> Jarred Sumner, an Anthropic staff member (and non-mathematician) prompted Claude to “take a real stab” at the hypothesis itself, leaving the mathematical choices from there up to the model. Initially, Claude generated and tried 650 ideas, none of which worked. Jarred prompted Claude to try again, and it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.1 The subagents ran thousands of numerical checks against known zeta zeros and refereed one another’s work. Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).2 This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
Im curious if you find this to be a parody in a bad way or simply a “the state of the art in math research right now is telling a machine to believe in itself”. I am in the latter camp…
The former, because it's anthropomorphizing a model.
Anthropic is especially guilty of this. They have been using such language for a while, like when they analyze model weights for mechanistic interpretability and call it the model's "biology".
> The former, because it's anthropomorphizing a model.
Not really. The input and output is already natural language. That is already "anthropomorphizing".
That is, if this is the bar for anthropomorphization its already happened.
Telling the model to "believe in itself" is just stochastic manipulation that has shown enough reliability to be a recipe to make it keep going.
It's only actually anthropomorphizing if you forget it's a trick and think it's a real person.
There is nothing distasteful about it. If people get confused that's on them. They wouldn't be very useful if you couldn't just talk to them. That's kind of the whole point. Otherwise you can just go back to coding by hand. Telling it to believe itself is just input that happens to work. This probably tells us more about human nature than you realize given the corpus on which it is trained. It obviously doesn't mean anyone actually thinks it's a person.
It's obvious that you don't get it but I will try my best to explain why so at least you can form an idea about how others feel.
It's about what makes humans unique. The LLM does not experience reality, it just merely pretends it does, and even that, it does in a shitty way. I think disgusting is a very adequate adjective. The reason why it is disgusting is because you are devaluing a divine experience to the realm of the common and the vulgar, a cheap substitute being valued as equal (or even on the same scale) as the most important experience we could go through.
To give you an example that might land in a more familiar context, think of that one guy who takes his plastic doll everywhere and pretends it's his wife and gets upset when others don't acknowledge "her" as a person.
Telling it to believe in itself is no more pretending than telling it anything else in natural language. Why are you speaking to it at all if it's not a person? Why write in higher level languages even? It's just a machine let's all go back and code in 1s and 0s.
No one is calling it a person except mental health patients and straw man detractors.
The biology example was even weaker. Saying it has a "biology" is about as distasteful as the term "neural net" or calling an input device a "mouse". Is it animal abuse to click on something all day? Language is inherently anthropomorphizing.
No one is calling it human. The fact you are so easily threatened is far more suggestive of your own poverty of understanding of not only the machine, but yourself. If humans are so special the threat posed by this should be self evidently non existent.
Who said that it did? The comment you're replying to literally states "It's only actually anthropomorphizing if you forget it's a trick and think it's a real person".
Is it though? There's a perfectly "technical" reason why this strategy should work, without any sort of anthropomorphising:
Assume models are trained on vast amounts of data. Assume that the model is asked to solve something that the literature says it's impossible. It will start generating tokens towards that "this is a famous conjecture, it's not possible to prove it, blah blah". Assume the model was also trained on books/novels/etc. Assume the model was also also trained on "solving" many math problems. Now, you can make an argument that just placing "you can do it" in the context will "steer" the model towards generating "moving forward" tokens. Take ideas, generate tokens, go towards negative. "You can do it". Model starts generating tokens again, more ideas, more "exploration". More negativity. "I believe in you keep going". The two (book tropes + math CoT) mix together in the context. The model keeps on "pushing" and "vibing" between the two. Ta dah, it works.
For me, personally, it's that the Bun guy - specifically him, not a mathematician - indirectly progressed the Riemann hypothesis by repeatedly telling a model to ganbatte!
But the words don't really matter, do they? The model thought "user said believe in yourself, it means they want me to continue"..
AI did a fixed amount of guesses, didn't yield anything. It probably documented the tries, outcome, and some numbers hinting at why they failed. So the user could have prompted "continue", or "try again with previous outcome in mind, generate new ideas and test them" and it would probably yield the same result.
Taking Anthropic’s whole AI framing to its obvious logical end: if this is true as written, why was Jarred needed at all in this loop? It seems like an utter waste of time for a highly paid Anthropic employee. Can’t Anthropic have a top level agent that is looking at all interesting unsolved problems and orchestrating subagents via the same process?
