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.
In transformer neural networks, discrete concepts reside as vectors inside high dimensional activation manifolds. When trained, linear combinations form consistent geometric relations. Subtracting Man from King extracts a royalty direction. Adding this offset to Woman projects directly toward Queen, closing the parallelogram with high cosine similarity. Selecting activation steering or adjusting alpha shifts states along a one-dimensional behavioral axis, modifying outputs directly without retraining the model.