RoPE attention is an exact forward-pass gradient step with softmax intact

We derive exact local responses for attention interventions, allowing candidate edits to be scored from a cached baseline and one backward pass. The starting point is the RoPE derivative $\partial_p z(p) = A z(p)$: its integral gives the finite positional displacement, which we carry through the softmax without linearising either rotation or normalisation. The resulting predictions achieve 95.36--96.52% sign accuracy across 92,160 executed positional edits on 768 held-out prompt sets, reducing answer-margin MAE by 73.6--82.5% against the positional Jacobian and by 36.2--50.9% against zero. For simultaneous key and value edits, the same divided-difference calculus isolates the interaction term $C_{KV} = \sum_j (p'_j - p_j)\,\varepsilon_j$, which is omitted by adding separate attributions. Retaining it reduces downstream margin MAE by more than a factor of nine in every setting of a 5,120-intervention sweep across two Qwen sizes, two tasks, and multiple layers; reductions against a quadratic interaction correction are 75.9--98.5%. Exactness concerns the edited attention write; downstream predictions contract that response with a baseline gradient and are evaluated by native execution. The calculus also yields a KL certificate for local approximation error, an exact query-conditioned gradient-step representation whose curvature identifies attention-preserving query directions, and minimum-norm query control. Sparse evaluation supports candidate ranking and cache decisions under explicit local distortion criteria.

Rotary Position Embedding (RoPE) encodes token positions by rotating each two-dimensional channel of the query and key vectors at a channel-specific frequency, making the attention logits invariant to a common shift of positions. However, this rotation is periodic, and it leads to position aliasing where relative positions separated by a full rotation period become hard to tell apart. To address this, we propose WavePrune, which restricts each channel to its first rotation period. We show that it removes the distractions in attention maps created by position aliasing and improves overall long-context performance. Specifically, WavePrune raises the HELMET score on four of five models we test without any extra tuning (e.g., 35.7 -> 40.0 on Qwen3-8B). When pretraining models from scratch, WavePrune also achieves lower validation loss at extrapolated lengths than pretraining without it. Because WavePrune restricts each channel to a sliding window, it induces a fine-grained sparsity that our hardware-aligned CUDA kernels exploit for 1.15x prefill and 1.24x decoding speedups over FlashAttention-2 at 32K context. Together, these results show that RoPE's periodic structure, widely regarded as essential, is largely redundant beyond the first rotation period.

Rotary Positional Encodings (RoPE) are currently the most popular positional encodings used in modern language models. RoPE rotates two-dimensional chunks of query and key vectors, operating as a function of their relative positional offset. The position-wise rates of rotation in RoPE typically follow a geometric sequence specified by a fixed base-frequency hyperparameter. Prior work has improved performance by either increasing this parameter to slow rotation or by applying RoPE to only a subset of QK dimensions. In this work we modify RoPE by learning a scalar per frequency, treating frequencies as learnable parameters rather than hyperparameters. We validate Learned RoPE by training a ladder of language models from scratch, ranging from 52M to 2.5B parameters. We observe and analyze the emergence of a high-norm, positional LeRoPE band. LeRoPE consistently outperforms RoPE and partial RoPE across all scales, with RoPE requiring 3.4% more compute (FLOPs) to match LeRoPE at the largest scale.