Attention Manifolds: Steering or Blocking Language Models by Editing Learned B-Spline Surfaces

arXiv:2610.00257v1 Announce Type: new Abstract: In standard transformer attention, a source token sends the same value vector to every receiver. The query determines \emph{how much} to attend but not \emph{what} to extract. This work introduces \textbf{attention manifolds}: learned 2D B-spline surfaces $S_d(q_d, k_d)$ that modulate each value dimension based on the query-key interaction. Each surface is a tensor-product cubic B-spline initialized to zero, preserving pretrained behavior. Applied to LLaMA 3.2-1B-Instruct and 3B-Instruct, attention manifolds reduce WikiText-2 validation perplexity by 2--2.5 points with 0.3% parameter overhead. Across 112 diverse prompts, surfaces change greedy-decoded output for 69% (1B) to 83% (3B) of cases, with the strongest effects on ambiguous and polysemous inputs (94--100% change rate). The surfaces improve output quality: correcting factual errors (\emph{the CAP theorem has three main components''} $\to$ \emph{ it is impossible to guarantee all three''}), increasing precision (\emph{impossible to know certain properties''} $\to$ \emph{ impossible to know both position and momentum''}), and adding specificity (a generic quote $\to$ an attributed Saint Augustine citation, consistently at both scales). The learned surfaces are also mechanically editable: inverting a layer's coefficients changes greedy output for 9/10 prompts (KL~0.010), providing a geometric mechanism for model steering. Setting surface coefficients to $-1$ creates ``attention walls'' that block value flow through specific dimensions. In a preliminary experiment, a layer-wide wall redirects an explosive-device prompt from specific instructions to general educational content, suggesting a path toward safety-oriented manifold shaping.