> Filip also told me that he asked Claude to continue on the even case after the odd case had been resolved. “But there after a while it seemed to get stuck. In the end, it was not even able to write and run explore programs correctly anymore, very weird. So I stopped the search.”
Interesting snippet towards the end. I wonder if they were using claude.ai or claude code. Sounds like they ran out of context and entered the "dumb zone."
It's fascinating to think about the space of problems which are amenable to RL scaling of these probability distributions.
Before, we didn't have a fast (we had to rely on human cognition) way to try problems - even if the techniques and workflows were known by someone. Now, we've baked these patterns into probability distributions - anyone can access them with the correct "summoning spell". Experts will naturally use these systems more productively, because they know how to coerce models into the correct conditional distributions which light up the right techniques.
One question this raises to me is how these models are going to keep up with the expanding boundary of science. If RL is required to get expert behavior into the models, what happens when experts start pushing the boundary faster? In 2030, how is Anthropic going to keep Claude "up-to-date" without either (a) continual learning with a fixed model (expanding context windows? seems hard) or (b) continual training (expensive)?
> In 2030, how is Anthropic going to keep Claude "up-to-date"
I think the majority of research, design and learning goes through LLMs and coding agents today, considering the large user base and usage it must be trillions of tokens per day. You can take a long research session or a series of them and apply hindsight - what idea above can be validated below? This creates a dense learning signal based on validation in real world with human in the loop and other tools, code & search.
Data sharing agreements permitting, today's inference runs can be tomorrow's training data. Presumably the models are good enough at labeling promising chains of thought already.
I could totally imagine "free" inference for researchers under the condition that the reasoning traces get to be used as future training data.
Agreed, there's no doubt this will happen. It's likely already happening (it feels safe to assume that Anthropic is curating data from the data they record from Claude Code?)
As far as I understand RL scaling (we've already maxxed out RLVR), these machines only get better as long as they have expert reasoner traces available.
Having an expert work with an LLM and successfully solve a problem is high signal data, it may be the only path forward?
My prior is that these companies will take this data without asking you as much as they can.
Exactly, or functionally equivalently, asking you in paragraph 37 of a 120-page PDF (bonus points: in an agreement update).
And importantly, this can be cross-lab/model too. I suspect there's a reason why e.g. Google has been offering me free Claude inference in Google Antigravity on a free plan...
> Data sharing agreements permitting, today's inference runs can be tomorrow's training data. Presumably the models are good enough at labeling promising chains of thought already.
This is the most fundamental argument that they are not, directly, an intelligence. They are not ever storing new information on a meaningful timescale. However, if you viewed them on some really large macro time scale where now LLMs are injecting information into the universe and the re-ingesting that maybe in some very philosophical way they are a /very/ slow oscillating intelligence right now. And as we narrow that gap (maybe with a totally new non-LLM paradigm) perhaps that is ultimately what gen AI becomes. Or some new insight that lets the models update themselves in some fundamental way without the insanely expensive training costs they have now.
That is a good area to explore. Their map of the past is fixed. They are frozen at some point in their psychological time. What has stopped working? Their hippocampus and medial temporal lobe. These are like the write-head that move data from the hippocampus to the neo cortex. Their "I" can no longer update itself. Their DMN is frozen in time. So if intelligence is purely the "I" telling a continuous coherent story about itself. The difference is that although they are fixed in time which is a characteristic shared by a specific LLM model. They can still completely activate their task positive network for problem solving and if their previous information stored is adequate to solve the problem they can. You could argue that is pretty similar to an LLM and what it does. So it is certainly a signifiant component of intelligence.
There is also the nature of the human brain, it is not just those systems of memory encoding, storage, and use of that in narratives. People with this type of amnesia still can learn physical skills and that happens in a totally different area of the brain with no need for the hippocampus->neocortex consolidation loop. So, the intelligence is significantly diminished, but not entirely. Other parts of the brain are still able to update themselves in ways an LLM currently cannot. The human with amnesia also has a complex biological sensory input mapping that is still active and integrating and restructuring the brain. So, I think when you get into the nuances of the human in this state vs. an LLM we can still say the human crosses some threshold for intelligence where the LLM does not in this framework.
So, they have an "intelligence", localized to the present in terms of their TPN and memory formation. LLMs have this kind of "intelligence". But the human still has the capacity to rewire at least some of their brain in real time even with amnesia.
A very good point. For anyone not familiar with anterograde amnesia, the classical case is patient H.M. (https://en.wikipedia.org/wiki/Henry_Molaison), whose condition was researched by Brenda Milner.
I thought maybe people would be curious to read about how we came to understand the condition and the history behind it, as well as any associated information. Forgive me for such a deep transgression as this assumption.
Sure, why can't both things be true? "Intelligence" is just what you call something and someone else knows what you mean. Why did AI discourse throw everyone back 100 years philosophically? Its like post-structuralism or Wittgenstein never happened..
It's so much less important or interesting to like nail down some definition here (I would cite HN discourse the past three years or so), than it is to recognize what it means to assign "intelligent" to something. What assumptions does it make? What power does it valorize or curb?
Each side of this debate does themselves a disservice essentially just trying to be Aristotle way too late. "Intelligence" did not precede someone saying it of some phenomena, there is nothing to uncover or finalize here. The point is you have one side that really wants, for explicit and implicit reasons, to call this thing intelligent, even if it looks like a duck but doesn't quack like one, and vice versa on the other side.
