Abstract: Video moment retrieval aims to localize a query-relevant temporal segment in an untrimmed video. Existing methods typically perform video-text alignment in Euclidean space, where representations are embedded in a flat geometry. However, accurate moment localization requires organizing query-relevant evidence across multiple grounding states, where target regions, boundary transitions, and contextual distractors exhibit distinct temporal structures and relevance patterns. Such Euclidean representations lack an explicit geometry to jointly capture these relationships, motivating us to explore structured temporal evidence organization from a geometric perspective. Specifically, we propose Hyperbolic Temporal Evidence Organization (HTEO), a geometric framework that organizes query-aware temporal memory in hyperbolic space. HTEO maps temporal representations into a Poincaré ball and organizes temporal evidence through Radial Evidence Organization (REO) and Orthogonal Angular Refinement (OAR). REO adjusts temporal tokens' radial positions to encode the structural differences among temporal evidence, while OAR enhances query-conditioned semantic alignment through orthogonal rotation. The organized representation is then projected back to the Euclidean tangent space and injected into the decoder through a lightweight residual memory update, enabling hyperbolic geometric cues to be incorporated into standard Transformer decoders while preserving the original cross-modal memory. Furthermore, we introduce radial structure regularization and angular relevance supervision to explicitly guide temporal evidence organization and query-conditioned semantic alignment, respectively. Extensive experiments and analyses across multiple benchmarks demonstrate that HTEO improves temporal localization while providing an interpretable geometric organization of temporal evidence.