{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06127",
  "author_total": 2,
  "authors": [
    {
      "name": "Xingting Wang"
    },
    {
      "name": "Min Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46rbxexq8z775.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06127",
  "fulltext_url": null,
  "id": "46rbxexq8z775",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06127",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.IR"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "1702|Logic, Reasoning, and Knowledge"
    ]
  ],
  "openalex": "W7212050899",
  "pdf_url": "https://arxiv.org/pdf/2609.06127",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46rbxexq8z775.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06127",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06127"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "PAGR: Proof-Carrying Algebraic-Geometric Retrieval: A Quiver-, Provenance-, and Sheaf-Theoretic Framework for Grounded LLM Retrieval",
  "title_text": "PAGR: Proof-Carrying Algebraic-Geometric Retrieval: A Quiver-, Provenance-, and Sheaf-Theoretic Framework for Grounded LLM Retrieval",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/46rbxexq8z775",
  "venue": "arXiv",
  "versions": []
}