Either way, we seem fundamentally incapable of being radical enough to reject AI on its own terms, or be proper champions of it. It is just tribal hypedom clinging to totem signifiers.
There's nothing to say that you can't build something intelligent out of them by bolting a memory on it, though.
Sure, it's not how we work, but I can imagine a system where the LLM does a lot of heavy lifting and allows more expensive, smaller networks that train during inference and RAG systems to learn how to do new things and keep persistent state and plan.
You aren't wrong and that is a fascinating area of research. I think the key thing is that the memory has to fundamentally influence the underlying model, or at least the response, in some way. Patching memory on top of an LLM is different from integrating it into the core model. To go back to human terms it is like an extra bit of storage, but not directly attached to our neo cortex. So it works more like a filter than a core part of our intelligence in the analogy. You think about something and assemble some thought and then it would go to this next filter layer and get augmented and that smaller layer is the only thing being updated.
It is still meaningful, but it narrows what the intelligence can be sufficiently that it may not meet the threshold. Maybe it would, but it is probably too narrow. This is all strictly if we ask that it meet some human-like intelligence and not the philosophy of "what counts as intelligence" but... we are humans. The strongest things or at least the most honest definitions of intelligence I think exist are around our metacognitive ability to rewire the grey matter for survival not based on immediate action-reaction but the psychological time of analyzing the past to alter the future.
Memory is not just bolted on top of the latest models. They under go training on how and when to effectively use memory and how to use compaction to avoid running out of context when working on problems.
Their consolidation of memory speed is what I was referring to. The model iterations are essentially their form of collective memory. In the sense of the human model of intelligence we have thoughts. Thoughts become memory. New thoughts use that memory and become recursively updated thoughts. LLMs cannot update their memory very fast.
Not an expert but surely it's only a matter of time until there's a way to update with the latest information without having to retrain on the entire corpus?
> Experts will naturally use these systems more productively, because they know how to coerce models into the correct conditional distributions which light up the right techniques.
Part of it comes down to “knowing” what questions to ask.
I see it like the relationship between a student and research advisor. The advisor will ideally know the terrain and suggest a fruitful line of attack (what to ask), and the student will follow through, learning along the way.
From my naive standpoint, LLMs like this seem to have some big strengths. One: possession of a superhuman expanse of knowledge. Two: making connections. Three: tireless trial and error.
If you put those three things together, you end up with some cool stuff from time to time. Perhaps the proof of P!=NP is tied to an obscure connection that humans don't easily see due to individual lack of knowledge or predisposition of bias.
Unless my understanding is incorrect about how these tools work that last point isn't really a quality of LLMs as such? It gets attributed because the lines are blurred but the tireless trial and error is actually just a quality of a regular programatic loop (agent/orchestrator) that happens to be doing the trickiest part of its work via an LLM.
I wonder how long we have until we start solving some truly hard problems with AI. How long until we throw AI at "connect general relativity and quantum physics", give the AI 6 months and a few data centers, and have it pop out a solution?
I think a very long time because part of our limit is experiment.
We need enough experimental results to explain to solve these theoretical mismatches and we don't and at present can't explore that frontier.
Once we have more results at that frontier we'd build a theory out from there that has two nearly independent limits for QFT and GR.
What we'd be asking if the AI is something that we can't expect a human to solve even with a lifetime of effort today.
It'll take something in par with Newton realising that the heavens and apples are under the same rules to do it. But at least Newton got to hold the apple and only had to imagine he could a star.
> I think a very long time because part of our limit is experiment.
Yes, maybe. But if you are smarter, you can think up better experiments that you can actually do. Or re-use data from earlier experiments in novel and clever ways.
What prevents us from giving this system access to other real systems that live in physical labs? I don't see much difference between parameterizing and executing a particle accelerator run and invoking some SQL against a provider. It's just JSON on the wire at some level.
In 1900 Henri Poincaré wrote that radiation (light) has an effective mass given by E/c^2.
So it really isn't far fetched. What intrigues me more is if it was capable of it would our Victorian conservative minded scientists have RLHF it out of that kind of thing?
If AGI will ever come, then. Currently, AI is only a statistical machines, and solutions like this are purely based on distribution and no logic/actual intelligence.
I swear that AI could independently develop a cure for cancer and people would still say that it's not actually intelligent, just matrix multiplications giving a statistically probable answer!
LLMs are at least designed to be intelligent. Our monkey brains have much less reason to be intelligent, since we only evolved to survive nature, not to understand it.
We are at this moment extremely deep into what most people would have been considered to be actual artificial intelligence a mere 15 years ago. We're not quite at human levels of intelligence, but it's close.
All the answers for all your questions is contained in randomness. If you have a random sentence generator, there is a chance that it will output the answer to this question every time it is invoked.
But that does not actually make it intelligent, does it?
Last week I put "was val kilmer in heat" into the search box on my browser. The AI answer came back with "No, Val Kilmer was not in heat. Val Kilmer played Chris Shiherlis in the movie Heat but the film did not indicate that he was pregnant or in heat. His performance was nuanced and skilled and represents a high point of the film." I was not curious about whether he was pregnant.
We are not only not close to human level of intelligence, we are not even at dog, cat, or mouse levels of intelligence. We are not actually at any level of intelligence. Devices that produce text, images, or code do not demonstrate intelligence any more than a printer producing pages of beautiful art demonstrate intelligence.
The issue to my mind is a lack of data at the meeting of QFT/GR.
Afterall few humans historically have been capable of the initial true leap between ontologies. But humans are pretty smart so we can't say that is a requirement for AGI.
