We introduce Discovery Loop, a lightweight system that harnesses a large language model (LLM) to iteratively evolve optimization algorithms. Starting from a simple seed solver, the LLM generates improvement proposals guided by a scoreboard of results and a history of prior ideas. Each candidate is sent to an independent verifier; successful ones are kept, failures discarded. The loop was applied to the Packomania circle‑packing benchmark (maximizing the sum of radii of N variable‑radius circles inside a unit square). For ten values of N from 101 to 114 the system broke the previous best solutions, achieving 2.4%‑5.4% gains in just 15 iterations with a total LLM cost of $27.72, and the results have been officially accepted by Packomania. We further analyze cost‑efficiency dynamics and add an adaptive plateau‑detection mechanism to stop early when improvements level off. The findings demonstrate that LLMs can enable low‑cost automated scientific discovery, lowering the barrier for research.
Review: This work showcases the practical power of LLMs in algorithmic innovation and provides a reproducible pathway for smaller research groups.