Hidden in the Nullspace: Behavioral Transfer Beyond the Training Interface

Subliminal learning shows that a student model can acquire a teacher's behavioral preference from apparently unrelated training data, even when the target behavior is never stated explicitly. While prior work establishes that such transfer can occur, the mechanism by which behavior is transmitted through an auxiliary training interface remains unclear. We study this question through the geometry of representations that are invisible, or weakly visible, to the declared carrier objective. In controlled two layer MLPs, we construct an auxiliary interface whose exact null space can be identified analytically. Although perturbations within this null space leave auxiliary logits unchanged, we find that a seed specific, input dependent component of the student reference residual lying in this space is strongly behaviorally active. In our controlled MNIST setting, removing this component reduces student accuracy to 11.3%, whereas a matched random null intervention retains 39.8% accuracy. Conversely, injecting the same component into the reference model raises accuracy from approximately 10% to 63.4%, while the matched random null control remains at 9.9%. The active residual is also highly compressible, a rank-8 approximation recovers 85.6% of the removal effect and 80.3% of the injection effect, increasing to 96.6% and 94.9%, respectively, at rank 16. We extend this geometric view to LLMs by replacing the exact auxiliary null space with empirically estimated low-sensitivity subspaces of the carrier objective, enabling analogous removal and injection interventions in transformer representations. Our findings support a mechanistic view of subliminal learning in which behavioral information can occupy representational degrees of freedom that are largely invisible to the objective through which training is observed.