Donald Knuth is an extremal outlier human and the problem is squarely in his field of expertise.
Claude, guided by a Filip Stappers, a friend of Knuth, solved a problem that Knuth and Stappers had been working on for several weeks. Unfortunately, it doesn't seem (from my quick scan) to have been stated how long (or how many tokens or $) it took for Claude + Stappers to complete the proof.
In response, Knuth said: "It seems that I’ll have to revise my opinions about “generative AI” one of these days."
Seems like good advice. From reading elsewhere in this comment section, the goalposts seem to be approaching the infrared and will soon disappear from the extreme redshift due to rate at which they are receding with each new achievement.
Connecting them is easy, one is the math of the exchange and one of the state machine.
A better question might be why no one is paying more attention to Barandes at Harvard. He's been publishing the answer to that question for a while, if you stop trying to smuggle a Markovian embedding in a non-Markovian process you stop getting weird things like infinities at boundaries that can't be worked out from current position alone.
But you could just dump a prompt into an LLM and pull the handle a few dozen times and see what pops out too. Maybe whip up a Claw skill or two
Unconstrained solution space exploration is surely the way to solve the hard problems
Ask those Millenium Prize guys how well that's working out :)
Constraint engineering is all software development has ever been, or did we forget how entropy works? Someone should remind the folk chasing P=NP that the observer might need a pen to write down his answers, or are we smuggling more things for free that change the entire game? As soon as the locations of the witness cost, our poor little guy can't keep walking that hypercube forever. Can he?
Maybe 6 months and a few data centers will do it ;)
Are not LLMs supposed to just find the most probable word that follows next like many people here have touted? How this can be explained under that pretense? Is this way of problem solving 'thinking'?
In some sense that is still correct, i.e. the words are taken from some probability distribution conditional on previous words, but the key point is that probability distribution is not just some sort of average across the internet set of word probabilities. In the end this probability distribution is really the whole point of intelligence. And I think the LLMs are learning those.
> just find the most probable word that follows next
Well, if in all situations you can predict which word Einstein would probably say next, then I think you're in a good spot.
This "most probable" stuff is just absurd handwaving. Every prompt of even a few words is unique, there simply is no trivially "most probable" continuation. Probable given what? What these machines learn to do is predicting what intelligence would do, which is the same as being intelligent.
No, it just predict what the next word would be if an intelligent entity translated its thoughts to words. Because it is trained on the text that are written by intelligent entities.
If it was trained on text written by someone who loves to rhyme, you would be getting all rhyming responses.
It imitates the behavior -- in text -- of what ever entity that generated the training data. Here the training data was made by intelligent humans, so we get an imitation of the same.
It is a clever party trick that works often enough.
Type "owejdpowejdojweodmwepiodnoiwendoinw welidn owindoiwendo nwoeidnweoind oiwnedoin" into ChatGPT and the response is "The text you sent appears to be random or corrupted and doesn’t form a clear question." because the prompt doesnt correlate to training data.
That description is really only fair for base models†. Something like Opus 4.6 has all kinds of other training on top of that which teach it behaviors beyond "predict most probable token," like problem-solving and being a good chatbot.
(†And even then is kind of overly-dismissive and underspecified. The "most probable word" is defined over some training data set. So imagine if you train on e.g. mathematicians solving problems... To do a good job at predicting [w/o overfitting] your model will have to in fact get good at thinking like a mathematician. In general "to be able to predict what is likely to happen next" is probably one pretty good definition of intelligence.)
I'd disagree, the other training on top doesn't alter the fundamental nature of the model that it's predicting the probabilities of the next token (and then there's a sampling step which can roughly be described as picking the most probable one).
It just changes the probability distribution that it is approximating.
To the extent that thinking is making a series of deductions from prior facts, it seems to me that thinking can be reduced to "pick the next most probable token from the correct probability distribution"...
Put a loop around an LLM and, it can be trivially made Turing complete, so it boils down to whether thinking requires exceeding the Turing computable, and we have no evidence to suggest that is even possible.
As typically deployed [1] LLMs are not turing complete. They're closer to linear bounded automaton, but because transformers have a strict maximum input size they're actually a subset of the weaker class of deterministic finite automaton. These aren't like python programs or something that can work on as much memory as you supply them, their architecture works on a fixed maximum amount of memory.
I'm not particularly convinced turing complete is the relevant property though. I'm rather convinced that I'm not turing complete either... my head is only so big after all.
[1] i.e. in a loop that appends output tokens to the input and has some form of sliding context window (perhaps with some inserted instructions to "compact" and then sliding the context window right to after those instructions once the LLM emits some special "done compacting" tokens).
[2] Common sampling procedures make them mildly non-deterministic, but I don't believe they do so in a way that changes the theoretical class of these machines from DFAs.
I think it's pretty likely that "intelligence" is emergent behavior that comes when you predict what comes next in physical reality well enough, at varying timescales. Your brain has to build all sorts of world model abstractions to do that over any significant timescale. Big LLMs have to build internal world models, too, to do well at their task.
That's the way many people reduce it, and mathematically, I think that's true. I think what we fail to realize is just far that will actually take you.
"just the most probable word" is a pretty powerful mechanism when you have all of human knowledge at your fingertips.
I say that people "reduce it" that way because it neatly packs in the assumption that general intelligence is something other than next token prediction. I'm not saying we've arrived at AGI, in fact, I do not believe we have. But, it feels like people who use that framing are snarkily writing off something that they themselves to do not fully comprehend behind the guise of being "technically correct."
