Reaping without sowing – Tobias J. Osborne's research notes

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Reaping without sowing

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tobiasosborne

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August 2, 2026

If you are at all connected with academic circles it would have been hard to miss the announcement: OpenAI has just published Ten advances in mathematics, ten results in mathematics and theoretical computer science, author listed simply as "OpenAI", accompanied by Lean certificates on GitHub, and, per Noam Brown, a token bill of under $2K for the lot (pretty cheap huh?). Sphere packing, non-sofic groups, a counterexample to Connes’ rigidity conjecture, parallel repetition for entangled games, and three Erdős problems.

Forgive the indulgence, but I called it: Back at the start of summer I warned that "you are probably going to have your LLM ‘oh f*** moment’ this summer", and I called the mechanics in Proof inflation and Conjecture sniping. Well, here we are. It is August, and the moment has arrived horribly on cue.

Let me say the important thing first, before receding into the criticism class: I do not want to diminish this. My first impression is that these results are legitimately impressive. I have flirted with two of these problems myself, and they are properly challenging. This is not conjecture sniping, and nobody stumbles onto a non-sofic group by prompting harder. Something real happened here.

Now let’s begin. I essentially want to make an observation, and then follow it to one rather uncomfortable conclusion.

Twenty pages of tex

Look at the shape of the artifacts. The manuscript is 249 pages, but that is ten separate results somewhat hastily and lovelessly stapled together: the individual chapters run from seven pages (the Ramsey bound) to fifty (the coding bounds), with most sitting in the ten-to-thirty band. Every single one of these proofs fits, with room to spare, inside a single context window. Twentyish pages of tex is comfortably under 100K tokens. That leaves a nice 900K for in-context learning.

This is telling, and it is suggestive (to me) of how these results may have been found.

Here is the thought experiment. If you gave me an infinite-token glitch, an OpenAI flamethrower with which to light tokens indiscriminately on fire, I might have attempted the following to get similar outcomes:

0. Use fast mode. If I can access inference on Cerebras-class hardware, I am definitely using it.

1. Swarm the entire mathematical literature with thousands of agents to build a database of high-value conjectures. Do sentiment analysis with hindcasting to validate the community importance of each one. Aim: at least 10000 conjectures.

2. Presort the conjectures by perceived difficulty. There is no point attempting Riemann. The goal is a good set of conjectures which might be provable in less than, say, twenty tex pages.

2′. Crucially: you do not want to build a mountain of theory to prove a conjecture (see below). Way too risky. Go for the ones which will fall within ten to twenty pages using tools already in the literature.

3. Spawn several orchestrator agents in parallel per conjecture, each overseeing, say, 64 subagents whose sole job is generating candidate proofs. Increase the temperature a bit on the provers to widen the output distribution.

4. Run an adversarial loop on every candidate. This is the "proof sampler" stage: we are hunting for approaches at the edge of the distribution. One might imagine 99% of all conjecture-approach pairs die here. Now we have, say, 100 warm conjectures.

5. Take the warmest 100 and iterate a proposer-critic loop on each until termination. This kills at least half of what remains. Call it 50.

6. We want to be hella cool, so we don’t stop there: farm the survivors out to subagents to build Lean 4 proofs. Here we encounter the uneven surface area of mathlib, so we audition the remaining conjectures for viability against it. (Note that the Lean certificates for the ten advances live in a separate repo. Catalogue keepers have already had to invent a new status for this genre of result: apparently "open (Lean)", means formally certified but humanly undigested.)

(7) Reach out to our math besties and ask what they think of the remaining proofs. Maybe ten respond 😉 (The acknowledgments sections suggest OpenAI did exactly this. Popa, Houdayer, Charles, and others are thanked as readers.)

(8) The rest is easy: get agents to write the papers, and it is time to announce.

The point is that all of this is embarrassingly parallel. If the probability of success on any one conjecture is 0.1%, you may well end up with ten good candidates. Want more? Deploy more compute. The advertised $2K is the price of the winning tokens, not the harvest. As several people immediately asked: how many conjectures went into the hopper to get ten out? Nobody outside the lab knows, and that undisclosed denominator is precisely where the recipe above hides.

Not entirely vibes

I would love to tell you the numbers above are pure vibes. They...