Semiclassical Geometry of Attention Switching

When one attention head is pushed toward a competing evidence span, do baseline quantities identify useful directions for controlling the answer margin? For a fixed-key, two-branch logit shift, the head's output change splits exactly into a branch-switching term and a total-evidence-mass term (a specialization of softmax differentiation, used as coordinates). Switching geometry identifies a task-enriched control subspace. Across eight Qwen3-8B examples (archived custom path), the three-dimensional switching normal subspace retains a mean $\text{ControlRetained}\%$ of squared task-gradient norm versus $\text{ControlRandomRetained}\%$ for the median matched-random subspace (paired difference $0.260$, 95% interval $[0.141,0.378]$; uses each example's gradient). Baseline-only prediction holds. On 50 ConflictQA examples and eight Gemma heads, finite attention recomputation from the baseline gives pooled $R^2=\text{FSRtwo}$ versus $\text{LinearRtwo}$ for local linearization. In an exploratory native Qwen3-8B cohort (a 32-fact prefix of a 200-fact ConflictQA test set, path gate bypassed), a predictor frozen from baseline forward and backward passes achieves small median local-regime error ($|\alpha |\le .05$) for all three steering arms—switching, fixed PCA basis, and static mean direction: geometrically grounded directions are collectively useful. A switching-specific advantage is not established. On those 32 facts, switching minus PCA at $\alpha =.20$ is $-0.020$ $[-0.068,0.014]$ and switching minus static is $-0.020$ $[-0.085,0.035]$ (not a test of equivalence); the 200-fact cohort favors switching over the fixed basis only. Switching geometry thus locates useful control directions but is not a privileged axis; path and attribution limits are in Section sec:discussion.