Reasoning over Simulations: Symbolic Intermediaries for LLM Design Agents with a Numerical Oracle

Learning to improve a mechanism from simulation requires connecting observed motion to changes in its design. For language agents, this is a representation problem: simulated trajectories provide numerical evidence, while design revisions are proposed through linguistic reasoning. We introduce symbolic intermediaries, compact analytical expressions fitted to simulated trajectories and supplied alongside the target description. They expose geometric structure that agents can compare, discuss, and use to formulate revisions. A design, simulation, critique, and memory loop accumulates this experience at inference time while the language model's parameters remain fixed. Our analysis characterizes the information needed for local revisions, conditions for their transfer, and boundaries of the feedback representation. We evaluate the approach on MSynth, covering six planar target-curve families and three language models. Critique supports sustained refinement across models, while symbolic feedback provides an additional benefit that depends on the model and target motion. With critique and memory matched, the share of runs improving after the first third of search rises from $44.4\%$ to $55.2\%$, and the median share of total improvement achieved in that later phase rises from $5.4\%$ to $15.6\%$. Critique transcripts show agents using the fitted expressions in geometric comparisons and calculations. These results connect inference-time learning to the representation of simulation experience, where analytical descriptions can help language agents use observed mechanism behaviour to guide subsequent design decisions.