STEER: Semantic Transformation for Embedding-space Exploration in Retrieval — An Honest Evaluation of Seventeen Algebraic Operations

What happens when you algebraically transform a query embedding before retrieval? We investigate this question systematically, evaluating seventeen algebraic operations across six sentence transformer architectures and five BEIR benchmarks. We present STEER (Semantic Transformation for Embedding-space Exploration in Retrieval), which formalizes algebraic retrieval as R(q, D) = argmax sim(T(q), d), where T is an algebraic transformation and standard RAG is the identity case. Key findings: (1) Isotropy governs efficacy — MiniLM (isotropy 0.13) improves, E5 (0.81) degrades. (2) Addition is the primary operation — 3.4x less degradation than NLERP at alpha=0.5. (3) Cross-domain rotation achieves 40% domain shift in top-10 results, enabling analogical retrieval. (4) int8 quantization is safe (r=0.9999, 0.4 degree angular deviation). (5) NLERP approximately equals SLERP (nDCG difference < 0.001 in 12/12 combinations). Null findings: SVD intersection, Conceptors (40 aperture configs), Mobius/Wasserstein/Gumbel operations all fail or are neutral. The systematic exploration supports a key finding: Euclidean operations are the natural operations for retrieval in normalized embedding spaces. Part of the Artisanal Intelligence Program — a Lakatosian research program on human-AI interaction.