# When Recursive Models Finish Computing

arXiv: https://arxiv.org/abs/2609.26487 · HTML: https://mlsanity.com/paper/2609.26487/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 35 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 23 of 23)

- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [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)
- [Looped Transformers as Programmable Computers](https://mlsanity.com/paper/2301.13196/) (2023)
- [Hierarchical Reasoning Model](https://mlsanity.com/paper/2506.21734/) (2025)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Can You Learn an Algorithm? Generalizing from Easy to Hard Problems with Recurrent Networks](https://mlsanity.com/paper/2106.04537/) (2021)
- [Recurrent Relational Networks](https://mlsanity.com/paper/1711.08028/) (2017)
- [Grokking at the Edge of Numerical Stability](https://mlsanity.com/paper/2501.04697/) (2025)
- [Reverse engineering recurrent networks for sentiment classification reveals line attractor dynamics](https://mlsanity.com/paper/1906.10720/) (2019)
- [Path Independent Equilibrium Models Can Better Exploit Test-Time Computation](https://mlsanity.com/paper/2211.09961/) (2022)
- [Lyapunov spectra of chaotic recurrent neural networks](https://mlsanity.com/paper/2006.02427/) (2020)
- [On Lyapunov Exponents for RNNs: Understanding Information Propagation Using Dynamical Systems Tools](https://mlsanity.com/paper/2006.14123/) (2020)
- [Non-normal Recurrent Neural Network (nnRNN): learning long time dependencies while improving expressivity with transient dynamics](https://mlsanity.com/paper/1905.12080/) (2019)
- [Sudoku-Bench: Evaluating creative reasoning with Sudoku variants](https://mlsanity.com/paper/2505.16135/) (2025)
- [Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning](https://mlsanity.com/paper/2605.21488/) (2026)
- [Are Your Reasoning Models Reasoning or Guessing? A Mechanistic Analysis of Hierarchical Reasoning Models](https://mlsanity.com/paper/2601.10679/) (2026)
- [Think Shallow, Solve Deep: Controlling Recurrent Dynamics for Reliable Test-Time Depth](https://mlsanity.com/paper/2608.18222/) (2026)
- [On Logical Extrapolation for Mazes with Recurrent and Implicit Networks](https://mlsanity.com/paper/2410.03020/) (2024)
- [Fractal basins trap latent reasoning](https://mlsanity.com/paper/2609.04963/) (2026)

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

- [Resample the Answer: Where to Spend Test-Time Compute in Recursive Reasoning](https://mlsanity.com/paper/2610.16427/) (2026)

## Abstract (arXiv)

Recursive models can continue updating their latent states beyond their nominal inference budget, so an incorrect output at that budget does not show whether computation is unfinished or has entered a persistently unsuccessful regime. We study the dynamics of completion in attention- and MLP-based Tiny Recursive Models (TRMs) on 1,000 hard Sudoku puzzles. Extending recurrence from the nominal 16 steps to 512 steps increases cumulative exact-solve accuracy from 59.2% to 87.5% for the attention model and from 74.4% to 91.9% for the MLP model, solving more than two-thirds of the puzzles unsolved in the nominal budget. Across both architectures, latent-state motion drops sharply after the first exact solution. Completed states are typically locally contractive along the trajectory direction, even though the same local Jacobian retains strongly expanding directions. We characterize this phenomenon as trajectory-conditioned anisotropic stability. Perturbation experiments confirm this directional stability across both models. The multi-step fate of the maximally expanding direction differs: it is absorbed within 16 steps in the attention model but persists longer in the MLP model. The anisotropic-stability pattern also holds for a second attention checkpoint. Together, these results distinguish nominal-budget failure from completed computation and identify a common dynamical signature of completion across two recurrent architectures.