AI Agent Rediscoveries a Blaschke-Curve Invariant in a Controlled Case Study

Summary

This paper presents a controlled case study of AI-assisted mathematical rediscovery using generalized Blaschke curves. For one fixed degree-four Blaschke product, an agent receives numerical coordinates for the six pair-lines associated with each of 80 boundary configurations, while the target theorem is withheld. Its research log records rejected geometric hypotheses and a homogeneous cubic fitted to polygon sides. Using frozen coefficients, the cubic predicts 480 lines from 80 unseen parameter values with a reported scale-free RMS residual of 8.88 × 10^-17. However, diagonals from the discovery set serve only as an out-of-fit consistency check, rather than a fully held-out test. A separate run using only one configuration finds insufficient evidence for invariance. A deterministic degree-search baseline added after review also recovers the same cubic, so the experiment does not show that the agent outperforms ordinary polynomial fitting. The authors frame the result as a single-instance protocol for distinguishing conjecture formation, numerical validation, and proof, while noting limitations involving agent metadata, prior knowledge, and reproducibility.