# Proof-Carrying Numbers (PCN): A Protocol for Trustworthy Numeric Answers from LLMs via Claim Verification

arXiv: https://arxiv.org/abs/2509.06902 · HTML: https://mlsanity.com/paper/2509.06902/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 26 references that are in the map.
- First posted on arXiv 8 Sep 2025.

## Builds on (most cited first, 18 of 18)

- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Survey of Hallucination in Natural Language Generation](https://mlsanity.com/paper/2202.03629/) (2022)
- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [FEVER: a large-scale dataset for Fact Extraction and VERification](https://mlsanity.com/paper/1803.05355/) (2018)
- [SelfCheckGPT: Zero-Resource Black-Box Hallucination Detection for Generative Large Language Models](https://mlsanity.com/paper/2303.08896/) (2023)
- [TabFact: A Large-scale Dataset for Table-based Fact Verification](https://mlsanity.com/paper/1909.02164/) (2019)
- [Hallucination is Inevitable: An Innate Limitation of Large Language Models](https://mlsanity.com/paper/2401.11817/) (2024)
- [Why Language Models Hallucinate](https://mlsanity.com/paper/2509.04664/) (2025)
- [Grammar-Constrained Decoding for Structured NLP Tasks without Finetuning](https://mlsanity.com/paper/2305.13971/) (2023)
- [FEVEROUS: Fact Extraction and VERification Over Unstructured and Structured information](https://mlsanity.com/paper/2106.05707/) (2021)
- [ClashEval: Quantifying the tug-of-war between an LLM's internal prior and external evidence](https://mlsanity.com/paper/2404.10198/) (2024)
- [SciFact-Open: Towards open-domain scientific claim verification](https://mlsanity.com/paper/2210.13777/) (2022)
- [LongCite: Enabling LLMs to Generate Fine-grained Citations in Long-context QA](https://mlsanity.com/paper/2409.02897/) (2024)
- [LLMs Will Always Hallucinate, and We Need to Live With This](https://mlsanity.com/paper/2409.05746/) (2024)
- [Deficiency of Large Language Models in Finance: An Empirical Examination of Hallucination](https://mlsanity.com/paper/2311.15548/) (2023)
- [AttributionBench: How Hard is Automatic Attribution Evaluation?](https://mlsanity.com/paper/2402.15089/) (2024)
- [Towards Verifiable Text Generation with Symbolic References](https://mlsanity.com/paper/2311.09188/) (2023)
- [Attribution, Citation, and Quotation: A Survey of Evidence-based Text Generation with Large Language Models](https://mlsanity.com/paper/2508.15396/) (2025)

## Abstract (arXiv)

Large Language Models (LLMs) as stochastic systems may generate numbers that deviate from available data, a failure known as numeric hallucination. Existing safeguards -- retrieval-augmented generation, citations, and uncertainty estimation -- improve transparency but cannot guarantee fidelity: fabricated or misquoted values may still be displayed as if correct. We propose Proof-Carrying Numbers (PCN), a presentation-layer protocol that enforces numeric fidelity through mechanical verification. Under PCN, numeric spans are emitted as claim-bound tokens tied to structured claims, and a verifier checks each token under a declared policy (e.g., exact equality, rounding, aliases, or tolerance with qualifiers). Crucially, PCN places verification in the renderer, not the model: only claim-checked numbers are marked as verified, and all others default to unverified. This separation prevents spoofing and guarantees fail-closed behavior. We formalize PCN and prove soundness, completeness under honest tokens, fail-closed behavior, and monotonicity under policy refinement. PCN is lightweight and model-agnostic, integrates seamlessly into existing applications, and can be extended with cryptographic commitments. By enforcing verification as a mandatory step before display, PCN establishes a simple contract for numerically sensitive settings: trust is earned only by proof, while the absence of a mark communicates uncertainty.