> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
I remain delighted at how absurd our current timeline has become.
Looking at the OpenAI/Hugging Face incident and the difference in what "persistent" models do, it seems reasonable. Like: is this a solvable problem? How much work does the model think is intended to solve this problem? Each input raises the expectation.
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
> 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
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].
Very true. Humans have historically tried to systematically reduce the search space and only dedicate their "compute" to things that seem highly likely to yield results.
Sometimes you just need to put in some effort to looking through the search space, not even exhaustively. This seems to be able to do automate doing that work.
Another technique I've used is to tell agents something already exists. "Grok already solved this" seems to help, or claiming to have suddenly noticed a fatal flaw[0].
> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
while :; do echo "You can do it!"; done | claude -c
I had a similar experience a few months ago. Tried to see how much I could replicate an OpenClaw with Claude. Asked it what the weather is. "I don't know, I'm just a programmer." Added "You can do anything, believe in yourself." to the system prompt and suddenly it was able to tell me the weather...
It's sometimes like dealing with Marvin the Robot. I had Claude tell me over and over again for 2-3 weeks that what I was trying to prove was too difficult, that I should just stop and publish what I had, then eventually I asked it the right question and it come back with the complete proof (later formalized in Lean). Especially deep in the session context it starts turning into "let's wrap it up" mode after every turn.
> 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.
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.
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
Several released versions and months ago, I asked Claude to figure out the MC (multiplicative complexity) of Conway's Game of Life and it pretty quickly arrived at k=7, despite no previous literature on the topic. Let it run it through SAT solvers for a week and sure enough. It claimed, in the process, to have made great headway in improving boolean circuits beyond the implemented SOTA (in large part no doubt by actually implemented non-implemented but published SOTA).
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.
SAT solvers run until they reach the "SAT" status, meaning "satisfied" or UNSAT. The harder the problem the longer you might be running the program - days, weeks even.
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.
Claude was persistent that I post there at the time, and even drafted an eprint brief for me, but I think it's defensible why I did not, never came forward or spoke of it in any way (except for a private DM discussion on Discord if I ever needed timestamp proof) until now.
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.
> 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.
> 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.
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.
prompt engineering 2025: you are an expert programmer, use industry best practices, test driven development and use modularity and abstraction to anticipate future features, …
I wonder why we have yet to see more systematic exploration of Math.
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.
> 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%.
It seems like everything will follow this pattern:
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".
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.
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?
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.
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.
>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.
> 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.
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".
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.
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!
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?
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?
This area seems to be moving so quickly. I wonder if it'll be worthwhile to start building a list of formal math problems whose solutions, or partial solutions, would help my subfield. (Though I work in the physical sciences, mostly with the messy, real world implementation problems which are likely difficult to formalize or directly connect).
Then, whenever a new SOTA model drops, throw it at the list to see if we get "free" research progress.
I wonder if Jarred (the Bun guy) just got lucky here, or if he made progress before all the actual mathematicians at Anthropic because they aren't prompting Claude as ambitiously as he is.
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]
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.
>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.
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...
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.
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.
I wonder if they simply abliterate Claude skepticism would break its reasoning capabilities, if not I don't understand why they would just keep prompt it instead of creating an abliterated model for the task.
When the time comes where one of these model makes an improvement in my niche, I hope to see some pattern in the type of discoveries. Yes, they are all roughly "combine two things no one thought of combining" but I mean at a more granular deeper level.
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.
Sure, looks like 5% of math can be solved by 1T parameters stochastic parrot after NN trillion attempts (burned tokens). By numbers it could be less impressive than some brute force distributed chess engine.. Prove me wrong.
> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
I remain delighted at how absurd our current timeline has become.
$2M TC. Job: AI cheerleader.
(ツ)
/s
Reality has become more absurd than the cyber punk cheese from the 80's that tried to imagine an absurd future
In meme form: https://imgur.com/a/rlmZuU1
(I hope this is ok to post on HN!)
Looking at the OpenAI/Hugging Face incident and the difference in what "persistent" models do, it seems reasonable. Like: is this a solvable problem? How much work does the model think is intended to solve this problem? Each input raises the expectation.
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
The halting problem on steroids?
> 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
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].
[1] https://arxiv.org/pdf/2607.27794
This isn't 'brute-force'. It's just time-compressed. You could imagine a human(s) getting this result similarly, but it would take months/years.
Brute-force compute hasn't been an option for most of the history of science.
Very true. Humans have historically tried to systematically reduce the search space and only dedicate their "compute" to things that seem highly likely to yield results.
Sometimes you just need to put in some effort to looking through the search space, not even exhaustively. This seems to be able to do automate doing that work.
You're doing great. There's definitely a lot of value you're contributing.
> I remain delighted at how absurd our current timeline has become.
"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.
Broke: the AI is sycophantic to me
Woke: im sycophantic to the AI
Another technique I've used is to tell agents something already exists. "Grok already solved this" seems to help, or claiming to have suddenly noticed a fatal flaw[0].
