$28 of LLM spend just beat 10 circle-packing records
Two results landed overnight that say something about who gets to do AI-assisted research now — and how much of it is actually new.
A single Python script, one consumer PC, and $27.72 in API spend broke ten standing records on Packomania, the circle-packing benchmark that has tracked best-known solutions since 1998. Wes Sander, writing as Practical Systems, posted the result on r/MachineLearning and on arXiv: his system, Discovery Loop, handed a language model the current best solver plus a scoreboard and a history of failed ideas, and asked it to write a complete replacement solver from scratch. Each candidate was scored by an independent verifier that shares no code with the solver, so the model can't quietly game its own evaluation. Fifteen iterations later it had improved the best-known sum of radii for N = 101 through 114 by 2.4% to 5.4%, and Packomania's maintainer, Eckard Specht, accepted the submissions.
The interesting number isn't the record — it's the cost curve. The first six iterations bought nearly all of the improvement for under $5. The remaining nine cost $22.76 and moved the total by 0.02, a 130× collapse in improvement per dollar. The paper's plateau rule would have stopped the run at iteration nine, saving half the spend for 0.01% of the result. That's the real finding: the technique is cheap to start and brutally fast to plateau, and anyone running one of these loops needs a stop rule before the budgets do it for them. It also reframes AlphaEvolve, DeepMind's version of the same idea, less as a capability moat than as an infrastructure convenience — Sander got there with 400 lines of Python, no evolutionary database, and no cluster. We covered the flagship version of this in OpenAI's Astra cuts the bounded prime gap record to 186, where the compute was anything but a laptop.
The mathematician who pushed the twin prime gap to 186 says the technique behind it is nearly maxed out. Su Weijie — OpenAI researcher, Wharton statistics professor and a COPSS Prize winner — walked through the result in an interview with China Science Daily, and he is careful about what it does and doesn't prove. The work still runs through the Maynard–Tao multidimensional sieve that produced the 246 bound in 2014; the model's contribution was a new constraint, "triple dense divisibility," that lets one set of divisors absorb large prime factors the other can't handle, widening the usable range of the sieve weights. Su calls it a substantive technical advance, not a breakthrough: 186 sits near what this sieve family can do, and going much lower runs into the parity problem, a structural wall that no amount of numerical polish will get past.
He is also precise about the verification, which matters more than the number. The main reasoning chain was formalized in Lean, but it rests on two existing finite-field exponential sum estimates and a set of integral and numeric bounds supplied as external inputs — checked by high-precision code outside Lean, not proved within it. "Mathematically this is strict, but formally it is not entirely unconditional," is the gist of his caveat. His framing of the division of labor is worth keeping: the better models get at local derivation and computation, the more human effort belongs in choosing the problem, building the model, and checking the result.
What to watch: whether Packomania-style benchmarks, which were built for humans working in months, hold up when a $30 run can move ten of them in a night.
If a laptop and an API key can break math records, what happens to the credit — do you think benchmark maintainers should start logging AI-assisted submissions separately? Tell us in the comments.
Sources: Discovery Loop on arXiv · r/MachineLearning discussion · Discovery Loop (GitHub) · Packomania csqv benchmark · China Science Daily interview with Su Weijie
