Uncovering Symmetry Transfer in Large Language Models via Layer-Peeled Optimization

The difficulty of training large language models (LLMs), together with their ubiquity, raises the threat of stegomalware, where malicious payloads are embedded into model weights. Recent work has demonstrated the use of permutation symmetry in model weights to mitigate these threats, but failed to show neutralization of stegomalware across all weights for LLMs. In this paper, we demonstrate the full potential of behavior-preserving symmetries as a defense against stegomalware, as well as the risks these symmetries pose when exploited by attackers. For stegomalware neutralization, we improve upon previous work, demonstrating that it is possible to select permutations which displace all model parameters. This contrasts with previous methods which left a significant percentage of weights unaltered in LLMs. When used in an attack, we show that permutation symmetries can encode malware into the weights of a model in a way that is theoretically lossless, requires no retraining after encoding, and needs no payload-specific information in the extraction script---a combination of characteristics not previously seen in any single method. While theoretically lossless, permutation can in practice alter model behavior due to the accumulation of numerical error. We therefore quantify the loss in model performance associated with applying these methods, for both attack and defense, showing it to be minimal.

While large language models (LLMs) are trained purely on textual data, prior work has shown that their internal representations can exhibit rich geometric structure in embedding space. Building on this line of work, we investigate whether such structure is similar to human perceptual organisation across different domains (e.g., color, pitch, emotion, and taste). Specifically, we study the layer-wise emergence of intrinsic geometrical structure corresponding to perceptual modalities within the residual streams of multiple open-weight transformer architectures. Our results reveal three key findings. First, we observe the emergence of layer-wise geometric structure across multiple perceptual domains, despite the absence of any direct perceptual supervision during training. Second, these perceptual domains exhibit distinct emergence profiles, with both geometric structure and its alignment with human baselines following domain- and model-specific trajectories across depth. Third, this emergence follows a consistent representational trajectory: geometry is weak or diffuse in early layers, becomes progressively organised in intermediate layers, and is attenuated in later layers, suggesting that perceptual geometry arises transiently as part of the model's internal transformation pipeline. This provides new insight into how and where human-like perceptual geometry arises in LLMs, offering a principled pathway for mechanistic analysis of internal representations.

A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symmetries differ from those of general matrix layers: embedding and LM head matrices, SwiGLU MLP projections, and MoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, and left-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whose equivariance matches its symmetry group. We corroborate this principle through pre-training experiments on dense and sparse MoE language models, including Qwen3-0.6B-style, Gemma 3 1B-style, OLMoE-1B-7B-style, and downsized gpt-oss architectures. Across these experiments, symmetry-compatible update rules consistently improve final validation loss, reduce expert load imbalance in sparse MoE models, and in several cases control final vocabulary-logit growth, improve router stability, and overall training stability over the corresponding AdamW updates.