Isotropy Cliffs: The Geometric Signature of Decision-Making in Large Language Models

Language models do not simply choose an answer at the output layer. In a 9,000-trajectory MMLU study across Qwen2.5-7B-Instruct, Llama-3.1-8B-Instruct, and Mistral-7B-Instruct-v0.3, the score of the answer moves across depth in structured ways. We describe each trajectory with three quantities: the current answer margin, the next-layer change in that margin, and the distance from a decision flip. The main empirical picture is that correctness and stability are different: the largest group is unstable-correct, not stable-correct. A traced subset then asks what moves the margin. In stable-correct cases, the average attention scalar points in the correct direction, while the average MLP scalar does not; span deletion shows that removing answer-supporting text hurts the margin and removing distractor-like text helps it. The result is not a full circuit explanation. It is a reproducible way to see which answers are settled, which remain fragile, and which measured sources move them.

Chain-of-thought (CoT) reasoning enables large language models (LLMs) to solve complex problems by generating intermediate reasoning steps. While much attention has been paid to the length and content of these reasoning chains, far less is known about their internal geometry. We study the \emph{geometry} of CoT trajectories in the hidden state space of transformer models, formalizing each reasoning chain as a discrete curve in $\mathbb{R}^d$ and characterizing it through spectral, positional, and kinematic geometric functionals. We introduce the effective dimension $d_ρ$ as a measure of trajectory complexity and show theoretically that trajectories with flatter eigenvalue spectra correspond to harder tasks, as they explore more of the hidden dimensions. Lastly, we explore how kinematic features of the trajectory, mean position, positional dispersion, initial and current hidden states, mean velocity, mean speed, and speed dispersion, can be used to predict solution correctness before generation is complete, and may inform future early-stopping strategies. Experimentally, on mathematical reasoning problems from the MATH500 dataset, $d_ρ$ achieves $0.93$ AUC in distinguishing easy from hard problems, while kinematic features potentially can predict correctness from only the first $20\%$ of generated tokens. These correctness signatures transfer across questions of varying difficulty, establishing that the shape of a model's internal reasoning trajectory is a principled window into both task hardness and solution quality.

Large language models learn similar behaviours, yet it remains unclear what structure they share or how to change one behaviour without disturbing others. The Fisher-Rao geometry of next-token probabilities connects these questions: behaviour determines this geometry up to output-preserving symmetries, whereas activation geometry depends on coordinates. Across transformer, state-space and recurrent models, output geometries agree more strongly than activation geometries, and shared geometry supports semantic-category transfer. Agreement with human word choices increases with predictive accuracy, scale and training, and improves further after model-only calibration. Token probabilities and read-out geometry jointly predict the spectrum and its effective dimension. Controlled language assignments show that geometry follows the language law across architectures. Pretraining corpus statistics predict held-out fact acquisition without recalibration, while randomised experiments show that deeper evidence substantially delays acquisition across every tested architecture and evidence construction. Finally, the geometry prescribes minimum-disturbance local interventions, predicts their relative cost, and supports reusable control: updates learned on donor prompts transfer to unseen prompts while better preserving behaviour on reference prompts than Euclidean control. The same geometric correction improves steering, editing, attribution, dictionary learning and fine-tuning.