# LayerNorm as Implicit Gain Control in Looped Transformers

arXiv: https://arxiv.org/abs/2607.10681 · HTML: https://mlsanity.com/paper/2607.10681/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 11 references that are in the map.
- First posted on arXiv 12 Jul 2026.

## Builds on (most cited first, 10 of 10)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [Adaptive Computation Time for Recurrent Neural Networks](https://mlsanity.com/paper/1603.08983/) (2016)
- [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)
- [Reasoning with Latent Thoughts: On the Power of Looped Transformers](https://mlsanity.com/paper/2502.17416/) (2025)
- [Parcae: Scaling Laws For Stable Looped Language Models](https://mlsanity.com/paper/2604.12946/) (2026)
- [Loop, Think, & Generalize: Implicit Reasoning in Recurrent-Depth Transformers](https://mlsanity.com/paper/2604.07822/) (2026)

## Abstract (arXiv)

In pre-LayerNorm looped transformers, LayerNorm inside the recurrent block acts as an implicit gain controller: by coupling the block's local Lipschitz constant inversely to the activation scale, it renders the recurrence Jacobian non-normal -- asymptotically contractive at every verified fixed point even where its operator norm exceeds 1 -- so the true stability budget is the spectral margin, not an operator-norm bound. That margin depletes as the carry ρ → 1, and a minority of initializations never converge to a fixed point at all, so the diagonal carry constraint ρ(barA) < 1 is necessary but not sufficient for convergence of the full recurrence. Training experiments across six tasks, including a controlled ablation, reveal that the linear carry is not the depth-memory mechanism: gradient descent routes memory through the block's more expressive nonlinear recurrence and leaves the stability-constrained carry at rest -- the carry's role is stabilization, not memory. We characterize the boundary of this claim: on tasks with axis-aligned per-channel structure, gradient descent does recruit the carry. All results are derived analytically and verified in a from-scratch, CPU-scale implementation; verification at larger scale is needed.