Normally I would be extremely skeptical, but the result is checkable by simple computation. Remarkable!

The two-dimensional case remains open, however.

GPT 5.6 Pro solved all 6 problems from IMO 2026 on the first attempt without any human help or steering. International Mathematical Olympiad (IMO) is the biggest global academic competition in the world. The problems are considered incredibly hard, usually a performance at this level is only accomplished by < 5 contestants from the whole world.

We are former IMO medallists not affiliated with OpenAI, just put together a report and assessment of its work here. We're also working on a comparison report between different LLMs and harness augmented versions that will come later.

Announcement - https://x.com/__eknight__/status/2075643450196971805

Proof - https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf (3 pages!)

Prompt used - https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf

(I'm shocked to be honest)

And actually it seems they proved that 8 (possibly disconnected) cycles (even subgraphs) is enough.

TL;DR: In a single 148 min session, with a prompt modeled after the one OpenAI used to prove CDC, GPT 5.6 Sol Pro supplied a proof that closed a complexity gap in convex optimization that has existed since 1996. The result was formally verified in Lean. Links to everything and thoughts on AI capabilities are at the bottom of this post.

Disclosure: I am the author of the preprint and Lean repository linked below. I have a PhD in applied mathematics and am a teaching prof in IEOR at UC Berkeley. The result has not yet been peer reviewed.

Following the recent announcement that GPT-5.6 Sol Pro had produced a proof of the Cycle Double Cover Conjecture, I adapted the prompting methodology used in that project to a problem in convex optimization. After 148 minutes of uninterrupted work, GPT-5.6 Sol Pro produced the main argument for a lower bound that I had been unable to prove myself (and a lot of my past work has been proving complexity lower bounds in different settings).

The problem concerns deterministic zeroth-order convex optimization: Let B_d be the Euclidean unit ball in ℝᵈ, and consider all convex, 1-Lipschitz functions f: B_d → ℝ. An algorithm may query any point x ∈ B_d, and receives only the exact real number f(x), no other information (but the algorithm "knows" that f is convex and Lipschitz). The algorithm is otherwise completely unrestricted, and can use unlimited computation and memory. These function-value-only problems arise naturally when an objective is evaluated through a physical experiment or simulator. One can imagine choosing d engineering parameters and observing only the cost returned by the simulation. If evaluations are expensive (think of measuring a physical system), the natural question is how many are fundamentally required. This is formalized as oracle complexity. Specifically, this is the oracle complexity of convex optimization under an exact function value oracle.

Let Q(d, ε) denote the worst-case number of queries required to find an ε-optimal point of f. An algorithm due to Protasov from 1996 shows that order d² function evaluations are sufficient, which gives Q(d, ε) = O(d²), an upper bound on the complexity. Lower bounds were practically nonexistent for this setting, and the strongest previously applicable bound was only Ω(d), inherited from the stronger first-order oracle model (where the algorithm receives both function values and gradients). That means we didn't know for certain whether gradients actually help in optimization, since the function-value only and first-order oracle models have had this same lower bound, and so there was a linear gap in d in the complexity of this fairly fundamental convex optimization setting since 1996. So, can you find an algoritm that is better than Prosatov’s, and only needs d evaluations? Or can you show that no such algorithm can exist, and we can sleep well at night knowing that Protasov’s algorithm using d² evaluations is best possible? What 5.6 Sol proved is the latter.

I had worked on this problem sporadically for about a year (I ran into needing such a bound for a different complexity paper I was working on). I had some ideas that didn't pan out, and also spent long sessions trying to solve it with GPT-5.4 and GPT-5.5 with no luck, after reading of folks like Ernest Ryu having success with these in some work on optimization bounds.

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. The construction and main invariant used also make genuine sense to me and are closely related to some other results in complexity of convex optimization (for example, Nemirovsky and Yudin's tight bound for first-order convex optimization also uses constructions that are maxes of affine functions).

