Judging What We Cannot Solve: A Consequence-Based Approach for Oracle-Free Evaluation of Research-Level Math

Guijin Son, Donghun Yang, Hitesh Laxmichand Patel, Hyunwoo Ko, Amit Agarwal, Sunghee Ahn, Kyong-Ha Lee, Youngjae Yu
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:114413-114435, 2026.

Abstract

Recent progress in reasoning models suggests that generating plausible attempts for research-level mathematics may be within reach, but verification remains a bottleneck, consuming scarce expert time. We hypothesize that a meaningful solution should contain enough method-level information that, when applied to a neighborhood of related questions, it should yield better downstream performance than incorrect solutions. Building on this idea, we propose Consequence-Based Utility, an oracle-free evaluator that scores each candidate by testing its value as an in-context exemplar in solving related yet verifiable questions. Our approach is evaluated on an original set of research-level math problems each paired with one expert-written solution and nine LLM-generated solutions. Notably, Consequence-Based Utility consistently outperforms reward models, generative reward models, and LLM judges on ranking quality. Specifically, for GPT-OSS-120B it improves Acc@1 from 67.2 to 76.3 and AUC from 71.4 to 79.6, with similarly large AUC gains on GPT-OSS-20B (69.0 to 79.2). Furthermore, compared to LLM-Judges, it also exhibits a larger solver–evaluator gap, maintaining stronger correct–wrong separation even on instances the underlying solver often fails to solve.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-son26a, title = {Judging What We Cannot Solve: A Consequence-Based Approach for Oracle-Free Evaluation of Research-Level Math}, author = {Son, Guijin and Yang, Donghun and Patel, Hitesh Laxmichand and Ko, Hyunwoo and Agarwal, Amit and Ahn, Sunghee and Lee, Kyong-Ha and Yu, Youngjae}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {114413--114435}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/son26a/son26a.pdf}, url = {https://proceedings.mlr.press/v306/son26a.html}, abstract = {Recent progress in reasoning models suggests that generating plausible attempts for research-level mathematics may be within reach, but verification remains a bottleneck, consuming scarce expert time. We hypothesize that a meaningful solution should contain enough method-level information that, when applied to a neighborhood of related questions, it should yield better downstream performance than incorrect solutions. Building on this idea, we propose Consequence-Based Utility, an oracle-free evaluator that scores each candidate by testing its value as an in-context exemplar in solving related yet verifiable questions. Our approach is evaluated on an original set of research-level math problems each paired with one expert-written solution and nine LLM-generated solutions. Notably, Consequence-Based Utility consistently outperforms reward models, generative reward models, and LLM judges on ranking quality. Specifically, for GPT-OSS-120B it improves Acc@1 from 67.2 to 76.3 and AUC from 71.4 to 79.6, with similarly large AUC gains on GPT-OSS-20B (69.0 to 79.2). Furthermore, compared to LLM-Judges, it also exhibits a larger solver–evaluator gap, maintaining stronger correct–wrong separation even on instances the underlying solver often fails to solve.} }
Endnote
%0 Conference Paper %T Judging What We Cannot Solve: A Consequence-Based Approach for Oracle-Free Evaluation of Research-Level Math %A Guijin Son %A Donghun Yang %A Hitesh Laxmichand Patel %A Hyunwoo Ko %A Amit Agarwal %A Sunghee Ahn %A Kyong-Ha Lee %A Youngjae Yu %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-son26a %I PMLR %P 114413--114435 %U https://proceedings.mlr.press/v306/son26a.html %V 306 %X Recent progress in reasoning models suggests that generating plausible attempts for research-level mathematics may be within reach, but verification remains a bottleneck, consuming scarce expert time. We hypothesize that a meaningful solution should contain enough method-level information that, when applied to a neighborhood of related questions, it should yield better downstream performance than incorrect solutions. Building on this idea, we propose Consequence-Based Utility, an oracle-free evaluator that scores each candidate by testing its value as an in-context exemplar in solving related yet verifiable questions. Our approach is evaluated on an original set of research-level math problems each paired with one expert-written solution and nine LLM-generated solutions. Notably, Consequence-Based Utility consistently outperforms reward models, generative reward models, and LLM judges on ranking quality. Specifically, for GPT-OSS-120B it improves Acc@1 from 67.2 to 76.3 and AUC from 71.4 to 79.6, with similarly large AUC gains on GPT-OSS-20B (69.0 to 79.2). Furthermore, compared to LLM-Judges, it also exhibits a larger solver–evaluator gap, maintaining stronger correct–wrong separation even on instances the underlying solver often fails to solve.
APA
Son, G., Yang, D., Patel, H.L., Ko, H., Agarwal, A., Ahn, S., Lee, K. & Yu, Y.. (2026). Judging What We Cannot Solve: A Consequence-Based Approach for Oracle-Free Evaluation of Research-Level Math. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:114413-114435 Available from https://proceedings.mlr.press/v306/son26a.html.

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