# Looped Reasoning Finishes Earlier Than You Think

arXiv: https://arxiv.org/abs/2610.18174 · HTML: https://mlsanity.com/paper/2610.18174/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 27 references that are in the map.
- First posted on arXiv 13 Oct 2026.

## Builds on (most cited first, 20 of 20)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Universal Transformers](https://mlsanity.com/paper/1807.03819/) (2018)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Reasoning with Latent Thoughts: On the Power of Looped Transformers](https://mlsanity.com/paper/2502.17416/) (2025)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Hierarchical Reasoning Model](https://mlsanity.com/paper/2506.21734/) (2025)
- [Looped Transformers for Length Generalization](https://mlsanity.com/paper/2409.15647/) (2024)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [CNNs on Surfaces using Rotation-Equivariant Features](https://mlsanity.com/paper/2006.01570/) (2020)
- [A Mechanistic Analysis of Looped Reasoning Language Models](https://mlsanity.com/paper/2604.11791/) (2026)
- [Fixed-Point Reasoners: Stable and Adaptive Deep Looped Transformers](https://mlsanity.com/paper/2606.18206/) (2026)
- [Stabilizing Recurrent Dynamics for Test-Time Scalable Latent Reasoning in Looped Language Models](https://mlsanity.com/paper/2605.26733/) (2026)
- [Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning](https://mlsanity.com/paper/2605.21488/) (2026)
- [Two-Scale Latent Dynamics for Recurrent-Depth Transformers](https://mlsanity.com/paper/2509.23314/) (2025)
- [Solve the Loop: Attractor Models for Language and Reasoning](https://mlsanity.com/paper/2605.12466/) (2026)
- [Dynamical phase selection controls compute scaling in looped transformers](https://mlsanity.com/paper/2608.26556/) (2026)

## Abstract (arXiv)

Looped reasoning with dynamic depth can be viewed as relaxation toward solution attractors in a learned dynamical landscape. In practice, however, latent trajectories often oscillate, cycle, or drift rather than cleanly converge to stable fixed points. How these non-stationary dynamics contribute to reasoning and constrain convergence remains unclear. We show that slowly decaying oscillatory modes can impose their convergence rate on coupled latent dynamics, creating a bottleneck. Yet damping these modes indiscriminately can disrupt the reasoning computation they support. This tension motivates an architectural approach: introducing rotational degrees of freedom along which symmetry-generated rotational motion leaves the reasoning-relevant quotient dynamics unchanged by construction. We propose QRR (Quotient Recurrent Reasoner), a gauge-equivariant recurrent architecture with an exact per-token phase-rotation symmetry. Motion along these symmetry directions leaves the quotient representation unchanged and is dynamically decoupled from reasoning-relevant dynamics. Empirically, QRR achieves stable and faster convergence to solution roots in quotient space, without forming a dominant slow oscillatory bottleneck, while matching or surpassing recent recurrent reasoning models on Sudoku, Maze, and Mini-ARC.