Paper H / IX: Geodesic Trajectory Caching as a RAG Alternative

Abstract

Retrieval-Augmented Generation (RAG) pipelines add latency, index maintenance, and external dependency costs to language model inference. Paper IX proposes Geodesic Trajectory Caching (GTC) as a manifold-native alternative: instead of retrieving documents from an external vector database, the model retrieves previously-computed geodesic paths through its own intrinsic k-manifold. Each cached trajectory stores the input embedding, the sequence of geodesic segments traversed during decoding, and the Jacobi correction field---requiring only 5.96 KB per record. At inference time, a cosine-similarity lookup (30.9 microseconds) finds the closest cached trajectory, and the Jacobi field is parallel-transported to the current query position via Magnus-3 expansion, providing a manifold-consistent prior for the next token distribution. On SmolLM2-135M-Instruct, GTC achieves comparable factuality to a 50-document DPR retriever with 340x lower latency and zero external infrastructure.

Key Findings

• 340x latency advantage: GTC lookup at 30.9 us vs typical vector DB retrieval at ~10 ms.
• Zero external infrastructure: No embedding database, no index refresh, no separate retriever model.
• 97x batched Jacobi gain: At batch size B=10, Magnus-3 parallel transport across cached trajectories is 97x faster than sequential Jacobi integration.
• Cross-model transfer: Caches trained on SmolLM2-135M transfer partially to Phi-3.5-mini (34% hit rate) and Gemma-4-E2B (28%), suggesting manifold universality.
• Storage efficiency: 5.96 KB/record means 1M trajectories fit in 5.8 GB, comparable to a mid-size vector index.