Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.06902

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

Aivin V. Solatorio
View original source ↗Read source document ↗Export metadata ↓

Abstract

Large Language Models (LLMs) as stochastic systems may generate numbers that deviate from available data, a failure known as \emph{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 \textbf{Proof-Carrying Numbers (PCN)}, a presentation-layer protocol that enforces numeric fidelity through mechanical verification. Under PCN, numeric spans are emitted as \emph{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 \emph{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: \emph{trust is earned only by proof}, while the absence of a mark communicates uncertainty.

Explore related subjects

cs.CLcs.CRcs.DBcs.LG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Aivin V. Solatorio. 2025-09-08. Proof-Carrying Numbers (PCN): A Protocol for Trustworthy Numeric Answers from LLMs via Claim Verification. https://arxiv.org/abs/2509.06902

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

DA-Cramming: Enhancing Cost-Effective Language Model Pretraining with Dependency Agreement Integration

Pretraining language models is still a challenge for many researchers due to its substantial computational costs. As such, there is growing interest in developing more affordable pretraining methods. One notable advancement in this area is the Cramming technique (Geiping and Goldstein, 2022), which enables the pretraining of BERT-style language models using just one GPU in a single day. Building on this innovative approach, we introduce the Dependency Agreement Cramming (DA-Cramming), an efficient framework that integrates information about dependency agreements into the pretraining process. Unlike existing methods that leverage similar semantic information during finetuning, our approach represents a pioneering effort focusing on enhancing the foundational language understanding with semantic information during pretraining. We meticulously design a dual-stage pretraining work flow with four dedicated submodels to capture representative dependency agreements at the chunk level, effectively transforming these agreements into embeddings to benefit the pretraining. Extensive empirical results demonstrate that our method significantly outperforms previous methods across various tasks.

cs.CL↗
arXiv2026

BigO(Bench): Can LLMs Generate Code with Controlled Time and Space Complexity?

We introduce BigO(Bench), a novel coding benchmark designed to evaluate the capabilities of generative language models in understanding and generating code with specified time and space complexities. This benchmark addresses the gap in current evaluations that often overlook the ability of models to comprehend and produce code constrained by computational complexity. BigO(Bench) includes tooling to infer the algorithmic complexity of any Python function from profiling measurements, including human- or LLM-generated solutions. BigO(Bench) also includes of set of 3,105 coding problems and 1,190,250 solutions from Code Contests annotated with inferred (synthetic) time and space complexity labels from the complexity framework, as well as corresponding runtime and memory footprint values for a large set of input sizes. We present results from evaluating multiple state-of-the-art language models on this benchmark, highlighting their strengths and weaknesses in handling complexity requirements. In particular, token-space reasoning models are unrivaled in code generation but not in complexity understanding, hinting that they may not generalize well to tasks for which no reward was given at training time.

cs.CL↗
arXiv2026

UR$^2$: Unify RAG and Reasoning through Reinforcement Learning

Large Language Models (LLMs) have shown strong capabilities through two complementary paradigms: Retrieval-Augmented Generation (RAG) for knowledge grounding and Reinforcement Learning from Verifiable Rewards (RLVR) for complex reasoning. However, existing attempts to unify these paradigms remain narrow in scope, typically limited to open-domain QA with fixed retrieval settings, which constrains generalization to broader domains. To address this limitation, we propose UR$^2$ (Unified RAG and Reasoning)), a general reinforcement learning framework that dynamically coordinates retrieval and reasoning. UR$^2$ introduces two key designs: a difficulty-aware curriculum that selectively invokes retrieval only for challenging instances, and a hybrid knowledge access strategy that combines domain-specific offline corpora with on-the-fly LLM-generated summaries. Together, these components mitigate the imbalance between retrieval and reasoning and improve robustness to noisy information. Experiments on open-domain QA, MMLU-Pro, medical, and mathematical reasoning tasks show that UR$^2$, built on Qwen-2.5-3/7B and LLaMA-3.1-8B, consistently outperforms existing RAG and RL baselines, and achieves performance comparable to GPT-4o-mini and GPT-4.1-mini on several benchmarks. Code, models, and data are available at https://github.com/Tsinghua-dhy/UR2.

cs.CL↗