arXiv:2609.38369v1 Announce Type: new
Abstract: We study generalized Blaschke curves as a controlled environment for AI-assisted mathematical rediscovery. For one fixed degree-four Blaschke product, an agent receives numerical coordinates of the six pair-lines determined by each of 80 boundary configurations. The target theorem is withheld from the task instructions. The saved research log reports rejected geometric hypotheses and a homogeneous cubic fitted to polygon sides. Its frozen coefficients predict 480 lines from 80 unseen parameter values, with a recorded RMS scale-free residual of $8.88\times10^{-17}$. Discovery-set diagonals provide an out-of-fit consistency check, not a fully held-out test. A separate one-configuration run reports insufficient evidence for invariance. A post-review deterministic degree-search baseline also recovers the cubic, so the experiment does not establish an advantage over polynomial fitting. We present this single-instance case study as a protocol for separating conjecture, numerical validation, and proof, with explicit limitations concerning agent metadata, prior knowledge, and reproducibility.
Abstract: We study generalized Blaschke curves as a controlled environment for AI-assisted mathematical rediscovery. For one fixed degree-four Blaschke product, an agent receives numerical coordinates of the six pair-lines determined by each of 80 boundary configurations. The target theorem is withheld from the task instructions. The saved research log reports rejected geometric hypotheses and a homogeneous cubic fitted to polygon sides. Its frozen coefficients predict 480 lines from 80 unseen parameter values, with a recorded RMS scale-free residual of $8.88\times10^{-17}$. Discovery-set diagonals provide an out-of-fit consistency check, not a fully held-out test. A separate one-configuration run reports insufficient evidence for invariance. A post-review deterministic degree-search baseline also recovers the cubic, so the experiment does not establish an advantage over polynomial fitting. We present this single-instance case study as a protocol for separating conjecture, numerical validation, and proof, with explicit limitations concerning agent metadata, prior knowledge, and reproducibility.
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