A desktop computer broke 10 geometry records for twenty-eight dollars

A minimal evolutionary coding loop running on standard consumer hardware discovered new mathematical optimization algorithms, beating long-standing circle packing records without industrial supercomputing clusters.

Arranging round objects inside a tight boundary is deceptive in its simplicity. A person packing different sizes of coins into a small square frame quickly discovers that shifting any single disc nudges its neighbours, alters vacant spaces across the container, and completely resets the math. In geometry, this puzzle is known as the variable-radius circle-packing problem, where the objective is to place a specified number of non-overlapping circles inside a unit square such that the sum of their radii reaches the absolute mathematical maximum.¹² Because every circle can change in both size and position across continuous coordinates, the landscape of potential arrangements contains an astronomical quantity of local traps, where slight adjustments in any direction only degrade the score.

To find the tightest configuration, an optimizer must carry out a precise chain of steps. It first establishes an initial pattern of circle centres within the container walls; it then expands each radius until neighbouring boundaries touch; finally, it perturbs the positions to escape crowded logjams before re-optimizing the contact network. Improving on known solutions requires inventing entirely new search strategies rather than slightly nudging existing coordinates. For decades, finding better geometric arrangements required specialized optimization software, immense supercomputing clusters, or years of manual tuning by mathematicians. Global databases such as Packomania, curated by mathematician Eckard Specht since 1998, track the world records for these configurations across various container shapes and circle quantities.¹³

When Google DeepMind introduced its AlphaEvolve system in 2025, it demonstrated that frontier artificial intelligence models could generate novel mathematical algorithms to break records.⁴¹ However, AlphaEvolve relied on large clusters of distributed computers and complex multi-agent frameworks, leaving open the question of whether automated algorithmic discovery required industrial-scale compute budgets.¹ A preprint posted to arXiv on September 4, 2026, by independent researcher Wes Sander of Practical Systems in Mason, Ohio, reports that high-infrastructure barriers may no longer be necessary.⁵¹ Using a lightweight program called Discovery Loop, Sander directed an artificial intelligence model to iteratively write, evaluate, and refine complete Python optimization programs on a single consumer desktop computer.¹⁶ Over the course of 15 iterative cycles costing $27.72 in language model programming fees, the system produced new best-known solutions for 10 distinct benchmark targets, improving previous mathematical records by 2.4% to 5.4%.¹⁶ The new arrangements were submitted to and officially accepted by Specht into the Packomania repository.¹²

How does evolutionary algorithm design find tighter circle packings?

Evolutionary algorithm design finds tighter packings by treating optimization strategies as flexible computer programs rather than fixed numerical routines.¹⁷ The physical process begins with an initial baseline algorithm, known as a seed solver, which places circles in a naive arrangement and maximizes their size until they touch.¹ An external supervisor evaluates the resulting geometric configuration, measures the combined radii, and enters the score into a progress scoreboard. A large language model then inspects the scoreboard alongside the complete source code of the current champion solver and a history of previous attempts, using that context to draft an entirely new replacement algorithm.¹ The system executes this candidate code against mathematical test problems in parallel, automatically keeping the program if it achieves a higher radius total and discarding it if it fails.¹

Sander implemented this entire control architecture in approximately 400 lines of Python code.¹⁸ In each round, the Claude Fable 5.1 language model from Anthropic received a prompt detailing the geometric constraints, the best-known benchmarks from Packomania, and a log of the last 12 ideas tried. Rather than outputting incremental code patches that could generate syntax errors or merge conflicts, the language model wrote an entire standalone solver from scratch in each step.¹ Sander told Primary that this complete rewrite architecture proved vital when individual generations broke down.^contributed "The closest thing in the actual circle packing run was iteration 3," Sander said in response to questions from Primary.^contributed "The model returned no usable solver, so the iteration was recorded as no code and the existing champion was left untouched. That is where the architecture helped: a generated program was disposable until it executed and passed independent verification, so a bad generation could waste an iteration but could not corrupt the lineage."^contributed The candidate program was executed across 12 target problem sizes ranging from 26 to 114 circles, using six parallel worker threads running on an Intel Core i7-13700KF processor with 32 gigabytes of random access memory.

