PAGR: Proof-Carrying Algebraic-Geometric Retrieval: A Quiver-, Provenance-, and Sheaf-Theoretic Framework for Grounded LLM Retrieval

High-capacity encoders in retrieval-augmented generation (RAG) can let the query dominate the latent state, leaving retrieved evidence functionally irrelevant. We call this failure mode query dominance. To address it, we introduce \textbf{GRIP} (Grounded Reasoning via Information-Restricted Premises), which imposes capacity asymmetry: the decoder keeps full-dimensional access to the query, while retrieved evidence passes through a severe stochastic bottleneck. This forces the evidence channel to encode only the residual information unavailable from the query. Across five reasoning benchmarks, GRIP outperforms strong iterative baselines, cuts a query--latent mutual-information diagnostic by roughly 30$\times$ (14.8 $\to$ 0.47 bits), and reduces hallucination by 73%. Residual-alignment analysis further shows that the bottleneck output occupies subspaces less aligned with the query than baseline representations.

Retrieval-Augmented Generation can improve factuality by grounding answers in external evidence, but Agentic GraphRAG complicates what it means for citations to be faithful. In these systems, an agent explores a knowledge graph before producing an answer and a small set of citations. We frame citation faithfulness as a trajectory-level problem: final citations should not only support the answer, but also account for the graph traversal, structure, and visited-but-uncited entities that may influence it. Through controlled ablation experiments, we compare the effects of isolating, removing, and masking cited and uncited graph entities. Our results show that cited evidence is often necessary, as removing it substantially changes answers and reduces accuracy. However, citations are not sufficient, because accurate answers can also depend on uncited traversal context and surrounding graph structure. These findings suggest that citation evaluation in Agentic GraphRAG should move beyond source support toward provenance over the broader retrieval trajectory.

Multi-hop question answering in retrieval-augmented gener?ation (RAG) often benefits from retrieving beyond the few candidates that will finally be read: narrow retrieval can miss an indispensable hop, while expanded retrieval introduces topical distractors. This challenge is not tied to a particu?lar knowledge-base format. Candidate pools may come from standalone retrievers, standard RAG backends, or graph-based retrieval pipelines. What is needed is a query-aware selection layer that can use relational structure to filter candidates be?fore generation. PAGE-RAG addresses this setting by using a graph as a temporary selection structure, rather than assum?ing a graph-structured knowledge base. It builds a query-local graph over retrieved candidates, records why candidates are connected, and treats each connection as a support hypothe?sis rather than support itself. We identify the resulting failure mode as a connectivity-support gap: connected candidates do not necessarily support the answer. We propose PAGE-RAG, a Provenance-Aware Graph Evidence promotion method that scores candidate paths with relevance, source-tracing meta?data, specificity, hubness, noise, and coherence signals, and applies minimal sufficient selection to promote supporting facts into a compact reader context. PAGE-RAG can serve as a complete retrieval-to-reading pipeline, and the same promo?tion stage can be inserted after existing retrieval or RAG sys?tems without replacing their upstream retrieval logic. Across three multi-hop QA benchmarks under the same final bud?get, PAGE-RAG improves support F1 and answer F1 by 10.4 and 3.3 points on a weighted average over a strong retriever. As a plug-in, PAGE-RAG further improves all reported RAG backends, including reasoning-oriented, compression-based, graph-based, and document/chunk-level systems.