I'm not saying all people do this. But I've noticed many do.
In some cases solving a problem is about restating the problem in a way that opens up a new path forward. “Why do planets move around the sun?” vs “What kind of force exists in the world that makes planets tethered to the sun with no visible leash?” (Obviously very simplified but I hope you can see what I am saying.) Given that a human is there to ask the right questions it isn’t just an LLM.
Further, some solutions are like running a maze. If you know all the wrong turns/next words to say and can just brute force the right ones you might find a solution like a mouse running through the maze not seeing the whole picture.
Whether this is thinking is more philosophical. To me this demonstrates more that we are closer to bio computers than an LLM is to having some sort of divine soul.
Thanks for your input. The way I saw this and how it looks Knuth interpreted it is that there were some reasoning steps taken by Claude independently. Some internal decisions in the model that made it try different things, finally succeeding.
>Are not LLMs supposed to just find the most probable word that follows next like many people here have touted?
The base models are trained to do this. If a web page contains a problem, and then the word "Answer: ", it is statistically very likely that what follows on that web page is an answer. If the base model wants to be good at predicting text, at some point learning the answer to common question becomes a good strategy, so that it can complete text that contains these.
NN training tries to push models to generalize instead of memorizing the training set, so this creates an incentive for the model to learn a computation pattern that can answer many questions, instead of just memorizing. Whether they actually generalize in practice... it depends. Sometimes you still get copy-pasted input that was clearly pulled verbatim from the training set.
But that's only base models. The actual production LLMs you chat with don't predict the most probable word according to the raw statistical distribution. They output the words that RLHF has rewarded them to output, which includes acting as an assistant that answers questions instead of just predicting text. RLHF is also the reason there are so many AI SIGNS [1] like "you're absolutely right" and way more use of the word "delve" than is common in western English.
Imagine training a chess bot to predict a valid sequence of moves or valid game using the standard algebraic notation for chess
Great! It will now correctly structure chess games, but we've created no incentive for it to create a game where white wins or to make the next move be "good"
Ok, so now you change the objective. Now let's say "we don't just want valid games, we want you to predict the next move that will help that color win"
And we train towards that objective and it starts picking better moves (note: the moves are still valid)
You might imagine more sophisticated ways to optimize picking good moves. You continue adjusting the objective function, you might train a pool of models all based off of the initial model and each of them gets a slightly different curriculum and then you have a tournament and pick the winningest model. Great!
Now you might have a skilled chess-playing-model.
It is no longer correct to say it just finds a valid chess program, because the objective function changed several times throughout this process.
This is exactly how you should think about LLMs except the ways the objective function has changed are significantly significantly more complicated than for our chess bot.
So to answer your first question: no, that is not what they do. That is a deep over simplification that was accurate for the first two generations of the models and sort of accurate for the "pretraining" step of modern llms (except not even that accurate, because pretraining does instill other objectives. Almost like swapping our first step "predict valid chess moves" with "predict stockfish outputs")
Are you feigning ignorance? The best way to answer a question, like completing a sentence, is through reasoning; an emergent behavior in complex models.
Thinking is a big word that sweeps up a lot of different human behavior, so I don't know if it's right to jump to that; HOWEVER, explanations of LLMs that depend heavily on next-token prediction are defunct. They stopped being fundamentally accurate with the rise of massive reinforcement learning and w/ 'reasoning' models the analogy falls apart when you try to do work with it.
Be on the lookout for folks who tell you these machines are limited because they are "just predicting the next word." They may not know what they're talking about.
time to use claude code to understand DEKs paper, in plain English. As someone who did a bit of formal verification in grad school. I feel like, there are a long tail of problems that can be solved by human-model collab like this one. The problems may not mean much but hopefully it can stack up understanding of intelligence.
I asked Claude to solve the pentominoes puzzle made famous by Arthur C. Clarke. It struggled mightily until I told it how I'd solved the problem using 64 bit unsigned integers to represent the board and pieces. Then, it created a C# program that solved the problem very quickly. However, in the 20x3 case it found four solutions when there are only two. Turns out it had incorrectly mapped one of the pentominoes. Sort of a silly mistake; the sort a human might make.
Regurgitation is pretty rare, and very difficult to coax out, if not even impossible, for things that aren't massively overrepresented in the training set relative to the size of the training set. Even the famous regurgitation paper showed this: while they got most of the models to regurgitate the first book of the Harry Potter series, only Claude 3.7 Sonnet was able to regurgitate any significant portion of any of the other books that had a high nv-recall rate, and basically all of them dropped off precipitously for works like GoT, The Catcher in the Rye, Beloved, and remembered almost nothing about the Da Vinci Code or Catch-22[0]. So you really need huge amounts of examples to get any kind of meaningful regurgitation on any kind of reliable basis. Thus, you'd have to prove that hypothesis.
TLDR (story, not math) - Knuth poses a problem, his friend uses Claude to conduct 30 some explorations, with careful human guidance, and Claude eventually writes a Python program that can find a solution for all odd values. Knuth then writes a proof of the approach and is very pleased by Claude's contribution. Even values remain an open question (Claude couldn't make much progress on them)
I think this is pretty clearly an overstatement of what was done. As Knuth says,
"Filip told me that the explorations reported above, though ultimately successful, weren’t really smooth.
