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

arXiv: https://arxiv.org/abs/2510.15987 · HTML: https://mlsanity.com/paper/2510.15987/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 9 of its 69 references that are in the map.
- First posted on arXiv 13 Oct 2025.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 9 of 9)

- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Phi-4 Technical Report](https://mlsanity.com/paper/2412.08905/) (2024)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Persona Vectors: Monitoring and Controlling Character Traits in Language Models](https://mlsanity.com/paper/2507.21509/) (2025)
- [Phi-4-reasoning Technical Report](https://mlsanity.com/paper/2504.21318/) (2025)
- [Physics of Language Models: Part 2.1, Grade-School Math and the Hidden Reasoning Process](https://mlsanity.com/paper/2407.20311/) (2024)
- [Can Large Language Models Explain Themselves? A Study of LLM-Generated Self-Explanations](https://mlsanity.com/paper/2310.11207/) (2023)
- [Inference-Time Scaling for Complex Tasks: Where We Stand and What Lies Ahead](https://mlsanity.com/paper/2504.00294/) (2025)
- [When a language model is optimized for reasoning, does it still show embers of autoregression? An analysis of OpenAI o1](https://mlsanity.com/paper/2410.01792/) (2024)

## Built on by (most cited first, 7 of 7)

- [Zipping the Thought: When and How Compressed Reasoning Data Works in LLM Post-Training](https://mlsanity.com/paper/2605.28008/) (2026)
- [A Compositional Framework for Open-ended Intelligence](https://mlsanity.com/paper/2606.15386/) (2026)
- [Reasoning Models Know What's Important, and Encode It in Their Activations](https://mlsanity.com/paper/2604.18307/) (2026)
- [Evolutionary Task Discovery: Advancing Reasoning Frontiers via Skill Composition and Complexity Scaling](https://mlsanity.com/paper/2605.11666/) (2026)
- [Distributed Sparse Interventions in Language Models](https://mlsanity.com/paper/2607.07128/) (2026)
- [A Calibrated Test of Internal Action Maps: State Signals Without Global Affine Closure](https://mlsanity.com/paper/2608.13626/) (2026)
- [A Pre-Specified Construction-Confirmation Test of Operation-Level Causal Transfer Across Finite Isomorphic Symbolic Domains](https://mlsanity.com/paper/2608.15809/) (2026)

## Abstract (arXiv)

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.