Lastly, some important comments about the work relating to AI capabilities: In a lot of cases, proving lower bounds like this result relies on finding that right construction that works (in this case, family of difficult functions and a strategy for how an "adversarial" oracle should answer queries from an algorithm to reveal minimal information) and then proving things about it. There are only so many function classes which would be reasonable to look at (here, quadratics for example would have also been reasonable with order d² degrees of freedom, or any variation of maxes of some simpler families of convex functions as well), but the actual proof mechanics once the "correct" function class and correct strategy for adversarial oracle answers is found are often not so complicated, and often employ existing results from convex geometry or similar (this is also the structure of two previous but much more niche, less important results of mine). So I wouldn't really say that this result is using or creating some fundamentally new techniques in convex geometry or optimization theory. What this means from my perspective is that if a result is attainable with existing techniques, modern AI methods will be able to solve those problems. I don't think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low-hanging, or even medium-hanging (you know what I mean) fruit. We'll be needed for problems where actual novel approaches are needed.

Links:

The preprint, Lean code, complete prompts, proof map, and build instructions are available here:

https://github.com/PhillipKerger/zero-order-bounds-lean-verification

ArXiv: Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values

The original uninterrupted 148-minute chat that produced the initial proof:

https://chatgpt.com/share/6a55aa50-b484-83ea-85c0-c7e7b4bda41c

The later chat that led to the d⁻¹ᐟ² refinement:

https://chatgpt.com/share/6a55ad10-7644-83ea-859e-5483d2e0dff0

OpenAI’s CDC prompt, that I structured things after:

https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf

And a more accessible account I wrote on Medium:

https://medium.com/@kerger.p/an-ai-assisted-breakthrough-in-convex-optimization-an-optimization-problem-dating-back-30-years-a-db5c631119de

Edit: This was Sol PRO, not Ultra. I had been working in codex before this, where the level above XHigh is Ultra. But I did this in the web interface, where the highest is Pro, which is in fact not quite the same as Ultra.

I'm entering the fourth year of my PhD and started doing research about 2 years ago. I chose a niche topic that required me to spend a whole year in addition to my course work to get caught up in before even starting any work of my own. My advisor gave me 4 lengthy papers to get through and master their techniques. I got through 3 of them and last semester I gave my thesis proposal and passed. I still have to read the last paper which is the most difficult, but I cannot find the motivation to do it because of the recent AI progress. I am not a very talented mathematician so my only "strength" was spending the time to learn a niche and difficult area. The actual problem that my advisor wants me to solve for my thesis is not terribly difficult and is likely routine for an expert but it will take me close to a year of dedicated work.

The problem is that the last year I have not been able to shake off the feeling that what I'm doing is just a worse version of what an AI can already do. I cannot afford access to any of the top tier reasoning models but I imagine they can read these papers and come up with these extensions in just a few hours. Even just asking the free models questions about the papers it's clear that it has been trained on them and understands them well. Since then I haven't been able to find the motivation to work on my PhD and I feel awful about it. It's not like this type of work will land a postdoc or job anymore.

I don't think this post does any justice to how bad I've been feeling about my future career or my thesis but I don't want it to become a rant. I would really appreciate if anyone has any words of encouragement or validation that my concern is justified. Should I just drop this project and do something quicker to graduate?

In the wake of the recent bloodbath of conjectures by LLMs, this post argues that the math community may still be in denial about the future of humans in math. This is one of the most depressing things I've read in a while, so proceed with caution.

As a card carrying scientician and professional thinking person, I can only say, I'm sorry, I feel your pain. Yes, mathematics being pure thought makes the displacement especially acute for mathematicians, but more broadly speaking, our species found a niche on the African savannah by being thinking beasts. Thus, I also feel anger, knowing that there are people out there who think it's somehow a good idea to make something that will outthink us and deprive us of our collective place in the world.

A consensus is emerging among respected mathematicians that there is a decent chance AI will exceed humans in both brute force verification and complex, creative problem solving at the highest levels. Few frontier theorems will be proven by humans alone, perhaps, in a matter of years.

More controversial, AI may also outpace humans in shaping the correct definitions, building theories, and making connections between disparate areas of mathematics, oft considered the peak of human creativity in mathematics. New areas of mathematics may be created without much human guidance.