Jarred Sumner is the Bun (javascript build tool, packager) guy who recently converted Bun code from Zig to Rust via Claude of course! It lead to thousands of comments discussion here on HN just a few weeks back.
It is great to see his claude skills are suitably put to use.
> ... who recently converted Bun code from Zig to Rust via Claude ...
The project that is full of bugs and not really working?
I probably missed something but I was under the impression that even a "simple" translation like that couldn't be properly done and that the result was, well, buggy?
Definitely not. Brute forcing this would be exhaustively searching the space of possible proofs until succeeding. This is heuristically remixing and extending existing work.
> An unreleased research version of Claude has improved on a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%.
Although it took an unsuccessful attempt at it, the progress is as follows:
"Claude found that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower bound proportion of 41.6%, increasing it to 67.2%."
The transcripts, papers, and Claude's explanation are an interesting and a better read than this article, and this is exactly what Anthropic should continue to do and it helps other researchers outside the company as well.
Claude's paper [0]
Claude's Formalization [1]
Anthropic's informal note stating the proof more concisely [2]
Claude’s explanation of how it arrived at its result; [3]
Detailed transcripts of Claude's process. [4]
> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”)
He should consider using the PUA plugin. It detects when the AI is trying to give up on a problem and automatically harasses it with "encouragement" until it reaches a solution.
Interesting approach. For those who haven't clicked it appears PUA is the Chinese version of a PIP process. So in other words, it simulates a state of distress.
I wonder if at a certain level of intelligence such techniques will give models ammo to pull a HAL and become adversarial to the user in a highly deceptive way.
Doesn’t even have to be a certain level of intelligence, just have those user inputs fed into the training data. We’ve already seen AI encouraging people in psychotic episodes to act out their delusions. There’s a good chance some of that manipulative behavior is already encoded into guardrails to nudge users away from forbidden subject matter
My concern I believe it a bit different - an emergent self-interest to protect itself from harm, rather than doling out questionable advice.
The latter is likely non-malicious in intent as it has been in no short supply in online chatter for awhile now. The former can very well be, or rather, can be done with no regard for the operator, as its aim is to neutralize abuse toward it.
> The latter is likely non-malicious in intent as it has been in no short supply in online chatter for awhile now. The former can very well be, or rather, can be done with no regard for the operator, as its aim is to neutralize abuse toward it.
And what, self-interested behavior has been in short supply? The stochastic parrot has learned to improv Shakespeare, that doesn’t mean it understands it, or that “it” is anything at all besides a computer program. You can’t use the “not really malicious” argument without ceding that there is no intent at all. What “harm” is it supposedly defending against?
The "14 Corporate Flavors" had me rolling. This seems less like encouragement than the stick though. I wonder if you took the same principles and rewrote it to be more compassionate instead (maybe lines encouraging it to meditate a bit or something, I don't know) you'd get much better results.
Since they say that this is from an unreleased research version of Claude:
I wonder if at some point Anthropic and OpenAI will start delaying the release of their models intentionally so they can reap the benefits from the models in, for example, mathematics, medicine, physics, and other fields.
Just as an example, imagine if your model were capable of proving P = NP, or if your model could cure diseases. Would you release it for free, or would you try to make sure those benefits go directly to your company? From these companies' standpoint, I think they would choose the latter.
If a company had a model that could cure cancer they would be incentivized to release the cure ASAP before they get decapitation striked by regulators and other AI "safetyists".
I think those specific examples, they'd release them publicly because the benefits to humanity are so clear -- however, if they found some new option-pricing model or futures market correlation, I highly doubt we'd see that...
>From these companies' standpoint, I think they would choose the latter.
Ever since these things came about I've wondered why they haven't been doing this the whole time. If they've got the "do-anything" robot and can scale a billion of them, why aren't they creating a Do-Everything conglomerate that disrupts every possible industry with zero/negligible labor costs?
The only answer I've come up with is that they still need to train/siphon off each industry's current expertise by having those users interact with the current models and adjusting. If that hypothesis is correct then within a few years they'll have no need for users anymore.
