[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
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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)
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