Dynamic streams from news feeds, social media, sensor networks, and financial markets challenge static RAG frameworks. Full-scale indices incur high memory costs; periodic rebuilds introduce latency that undermines data freshness; naive sampling sacrifices semantic coverage. We present Streaming RAG, a unified pipeline that combines multi-vector cosine screening, mini-batch clustering, and a counter-based heavy-hitter filter to maintain a compact prototype set. We further prove an approximation bound $E\[R(K\_t)] \ge R^\* - L \Delta$ linking retrieval quality to clustering variance. An incremental index upsert mechanism refreshes prototypes without interrupting queries. Experiments on eight real-time streams show statistically significant gains in Recall\@10 (up to 3 points, p < 0.01), end-to-end latency below 15 ms, and throughput above 900 documents per second under a 150 MB budget. Hyperparameter sensitivity analysis over cluster count, admission probability, relevance threshold, and counter capacity validates default settings. In open-domain question answering with GPT-3.5 Turbo, we record 3.2-point gain in Exact Match and 2.8-point gain in F1 on SQuAD; abstractive summarization yields ROUGE-L improvements. Streaming RAG establishes a new Pareto frontier for retrieval augmentation.