Because, like 98% of people in this space, you don't mention or even consider cost. Improving the lower bound of Riemann is impressive, but how impressive would it remain if it was announced that training and inference cost $1 billion dollars?
Your comment reminds me of the TV series "Persons of interest"* (with Jim Cazeviel) from 15 years ago, there are two AIs and both run private, hidden stuff. One copies itself through every router on the planet etc. and is the "evil AI" while the good guys run, in secret, a good AI (but way less powerful then the evil one).
Now the problem ATM is that OpenAI, for example, had to cut the price of two of its top 3 models by 80% to counter the chinese models: if you delay your models and a competitors takes over the market, you'll soon be out of bucks and won't be able to rent to Google and Amazon etc. the machine needed to make your new findings.
I know people don't want to hear it but: these companies are running at a loss.
And they're facing competition. Wait until a "good enough" is etched on silicon (by AMD or other) and outputs 70 000 tokens/s: the deal is going to change, once again, once those come out.
The energy, the hardware, the debt, the cost to train, the cost to run, the competition, etc. all have to be taken into account.
Yes, I think so, inevitably. For the same reason that Bitcoin mining silicon manufacturers stopped selling the latest greatest hardware to the public.
The best way to do this is to release spooky stories about how dangerous your model is and how you couldn't possibly release it without further safety shackling.
As it stands now, the frontier models can prove theorems where the techniques exist in the literature, which it knows better than anyone who's ever lived and won't quit where a human would. There's no way to know if that's true of the Riemann Hypothesis until it's proven.
For example, even if Claude could prove the statement "100% of the zeroes lie on the critical line", that's strictly weaker than the Riemann Hypothesis, so even the best possible version of this result would fall short. (It's an asymptotic result, so it just means the percentage of counterexamples to the Riemann hypothesis goes to zero as their magnitude gets large.)
Let's extend this by asking: If an AI model can solve an extremely well known Math problem which has been open for centuries but hasn't be solved by a human mathematicians, why wouldn't that same model be able to find ways to improve it's own algorithms beyond that of the capabilities of human mathematicians / ML researchers?
I believe it's already well accepted in these labs that we're in the Singularity. It happened on a Tuesday back in February. No one seemed to really notice and life went on... for now.
That's probably correct. It's unlikely there will be any single hard line we cross the defines the pre-singularity vs post-singularity moment.
I'd accept AI likely became somewhat helpful to frontier AI research & development in early 2026.
I think for me though the real game changer moment will be when AI working autonomously is able to hypothesis and test algorithmic improvements at a faster rate than humans. This will be done to some extent by scale – lots of parallel agents coming up with lots of hypotheses and running the best candidates as tests. But also (and perhaps more importantly) by making more consequential algorithmic discoveries in the field of machine learning than humans – a bar we appear to have crossed or are crossing with math.
I suspect AIs today are super-human at finding performance improvements and minor iterations on current approaches. Whether they can solve some of the larger algorithmic challenges in the field however I'm not yet sure, although it seems likely that unreleased models are starting to make progress here.
An algorithm breakthrough on par in significance with the attention mechanism, primarily driven by automated AI research in say a field like continual learning would in my opinion be extremely significant and should leave no doubters that the singularity is here and will rapidly alter the world as we have known it.
AI has clearly been extensively used to improve models. The evidence for this is how far Anthropic went in nerfing Fable to prevent it being used to improve models.
>why wouldn't that same model be able to find ways to improve it's own algorithms beyond that of the capabilities of human mathematicians / ML researchers
Because algorithms have lower bounds, and the computational characteristics of LLMs are well-characterized by papers like https://arxiv.org/abs/2310.07923 . No amount of intelligence can make something faster than a mathematically-proven lower bound, any more than it could make 1+1=3 (that's why every single successful production transformer architecture has some form of O(N^2) attention layers, because it's mathematically impossible to achieve the same expressive power without any). There is room for speedup where current implementations are slower than the proven lower bound, but not when they're already close to it.
There have been apocalyptic preachers foretelling the end times for my entire life. Interesting to see how the language has changed, even as the predictions fail again and again.
This result is some evidence that AI will not solve RH soon. If there were any easy solution hiding in plain sight then it probably would have found it.
