# Rotate2Think: Geometric Priming via Orthogonal Rotation to Improve Language Model Reasoning

arXiv: https://arxiv.org/abs/2606.09873 · HTML: https://mlsanity.com/paper/2606.09873/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 12 of its 29 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Placed in

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

## Builds on (most cited first, 12 of 12)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Phi-4 Technical Report](https://mlsanity.com/paper/2412.08905/) (2024)
- [The Platonic Representation Hypothesis](https://mlsanity.com/paper/2405.07987/) (2024)
- [Symmetry in language statistics shapes the geometry of model representations](https://mlsanity.com/paper/2602.15029/) (2026)
- [The Markovian Thinker: Architecture-Agnostic Linear Scaling of Reasoning](https://mlsanity.com/paper/2510.06557/) (2025)
- [Therefore I am. I Think](https://mlsanity.com/paper/2604.01202/) (2026)

## Abstract (arXiv)

Reasoning models achieve strong performance on challenging tasks by generating explicit intermediate reasoning traces before producing a final answer. Yet the internal structure of representation space when reasoning remains poorly understood: how do a model's hidden representations differ during thinking versus the embeddings of the input prompt, and can this structure be exploited to elicit stronger reasoning at inference time? We show that both input embeddings and thinking embeddings (mean-pooled last-layer hidden states over the prompt and reasoning trace, respectively) exhibit extremely high conicity, with all vectors clustering tightly around a single mean direction. Crucially, these mean input and thinking directions are non-collinear, with thinking embeddings occupying a geometrically distinct region of embedding space across many different models and benchmark tasks. This observation motivates casting the input-to-thinking transition as a rotation problem admitting a closed-form solution via orthogonal Procrustes analysis. We propose Rotate2Think, a training-free method that estimates this rotation from a small set of correctly solved examples and injects the resulting synthetic thinking vector between thinking delimiters at inference time, providing a geometric primer at the onset of the reasoning trace. Evaluated across multiple benchmarks and model families, Rotate2Think improves accuracy in 30 of 32 model-benchmark configurations across mathematics, science, and code tasks, and generalizes zero-shot to multimodal reasoning on MATH-Vision.