cs.AISep 29, 2026

Can an AI Agent Rediscover a Blaschke-Curve Invariant?

Authors: Yunus E. Zeytuncu

Organizations: University of Michigan-Dearborn

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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 8.88×10−178.88\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.

Figures & tables

Figure 1: The 480 stored discovery lines from exactly 80 values of λ\lambda . Blue lines are sides and red lines are diagonals; black points are the zeros of BB . This post-review visualization uses the experimental discovery CSV, not an additional sampling grid. No envelope figure was supplied in the original discovery task.
ProcedureEvaluation RMSEvaluation maximum
Frozen agent cubic (archived report)8.88×10−178.88\times 10^{-17}3.18×10−163.18\times 10^{-16}
Deterministic degree search (new control)1.24×10−151.24\times 10^{-15}3.18×10−153.18\times 10^{-15}
Table 1: Original agent outcome and post-review deterministic comparison. Both polynomials use coefficient −1-1 for w3w^{3} and residual ( 3 ). Roundoff-level differences are not evidence of superior discovery performance.
Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

RunPhaseElapsed secondsOutput tokens
Full dataDiscovery143.16,219
Full dataEvaluation43.32,129
Sparse dataDiscovery122.75,209
Sparse dataEvaluation100.84,815
Appendix
Table 2: Recovered original-run accounting. Elapsed times span recorded task start to completion. Output-token counts are the harness-reported field, not a prespecified budget or a cost estimate.
DegreeNumber of monomialsNumerical nullityσmin⁡/σmax⁡\sigma_{\min}/\sigma_{\max}
1309.46×10−19.46\times 10^{-1}
2608.71×10−38.71\times 10^{-3}
31012.75×10−162.75\times 10^{-16}
41531.37×10−161.37\times 10^{-16}
52168.17×10−178.17\times 10^{-17}
Appendix
Table 3: Numerical degree search on the 320 discovery side rows.
ConfigurationsUnique side rowsCubic nullityEvaluation RMS
146Not uniquely identified
282Not uniquely identified
52012.00×10−152.00\times 10^{-15}
104012.22×10−152.22\times 10^{-15}
208012.00×10−152.00\times 10^{-15}
4016017.78×10−167.78\times 10^{-16}
Appendix
Table 4: Post-review deterministic controls, not additional agent ablations. Residuals here use unit Euclidean coefficient norm, unlike Table 1 .

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