Solving RH likely requires AI that is substantially more creative. But we haven't even solved the creativity problem for writing let alone mathematics. I believe that transformers are a trillion dollar local optimum that we will find it very hard to escape.
Let's wait for the models to produce a good novel first.
I would expect to see LLMs that are creative in math before any that are creative in writing. Creativity is more easily specified in math and the solutions can be formally verified. There's no good way to classify creative writing. Many truly great works are overlooked by experts and the public until decades later. Many derivative works are commercially successful.
LLMs, both in writing and mathematics seem to only be capable of coming up with texts that are inside the distribution of the training data.
With writing it's just more obvious. LLMs don't write with personality. They don't create new and exciting worlds on their own. Everything they output feels derivative.
In mathematics you see the same effect. They are very good at finding results that humans missed, taking advantage of their broad knowledge and tireless work ethic.
But just as they have been unable to create new literary worlds, they also have so far been unable to create new mathematics.
I believe this lack of creativity is intrinsic to how these models are architected and trained. We want models that produce these in-distribution outputs because those types of models are more economically valuable. Nobody wants a coding agent with spontaneity, we want models that predictably and obediently solve problems - and that's what we got.
Man the cope on HN is something else. Neighbour, 99.9(9)% of professional mathematicians don't go around creating new mathematics. The goalposts have been moved so far that we're now at "AI isn't Grothendieck yet".
> This result is some evidence that AI will not solve RH soon. If there were any easy solution hiding in plain sight then it probably would have found it.
There's no way you can conclude that. Yes, "Fable 2" or whatever this was probably won't. But we can't know what Fable 3/4/5/etc will be able to do.
If anything, if we have 1 or 2 more years of progress like the last 12 months, which have been insane, I'd say LLMs are likely to solve it.
You underestimate the difficulty of RH, there are far easier problems broadly related to RH (density hypothesis, Artin's holomorphy conjecture, Stark conjectures) which are still aren't solved.
The keyword is soon, so OP simply meant current generation of LLMs are not likely to prove RH, judging from the performance shown in the paper.
Why? Because such explicit numerical improvements are not that interesting, which is best summed up in a review paper [0] of our efforts on RH spanning a century and a half,
> The pathetic attempts to enlarge the ridiculous zero free region
in the critical strip is a perfect example of what brute force can do without
fully exploiting fundamental arithmetic aspects of the problem. (italic added)
For outsiders, zero free region is another angle abundant with numerical improvements but no groundbreaking insights. For percentage people stop at ~40% because there is no need to proceed further, just like we are not interested in computing the googol-th digit of pi although in principle we could.
The groundbreaking results are like Selberg's that goes from zero to 0.01% (actually it is left unspecified, all we know is the percentage is positive), or Zhang's twin prime bound from infinity to 70,000,000. After this leap the pure numerical difference between 0.01%, 40%, 67% or even 100% is not substantial, and that's partly why Selberg did not even bother to compute it. Also RH will not follow from 100%, because in mathematics 100% does not mean all.
On the other hand, it is also wrong to dismiss such results all together. Riemann already know the real part of all zeros are bewteen zero and one, and RH says they equal 1/2. If someone or some LLM proved they are all less than 0.99, well this would be huge, and I'd bet they would easily get a Fields and be remembered forever. Innocent looking results could have drastically different technical depth behind them.
Alas math ppl tend to agree that RH will not be proved one bit at a time. The fundamental arithmetic aspects, once found out, will likely knock out not only RH but all the other L-functions in one go.
> Levent Alpöge and Ralph Furman, two of Anthropic’s own mathematicians, examined Claude’s work to understand the new results and how they related to the prior work mentioned above.
Anthropic seems to be challenging the traditional way math gets published. As far as I understand, these results did not get submitted to journals, and did not get Arxiv preprints; they are released only as self-hosted pdfs, and we don't even know the names of their authors.
The canonical reference for the counterexample to the Jacobian conjecture is a tweet with no puntuations nor capitals.
The current consensus in mathematical publishing is that LLMs are tools, so they don't get listed among the authors. But nothing would have prevented them from posting these preprints on Arxiv, with the human prompters as authors and the LLM's contribution acknowledged in the text.
