> 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.
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=6, 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 tried out Claude's mathematical prowess. I've been working with boolean circuits, FHE, and lean proofs ever since.
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 AI is dismissed because an expert in a particular field finds an outdated model's outputs sub-par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".
> 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.
> 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.
“ Ralph Furman
and Levent Alp¨oge studied the result in detail after the session, placed it in the context of the existing literature, checked the argument independently, and have taken responsibility for
its communication […]”
I could well imagine that writing the informal note is part of them “taking responsibility for [the argument’s] communication” but this is just speculation on my part.
prompt engineering 2025: you are an expert programmer, use industry best practices, test driven development and use modularity and abstraction to anticipate future features, …
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.
> 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%.
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.
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?
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.)
> 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.
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.
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!
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?
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.
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".
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”). 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.
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=6, 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 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.
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".
> 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.
> 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.
Doesn’t seem to say so explicitly.
From the Acknowledgements section of the paper:
“ Ralph Furman and Levent Alp¨oge studied the result in detail after the session, placed it in the context of the existing literature, checked the argument independently, and have taken responsibility for its communication […]”
I could well imagine that writing the informal note is part of them “taking responsibility for [the argument’s] communication” but this is just speculation on my part.
This is a beyond remarkable achievement. Finding this lower bound within a few days of prompting is absolutely crazy.
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 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.
> 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%.
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.
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.
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.)
2 years until Riemann is solved I guess.
> 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.
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.
I mean people beat deceases by encouragement and some sugar water (placebo)
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.
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/
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.
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?
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.
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.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".
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.
No more "stochastic parrots" and "LLM's can never produce anything novel, just regurgitate" comments anymore huh?