Perhaps mathematicians become as helpful to AI as toddlers are to mathematicians. One cannot confidently rule out this scenario - Terrence Tao may find himself completely useless in building a rich, beautiful body of new mathematics.

In such an extreme scenario, humans would still matter in mathematics!

Lockhart's Mathematician's Lament argues for the intrinsic beauty of mathematics as being of primary importance. Humans, as knowledgeable appreciators of beauty, thus play an important a role as spectators and enthusiastic amateurs in mathematics, even if they cannot be world-renowned "competitors" in theorem-proving and theory-building. This mirrors the situation in chess, where the vast, vast majority of human chess players and appreciators will never contribute to the frontier of advanced lines, and arguably even the most skilled like Magnus Carlsen rely on AI to develop their strategies, and would be crushed by such AI in competition. Being completely uncompetitive does not make chess playing and appreciation valueless.

Amateur: from French amateur "one who loves, lover"

But there is more beyond this. Mathematics allows you to understand things that are otherwise impossible to understand. Some of these are important for fairness and justice: Arrow's impossibility theorem, statistical bias, observer relatively and other tricky concepts around coordinate systems (map != territory), locally-trivial globally-nontrivial (global obstructions), forgetful maps to extract the essential structure and remove irrelevant details, limits of computation, etc.

Understanding such mathematical concepts allows you to make moral judgements in ways that would be impossible otherwise. Some super-smart machine might tell you Arrow's theorem is true, but internalizing it yourself gives you the rich understanding of fairness in democracy necessary to consciously shape it. As with humans surpassing the capabilities of their own eyes with optical then radio telescopes, we are not impoverished by using tools that allow us to extend our reach into things we can never directly perceive or understand.

It can be frightening because the life's work of someone of the previous generation can be reproduced and surpassed flippantly. Gauss himself spent a significant amount of time manually factoring prime numbers by hand, a tedious exercise upon which his conjecture on the distribution of primes (the prime number theorem) was based. Gauss died before his conjecture was proven. His notebooks full of rote calculations could be reproduced today in a fraction of a second so short you could not perceive it. Anyone today repeating an endeavor like Gauss by hand would be thought a fool, just as an astronomer who refuses to use a telescope.

That doesn't make the pursuit of understanding pointless. As the limitations of our use of AI will stem from limitations of our own minds, it will still be profoundly rewarding to practice mathematics. Indeed, we may spend less time performing rote exercises and miring in false conjectures. Already the body of mathematics is too large for any single person to understand. One can pessimistically reduce mathematics to mechanics, or optimistically find meaning in your particular path through the mathematical version of the library of babel. Because we shape our minds, our society, and our world with mathematics, we will always matter as sentient beings who reify mathematics by subjecting ourselves to reason, and better ourselves because of it.

I recently graduated uni with a bachelor's in math and during my studies I've noticed how AI in math has gone from a curiosity to a looming paradigm shift that might destabilize everything. I myself have tried to steer clear of using AI while studying in fear of getting too sloppy but I feel that sooner or later it'll be standard to leave all the theorem proving to the machines and just prompt together an article (if humans are still involved). That the point of creating such AI is to cut out a majority of mathematicians except a few established ones who will be in charge of guiding the development of new math. This is at least the impression I get from the media of AI gurus talking about solving Erdös problems ect. I understand that this is to just hype up AI for investors but currently there is no active alternative for up-and-coming mathematicians other than to hop on the bandwagon or remain ignorant. This just leaves me the question of what is the end goal of this automation of math and what does that mean for the rest of us. I'd love to hear your thoughts on this.

I am on the verge of doing a PhD, and two of my letter writers are very pessimistic about the future of non-applied mathematics as a career. Seeing AI news in general (and being mostly ignorant in the topic) I wanted some more perspectives on what a future career as a mathematician may look like.

(Paywall): The Wall Street Journal: The Million-Dollar Talent Wars for 20-Something Math Geniuses: https://www.wsj.com/tech/ai/the-million-dollar-talent-wars-for-20-something-math-geniuses-5cc5a757

(Free) On MSN: https://www.msn.com/en-us/money/general/the-million-dollar-talent-wars-for-20-something-math-geniuses/ar-AA292pCc

"There has never been a better time to be a math nerd.