He had to do some restarts when Claude stopped on random errors; then some of the previous search results
were lost. After every two or three test programs were run, he had to remind Claude again and again that
it was supposed to document its progress carefully. "
That doesn't look like careful human guidance, especially not the kind that would actually guide the AI toward the solution at all, let alone implicitly give it the solution — that looks like a manager occasionally checking in to prod it to keep working.
> Shock! Shock! I learned yesterday that an open problem I’d been working on for several weeks had just
been solved by Claude Opus 4.6— Anthropic’s hybrid reasoning model that had been released three weeks
earlier! It seems that I’ll have to revise my opinions about “generative AI” one of these days. What a joy
it is to learn not only that my conjecture has a nice solution but also to celebrate this dramatic advance in
automatic deduction and creative problem solving.
I encourage you to look at what the current models with a bit of harnessing are capable of, e.g. Opus 4.6 and Claude Code. Try to make it solve some mathematics-heavy problem you come up with. If only to get a more accurate picture of whats going on.
Unfortunately, these tools generalize way beyond regurgitating the training set.
I would not assume they stay below human capabilities in the next few years.
Why any moral person would continue building these at this point I don't know.
I guess in the best case the future will have a small privileged class of humans having total power, without need for human workers or soldiers.
Picture a mechanical boot stomping on a human face forever.
I would like to note that it would be trivial to definitively prove or disprove such things if we had a searchable public archive of the training data. Interestingly, the same people (and corporate entities) who loudly claim that LLMs are creating original work seem to be utterly disinterested in having actual, definitive proof of their claims.
Was it? It was an open problem to Knuth - who generally knows how to search literature. However there is enough literature to search that it wouldn't be a surprise at all to discover it was already solved but he just used slightly different terms and so didn't find it. Or maybe it was sovled because this is a specialization of something that looks unrelated and so he wouldn't have realized it when he read it. Or...
Overall I'm going with unsolved, because Knuth is a smart person who I'd expect to not miss the above. I'm also sure he falls for the above all the time even though the majority of the time he doesn't.
Agreed with all of that, but with the added point that Knuth has done a lot of work in this exact area in The Art of Computer Programming Volume 4. If he considers this conjecture open given his particular knowledge of the field, it likely is (although agreed, it's not guaranteed).
> If he considers this conjecture open given his particular knowledge of the field, it likely is (although agreed, it's not guaranteed).
It is as good as guaranteed. If Knuth says it doesn't know how to solve the problem, and if anyone knows, then they will inform Knuth about it. Knuth not just a very knowledgeable person, but a celebrity also.
> Filip also told me that he asked Claude to continue on the even case after the odd case had been resolved. “But there after a while it seemed to get stuck. In the end, it was not even able to write and run explore programs correctly anymore, very weird. So I stopped the search.”
Interesting snippet towards the end. I wonder if they were using claude.ai or claude code. Sounds like they ran out of context and entered the "dumb zone."
> I think Claude Shannon’s spirit is probably proud to know that his name is now being associated with such advances. Hats off to Claude!
I didn't realize Claude was named after Claude Shannon!
https://en.wikipedia.org/wiki/Claude_Shannon
It's fascinating to think about the space of problems which are amenable to RL scaling of these probability distributions.
Before, we didn't have a fast (we had to rely on human cognition) way to try problems - even if the techniques and workflows were known by someone. Now, we've baked these patterns into probability distributions - anyone can access them with the correct "summoning spell". Experts will naturally use these systems more productively, because they know how to coerce models into the correct conditional distributions which light up the right techniques.
One question this raises to me is how these models are going to keep up with the expanding boundary of science. If RL is required to get expert behavior into the models, what happens when experts start pushing the boundary faster? In 2030, how is Anthropic going to keep Claude "up-to-date" without either (a) continual learning with a fixed model (expanding context windows? seems hard) or (b) continual training (expensive)?
Crazy times.
> In 2030, how is Anthropic going to keep Claude "up-to-date"
I think the majority of research, design and learning goes through LLMs and coding agents today, considering the large user base and usage it must be trillions of tokens per day. You can take a long research session or a series of them and apply hindsight - what idea above can be validated below? This creates a dense learning signal based on validation in real world with human in the loop and other tools, code & search.
> In 2030, how is Anthropic going to keep Claude "up-to-date"
In 2030 Anthropic hopes Claude will keep Anthropic "up-to-date" on its progress on itself.
I'm only half joking here.
Data sharing agreements permitting, today's inference runs can be tomorrow's training data. Presumably the models are good enough at labeling promising chains of thought already.
I could totally imagine "free" inference for researchers under the condition that the reasoning traces get to be used as future training data.
Agreed, there's no doubt this will happen. It's likely already happening (it feels safe to assume that Anthropic is curating data from the data they record from Claude Code?)
As far as I understand RL scaling (we've already maxxed out RLVR), these machines only get better as long as they have expert reasoner traces available.
Having an expert work with an LLM and successfully solve a problem is high signal data, it may be the only path forward?
My prior is that these companies will take this data without asking you as much as they can.
Exactly, or functionally equivalently, asking you in paragraph 37 of a 120-page PDF (bonus points: in an agreement update).
And importantly, this can be cross-lab/model too. I suspect there's a reason why e.g. Google has been offering me free Claude inference in Google Antigravity on a free plan...
> Data sharing agreements permitting, today's inference runs can be tomorrow's training data. Presumably the models are good enough at labeling promising chains of thought already.
Wouldn't this lead to model collapse?
Not necessarily, as exhibited by the massive success of artificial data.