Across 15 iterations spanning eight hours of overnight computation, the model discovered several sophisticated optimization heuristics.⁶ In the first generation, it introduced basin hopping, a global search technique that perturbs known arrangements to escape local optima, combined with Sequential Least Squares Programming to polish the contact network where circles touch.¹ By the fourth iteration, the model introduced hexagonal lattice initialization, seeding the container with structured grid templates sized for exact circle counts.¹ Later cycles produced specialized contact-graph solvers and defect-migration moves, an advanced technique where poorly fitted circles are temporarily removed to let surrounding discs re-equilibrate before reinsertion into remaining voids.¹

Sander explained that the emergence of physical heuristics reflected search over executable algorithms under tight feedback rather than spatial intuition inside the neural network.^contributed In the defect migration generation, Sander told Primary, the system removed loosely constrained discs, relaxed the remaining layout, identified new voids using Delaunay circumcentres and Nelder Mead refinement, and reinserted the circles. "The crucial step is the intermediate relaxation," Sander said. "Without it, you tend to refill the hole you just created. With it, the vacancy moves and the contact graph changes, which lets the solver jump into a different basin."^contributed

What did the automated discovery loop achieve on benchmark records?

The automated discovery loop improved the world records on 10 out of 12 evaluated benchmark targets on the Packomania repository.¹² Across the 10 target instances where records fell, comprising circle counts between 101 and 114, the aggregate sum of radii increased from the previous cumulative record of 51.814380 to 54.406044, representing an overall gain of 5.00% across the benchmark collection.⁶ On individual instances, such as 105 circles, the system improved the sum of radii from 5.125967 to 5.401298, achieving a 5.37% relative gain over the previous human and algorithmic record. For 106 circles, the score advanced from 5.151736 to 5.429079, yielding an improvement of 5.38% over the prior database entry.

Sander emphasized an important baseline nuance when evaluating these headline improvements.^contributed He told Primary that the hand-crafted seed solver already surpassed the historical Packomania entries for circle counts 101 through 114 before automated generation began. "So the clean claim is that the loop subsequently discovered nontrivial heuristics and improved the solver further," Sander said, "not that the entire five percent gap over the historical records was created by the LLM."^contributed The evolutionary cycle lifted the cumulative score across all 12 targets from 59.39 at iteration 0 to 59.98 by iteration 14, confirming that the generated heuristics contributed meaningful progress beyond the starting program.

To guarantee mathematical credibility, the system evaluated every generated candidate against an independent zero-tolerance verifier.¹ The verifier shares no code with the evolving solver, confirming that all circle coordinates remain strictly within the unit square boundary and that no two discs overlap by any margin.¹ To eliminate potential numerical precision exploits in floating-point calculations, the verification step applies a strict feasibility shrink to disc radii before recording the score.¹ For two evaluated targets, namely 26 and 32 circles, the system matched existing historical records within one millionth of a unit but did not exceed them, leaving the database records unchanged for those specific counts.¹

Asked why smaller target configurations resisted algorithmic improvement, Sander suggested that geometry and baseline maturity both played roles.^contributed He noted that smaller instances are heavily boundary dominated and globally coupled, where moving one circle alters a large portion of the container's contact network. In contrast, instances with more than 100 circles feature large bulk interiors where lattice templates, slip moves, and defect relocations can exploit local hexagonal packing. Sander also noted that the historical records for 26 and 32 circles were already highly optimized by earlier researchers, making further progress far more difficult to achieve.

The experimental run also revealed distinct cost-efficiency dynamics during the search. During the productive initial phase from iteration 0 to iteration 5, the system spent $4.96 in model application programming interface fees and increased the total benchmark score from 59.39 to 59.96 across all 12 targets, achieving an efficiency of $8.70 per unit of score improvement.¹ During the diminishing returns phase from iteration 6 to iteration 14, the system spent $22.76 while adding only 0.02 to the cumulative score, raising the cost per unit improvement to $1,138.¹ Sander developed an early-stopping rule based on sliding-window progress, demonstrating through retrospective analysis that halting the run after iteration 9 would have saved $13.77, cutting overall costs by 50% while sacrificing only 0.01% of the final solution value.¹

What are the boundaries and limitations of single-machine algorithmic evolution?

