You want to know how long it takes to solve an optimization problem, in this case over convex, lipschitz functions. (The restriction to a spherical domain is not really a restriction, you can just change variables for any bounded domain.) Anyway, showing upper bounds on time complexity is "easy" because it's just the runtime of your algorithm. Showing (nontrivial) lower bounds is usually much harder because it requires constraining all algorithms.

This proof apparently shows that the lower bound time complexity is equal to the time complexity of an existing 30-year old algorithm: it requires Omega(d^2) function evaluations to solve over this class of functions.

My gut says likely implies that d is the minimal number of evaluations if you have a gradient oracle because you can approximate a gradient with d function evaluations, but I'm not sure how hard it is to make that rigorous.

I wonder how this compares to what we see happening with "juniors" in software development? In math research, do you also get the training for the profession from working on the low hanging fruits for a while, to then move to the medium-hanging, and later go on to work on previously unsolved stuff?

My understanding is that ChatGPT Pro is effectively a multi agent system, or somehow uses multiple LLMs in parallel and selects a best answer. And Ultra is more similar to Claude-Code UltraCode where the main agent can choose to create a dynamic JS workflow that deterministically orchestrates multiple agents to handle different parts of a task and have adversarial checkers etc.

Is that more or less the difference? Any substantiating sources would be great to see.

One solution is to ban LLM’s, to artificially create a demand for human thought, that just feels like living in an artificially constructed zoo.

Another solution is humans don’t do anything that AI can do better , / doesn’t need the human touch. So I suppose we will all become artists, sportsmen or politicians, the only jobs that will remain except for select few. Maybe this is ok, I don’t know.

Another solution is we find a way to mind-meld with AI so that human + Ai >> AI alone. This is dystopian, who gets to decide who mind melds with AI, how much will it cost etc etc.

For the stupid copes that the prompt required human ingenuity, let me first add that the author used GPT5.6 to write most of the prompt. He just gave some mild direction. That amount of direction does not require deep expertise and the expertise required will keep falling with time, eventually an undergrad can create this loop and then maybe a high school student.

  And prompt engineering / loop engineering nonsense is not real. Calling it engineering is a psy-op because it is something simple, imprecise and future models will be much better at it than you.

- Hasn't been peer reviewed yet, so take with a grain of salt. This applies to all claimed proofs, not just AI-generated ones. Even humans hallucinate proofs too!

- The prompt is on page 27 here[1]. It is ten pages of advanced mathematics priming the model in the right direction, apparently informed by a year of prior research. That doesn't invalidate the result if it is genuine, but it is worth noting that this wasn't a matter of "ChatGPT, solve this unsolved problem. Make no mistakes." and required substantial domain expertise and human research beforehand.

[1]https://arxiv.org/pdf/2607.13335

It's not that AI brings equality, but rather that the output varies depending on how much background knowledge you have. You could call it a stratification of input

I'm starting to feel like there's no place left for programmers like me who focus on quickly churning out MVPs.

this is changing my mind, at least about experts using advanced tools like any profession where it's like the magic of watching a lifetime of hard-earned skill at work

> After seeing OpenAI’s CDC result, I wrote a much more elaborate prompt following the same general methodology. My prompt is about ten pages long and attached at the end of the preprint (see collection of links below). There is a lot baked into this prompt, on approaches to try and also on how exactly the model should proceed, but it's built exactly in the style of OpenAI's CDC prompt. One note is that I gave it a relatively small error requirement, to prove the quadratic lower bound under order d⁻⁴ accuracy.

> After 148 minutes, GPT-5.6 Sol Pro returned a proposed proof resolving the quadratic dimension dependence at accuracy of order d⁻³. After checking things myself, I formally verified the proof in Lean, and it passed the formal verification check.

I'm very curious how people reconcile their fear/hatred of AI with actual objective reality. This is actually what interests me most about the whole AI thing. How we tell ourselves what we tell ourselves.

The most interesting thing in research is finding new questions, that we understand and that we know why they are important. And that's something that humans need to do (by definition)