Multi-hop retrieval-augmented generation (RAG) requires evidence that remains relevant to a query while introducing enough novelty to bridge semantic gaps. Dense retrieval tends to concentrate on semantically similar documents, whereas graph-based alternatives often depend on costly Large Language Model (LLM) entity extraction and may propagate through noisy connections. We introduce Geometric Gain Graph RAG (G$^3$RAG), a document-only framework whose offline graph construction uses no LLM calls or generated tokens. G$^3$RAG assigns each edge a geometric gain score, $\cosθ\cdot \sinθ$, that jointly captures directional consistency and orthogonality between document representations. A density-aware topological penalty suppresses highly connected hubs, while single-step controlled diffusion expands from filtered query seeds toward complementary evidence. We evaluate G$^3$RAG on MusiQue, 2WikiMultiHopQA, and HotpotQA using Nv-embed-v2 and Qwen3-8B-embed. G$^3$RAG obtains the best average F1 and answer-document hit rate among the evaluated graph-based baselines in both embedding settings, with gains of up to 4.26 F1 points in average performance and 5.76 points on MusiQue. It also removes the graph-construction token cost incurred by entity-based graph methods. These results show that geometric structure can support efficient multi-hop evidence discovery without LLM-based graph construction. Code is available at https://anonymous.4open.science/r/G3RAG-99D9/

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