A bit related: open weights models are basically time capsules. These models have a knowledge cut off point and essentially forever live in that time.
This is the most fundamental argument that they are not, directly, an intelligence. They are not ever storing new information on a meaningful timescale. However, if you viewed them on some really large macro time scale where now LLMs are injecting information into the universe and the re-ingesting that maybe in some very philosophical way they are a /very/ slow oscillating intelligence right now. And as we narrow that gap (maybe with a totally new non-LLM paradigm) perhaps that is ultimately what gen AI becomes. Or some new insight that lets the models update themselves in some fundamental way without the insanely expensive training costs they have now.
Would you consider someone with anterograde amnesia not to be intelligent?
That is a good area to explore. Their map of the past is fixed. They are frozen at some point in their psychological time. What has stopped working? Their hippocampus and medial temporal lobe. These are like the write-head that move data from the hippocampus to the neo cortex. Their "I" can no longer update itself. Their DMN is frozen in time. So if intelligence is purely the "I" telling a continuous coherent story about itself. The difference is that although they are fixed in time which is a characteristic shared by a specific LLM model. They can still completely activate their task positive network for problem solving and if their previous information stored is adequate to solve the problem they can. You could argue that is pretty similar to an LLM and what it does. So it is certainly a signifiant component of intelligence.
There is also the nature of the human brain, it is not just those systems of memory encoding, storage, and use of that in narratives. People with this type of amnesia still can learn physical skills and that happens in a totally different area of the brain with no need for the hippocampus->neocortex consolidation loop. So, the intelligence is significantly diminished, but not entirely. Other parts of the brain are still able to update themselves in ways an LLM currently cannot. The human with amnesia also has a complex biological sensory input mapping that is still active and integrating and restructuring the brain. So, I think when you get into the nuances of the human in this state vs. an LLM we can still say the human crosses some threshold for intelligence where the LLM does not in this framework.
So, they have an "intelligence", localized to the present in terms of their TPN and memory formation. LLMs have this kind of "intelligence". But the human still has the capacity to rewire at least some of their brain in real time even with amnesia.
A very good point. For anyone not familiar with anterograde amnesia, the classical case is patient H.M. (https://en.wikipedia.org/wiki/Henry_Molaison), whose condition was researched by Brenda Milner.
Or you could have just said "they can't form new memories."
I thought maybe people would be curious to read about how we came to understand the condition and the history behind it, as well as any associated information. Forgive me for such a deep transgression as this assumption.
Sure, why can't both things be true? "Intelligence" is just what you call something and someone else knows what you mean. Why did AI discourse throw everyone back 100 years philosophically? Its like post-structuralism or Wittgenstein never happened..
It's so much less important or interesting to like nail down some definition here (I would cite HN discourse the past three years or so), than it is to recognize what it means to assign "intelligent" to something. What assumptions does it make? What power does it valorize or curb?
Each side of this debate does themselves a disservice essentially just trying to be Aristotle way too late. "Intelligence" did not precede someone saying it of some phenomena, there is nothing to uncover or finalize here. The point is you have one side that really wants, for explicit and implicit reasons, to call this thing intelligent, even if it looks like a duck but doesn't quack like one, and vice versa on the other side.
Either way, we seem fundamentally incapable of being radical enough to reject AI on its own terms, or be proper champions of it. It is just tribal hypedom clinging to totem signifiers.
Good luck though!
There's nothing to say that you can't build something intelligent out of them by bolting a memory on it, though.
Sure, it's not how we work, but I can imagine a system where the LLM does a lot of heavy lifting and allows more expensive, smaller networks that train during inference and RAG systems to learn how to do new things and keep persistent state and plan.
You aren't wrong and that is a fascinating area of research. I think the key thing is that the memory has to fundamentally influence the underlying model, or at least the response, in some way. Patching memory on top of an LLM is different from integrating it into the core model. To go back to human terms it is like an extra bit of storage, but not directly attached to our neo cortex. So it works more like a filter than a core part of our intelligence in the analogy. You think about something and assemble some thought and then it would go to this next filter layer and get augmented and that smaller layer is the only thing being updated.
It is still meaningful, but it narrows what the intelligence can be sufficiently that it may not meet the threshold. Maybe it would, but it is probably too narrow. This is all strictly if we ask that it meet some human-like intelligence and not the philosophy of "what counts as intelligence" but... we are humans. The strongest things or at least the most honest definitions of intelligence I think exist are around our metacognitive ability to rewire the grey matter for survival not based on immediate action-reaction but the psychological time of analyzing the past to alter the future.
Memory is not just bolted on top of the latest models. They under go training on how and when to effectively use memory and how to use compaction to avoid running out of context when working on problems.
But they're not "slow"! Unlike biological thinking, which has a speed limit, you can accelerate these chains of thought by orders of magnitude.
Their consolidation of memory speed is what I was referring to. The model iterations are essentially their form of collective memory. In the sense of the human model of intelligence we have thoughts. Thoughts become memory. New thoughts use that memory and become recursively updated thoughts. LLMs cannot update their memory very fast.
I assure you that LLM thinking also has a speed limit.
Not an expert but surely it's only a matter of time until there's a way to update with the latest information without having to retrain on the entire corpus?
> Experts will naturally use these systems more productively, because they know how to coerce models into the correct conditional distributions which light up the right techniques.
Part of it comes down to “knowing” what questions to ask.
I see it like the relationship between a student and research advisor. The advisor will ideally know the terrain and suggest a fruitful line of attack (what to ask), and the student will follow through, learning along the way.
