that support this behavior. However, because in-context computation -- regardless of its algorithmic form -- relies on transformations in high-dimensional representation space, it remains unclear how the geometry of that space shapes ICL effectiveness. Motivated by the neuroscience view of classification as the untangling of neural representations, we hypothesize that ICL depends on the successful online untangling of task-relevant representations. To test this idea, we study how LLMs classify in-context examples whose labels are defined by the model's own internal representations with known structure. We show that ICL performance correlates systematically with the representational structure of the underlying classification task and that successful ICL is accompanied by geometric reorganization that increases online separability. We further find that LLM behavior is well described by a prototype-like algorithm that integrates evidence while reshaping representations to support classification. These findings offer a geometric account of ICL in pretrained LLMs, establish representational geometry as a mechanistic constraint on ICL, and quantify the gap between what pretrained representations afford and what in-context learning can exploit.