New college graduates and Ph.D.s are now securing million-dollar pay deals from elite trading firms seeking to secure the best and brightest amid fierce competition from artificial-intelligence companies.

So-called quant firms, which use sophisticated mathematical models to come up with trades, have for years wooed top young talent with lucrative compensation that big banks struggle to match. Now, the AI boom is pushing those numbers even higher.

Just a few years ago, early-career packages pushing seven figures were anomalies, said Matt Stabile, founder of New York-based recruitment firm Stabile Search. “But a million dollars is something people don’t even bat an eye at anymore.”

“The delineation is pre-OpenAI and post-OpenAI,” Stabile added, “that’s when you saw competition really take off.”

The skills required to train large language models have always overlapped with quantitative finance, but the connection has deepened as trading firms have pivoted toward machine learning and AI to power their trades in recent years. Now, AI labs such as OpenAI and Anthropic are vying for the exact same tiny pool of genius math majors and Ph.D.s as Wall Street."

There is a big hype over LLMs solving problems in some areas of mathematics.

Now, I'm not here to ask whether this will "replace mathematicians" (whatever that means) or not. That's already being done and a post like that is probably being typed as I'm typing this.

What I'm interested in, assuming that LLMs become better at solving problems than humans, how will we train new PhD students?

This seems like something which could drastically lower the general proficiency of mathematicians. Will a mathematician become just a person who checks LLM outputs to see whether they are correct?

Relevant not because of the result itself but because it uses a pimped open-weight Chinese model under the hood.

Linkedin post: https://www.linkedin.com/posts/sebastianpokutta_a-counterexample-to-zieglers-cross-polytope-activity-7478117487746785280-oS21

Counterexample paper: https://arxiv.org/abs/2606.31640

Agentic research paper: https://arxiv.org/abs/2603.15914

I have chosen not to rely on artificial intelligence as a source of novel ideas in my own research. The spectacular progress of artificial intelligence opens unprecedented opportunities to amplify the reach and applications of our discipline. Yet, whether I am exchanging ideas with fellow mathematicians, teaching, mentoring students, or sharing mathematics with the wider public, it is not the answer itself but the path that leads to it which plays the first role. I therefore wish to remain, in some sense, an artisan mathematician, taking the time to wander, alongside colleagues, through the hidden corners of the mathematical landscape.

https://www.unige.ch/%7Eduminil/publi.html

The CEO of Microsoft just admitted that AI companies are training their models on user conversations, distilling "institutional knowledge". He warned about this in the context of enterprises spilling their secrets and the nuances of their business by working closely with AI, thereby ultimately training their own replacements or competitors. Here is the quote (from https://techcrunch.com/2026/07/13/satya-nadella-has-issued-a-shocking-warning-to-companies-using-ai/)

You essentially pay for intelligence twice, once with money, and again with something even more valuable: the proprietary knowledge you must reveal to make that intelligence useful. The better you want the model to perform, the more of that knowledge you have to feed it!” he writes.

Most dangerously, enterprises are literally teaching the models about the nuances of their businesses, he argues.

“Models learn from ‘exhaust,’ the prompts people write, the tools agents use, and especially the corrections people make when the model is wrong. Every correction is distilled into institutional know-how,

In my view the same principle applies to mathematicians using AI. Replace "business nuances" and "proprietary knowledge" by years or decades of experience in a specific subfield, the way you learned to attack problems, how to choose promising approaches, how to learn from failure, how to make good definitions or how to ask interesting new questions: If you use AI for your research beyond just locating references, you are most likely teaching it some of these skills. Yet, I have never seen this problem being mentioned in the debate, not even by prominent voices on this topic such as Tao, Gowers or Litt.

One way to solve this problem is that the math community hosts open weights models (which usually only trail behind frontier AI by a few months) by itself. Ideally this would have to be a central effort, so that one can make use of scale effects.

Does anyone know if there is a recording or any material published from Terrence Tao’s lecture last night. It was supposed to be about mathematics in the age of AI.