# Rotate the Loop: Blockwise Conformality for Looped Transformers

arXiv: https://arxiv.org/abs/2610.14030 · HTML: https://mlsanity.com/paper/2610.14030/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 41 of its 45 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 30 of 41)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Universal Transformers](https://mlsanity.com/paper/1807.03819/) (2018)
- [On the Parameterization and Initialization of Diagonal State Space Models](https://mlsanity.com/paper/2206.11893/) (2022)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [SliceGPT: Compress Large Language Models by Deleting Rows and Columns](https://mlsanity.com/paper/2401.15024/) (2024)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Reasoning with Latent Thoughts: On the Power of Looped Transformers](https://mlsanity.com/paper/2502.17416/) (2025)
- [The Parallelism Tradeoff: Limitations of Log-Precision Transformers](https://mlsanity.com/paper/2207.00729/) (2022)
- [The Illusion of State in State-Space Models](https://mlsanity.com/paper/2404.08819/) (2024)
- [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)
- [Saturated Transformers are Constant-Depth Threshold Circuits](https://mlsanity.com/paper/2106.16213/) (2021)
- [Mixture-of-Recursions: Learning Dynamic Recursive Depths for Adaptive Token-Level Computation](https://mlsanity.com/paper/2507.10524/) (2025)
- [What Formal Languages Can Transformers Express? A Survey](https://mlsanity.com/paper/2311.00208/) (2023)
- [Parameter-Efficient Orthogonal Finetuning via Butterfly Factorization](https://mlsanity.com/paper/2311.06243/) (2023)
- [Unlocking State-Tracking in Linear RNNs Through Negative Eigenvalues](https://mlsanity.com/paper/2411.12537/) (2024)
- [Looped Transformers for Length Generalization](https://mlsanity.com/paper/2409.15647/) (2024)
- [On orthogonality and learning recurrent networks with long term dependencies](https://mlsanity.com/paper/1702.00071/) (2017)
- [DeltaProduct: Improving State-Tracking in Linear RNNs via Householder Products](https://mlsanity.com/paper/2502.10297/) (2025)
- [A Clockwork RNN](https://mlsanity.com/paper/1402.3511/) (2014)
- [Tighter Bounds on the Expressivity of Transformer Encoders](https://mlsanity.com/paper/2301.10743/) (2023)
- [Tunable Efficient Unitary Neural Networks (EUNN) and their application to RNNs](https://mlsanity.com/paper/1612.05231/) (2016)
- [Parcae: Scaling Laws For Stable Looped Language Models](https://mlsanity.com/paper/2604.12946/) (2026)
- [A Little Depth Goes a Long Way: The Expressive Power of Log-Depth Transformers](https://mlsanity.com/paper/2503.03961/) (2025)
- [The Expressive Capacity of State Space Models: A Formal Language Perspective](https://mlsanity.com/paper/2405.17394/) (2024)
- [Path Independent Equilibrium Models Can Better Exploit Test-Time Computation](https://mlsanity.com/paper/2211.09961/) (2022)
- [On Expressive Power of Looped Transformers: Theoretical Analysis and Enhancement via Timestep Encoding](https://mlsanity.com/paper/2410.01405/) (2024)

## Abstract (arXiv)

Looped transformers reuse one shared block across many recurrent steps, so depth becomes a quantity that can be spent at runtime rather than a stack of distinct layers. Those extra steps help only if a write made on one recurrent step is still readable on later ones, and that depends on the linear map applied to the residual between recurrent steps. We show that whether the loop learns to track state is a geometric question about the form of this map, and we propose blockwise conformal injections: the residual splits into coordinate-aligned groups and rotates inside each one, so every group has its own timescale while the geometry stays exact within it. Our experiments show a performance increase on a demanding form of state tracking - last-token accuracy when the rest of the prefix is already correct - and on language modelling, where a signal written earlier has to be retrieved at the end of the sequence.