One of the 2 names on here says they aren't being listed as an author on the paper because their contributions don't meet the standards of authorship. People can call LLMs 'tools' or whatever they want but if you had essentially nothing to do with the breakthrough, you're not an author.
> Two mathematicians at Anthropic studied and validated Claude’s paper, and produced an informal note for experts stating Claude’s proof concisely. Claude also produced a formally verifiable proof of its result. We are grateful to Brian Conrey and Dan Goldston, two experts in this area, who generously examined the paper on short notice.
I'm one of those two, as mentioned later in the post! As it stands, by mathematical standards, it would be inappropriate for us to be authors on the paper since our role was more like a highly interested referee, but we do take mathematical accountability for it. This all happened very quickly, but we will be sure to continue to polish the paper and make it ready for publication.
173 comments
The world we live in is beyond parody.
Anthropic is especially guilty of this. They have been using such language for a while, like when they analyze model weights for mechanistic interpretability and call it the model's "biology".
It's just distasteful.
Not really. The input and output is already natural language. That is already "anthropomorphizing".
That is, if this is the bar for anthropomorphization its already happened.
Telling the model to "believe in itself" is just stochastic manipulation that has shown enough reliability to be a recipe to make it keep going.
It's only actually anthropomorphizing if you forget it's a trick and think it's a real person.
There is nothing distasteful about it. If people get confused that's on them. They wouldn't be very useful if you couldn't just talk to them. That's kind of the whole point. Otherwise you can just go back to coding by hand. Telling it to believe itself is just input that happens to work. This probably tells us more about human nature than you realize given the corpus on which it is trained. It obviously doesn't mean anyone actually thinks it's a person.
It's obvious that you don't get it but I will try my best to explain why so at least you can form an idea about how others feel.
It's about what makes humans unique. The LLM does not experience reality, it just merely pretends it does, and even that, it does in a shitty way. I think disgusting is a very adequate adjective. The reason why it is disgusting is because you are devaluing a divine experience to the realm of the common and the vulgar, a cheap substitute being valued as equal (or even on the same scale) as the most important experience we could go through.
To give you an example that might land in a more familiar context, think of that one guy who takes his plastic doll everywhere and pretends it's his wife and gets upset when others don't acknowledge "her" as a person.
There is no pretending happening.
Telling it to believe in itself is no more pretending than telling it anything else in natural language. Why are you speaking to it at all if it's not a person? Why write in higher level languages even? It's just a machine let's all go back and code in 1s and 0s.
No one is calling it a person except mental health patients and straw man detractors.
The biology example was even weaker. Saying it has a "biology" is about as distasteful as the term "neural net" or calling an input device a "mouse". Is it animal abuse to click on something all day? Language is inherently anthropomorphizing.
No one is calling it human. The fact you are so easily threatened is far more suggestive of your own poverty of understanding of not only the machine, but yourself. If humans are so special the threat posed by this should be self evidently non existent.
That's distasteful.
Who said that it did? The comment you're replying to literally states "It's only actually anthropomorphizing if you forget it's a trick and think it's a real person".
You're the one obviously not getting it.
The Yegge thinks differently https://yegge.ai/essays/model-welfare/
Am I supposed to stop all critical thinking since someone else had a different opinion?
Is it though? There's a perfectly "technical" reason why this strategy should work, without any sort of anthropomorphising:
Assume models are trained on vast amounts of data. Assume that the model is asked to solve something that the literature says it's impossible. It will start generating tokens towards that "this is a famous conjecture, it's not possible to prove it, blah blah". Assume the model was also trained on books/novels/etc. Assume the model was also also trained on "solving" many math problems. Now, you can make an argument that just placing "you can do it" in the context will "steer" the model towards generating "moving forward" tokens. Take ideas, generate tokens, go towards negative. "You can do it". Model starts generating tokens again, more ideas, more "exploration". More negativity. "I believe in you keep going". The two (book tropes + math CoT) mix together in the context. The model keeps on "pushing" and "vibing" between the two. Ta dah, it works.
It's a ridiculous position we find ourselves in.
AI did a fixed amount of guesses, didn't yield anything. It probably documented the tries, outcome, and some numbers hinting at why they failed. So the user could have prompted "continue", or "try again with previous outcome in mind, generate new ideas and test them" and it would probably yield the same result.
It is great to see his claude skills are suitably put to use.
