Coupled Harness–Model Evolution for Scientific Equation Discovery

Large language models (LLMs) are increasingly used to discover scientific equations from experimental data. Such a discovery system has two parts that must work together: a library of executable computational tools and the model's tool-use policy. The tools supply the solving capability, from diagnosing structure in the data to fitting candidate equations; the policy plans and orchestrates their use. Yet existing systems adapt only one part, keeping either the toolset or the model fixed, which leaves the corresponding failures uncorrectable. We develop a coupled evolution framework that improves both parts from the same verifiable fit-error signal. In the tool-evolution phase, an agent grows an executable tool library from a small set of generic numerical operations by diagnosing its own failures: it authors a new tool when a capability is missing and revises its selection rules when the right tool went unused. In the policy-evolution phase, a small open model (roughly 3B active parameters) is fine-tuned on successful tool-use trajectories collected with the frozen library, on training formulas screened to be disjoint from all evaluation benchmarks. Ablations show the two phases are complementary: with no tool library at all the same model solves almost nothing, the tools account for most of the gain, and trajectory training adds a further gain that is largest on the most difficult problems. The full system solves 79.1/71.3% of LSR-Synth problems at fit tolerances $10^{-2}$/$10^{-3}$, reaches 63.8% pooled accuracy on the distributionally different SRSD-Feynman panel, and obtains a normalized CP3-28 aggregate of $-7.27$. These results suggest a broader recipe for scientific agents: let the model and its instruments improve each other under verifiable feedback.