Skills, as a useful abstraction for the procedural capabilities of large language models (LLMs), capture how models perform structured, multi-step reasoning and program execution. Existing approaches typically treat skills as explicit, surface-level constructs specified through prompts or programs, leaving open the question of how such procedural capabilities are represented inside the model and whether they can be manipulated as structured objects in latent space. In this empirical study, we investigate whether procedural LLM skills can be represented as directions in activation space and whether vector-space operations over these directions can express skill-level behaviors. We find that procedural skills admit a vector-space representation: individual skill directions can be activated to shift model behavior; independently extracted directions can compose to form higher-level skills. Contrastive directions yield context-conditioned algorithmic personalization and optimization trajectories over skill directions evolve non-monotonically, with intermediate states often surpassing fully optimized solutions. These results support a representation-level view of procedural LLM skills: they admit a latent vector-space organization that allows direct manipulation through internal interventions.

Recent advances in chain-of-thought (CoT) prompting have demonstrated the ability of large language models (LLMs) to perform multi-step reasoning. While prior work focuses on improving CoT generation quality or attributing token-level importance, we propose a novel framework to structurally analyze the latent dynamics of CoT trajectories for interpretability. Our method segments generated CoT into discrete reasoning steps, abstracts each step into a spectral embedding based on the eigenvalues of token-level Gram matrices, and clusters these embeddings into semantically meaningful latent states. We model the global evolution of reasoning as a first-order Markov chain over latent clusters, yielding interpretable transition structures. Through t-SNE visualizations and Monte Carlo rollouts, we uncover consistent trajectories across tasks and models, supporting the hypothesis that LLM reasoning follows globally coherent yet abstract paths.

Recent MLLMs have demonstrated strong visual understanding and reasoning after large-scale multimodal pre-training. However, instruction-tuning is typically text-driven with limited visual supervision, leading to significant visual forgetting and degradation of pre-trained visual knowledge. Existing fine-tuning and continual learning methods compress visual representations and emphasize task alignment over visual retention, failing to address this challenge. We present a novel perspective using effective rank to quantify the loss of visual representation richness, framing visual forgetting as excessive compression under the information bottleneck principle. To address this, we propose modality-decoupled gradient descent (MDGD), which regulates gradient updates to preserve the effective rank of visual features and explicitly disentangles visual learning from task-specific alignment. We further introduce a memory-efficient fine-tuning variant using gradient masking for parameter-efficient adaptation. Extensive experiments show that MDGD effectively mitigates visual forgetting across downstream tasks and models, maintaining pre-trained visual knowledge while supporting strong task adaptation.