# Right Direction, Wrong Step: Geometric Analysis of Finite-Step Failure in Looped Transformers

arXiv: https://arxiv.org/abs/2609.16665 · HTML: https://mlsanity.com/paper/2609.16665/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Prompting and chain-of-thought reasoning, one of 17 directions in LLM reasoning.
- Builds on 31 of its 35 references that are in the map.
- First posted on arXiv 15 Sep 2026.

## Placed in

- LLM reasoning / [Prompting and chain-of-thought reasoning](https://mlsanity.com/lr/prompting-and-chain-of-thought-reasoning/)

## Builds on (most cited first, 30 of 31)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [CommonsenseQA: A Question Answering Challenge Targeting Commonsense Knowledge](https://mlsanity.com/paper/1811.00937/) (2018)
- [Universal Transformers](https://mlsanity.com/paper/1807.03819/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Adaptive Computation Time for Recurrent Neural Networks](https://mlsanity.com/paper/1603.08983/) (2016)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [Reasoning with Latent Thoughts: On the Power of Looped Transformers](https://mlsanity.com/paper/2502.17416/) (2025)
- [Looped Transformers as Programmable Computers](https://mlsanity.com/paper/2301.13196/) (2023)
- [Depth-Adaptive Transformer](https://mlsanity.com/paper/1910.10073/) (2019)
- [PonderNet: Learning to Ponder](https://mlsanity.com/paper/2107.05407/) (2021)
- [Scaling Latent Reasoning via Looped Language Models](https://mlsanity.com/paper/2510.25741/) (2025)
- [Looped Transformers are Better at Learning Learning Algorithms](https://mlsanity.com/paper/2311.12424/) (2023)
- [Can You Learn an Algorithm? Generalizing from Easy to Hard Problems with Recurrent Networks](https://mlsanity.com/paper/2106.04537/) (2021)
- [Looped Transformers for Length Generalization](https://mlsanity.com/paper/2409.15647/) (2024)
- [Relaxed Recursive Transformers: Effective Parameter Sharing with Layer-wise LoRA](https://mlsanity.com/paper/2410.20672/) (2024)
- [Stabilizing Equilibrium Models by Jacobian Regularization](https://mlsanity.com/paper/2106.14342/) (2021)
- [Can Looped Transformers Learn to Implement Multi-step Gradient Descent for In-context Learning?](https://mlsanity.com/paper/2410.08292/) (2024)
- [LoopFormer: Elastic-Depth Looped Transformers for Latent Reasoning via Shortcut Modulation](https://mlsanity.com/paper/2602.11451/) (2026)
- [Teaching Pretrained Language Models to Think Deeper with Retrofitted Recurrence](https://mlsanity.com/paper/2511.07384/) (2025)
- [On Expressive Power of Looped Transformers: Theoretical Analysis and Enhancement via Timestep Encoding](https://mlsanity.com/paper/2410.01405/) (2024)
- [Mitigating Overthinking in Large Reasoning Models via Manifold Steering](https://mlsanity.com/paper/2505.22411/) (2025)
- [Stabilizing Recurrent Dynamics for Test-Time Scalable Latent Reasoning in Looped Language Models](https://mlsanity.com/paper/2605.26733/) (2026)
- [Latent Thinking Optimization: Your Latent Reasoning Language Model Secretly Encodes Reward Signals in Its Latent Thoughts](https://mlsanity.com/paper/2509.26314/) (2025)
- [LoopUS: Recasting Pretrained LLMs into Looped Latent Refinement Models](https://mlsanity.com/paper/2605.11011/) (2026)
- [Two-Scale Latent Dynamics for Recurrent-Depth Transformers](https://mlsanity.com/paper/2509.23314/) (2025)
- [Think Shallow, Solve Deep: Controlling Recurrent Dynamics for Reliable Test-Time Depth](https://mlsanity.com/paper/2608.18222/) (2026)
- [Efficient Post-Training Refinement of Latent Reasoning in Large Language Models](https://mlsanity.com/paper/2506.08552/) (2025)
- [Thinking on the Fly: Test-Time Reasoning Enhancement via Latent Thought Policy Optimization](https://mlsanity.com/paper/2510.04182/) (2025)

## Built on by (most cited first, 1 of 1)

- [SanSi: A Looped Typed Decision Model for System 1.5 Thinking](https://mlsanity.com/paper/2610.07730/) (2026)

## Abstract (arXiv)

Looped Transformers offer a parameter-efficient route to test-time scaling by reusing shared layers for iterative latent reasoning. However, additional iterations can reduce support for a reference answer, leaving unclear whether an update's direction is locally unhelpful or its full displacement moves too far. We study this distinction by analysing reference utility, which measures this support, along the model's own update direction, varying the fraction of the proposed displacement supplied to the readout. This reveals finite-step failures in which a locally improving direction produces a harmful full update. A pathwise curvature decomposition characterises how initial progress is lost, while a local quadratic model predicts full-step gains and useful step scales. Bounds based on accumulated curvature variation characterise the approximation error of these predictions. Experiments across two model families reveal this separation on mathematical and commonsense tasks. A fixed quarter step produces positive gains in reference utility for 72.2--83.2% of selected failures across four settings. These findings identify a mismatch between update direction and step scale as a mechanism of lost progress, explaining how some harmful updates retain useful computation.