AlgoTrace: Algorithmic Primitives and Compositional Geometry of Reasoning in Language Models

How do inference time and latent computations enable large language models (LLMs) to solve multi-step reasoning problems? We introduce AlgoTrace, a framework for tracing and steering algorithmic operations in the model latent space for multi-step reasoning. We operationalize primitives by clustering latent activations of the model when solving four benchmarks: Traveling Salesperson Problem (TSP), 3SAT, AIME, and Graph Navigation. We annotate the clusters using their corresponding tokens in the reasoning trace. We then apply function vector methods to extract primitive vectors as reusable compositional building blocks of reasoning. We find that a) injecting a primitive vector into models (Phi, Qwen, Llama) elicits the associated algorithmic operation in the reasoning trace, b) injecting primitives can steer behavior across tasks, c) primitive vectors can be composed through algebraic operations, revealing a geometric logic in activation space, and d) a fine-tuned model exhibits improved composition of primitives (Phi-4-Reasoning vs. Phi-4). These findings demonstrate that LLM reasoning can be understood as a walk through algorithmic primitives in the latent space governed by compositional geometry. These primitives transfer across tasks, and reasoning finetuning strengthens algorithmic generalization and composition across domains.