AI サービスが一時的に利用できないため、復旧後に翻訳を補完します。
[Submitted on 1 Oct 2026] Title:Counterexample Generation via Per-Theorem Symbolic Verifiers: When Imitation Hurts and Reinforcement Repairs View a PDF of the paper titled Counterexample Generation via Per-Theorem Symbolic Verifiers: When Imitation Hurts and Reinforcement Repairs, by Omar Farouk Zouak and 4 other authors View PDF HTML (experimental) Abstract:Large language models often solve a theorem forward yet fail to disprove a closely related false one: a falsification gap that supervised fine-tuning does not close and can actively worsen. We frame counterexample generation as constrained witness emission against a deterministic per-theorem Python verifier, and release SymCE, a corpus of 4,707 false undergraduate-algebra and real-analysis conjectures, each paired with executable verifiers. The verifier also serves as the reward function, making SymCE a training environment. Training Qwen3-4B with SFT followed by GRPO under this oracle reveals an imitation trap: counterexample-only SFT collapses true-theorem recognition from 0.27 to 0.00, while RLVR with a sparse outcome-only reward repairs this and exceeds the base, to 0.66. The collapse replicates across four seeds and on Gemma-3-4B. Sparse and dense rewards yield statistically indistinguishable in-domain success yet diverge by 33 points on a held-out calibration probe, a dissociation we trace to the partial-credit term. Our 4B model outperforms every evaluated 7B open-weights math specialist, remains competitive with six frontier commercial APIs, and transfers under unchanged prompting to GSM8K, MATH-500 and MMLU-college-math. A human audit of 177 verifier decisions finds 97.7% accuracy. Code, data, verifier modules and annotations: this https URL. Comments: Accepted at EMNLP 2026 Findings Subjects: Computation and Language (cs.CL); Artificial Intelligence (cs.AI) Cite as: arXiv:2610.02444 [cs.CL] (or arXiv:2610.02444v1 [cs.CL] for this version) https://doi.org/10.48550/arXiv.2610.02444 arXiv-issued DOI via DataCite (pending registration) Submission history From: Omar Farouk Zouak [view email] [v1] Thu, 1 Oct 2026 20:13:14 UTC (105 KB) Full-text links: Access Paper: View a PDF of the paper titled Counterexample Generation via Per-Theorem Symbolic Verifiers: When Imitation Hurts and Reinforcement Repairs, by Omar Farouk Zouak and 4 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.CL new | recent | 2026-10 Change to browse by: cs cs.AI References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)