All You Need Is Non-Commutative Words

Gradient updates on different data generally do not commute: training a language model on two data sources in opposite orders gives different weights, even with the same data and total exposure. Loss or benchmark deltas show that the models differ, not where. We ask whether this path dependence leaves a parametric training-history memory: a weight component that flips sign when the two sources are swapped, is localized in output space, changes the held-out loss gap between the two orders under targeted interventions, and reveals which trained model came from which order. For one small SGD step of size $η$ on each of sources $A$ and $B$, the weight difference $θ_{AB}-θ_{BA}$ is, to leading order, $η^2 b_{AB}$, where $b_{AB}=H_Bg_A-H_Ag_B$ is the Lie bracket of the two gradient fields at the base model. We define commutator memory by projecting the bracket through the logits into one score per vocabulary token; the scores sum to the bracket's prediction of the gap. The scores are localized: on three models, the same readout of the measured $θ_{AB}-θ_{BA}$, or of a bracket from disjoint batches, shares 82-99% of the original top-20 tokens, versus 35-49% for norm-matched random directions. They are causally actionable: in Qwen-3-4B SFT, downweighting the ten tokens with the largest predicted share of the gap closes a median 32% of the measured gap, while frequency-matched tokens with near-zero scores have almost no effect. The weights themselves carry the component: projecting the difference between the two trained models onto $b_{AB}$ identifies which came from which order in 92% of cases across four LLMs (chance 50%). Controlled tests also cover matched-batch DPO, a frozen-rollout GRPO-style objective, and an AdamW endpoint check. The memory is defined per source pair, not per example, and its projection on $b_{AB}$ decays with further training.

Fast matrix multiplication algorithms keep the product fixed and search for a cheaper way to evaluate it. We instead ask whether a Transformer's learned projections can use a different, cheaper product altogether. Building on an associative-algebra construction that replaces ordinary matrix multiplication with a sparser interaction table over the same weight blocks, we construct a family with quadratic arithmetic in the matrix dimension when the physical block size remains fixed, and derive finite-shape constraints for GPU execution. The construction is provably optimal for its bilinear rank by the Alder--Strassen bound and can be realized as row-typed rectangular projections compatible with causal masking and KV-cached decoding. We provide an empirical test of this approach by training two approximately 110M-parameter decoder-only Transformer LMs from the same recipe and 12.3B-token budget, differing only in their feed-forward layer: one uses ordinary dense matrix multiplication and the other uses the associative-algebra product. Across four prompt domains, the algebraic model achieves a 6.2--7.8% increase in end-to-end generation throughput, while obtaining lower scores on all three reported downstream metrics. We treat these results as a feasibility and trainability check for the proposed approach at small scale, leaving further investigation to future work.

Standard attention scales quadratically with sequence length. Efficient attention methods reduce this O(n^2) cost, but when retrofitted into pretrained models, they often degrade perplexity, downstream accuracy, or both. We introduce Focus, a method that learns which token pairs matter. Focus adds a small set of learnable centroids--as few as 148K parameters per layer--that act as gates: only token pairs belonging to the same centroid group attend to each other over long ranges. Focus is composable: it can be added to any pretrained model by training only the centroids while keeping all original weights frozen. Experiments show that composing Focus onto pretrained models yields zero degradation on downstream benchmarks across model sizes from 124M to 70B parameters and five attention architectures. Surprisingly, sparse Focus attention outperforms full attention at 124M scale (30.3 vs. 31.4 perplexity) and matches full attention when trained from scratch at 7B scale (13.82 vs. 13.89). Focus is also fast: top-k group membership gives a 2x speedup with better quality than the original pretrained model. Using our FlashAttention decomposition, Focus achieves an 8.6x speedup at 1M tokens without custom kernels.