The attention interaction matrix QK⊤ contains two entangled computations: a skew-symmetric component that redistributes information between positions (routing) and a symmetric component that scales mutual relevance (filtering). We decompose 1776 heads across five pretrained transformers and find routing operating at low rank, well below the routing capacity allocated by the weight kernel. We introduce S-D attention as a diagnostic parameterization that disentangles routing from filtering by construction with guaranteed stability (Re(λ)≤0) and trains stably without layer normalization. When disentangled and unnormalized, routing self-organizes into a spectral cascade, effective rank 2 at the first layer, expanding with depth across six scales from 7M to 355M parameters. The cascade predicts where attention can be simplified: linearizing the first seven layers of 125M S-D attention costs <5% perplexity, whereas standard attention collapses under the same intervention. The linearizable region widens with depth. Replacing the first four layers with ELU+1 linear attention reaches within 1.4% of baseline at full head dimension. Cascade-allocated architectures trade attention parameters for perplexity (47%−65% fewer attention parameters at +3.9% to +8.4% PPL). The routing-filtering decomposition makes the spectral budget legible; the cascade makes it actionable.
Feed-forward networks hold two thirds of a transformer's non-embedding parameters, yet the architecture has not received a necessity test that controls parameters, compute, and depth at once. We pretrain attention-only decoder transformers (Simple Attention Networks, SANs) against standard transformers matched separately for parameter count, training FLOPs, and depth (2 to 48 layers), for up to 105B tokens at 6M to 87M parameters. Deleting feed-forward layers in place is costly: the standard transformer leads by 0.47 nats at matched depth and 0.26 nats at matched FLOPs. Reallocating the freed budget into attention depth closes the gap: at matched parameters the difference is 0.006 nats (0.27 percent of loss), reproducible to one part in ten thousand across seed pairs, shrinking across 5B, 30B, and 105B budgets, and holding near 0.02 nats across a 29x size range. Three measurements localize the remaining gap to parametric recall: attention-only models are better on context-grounded answers and worse where knowledge must come from weights. Weight spectra show why: routing matrices (Q/K) crystallize early, content matrices accumulate rank slowly, and removing feed-forward layers relocates this accumulation to the attention output projection. QK-normalization, not feed-forward layers or residual gating, keeps 48-layer attention-only stacks trainable. The deficit concentrates on low-context query prediction and localizes there entirely by the largest budget. A pre-registered test confirms the account: it predicts a 0.02 to 0.05 nat gap on knowledge-dense web text; a matched pair trained on fineweb-edu measures 0.040. Within the tested regime, attention does the rest.
Self-attention selects information freely across the sequence, but across depth, Transformers merely add each layer's output to the residual stream, so later layers cannot selectively reuse earlier-layer representations. Recent cross-layer methods improve this flow but operate on hidden states outside attention, adding state beyond the key-value cache at inference--a cost that becomes increasingly salient as modern LLMs compress the cache with grouped-query and multi-head latent attention. We introduce Depth-Attention, which performs this selection inside the attention module itself: before a layer attends over the sequence, its query attends over the keys of earlier layers at the same token position and mixes their values into the value that self-attention then reads. Because Depth-Attention reuses the standard attention queries, keys, and value-cache slots, storing depth-mixed values in place of the original values, it adds no parameters and introduces no persistent inference state beyond the standard key-value cache--the same cache size as a vanilla decoder and less than hidden-state-based cross-layer methods. On Qwen3-style decoders at 1.5B and 3B parameters, Depth-Attention attains the lowest perplexity and the highest average downstream accuracy, improving over the vanilla Transformer by up to 2.3 accuracy points and surpassing strong cross-layer baselines in perplexity and average accuracy, while adding under 0.01% extra arithmetic FLOPs and no additional persistent inference state. The gains hold from 360M to 3B parameters and extend to looped Transformers.
Standard transformer architectures apply a single attention mechanism uniformly across all tokens and sequence positions, irrespective of local context or computational budget. We propose Meta-Attention, a framework that dynamically routes each token to the most appropriate attention strategy -- full softmax attention, linear (kernel) attention, or sliding-window local attention -- via a Bayesian Meta-Controller. Unlike prior routing approaches that use deterministic or prior-free learned routing, the Meta-Controller treats per-token mechanism selection as posterior inference under a compute-aware Dirichlet prior: routing weights are the output of an amortised variational posterior q(alpha | x_t; phi) trained with an Evidence Lower Bound (ELBO) objective that jointly encodes task performance and attention-mechanism cost. This design produces principled routing uncertainty estimates that govern the soft-to-hard routing transition, mitigates routing collapse without ad hoc load-balancing losses, and yields better compute-performance trade-offs than deterministic or prior-free learned routing at negligible overhead. Phase 1 empirical results on a Tiny LM benchmark confirm core predictions: the Bayesian controller's learned routing distribution implies a projected normalised FLOP cost of 25.1% under hard routing, vs. 59.3% for the prior-free baseline (-34.2 pp), and reduces routing entropy from 55.8% to 43.3% (-12.5 pp), demonstrating that the Dirichlet prior prevents routing collapse while the non-Bayesian model defaults to full attention. We present the Bayesian architecture, ELBO training objective, and a Phase 1 PyTorch prototype validating forward-pass correctness, posterior diversity, and a controlled ablation against a prior-free baseline. Code available at: https://github.com/KFEAL/meta-attention