Towards Typed Program Synthesis of Geometric Scorers for Knowledge Graph Embeddings

Knowledge graph embedding (KGE) models are often distinguished by a short, hand-designed scoring formula and the geometric space in which that formula is interpreted. This couples two design decisions: \emph{which representation algebra to use} and \emph{which program should score a triple}. We recast their joint design as typed program synthesis. AlphaKGE defines an algebra-indexed, role-refined language whose programs compose distance, bilinear, relation-aware, and Clifford operations. Types jointly track algebra and dependence on the head, relation, and tail, so accepted programs are scalar, algebra-valid, and relation-complete. Search is replaceable: we instantiate MCTS/PUCT, canonicalize equivalent programs, and use one promotion/tuning contract. The type boundary also lets an LLM act as an untrusted proposer or prior; only well-typed programs reach evaluation. On four controlled small-graph benchmarks, a single synthesis configuration improves over a curated 20-scorer pool and fixed-template Clifford and block-bilinear search baselines. Ablations show that neither Clifford structure nor MCTS alone explains the result: the useful inductive bias is dataset dependent, and the real algebra is sufficient on one benchmark. These experiments are a proof of concept for treating geometry as a typed, searchable model component, not evidence of large-scale or cross-graph generalization.