A paper submitted to arXiv in September 2026 demonstrates that a large language model can evolve optimization algorithms well enough to break established mathematical records, and do it for less than the cost of a nice dinner. The system, called Discovery Loop, uses an LLM to iteratively improve a circle-packing solver, producing results that surpassed the best known solutions for 10 values on the Packomania benchmark. The improvements ranged from 2.4% to 5.4% over prior records, all achieved within 15 iterations at a total LLM cost of $27.72.
The circle-packing problem
The Packomania benchmark, known as csqv, asks a deceptively simple question: given N variable-radius circles, how large can you make their combined radii if they must all fit inside a unit square without overlapping? The problem has been studied for decades, and the best known solutions for many values of N are the result of careful manual optimization and computational search. For values in the 101 to 114 range, the records had held until this work came along.
Circle packing is not just an aesthetic curiosity. The problem appears in logistics, materials science, and chip design, anywhere objects need to be arranged with maximum efficiency in a constrained space. The difficulty scales combinatorially with N, making it a useful testbed for optimization methods.
How Discovery Loop works
The system starts with a simple seed solver and asks the LLM to propose algorithmic improvements. Each proposal is evaluated against an independent verifier that checks whether the solution actually improves the objective function. Successful improvements are kept and fed back into the next iteration. Failures are discarded. The LLM sees a scoreboard of current results and a history of prior ideas, which lets it avoid repeating approaches that did not work.
The key design choice is separating the proposal step from the verification step. The LLM generates code, but the verifier runs it independently and measures the outcome. This means the system does not trust the LLM's claims about what will work. It only trusts measured results. The LLM acts as a search heuristic over the space of possible algorithmic improvements, guided by concrete feedback rather than abstract reasoning about the problem.
An adaptive plateau-detection mechanism monitors progress across iterations. When the system stops making meaningful gains, the mechanism adjusts the search strategy, either by encouraging the LLM to try more radical changes or by narrowing the focus to refinements of the current best approach. This prevents the system from getting stuck in local optima while also avoiding wasted iterations on unpromising directions.
Breaking records with modest compute
The results are striking not just for their quality but for their cost. Ten Packomania records fell, each independently accepted by the benchmark maintainers. The total LLM cost was $27.72, spread across 15 iterations. For context, running a single large-scale computational search for circle packing can consume far more in compute credits alone, without any guarantee of finding improvements.
The 2.4% to 5.4% gains over prior best-known solutions are meaningful in a domain where incremental improvements often require significant human effort. These are not marginal improvements that would be found by any optimizer. They represent genuine advances in the state of the art for specific values of N, achieved by a system that started with a basic solver and refined it through language-model-guided evolution.
What this means for automated discovery
The paper frames its contribution around democratizing automated scientific discovery. Traditionally, breaking computational records requires specialized expertise in both the problem domain and the optimization methods. Discovery Loop lowers that barrier by using an LLM as the reasoning engine, with the human providing only the initial seed solver and the problem specification.
This is a different model from using LLMs to generate solutions directly. The LLM here is not solving the circle-packing problem itself. It is writing code that solves the problem, and that code is being evaluated empirically. The distinction matters because it means the approach can in principle be applied to any optimization problem where solutions can be verified automatically. The LLM provides the creative leaps in algorithm design, while the verifier provides the ground truth.
The $27.72 price point is the headline, but the real signal is that LLM-guided program evolution can produce results that stand up to independent verification in a well-established benchmark. As LLM costs continue to drop and model capabilities improve, systems like Discovery Loop could become a standard tool in the optimization researcher's toolkit, turning what used to be months of manual tuning into hours of automated exploration.