Geometry
What the low-rank alignment control surface looks like.
Preference / alignment post-training concentrates its weight update. Measured by
stable rank — ‖ΔW‖_F² / ‖ΔW‖₂², a soft, continuous count of effective
directions — the update to the attention QKV projection is low and, strikingly,
flat: it does not grow with model width.
A task-intrinsic stable-rank floor
The lazy-rudder study trains DPO LoRA adapters across a width sweep and finds the QKV-update stable rank sits at a low constant — the task-intrinsic floor ≈ 3.65 on the GPT-NeoX (Pythia) family. The rank is set by the preference-learning task, not by parameter count.
It replicates across architecture (LRS1)
LRS1 re-measures the floor on Qwen2.5 — a Llama-style architecture (split q/k/v projections, grouped-query attention, gated SwiGLU). The floor holds:
| Model | d_model | QKV stable rank |
|---|---|---|
| Qwen2.5-0.5B | 896 | 3.4351 |
| Qwen2.5-1.5B | 1536 | 3.9625 |
| Qwen2.5-3B | 2048 | 3.9387 |
Across a 4× width range the stable rank stays inside a spread of just 0.5274 — verdict REPLICATES_FLAT_FLOOR. The Qwen2.5 floor agrees with the Pythia reference to within ≈ 0.13. The low-rank alignment control surface is architecture-robust and scale-invariant.
That is the geometry. The natural next question — answered on the Function page — is what a surface this compact actually does.
Source: cross-check/preregistry/lrs1_srank_scaling_qwen25_2026-05-17/; the
lazy-rudder paper for the Pythia sweep.