I (author of the paper) can't speak for others, but I have no affiliation whatsoever with OpenAI and have not received anything from them (I even pay for my subscription lol). I've thrown this problem at each model with new model releases, and 5.6 was the first one to solve it, so that's my side of that story.

Thanks for responding. I don't personally believe you have affiliation, more that OpenAI saw the problem was being worked on, gathered training data for solving it and training the model on it, then eventually you found it through the model (and now it's a big impressive thing for OpenAI)

Part of me agrees with the other comments that it sounds absurd, especially because of how involved/intricate your prompting was. But given you have been prompting this problem for a year, OpenAI could easily have seen your attempts/progress, and worked towards solving it with human intelligence that the agent was then trained on. Given the lengths these companies go to for training these things (Meta literally reorganizing their engineering department to provide training data for engineering problems), and also how much difficulty these models have with problems outside their dataset, I really have to wonder if it's the model being "intelligent" or if it has been trained on it

clear specifications for what counts as completing the tasks, and an explicit list of what does not count as completing the task, and clearly stating to not return until the task has been completed. I've had agents run for almost 12h!

I (author of the original post and paper) can add a few things here: 1. My previous approaches with GPT 5.5 were really not very sophisticated in terms of my input. I threw the problem at it, and just kept encouraging it to go iterate through ideas without any success. 2. The approaches that are in the prompt, though they will seem cryptic to someone not in the field, are relatively natural ideas. In fact, the construction that worked was something that even 5.5 initially looked at but was just too weak to see how to make it work. From my view, I would have never gotten this result myself. Imagine you are telling a contractor to build the empire state building, and you say: "You should explore approaches that can include building materials like steel, wood, concrete, or clay, and any combinations of those. You can use arcs, columns, supportive beams, and anything else you can think of to solve load-bearing issues. Do not stop until you've completed a viable plan to construct the empire state building." And then the contractor shows you the finished empire state building using reinforced concrete and steel beams with all kinds of crazy ways of making everything stable; that's kinda how I feel.

Thank you for replying and for reporting your results. My schtick is AI (i.e. I publish in AI conferences and journals), not maths and I'm interested in the question of autonomy from a professional point of view: how close are we to a machine that can carry out the job of a mathematician like yourself, by itself?

If you, e.g. got an LLM (any one) to one-shot the problem with a prompt that said "solve this problem", then we're much closer to that, than if you spent a year trying with different LLMs and then finally got it to work with a lot of hand-holding and even suggesting the ultimate solution. An autonomous system can't succeed once a year, if it's going to be of any use. It should also be able to identify the right tools on its own, not rely on a human to tell it what to do.

This should also go to address some of the questions you pose on reddit, about the future of mathematics. If LLMs can already do your job fully autonomously (as I would explain the term) then ... you're out of a job. You and all other mathematicians, young or old.

I personally don't think we're there yet.

I hear what you say, btw, about never being able to get there by yourself etc. Maybe you would, maybe you wouldn't. What we know is you used a tool to get there, in fact a series of tools, and it took you many tries before you did. The fact that an earlier LLM tried and failed is interesting, but that may just mean you were capable of crafting a better prompt after your interaction with the earlier LLMs.

There is a firm opinion in some circles that an AI can never do anything but recombine already known things despite a rising number of cases where that very much appears to be an inaccurate view.

Any slim possibility that vital information was given to the AI to make the task just one of recombination, rather than coming up with anything original, is grabbed with both hands no matter how tenuous.

I can understand why, nobody likes the idea of a machine capable of doing the job of a human, and jobs requiring the mind are especially sacrosanct. But like farm workers in the early part of the C20th, not liking the incoming technology won't change the effect it will have when companies get the idea they can make more money and hire less people.

I would agree with your take. I (author of the post & paper) learned a ton from working on small parts of problems my PhD advisor was doing a lot of the heavy lifting on, and later also from getting some results that were essentially putting together the right pieces that already existed followed by some deep-in-the-weeds analysis.

Yes, order d is the minimal number of evaluations of gradients needed for the same problem! That has actually been known since 1979 (Nemirovsky and Yudin showed that), and there are methods with the same complexity so this question in the gradient model has been solved for a long time. "because you can approximate a gradient with d function evaluations" was exactly why d^2 made sense as a lower bound for this case! Basically, the lower bound question can also be thought about as "can you do better than approxing a gradient?", so this result says no.