They can use LORA.
From my naive standpoint, LLMs like this seem to have some big strengths. One: possession of a superhuman expanse of knowledge. Two: making connections. Three: tireless trial and error.
If you put those three things together, you end up with some cool stuff from time to time. Perhaps the proof of P!=NP is tied to an obscure connection that humans don't easily see due to individual lack of knowledge or predisposition of bias.
Unless my understanding is incorrect about how these tools work that last point isn't really a quality of LLMs as such? It gets attributed because the lines are blurred but the tireless trial and error is actually just a quality of a regular programatic loop (agent/orchestrator) that happens to be doing the trickiest part of its work via an LLM.
This is why the whole "LLMs for mass surveillance" thing is scary imo.
I wonder how long we have until we start solving some truly hard problems with AI. How long until we throw AI at "connect general relativity and quantum physics", give the AI 6 months and a few data centers, and have it pop out a solution?
I think a very long time because part of our limit is experiment.
We need enough experimental results to explain to solve these theoretical mismatches and we don't and at present can't explore that frontier.
Once we have more results at that frontier we'd build a theory out from there that has two nearly independent limits for QFT and GR.
What we'd be asking if the AI is something that we can't expect a human to solve even with a lifetime of effort today.
It'll take something in par with Newton realising that the heavens and apples are under the same rules to do it. But at least Newton got to hold the apple and only had to imagine he could a star.
> I think a very long time because part of our limit is experiment.
Yes, maybe. But if you are smarter, you can think up better experiments that you can actually do. Or re-use data from earlier experiments in novel and clever ways.
What prevents us from giving this system access to other real systems that live in physical labs? I don't see much difference between parameterizing and executing a particle accelerator run and invoking some SQL against a provider. It's just JSON on the wire at some level.
Nothing, we can give it all the data we have and have it lead experiments.
But we can not yet experiment at the GR/QFT frontier.
To do so with a particle accelerator it would need to be the size of the milky way.
The question is, if you trained an LLM on everything up until 1904, could it come up with E=MC² or not?
In 1900 Henri Poincaré wrote that radiation (light) has an effective mass given by E/c^2.
So it really isn't far fetched. What intrigues me more is if it was capable of it would our Victorian conservative minded scientists have RLHF it out of that kind of thing?
If AGI will ever come, then. Currently, AI is only a statistical machines, and solutions like this are purely based on distribution and no logic/actual intelligence.
I swear that AI could independently develop a cure for cancer and people would still say that it's not actually intelligent, just matrix multiplications giving a statistically probable answer!
LLMs are at least designed to be intelligent. Our monkey brains have much less reason to be intelligent, since we only evolved to survive nature, not to understand it.
We are at this moment extremely deep into what most people would have been considered to be actual artificial intelligence a mere 15 years ago. We're not quite at human levels of intelligence, but it's close.
>AI could independently develop a cure for cancer
All the answers for all your questions is contained in randomness. If you have a random sentence generator, there is a chance that it will output the answer to this question every time it is invoked.
But that does not actually make it intelligent, does it?
Last week I put "was val kilmer in heat" into the search box on my browser. The AI answer came back with "No, Val Kilmer was not in heat. Val Kilmer played Chris Shiherlis in the movie Heat but the film did not indicate that he was pregnant or in heat. His performance was nuanced and skilled and represents a high point of the film." I was not curious about whether he was pregnant.
We are not only not close to human level of intelligence, we are not even at dog, cat, or mouse levels of intelligence. We are not actually at any level of intelligence. Devices that produce text, images, or code do not demonstrate intelligence any more than a printer producing pages of beautiful art demonstrate intelligence.
That's wrong. Humans were evolved to have big brains so they can better understand the env and use it to their advantage.
I still see AI making stupid silly mistakes. I rather think and not waste time on something that only remembers data, and doesn't even understand it.
Reasoning in AI is only about finding contradictions between his "thoughts", not actually understand it.
> I still see AI making stupid silly mistakes.
Humans also make silly mistakes.
> I still see AI making stupid silly mistakes.
In contrast with humans, who are famously known for never making stupid silly mistakes...
I don't even think that's the issue.
The issue to my mind is a lack of data at the meeting of QFT/GR.
Afterall few humans historically have been capable of the initial true leap between ontologies. But humans are pretty smart so we can't say that is a requirement for AGI.
When it comes to revolutionary/unsolved subjects, there will never be enough data. That's why its revolutionary/unsolved.
Did you read the linked paper? Claude out-reasoned humans on a challenging (or at least, unsolved) math problem.
"humans"
Donald Knuth is an extremal outlier human and the problem is squarely in his field of expertise.
Claude, guided by a Filip Stappers, a friend of Knuth, solved a problem that Knuth and Stappers had been working on for several weeks. Unfortunately, it doesn't seem (from my quick scan) to have been stated how long (or how many tokens or $) it took for Claude + Stappers to complete the proof.
In response, Knuth said: "It seems that I’ll have to revise my opinions about “generative AI” one of these days."
Seems like good advice. From reading elsewhere in this comment section, the goalposts seem to be approaching the infrared and will soon disappear from the extreme redshift due to rate at which they are receding with each new achievement.
Merely luck in my opinion. There could be also multiple times where it didn't solve it.
Connecting them is easy, one is the math of the exchange and one of the state machine.
A better question might be why no one is paying more attention to Barandes at Harvard. He's been publishing the answer to that question for a while, if you stop trying to smuggle a Markovian embedding in a non-Markovian process you stop getting weird things like infinities at boundaries that can't be worked out from current position alone.
