Projection: PCA 2D
In high-dimensional space, relations are encoded as linear direction vectors: vec("Man") → vec("King") is nearly identical in orientation and magnitude to vec("Woman") → vec("Queen"). Neural networks learn linear representations because matrix multiplication naturally isolates orthogonal semantic factors.
Behaviors such as "honesty," "refusal," or "sycophancy" are represented as 1D direction vectors in transformer activation space. Injecting an activation steering vector h' = h + α·v directly shifts outputs along that behavioral line without retraining the model.
When learning modular arithmetic (e.g. (a + b) mod p), neural networks exhibit "grokking", transitioning from memorization to discovering Fourier features where numbers coil into circular periodic manifolds and 3D helices.