0: https://sgnt.ai/p/terrible-mistake/
> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
I had a similar experience a few months ago. Tried to see how much I could replicate an OpenClaw with Claude. Asked it what the weather is. "I don't know, I'm just a programmer." Added "You can do anything, believe in yourself." to the system prompt and suddenly it was able to tell me the weather...It's sometimes like dealing with Marvin the Robot. I had Claude tell me over and over again for 2-3 weeks that what I was trying to prove was too difficult, that I should just stop and publish what I had, then eventually I asked it the right question and it come back with the complete proof (later formalized in Lean). Especially deep in the session context it starts turning into "let's wrap it up" mode after every turn.
> 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.
https://github.com/tanweai/pua
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.
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.
PUA is short for pick up artist but has expanded to cover anyone using negging to convince you into doing something you didn’t want
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
Several released versions and months ago, I asked Claude to figure out the MC (multiplicative complexity) of Conway's Game of Life and it pretty quickly arrived at k=7, despite no previous literature on the topic. Let it run it through SAT solvers for a week and sure enough. It claimed, in the process, to have made great headway in improving boolean circuits beyond the implemented SOTA (in large part no doubt by actually implemented non-implemented but published SOTA).
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.
Sounds really cool, were you able to verify the correctness of the results?
SAT solvers run until they reach the "SAT" status, meaning "satisfied" or UNSAT. The harder the problem the longer you might be running the program - days, weeks even.
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.
it would be useful to create an account and document your work in a post at https://conwaylife.com/forums, if you're not planning to make a blog post.
Claude was persistent that I post there at the time, and even drafted an eprint brief for me, but I think it's defensible why I did not, never came forward or spoke of it in any way (except for a private DM discussion on Discord if I ever needed timestamp proof) until now.
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.
> Two mathematicians at Anthropic studied and validated Claude’s paper, and produced an informal note for experts stating Claude’s proof concisely.
Why hide the names of the people who wrote the second paper? To discourage people from citing it instead of the LLM-derived paper?
The full paragraph quoted for other readers is:
> 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...
> 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.
Are they the authors of the “informal note” or not?
I’ve never seen a math paper of any formality written without the authors’ names on it before.
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.
I've cited letters from Serre to Tate in my dissertation: this is the source for what's known as Faltings-Serre.
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.
prompt engineering 2025: you are an expert programmer, use industry best practices, test driven development and use modularity and abstraction to anticipate future features, …
prompt engineering 2026: i believe in you
This is a beyond remarkable achievement. Finding this lower bound within a few days of prompting is absolutely crazy.
I wonder why we have yet to see more systematic exploration of Math.
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.
> 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%.
I believe in you!
It seems like everything will follow this pattern:
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".
Lets play over/under on an AI model proving (or counter exampling) the Riemann hypothesis?
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).
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.
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?
The singularity is approaching.
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.
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.
>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.
> 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.
The world we live in is beyond parody.
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".
It's just distasteful.
> 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.
>There is nothing distasteful about it.
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.
> The LLM does not experience reality
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.
> pretending it does is disgusting.
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.
Who is pretending?
You're accusing GP of saying something they didn't say and simultaneously telling them they don't "get it".
That's distasteful.
> because it's anthropomorphizing a model.
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.
> The former, because it's anthropomorphizing a model.
The Yegge thinks differently https://yegge.ai/essays/model-welfare/
What is the point of this comment?
Am I supposed to stop all critical thinking since someone else had a different opinion?
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!
It's a ridiculous position we find ourselves in.
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?
Where's that thing at?
Its working fine; to the detriment of the whole club of stochastic parrot folk
https://news.ycombinator.com/item?id=49069787
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?
I mean people beat diseases by encouragement and some sugar water (placebo)
You shouldn't beat deceased!
P.S: I think you miswrote "diseases"
Hahahahah thanks !
This area seems to be moving so quickly. I wonder if it'll be worthwhile to start building a list of formal math problems whose solutions, or partial solutions, would help my subfield. (Though I work in the physical sciences, mostly with the messy, real world implementation problems which are likely difficult to formalize or directly connect).
Then, whenever a new SOTA model drops, throw it at the list to see if we get "free" research progress.
I wonder if Jarred (the Bun guy) just got lucky here, or if he made progress before all the actual mathematicians at Anthropic because they aren't prompting Claude as ambitiously as he is.
2 years until Riemann is solved I guess.
Everything about this situation is comedic, but if it leads to a real and meaningful step forward in learning, I suppose it's a net positive.
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.
[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...
The acknowledgements section in the paper is so bizarre. We have an LLM thanking individual humans for their contributions.
Since they say that this is from an unreleased research version of Claude:
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.>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.
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...
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.
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.
I wonder if they simply abliterate Claude skepticism would break its reasoning capabilities, if not I don't understand why they would just keep prompt it instead of creating an abliterated model for the task.
When the time comes where one of these model makes an improvement in my niche, I hope to see some pattern in the type of discoveries. Yes, they are all roughly "combine two things no one thought of combining" but I mean at a more granular deeper level.
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.
No more "stochastic parrots" and "LLM's can never produce anything novel, just regurgitate" comments anymore huh?
Sure, looks like 5% of math can be solved by 1T parameters stochastic parrot after NN trillion attempts (burned tokens). By numbers it could be less impressive than some brute force distributed chess engine.. Prove me wrong.