The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason

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 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 reasoning models (LRMs) generate extended solutions, yet it remains unclear whether these traces reflect substantive internal computation or merely verbosity and overthinking. Although recent hidden-state analyses suggest that internal representations carry correctness-related signals, their coarse aggregations may obscure the token and layer structure underlying reasoning computation. We investigate hidden-state transitions across decoding steps and layers, and identify a distinct spatiotemporal pattern in LRMs: successful trajectories exhibit broad temporal dynamics with localized layer-wise concentration, while this structure is weaker in non-reasoning models and knowledge-heavy domains. We formalize this characteristic as Spatiotemporal Amplitude of Latent Transition (StALT), a training-free trajectory statistic that summarizes temporal changes between adjacent tokens weighted by within-token layer saliency. Across diverse models and benchmarks, StALT reliably separates correct from incorrect trajectories in reasoning-intensive regimes, providing a competitive label-free correctness signal alongside strong output-space and length-based baselines. Intervention analyses further show that this spatiotemporal amplitude responds systematically to manipulations that increase or reduce the demand for internal reasoning, supporting its association with latent reasoning dynamics in LRMs. These findings provide empirical evidence that LRMs exhibit measurable hidden-state dynamics and offer a practical probe for understanding internal computation beyond output-based evaluation.

We consider the stability of multi-step reasoning processes, which have extensive applications in language models, including chain-of-thought and algorithmic reasoning. While longer sequences of reasoning can improve a model's generation capability at test time, the errors due to intermediate reasoning steps can accumulate in autoregressive generation, and thus grow substantially at the end. In this paper, we ask: What are the key factors determining the stability of multi-step reasoning? First, we show an inference error bound governed by the product of spectral norms of the Jacobians taken through the input space across generation steps. This product can be viewed as an error amplification factor, which could scale exponentially with the number of reasoning steps, serving as a quantitative measure of reasoning stability. Second, we analyze this measure in transformer models trained to predict simple tasks like linear and quadratic functions. We theoretically prove that the transformer model converges to a solution where the stability measure decays, thus yielding nearly zero inference loss over (arbitrarily) long steps. Finally, the stability analysis leads to several algorithmic implications for controlling the stability, through (i) chain-of-thought length compression that reduces the sensitivity of each step, and (ii) quantization-aware training that regularizes the input Jacobian norms. We validate the proposed algorithms by fine-tuning language models on graph-algorithmic reasoning tasks and symbolic state-tracking tasks. Across seven evaluations, our algorithms improve over baseline comparisons by 3.5% on average, and by 8.2% for longer-length inputs. Ablation analysis validates that the stability measure is drastically reduced by 3-8$\times$, confirming the regularization effect on the spectral norms of the (input space) Jacobians.