LLM-Guided Program Evolution for Circle Packing: Breaking 10 Packomania Records for $28

The paper presents Discovery Loop, a roughly 400-line system that uses Anthropic’s Claude Fable 5.1 to evolve complete optimization solvers rather than isolated code patches. Each iteration supplies the current champion, a scoreboard, and the last 12 ideas; the model proposes a replacement Python solver, which is evaluated in parallel and checked by an independent zero-tolerance verifier. Starting from a multi-start penalty L-BFGS-B solver with LP-optimized radii, the system applied basin hopping, hexagonal-lattice initialization, island-model parallelism, KKT-Newton polishing, and defect migration to the Packomania csqv benchmark, which maximizes the sum of variable circle radii in a unit square. It improved 10 of 12 targets for N values from 101 to 114, achieving gains of 2.4% to 5.4% over accepted records, and completed 15 iterations for a total LLM cost of $27.72 on a consumer PC. The approach is a lightweight counterpart to Google DeepMind’s Gemini-powered AlphaEvolve and FunSearch: it uses one model call per iteration, no distributed cluster, and per-target best tracking. Cost-efficiency declined sharply after early gains, but retrospective plateau detection would have stopped at iteration 9, reducing spending to $13.95 while sacrificing only 0.01% of final value. The results suggest that independently verifiable, LLM-guided algorithm discovery can be performed at laptop scale, though performance remains sensitive to problem structure and code-generation reliability.