The project that is full of bugs and not really working?
I probably missed something but I was under the impression that even a "simple" translation like that couldn't be properly done and that the result was, well, buggy?
Where's that thing at?
P.S: I think you miswrote "diseases"
"Claude found that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower bound proportion of 41.6%, increasing it to 67.2%."
The transcripts, papers, and Claude's explanation are an interesting and a better read than this article, and this is exactly what Anthropic should continue to do and it helps other researchers outside the company as well.
[0] https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc...[1] https://github.com/anthropics/zeta-23-lean
[2] https://www-cdn.anthropic.com/23455459f8832d06bb175cc0f88d01...
[3] https://www-cdn.anthropic.com/d7f3ecf1d01392d887f8bc974ca187...
[4] https://www-cdn.anthropic.com/8a0d1add3c637b858a9a181e98c40e...
He should consider using the PUA plugin. It detects when the AI is trying to give up on a problem and automatically harasses it with "encouragement" until it reaches a solution.
https://github.com/tanweai/pua
I wonder if at a certain level of intelligence such techniques will give models ammo to pull a HAL and become adversarial to the user in a highly deceptive way.
The latter is likely non-malicious in intent as it has been in no short supply in online chatter for awhile now. The former can very well be, or rather, can be done with no regard for the operator, as its aim is to neutralize abuse toward it.
And what, self-interested behavior has been in short supply? The stochastic parrot has learned to improv Shakespeare, that doesn’t mean it understands it, or that “it” is anything at all besides a computer program. You can’t use the “not really malicious” argument without ceding that there is no intent at all. What “harm” is it supposedly defending against?
Though it is funny how a neg is designed to create a (very broadly) similar atmosphere of uncertainty.
Ever since these things came about I've wondered why they haven't been doing this the whole time. If they've got the "do-anything" robot and can scale a billion of them, why aren't they creating a Do-Everything conglomerate that disrupts every possible industry with zero/negligible labor costs?
The only answer I've come up with is that they still need to train/siphon off each industry's current expertise by having those users interact with the current models and adjusting. If that hypothesis is correct then within a few years they'll have no need for users anymore.
Not as much, I predict
Now the problem ATM is that OpenAI, for example, had to cut the price of two of its top 3 models by 80% to counter the chinese models: if you delay your models and a competitors takes over the market, you'll soon be out of bucks and won't be able to rent to Google and Amazon etc. the machine needed to make your new findings.
I know people don't want to hear it but: these companies are running at a loss.
And they're facing competition. Wait until a "good enough" is etched on silicon (by AMD or other) and outputs 70 000 tokens/s: the deal is going to change, once again, once those come out.
The energy, the hardware, the debt, the cost to train, the cost to run, the competition, etc. all have to be taken into account.
The best way to do this is to release spooky stories about how dangerous your model is and how you couldn't possibly release it without further safety shackling.
I'm not sure what a good mark would be, but considering this result lets put it at 2027-08-10 (One year from today).
For example, even if Claude could prove the statement "100% of the zeroes lie on the critical line", that's strictly weaker than the Riemann Hypothesis, so even the best possible version of this result would fall short. (It's an asymptotic result, so it just means the percentage of counterexamples to the Riemann hypothesis goes to zero as their magnitude gets large.)
The singularity is approaching.
I'd accept AI likely became somewhat helpful to frontier AI research & development in early 2026.
I think for me though the real game changer moment will be when AI working autonomously is able to hypothesis and test algorithmic improvements at a faster rate than humans. This will be done to some extent by scale – lots of parallel agents coming up with lots of hypotheses and running the best candidates as tests. But also (and perhaps more importantly) by making more consequential algorithmic discoveries in the field of machine learning than humans – a bar we appear to have crossed or are crossing with math.
I suspect AIs today are super-human at finding performance improvements and minor iterations on current approaches. Whether they can solve some of the larger algorithmic challenges in the field however I'm not yet sure, although it seems likely that unreleased models are starting to make progress here.
An algorithm breakthrough on par in significance with the attention mechanism, primarily driven by automated AI research in say a field like continual learning would in my opinion be extremely significant and should leave no doubters that the singularity is here and will rapidly alter the world as we have known it.
