How AI Agents Discover Scientific Equations: From Hydrotope Rediscovery to New Water-Wave Amplitudes

We study how AI agents discover and validate scientific formulas using a controlled case study of the hydrotope, a recently discovered geometric formula that combines the different polynomial pieces of nonlinear surface-wave scattering into one global expression. This problem is deceptively difficult: simple formulas can hold within individual frequency regions, but the global result must identify their boundaries and combine exponentially many potentially active terms. We reconstruct how the formula was originally discovered through human--agent collaboration and analyze 18 single-prompt rediscovery runs under no hint and two forms of human guidance: a false hint representing an incorrect prior and a true hint representing domain-informed insight. Only four recover the formula across all kinematic chambers (i.e., regions in which a single polynomial form applies), while most unsuccessful runs find correct chamber polynomials but fail to combine them or test their full domain. Conventional and LLM-assisted symbolic regression and standard machine-learning regressors likewise fail to recover the global formula in our experiments. Guided by these failure modes, we test a PI+two-student workflow in which a coordinating lead agent assigns complementary analytic and numerical tasks to two research agents and independently evaluates their results. The PI+two-student team successfully rediscovers the complete hydrotope formula, while the same workflow applied to the harder three negative wavenumber problem discovers a new independent verified analytic expression for the six-point amplitude A6.