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

arXiv: https://arxiv.org/abs/2609.06127 · HTML: https://mlsanity.com/paper/2609.06127/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Knowledge-graph and structured-knowledge RAG, one of 15 directions in Retrieval-augmented generation.
- Builds on 22 of its 61 references that are in the map.
- First posted on arXiv 5 Sep 2026.

## Placed in

- Retrieval-augmented generation / [Knowledge-graph and structured-knowledge RAG](https://mlsanity.com/rag/knowledge-graph-and-structured-knowledge-rag/)

## Builds on (most cited first, 22 of 22)

- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [From Local to Global: A Graph RAG Approach to Query-Focused Summarization](https://mlsanity.com/paper/2404.16130/) (2024)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space](https://mlsanity.com/paper/1902.10197/) (2019)
- [Enabling Large Language Models to Generate Text with Citations](https://mlsanity.com/paper/2305.14627/) (2023)
- [HippoRAG: Neurobiologically Inspired Long-Term Memory for Large Language Models](https://mlsanity.com/paper/2405.14831/) (2024)
- [Inductive Relation Prediction by Subgraph Reasoning](https://mlsanity.com/paper/1911.06962/) (2019)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Retrieval-Augmented Generation with Graphs (GraphRAG)](https://mlsanity.com/paper/2501.00309/) (2024)
- [GRAG: Graph Retrieval-Augmented Generation](https://mlsanity.com/paper/2405.16506/) (2024)
- [InGram: Inductive Knowledge Graph Embedding via Relation Graphs](https://mlsanity.com/paper/2305.19987/) (2023)
- [The Representation Theory of Neural Networks](https://mlsanity.com/paper/2007.12213/) (2020)
- [Knowledge Sheaves: A Sheaf-Theoretic Framework for Knowledge Graph Embedding](https://mlsanity.com/paper/2110.03789/) (2021)
- [Heterogeneous Sheaf Neural Networks](https://mlsanity.com/paper/2409.08036/) (2024)
- [Generalizing Knowledge Graph Embedding with Universal Orthogonal Parameterization](https://mlsanity.com/paper/2405.08540/) (2024)
- [What relations are reliably embeddable in Euclidean space?](https://mlsanity.com/paper/1903.05347/) (2019)
- [Convergence of Datalog over (Pre-) Semirings](https://mlsanity.com/paper/2105.14435/) (2021)
- [Quiver neural networks](https://mlsanity.com/paper/2207.12773/) (2022)
- [Knowledge Graph Completion with Mixed Geometry Tensor Factorization](https://mlsanity.com/paper/2504.02589/) (2025)
- [Proof-Carrying Certificates for LLM Pipelines: A Trust-Boundary Architecture](https://mlsanity.com/paper/2605.16407/) (2026)
- [NeSy-RAG: Neuro-Symbolic RAG for Explainable Question Answering](https://mlsanity.com/paper/2608.06292/) (2026)

## Abstract (arXiv)

Retrieval-augmented generation is usually formulated as a statistical information-retrieval problem. Graph-based variants add relational structure, but the mathematical status of that structure is often left underspecified. Three distinct questions tend to be conflated: which statements are certified as knowledge, which latent representations are useful for retrieval, and which multi-hop compositions are semantically admissible. We propose Proof-Carrying Algebraic-Geometric Retrieval (PAGR), a framework that separates these questions mathematically. Its symbolic layer is a many-sorted relational theory generated by a typed quiver, path equations, and positive Horn inclusions. A quiver representation assigns inner-product spaces to entity types and linear operators to relations. A cellular sheaf measures local-to-global consistency. Semiring provenance records derivations and supports machine-checkable certificates. The central principle is epistemic separation: learned geometry may rank and organize evidence, but cannot promote a hypothesis to certified ground truth. We show the certification criterion is invariant under arbitrary replacement of learned components. Further results include a conditional completeness bound, identification of the isometry group as the relevant symmetry for residual-based retrieval, a cohomological consistency diagnostic, and a bounded-bisimulation index for admissible-path expansion. PAGR is a mathematical architecture for separating where a system should look from what it is allowed to treat as knowledge.