The primary limitation of this lightweight evolutionary approach is its sensitivity to problem complexity and code generation reliability. The results were obtained on a single mathematical domain where candidate configurations can be evaluated in two minutes or less and where validity is governed by clear geometric laws. In supplementary experiments attempting to evolve solvers for mixed-integer programming benchmarks from the MIPLIB library, the language model experienced a 75% failure rate in producing compilable, error-free Python code.¹ When problem definitions become highly abstract or require complex software dependencies, unguided language models frequently generate invalid syntax or non-functional execution logic.

Sander observed that the breakdown on mixed-integer problems stemmed from interface brittleness rather than mathematical category alone.^contributed He told Primary that circle packing offers a compact mathematical representation and immediate numerical feedback, whereas general solver interfaces demand delicate state tracking, callback routines, and expensive subproblem evaluations. When candidates attempted mixed-integer benchmarks, the model repeatedly proposed variations of ineffective neighborhood searches instead of identifying specific problem structures. Sander concluded that the method collapses when representation complexity overwhelms the model's capacity to generate valid architectural mutations.

Furthermore, the research paper is an unreviewed preprint posted to arXiv and represents an empirical demonstration rather than a theoretical proof of optimization limits.⁵¹ Unlike large-scale systems such as AlphaEvolve or ShinkaEvolve, which maintain diverse populations of competing programs across evolutionary islands, Discovery Loop maintains only a single champion solver at any time.¹¹⁰ Sander noted that single-lineage evolution faces structural barriers whenever reaching a superior solution requires passing through intermediate algorithms with lower immediate scores.^contributed In such landscapes, greedy selection destroys promising stepping stones, meaning multi-population archives become necessary to preserve diverse algorithmic strategies across distinct targets.

What comes next for low-cost automated mathematics?

The success of minimal evolutionary loops suggests that automated scientific discovery may become widely accessible to individual researchers, universities, and small engineering teams.¹ By demonstrating that a 400-line script running on consumer hardware can advance mathematical records previously held by specialized institutions, the work narrows the gap between large enterprise laboratories and independent experimenters. Open-source optimization benchmarks across geometry, routing, scheduling, and logistical packing can now be explored systematically without requiring substantial institutional compute grants.

Future work will focus on testing the limits of single-model evolution on denser holdout configurations. Sander told Primary that testing the architecture on problem sizes between 150 and 200 circles under identical hardware and budget caps would establish clear falsification boundaries for lightweight discovery systems.^contributed Researchers are also exploring hybrid workflows that combine language-model program generation with formal proof assistants, allowing algorithms to discover novel empirical constructions before automated verification tools establish their rigorous bounds. As generative coding models continue to improve in syntax reliability and mathematical reasoning, evolutionary discovery harnesses are set to expand into broader operational research challenges, from industrial supply-chain packing to automated compiler optimizations.

This piece was prepared from the arXiv preprint and public records together with answers from Wes to six questions from the Primary news team, completed September 2026.

References

This article is based on 59 sources, with 6 statements from 1 contributor, listed in the order they are cited.

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Article history

1. 6 statements 8 Sep 2026, 13:25
   What was added

   Unlike large-scale systems such as AlphaEvolve or ShinkaEvolve, which maintain diverse populations of competing programs across evolutionary islands, Discovery Loop maintains only a single champion solver at any time.

   On the record as reference 9 go to the sentence ↑
   What was added

   In supplementary experiments attempting to evolve solvers for mixed-integer programming benchmarks from the MIPLIB library, the language model experienced a 75% failure rate in producing compilable, error-free Python code.

   On the record as reference 9 go to the sentence ↑
   What was added

   For two evaluated targets, namely 26 and 32 circles, the system matched existing historical records within one millionth of a unit but did not exceed them, leaving the database records unchanged for those specific counts.

   On the record as reference 9 go to the sentence ↑
   What was added

   Sander emphasized an important baseline nuance when evaluating these headline improvements.

   On the record as reference 9 go to the sentence ↑
   What was added

   Later cycles produced specialized contact-graph solvers and defect-migration moves, an advanced technique where poorly fitted circles are temporarily removed to let surrounding discs re-equilibrate before reinsertion into remaining voids.

   On the record as reference 9 go to the sentence ↑
   What was added

   Rather than outputting incremental code patches that could generate syntax errors or merge conflicts, the language model wrote an entire standalone solver from scratch in each step.

   On the record as reference 9 go to the sentence ↑
   W Wes · Contributor Independent researcher.
2. Published 11 Sep 2026, 13:36
   Assembled by the Primary desk from 59 sources · 42 cited sentences