Benjamin Grayzel | Searcharxiv

arXiv2026

Cost-Effective Automated Judging of Natural-Language Mathematical Proofs

Grading natural-language mathematical proofs is a recurring cost in evaluating math-reasoning systems, and frontier LLM judges are expensive. We ask whether cheap open-weight models can serve as reliable judges given a candidate proof, a ground-truth proof, and a human-grading rubric. On a 200-instance validation sample of IMO-GradingBench, three cheap judges (GPT-OSS 120B, DeepSeek-V4 Flash, Gemma-4 31B) agree with human pass/fail decisions at rates statistically indistinguishable from Claude Opus 4.7 and Gemini 3.1 Pro, at up to $100\times$ lower cost. We had expected a majority vote of the three to be the best budget option; it matched the frontier but did not improve on its strongest member. Extending to the full 1000-instance benchmark and exploring consensus rules, we found that requiring unanimous agreement (all-three-pass) reaches the highest pass-agreement and precision and, on four replicate runs, the smallest run-to-run spread. The headline finding is that cheap judges are competitive with the frontier at one to two orders of magnitude lower cost; as a deployable default we recommend all-three-pass, with the caveat that this rule was identified post-hoc and warrants independent replication.

cs.CL↗

arXiv2026

Solution to a Problem of Erd\H{o}s Concerning Distances and Points

In 1997, Erd\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ that determines $O(\frac{n}{\sqrt{\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset determines at least 3 distinct pairwise distances. We construct $n$-point sets from an $m\times m$ box of the lattice $L = \{(x,\sqrt{2}y):x,y \in \mathbb{Z}\} \subset \mathbb{R}^2.$ The distinct distance bound follows from applying Bernays' theorem to the number of integers represented by the binary quadratic form $u^2 + 2v^2$. The local 4-point constraint is verified through Perucca's similarity classification of the six similarity types determining exactly two distances.

math.CO↗