TeCES: Collaborative Geometric Knowledge Representation Framework under Evolving Fact Snapshots

Existing knowledge graph completion research is gradually shifting from representing logical semantics of static facts to modeling evolving semantics of temporal facts, yet lacks collaborative modeling of both within a unified framework. To this end, we use concept of snapshots to decompose fact features into two complementary mechanisms: (a) intra-snapshot semantic coupling, where entities and relations exhibit snapshot-specific meanings through multidimensional interactions; (b) trans-snapshot evolutionary synergy, where relations between entities evolve across snapshots and manifest varying states. These snapshot mechanisms jointly reveal underlying logic of facts. To track them, we propose TeCES, a framework for high-fidelity modeling of evolving snapshots. TeCES embeds facts into a 2-grade geometric algebra (GA) system to capture complex semantics via multilevel structures. Temporal information is attached to each entity for mapping into snapshot spaces, while relations and timestamps are reconfigured into composite GA representations. Geometric products enable multidimensional interactions, revealing relation state changes over time. Lastly, the head entity at each snapshot combines with fused temporalrelational representation via geometric product to approximate the target tail entity at multiple levels. Overall, TeCES supports joint modeling of evolving snapshots within a lightweight GA system and significantly outperforms SOTA models on six benchmarks.