Learning more about Claude's mathematical capabilities
anthropic.com
anthropic.com
I remain delighted at how absurd our current timeline has become.
I would love to see the breakdown on token spend between each or Jared’s “go on spend more tokens, continue experiment, believe in yourself”
The world collectively spends trillions training, and entertaining and then watching people kick and hit balls around but somehow working with AI is preposterous for less money.
I don’t understand why people think describing something in some ostensibly dismissive way constitutes making a point. But some people spend their time banging keys with letters on them (or not with letters on them) into input fields and then pressing Return, so I guess some subset of those people will do that.
Text is an imperfect distillation of the human experience, but it doesn’t surprise me that a machine trained on human text responds to human encouragement.
And then finally both model output and human input become one world frame for the model, and the human adding a "you can do it!" isn't just input but a frame that colors not just the next step for the model, but also all previous steps (since at each step the model is viewing the totality of the transcript).
That this makes sense only makes it all the more absurd
If there is anything to learn from the history of science, it is that breakthroughs happen via better or new theory and not by brute-force compute [1].
Moreover, frontier models have some degree of reasoning. So, I don’t think the gap is all that big.
[1] https://en.wikipedia.org/wiki/The_Structure_of_Scientific_Re...
that might just be an illusion. humanity as a whole applies brute force by just trying all the reasonable new theories until one of us discovers something that works and declares it a breakthrough
(I hope this is ok to post on HN!)
Woke: im sycophantic to the AI
"delighted" is doing a LOT of work there, tbh ¯\_(ツ)_/¯
I do share @simonW's skepticism though. (His blog is my essential reading, FWIW)
On the actual blog post, I'd would be more enthusiastic if Anthropic showed us if the results were repeatable, reproducible, and consistent.
"Sometimes, magic is just someone spending more time on something than anyone else might reasonably expect." -- Teller (of Penn & Teller)
"Sometimes, any sufficiently advanced technology is just spending more time on something than anyone else might reasonably expect." -- an LLM's original thought, probably
LLMs have enabled brute forcing of math in a way that classical computing somehow has not, which is fascinating in and of itself.
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.
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.
Any human mathematician would be thrilled to prove a result like this, and it's not even big news anymore that an LLM can do it.
So far I only found decent discussions about it in Chinese.
https://www.zhihu.com/question/2070336637360518307/answer/20...
The general public is not interested in the mathematical details, it is interested in the fact this is unique. And it is no longer unique.
prompt engineering 2026: i believe in you
First of all, it will convince itself that a task is just too hard and find excuses not to try hard enough.
Then when it struggles on something it loves to write down confusing notes about things which it believes cases the problem. Then next iteration it reads its own note, misinterprets it and completely messes up by trying to avoid some imaginary problem.
it sounds like this might benefit from a ui that helps to edit/re-write the history
I also think this would make sense for programming but there it is a bit harder to justify the effort
when you are working on something that really matters this can make the difference though
ty for sharing!
And that was just the first time I really tried out Claude's mathematical prowess. I've been working with boolean circuits, FHE, and lean proofs ever since.
So none of this suprises me.
Ideally, what you want is a single SAT value among a remainder universe of UNSATs.
Sometimes the best you can achieve at any given point is a lower bound and an upper bound range, like "greater than 3 but less than 9."
Of course I simplified in my post but it started out with a pretty broad range of a lower and upper bound, then narrowed further, then narrowed further, then narrowed further, etc...until the specific final result achieved K=7=SAT while every K<7=UNSAT & every K>7=UNSAT. I think it ran for a full week alone on K between 6 and 7.
As amazing as Claude is to seemingly make unprecedented progress, it is even more likely to blow the most insane levels of smoke up your ass before you've legitimately reached that point.
"You should publish right now! Don't wait! There is no reason to wait!"
Like seriously, Claude was outputting something closely resembling (non?)peer pressure on me to not just keep this information to myself - and this was before all the recent math-related breakthroughs started becoming public.
It was also - most notably - before it had actually verified what it was saying it had calculated. I was the one pushing for more verification, more contemplation, more proofs of claims. And though Claude is better at this stuff now, it's definitely not not still happening.