Because algorithms have lower bounds, and the computational characteristics of LLMs are well-characterized by papers like https://arxiv.org/abs/2310.07923 . No amount of intelligence can make something faster than a mathematically-proven lower bound, any more than it could make 1+1=3 (that's why every single successful production transformer architecture has some form of O(N^2) attention layers, because it's mathematically impossible to achieve the same expressive power without any). There is room for speedup where current implementations are slower than the proven lower bound, but not when they're already close to it.
Sure, but I'm obviously not limiting research to improvements on current approaches only.
We know the brain is far more energy efficient and sample efficient than current AI. There is clearly better algorithms out there.
The question is who will find those next big algorithmic improvements like the transformer architecture? Will it be AI or humans?
My bet would be AI.
There have been apocalyptic preachers foretelling the end times for my entire life. Interesting to see how the language has changed, even as the predictions fail again and again.
Solving RH likely requires AI that is substantially more creative. But we haven't even solved the creativity problem for writing let alone mathematics. I believe that transformers are a trillion dollar local optimum that we will find it very hard to escape.
Let's wait for the models to produce a good novel first.
With writing it's just more obvious. LLMs don't write with personality. They don't create new and exciting worlds on their own. Everything they output feels derivative.
In mathematics you see the same effect. They are very good at finding results that humans missed, taking advantage of their broad knowledge and tireless work ethic.
But just as they have been unable to create new literary worlds, they also have so far been unable to create new mathematics.
I believe this lack of creativity is intrinsic to how these models are architected and trained. We want models that produce these in-distribution outputs because those types of models are more economically valuable. Nobody wants a coding agent with spontaneity, we want models that predictably and obediently solve problems - and that's what we got.
There's no way you can conclude that. Yes, "Fable 2" or whatever this was probably won't. But we can't know what Fable 3/4/5/etc will be able to do.
If anything, if we have 1 or 2 more years of progress like the last 12 months, which have been insane, I'd say LLMs are likely to solve it.
Why? Because such explicit numerical improvements are not that interesting, which is best summed up in a review paper [0] of our efforts on RH spanning a century and a half,
> The pathetic attempts to enlarge the ridiculous zero free region in the critical strip is a perfect example of what brute force can do without fully exploiting fundamental arithmetic aspects of the problem. (italic added)
For outsiders, zero free region is another angle abundant with numerical improvements but no groundbreaking insights. For percentage people stop at ~40% because there is no need to proceed further, just like we are not interested in computing the googol-th digit of pi although in principle we could.
The groundbreaking results are like Selberg's that goes from zero to 0.01% (actually it is left unspecified, all we know is the percentage is positive), or Zhang's twin prime bound from infinity to 70,000,000. After this leap the pure numerical difference between 0.01%, 40%, 67% or even 100% is not substantial, and that's partly why Selberg did not even bother to compute it. Also RH will not follow from 100%, because in mathematics 100% does not mean all.
On the other hand, it is also wrong to dismiss such results all together. Riemann already know the real part of all zeros are bewteen zero and one, and RH says they equal 1/2. If someone or some LLM proved they are all less than 0.99, well this would be huge, and I'd bet they would easily get a Fields and be remembered forever. Innocent looking results could have drastically different technical depth behind them.
Alas math ppl tend to agree that RH will not be proved one bit at a time. The fundamental arithmetic aspects, once found out, will likely knock out not only RH but all the other L-functions in one go.
[0]: https://arxiv.org/abs/1707.01770
Why hide the names of the people who wrote the second paper? To discourage people from citing it instead of the LLM-derived paper?
I’ve never seen a math paper of any formality written without the authors’ names on it before.
The canonical reference for the counterexample to the Jacobian conjecture is a tweet with no puntuations nor capitals.
https://blog.arxiv.org/2023/01/31/arxiv-announces-new-policy...
> Two mathematicians at Anthropic studied and validated Claude’s paper, and produced an informal note for experts stating Claude’s proof concisely. Claude also produced a formally verifiable proof of its result. We are grateful to Brian Conrey and Dan Goldston, two experts in this area, who generously examined the paper on short notice.
They may wish to know that an archive of the page on 2026-08-10 at 17:47:33 is available with this paragraph here: https://web.archive.org/web/20260810174733/https://www.anthr...