I think I made the right choice then and I will consider being more open now that others have taken the burden of proving that, no it can actually sometimes do the incredible things its claimed its done for you.
My wife remains skeptical - she is/was seriously concered that I was under AI psychosis for believing that I had made such progress - and I can't even fault her for that. It sounds crazy to say it.
If anyone is reading this and is actively involved with FHE, especially someone from Zama or related group, I'd very much love to chat privately. I have many other "innovations" I've been working on since.
someone who loves Game of Life and is also technically capable if they were so inclined, is more likely to want to collect stuff like this for GoL specifically and build a system for that niche.
the combined GoL/technical community can vouch for things - the greater populace can see what the technical GoL community has vouched for/identified as serious work.
just my two cents on top of your thoughtful comment.
"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]
[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...
Anthropic describes that Claude identified a set of possibilities and then explored them using sub-agents. The human saying "I believe in you" could literally just be something along lines of a harness with a /goal loop.
We all identify this as absurd because... it's so lacking in rigor despite making major progress. What if we just applied a little more rigor? Ask the model to identify many possibilities, encode them, fan it out to other agents, loop them all, collect the results, etc. Then what happens? It feels like we have weak AGI and a decent system for discovery could transform it into weak ASI. That in turn could yield strong AGI and so on. I suppose that's what the Discovery Loop announcement was all about.
It just shifts the work from
> reading the (natural language) proof and confirming it has no errors
to
> reading the Lean code and confirming it correctly encodes the theorem
For example here is a statement of the Pythagorean theorem in Lean:
theorem EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ : P) : dist p₁ p₃ * dist p₁ p₃ = dist p₁ p₂ * dist p₁ p₂ + dist p₃ p₂ * dist p₃ p₂ <-> angle p₁ p₂ p₃ = Real.pi / 2
This is just one possible way of formalizing it and it depends on other definitions, wherein you also need to understand the assumptions they make, etc.
Answering the question of “whether proving this theorem in Lean proves the Pythagorean theorem” thus requires expert judgement as well as domain knowledge of Lean’s libraries.
So if the AI says “this theorem is true, here is the proof in Lean” it’s still possible that it’s not correct, even if the Lean code compiles. The result will still be in question until a human expert reviews it.
> 60 Claude subagents
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"
while :; do echo "You can do it!"; done | claude -c
My thinking here is that Claude's "self concept" for what is easy and what is hard comes from human training data. Its ideas about what is hard and easy come from humans, and much of that doesn't apply to transformers at all.
I also often have the opposite problem, where I'll use AI to compensate for the fact that I can't process a lot of information simultaneously, but they'll treat me like a transformer and give me 17 research projects in response to a single question.
They seem to be oblivious to the fact that humans don't have infinite working memory.
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...
Then, whenever a new SOTA model drops, throw it at the list to see if we get "free" research progress.
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.
It is really the same with any art form.
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.
1. AI is dismissed because an expert in a particular field finds an outdated model's outputs sub-par
2. New model, released or unreleased, makes a major stride in that field
3. Expert either recants and becomes AI-pilled, or claims it is just an artifact of the broad search space available to AI, and "no new knowledge was created".
We went from AI being human sycophants to humans becoming AI sycophants.
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.
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.
It didn't do this whole 41.6->67.2% jump by itself, humans had done most of the work and it came in at the end and found a way to remove the condition. Impressive, but not as massively impressive as when it sounds like it did the jump by itself.
This isn't goalpost moving, it's clarifying what exactly happened bc at first I thought it had made the jump by itself. The blog post is written in a technically correct, but misleading way where it takes credit for the whole jump.
Edited to add: Found a conversation between actual mathematicians that probably could be used to track this down, but at the moment I'd rather go to sleep than pursue it further. https://xcancel.com/ElliotGlazer/status/2086950873523986440
I want to dive into the "data" and then see if it's possible to distill this skill into small models that are "benchmaxxed" for this type of work, maybe in limited domains, similar to small models being benchmaxxed(I don't mean this in a bad way) for coding these days.
That's hilarious. Maybe I do need to glaze the LLM a bit more in the AGENTS.md
Also, I am not expert in this specific topic, but base on comments looks like claude parroted few recent works?
the novelty remains a structural problem with how they work and what the weights try to predict