But you could just dump a prompt into an LLM and pull the handle a few dozen times and see what pops out too. Maybe whip up a Claw skill or two
Unconstrained solution space exploration is surely the way to solve the hard problems
Ask those Millenium Prize guys how well that's working out :)
Constraint engineering is all software development has ever been, or did we forget how entropy works? Someone should remind the folk chasing P=NP that the observer might need a pen to write down his answers, or are we smuggling more things for free that change the entire game? As soon as the locations of the witness cost, our poor little guy can't keep walking that hypercube forever. Can he?
Maybe 6 months and a few data centers will do it ;)
Are not LLMs supposed to just find the most probable word that follows next like many people here have touted? How this can be explained under that pretense? Is this way of problem solving 'thinking'?
In some sense that is still correct, i.e. the words are taken from some probability distribution conditional on previous words, but the key point is that probability distribution is not just some sort of average across the internet set of word probabilities. In the end this probability distribution is really the whole point of intelligence. And I think the LLMs are learning those.
> just find the most probable word that follows next
Well, if in all situations you can predict which word Einstein would probably say next, then I think you're in a good spot.
This "most probable" stuff is just absurd handwaving. Every prompt of even a few words is unique, there simply is no trivially "most probable" continuation. Probable given what? What these machines learn to do is predicting what intelligence would do, which is the same as being intelligent.
>Probable given what?
The training data..
>predicting what intelligence would do
No, it just predict what the next word would be if an intelligent entity translated its thoughts to words. Because it is trained on the text that are written by intelligent entities.
If it was trained on text written by someone who loves to rhyme, you would be getting all rhyming responses.
It imitates the behavior -- in text -- of what ever entity that generated the training data. Here the training data was made by intelligent humans, so we get an imitation of the same.
It is a clever party trick that works often enough.
> The training data
If the prompt is unique, it is not in the training data. True for basically every prompt. So how is this probability calculated?
The prompt is unique but the tokens aren't.
Type "owejdpowejdojweodmwepiodnoiwendoinw welidn owindoiwendo nwoeidnweoind oiwnedoin" into ChatGPT and the response is "The text you sent appears to be random or corrupted and doesn’t form a clear question." because the prompt doesnt correlate to training data.
Just using a scaled up and cleverly tweaked version of linear regression analysis...
That description is really only fair for base models†. Something like Opus 4.6 has all kinds of other training on top of that which teach it behaviors beyond "predict most probable token," like problem-solving and being a good chatbot.
(†And even then is kind of overly-dismissive and underspecified. The "most probable word" is defined over some training data set. So imagine if you train on e.g. mathematicians solving problems... To do a good job at predicting [w/o overfitting] your model will have to in fact get good at thinking like a mathematician. In general "to be able to predict what is likely to happen next" is probably one pretty good definition of intelligence.)
I'd disagree, the other training on top doesn't alter the fundamental nature of the model that it's predicting the probabilities of the next token (and then there's a sampling step which can roughly be described as picking the most probable one).
It just changes the probability distribution that it is approximating.
To the extent that thinking is making a series of deductions from prior facts, it seems to me that thinking can be reduced to "pick the next most probable token from the correct probability distribution"...
Put a loop around an LLM and, it can be trivially made Turing complete, so it boils down to whether thinking requires exceeding the Turing computable, and we have no evidence to suggest that is even possible.
What are you doing in your loop?
As typically deployed [1] LLMs are not turing complete. They're closer to linear bounded automaton, but because transformers have a strict maximum input size they're actually a subset of the weaker class of deterministic finite automaton. These aren't like python programs or something that can work on as much memory as you supply them, their architecture works on a fixed maximum amount of memory.
I'm not particularly convinced turing complete is the relevant property though. I'm rather convinced that I'm not turing complete either... my head is only so big after all.
[1] i.e. in a loop that appends output tokens to the input and has some form of sliding context window (perhaps with some inserted instructions to "compact" and then sliding the context window right to after those instructions once the LLM emits some special "done compacting" tokens).
[2] Common sampling procedures make them mildly non-deterministic, but I don't believe they do so in a way that changes the theoretical class of these machines from DFAs.
I think it's pretty likely that "intelligence" is emergent behavior that comes when you predict what comes next in physical reality well enough, at varying timescales. Your brain has to build all sorts of world model abstractions to do that over any significant timescale. Big LLMs have to build internal world models, too, to do well at their task.
That's the way many people reduce it, and mathematically, I think that's true. I think what we fail to realize is just far that will actually take you.
"just the most probable word" is a pretty powerful mechanism when you have all of human knowledge at your fingertips.
I say that people "reduce it" that way because it neatly packs in the assumption that general intelligence is something other than next token prediction. I'm not saying we've arrived at AGI, in fact, I do not believe we have. But, it feels like people who use that framing are snarkily writing off something that they themselves to do not fully comprehend behind the guise of being "technically correct."
I'm not saying all people do this. But I've noticed many do.
In some cases solving a problem is about restating the problem in a way that opens up a new path forward. “Why do planets move around the sun?” vs “What kind of force exists in the world that makes planets tethered to the sun with no visible leash?” (Obviously very simplified but I hope you can see what I am saying.) Given that a human is there to ask the right questions it isn’t just an LLM.
Further, some solutions are like running a maze. If you know all the wrong turns/next words to say and can just brute force the right ones you might find a solution like a mouse running through the maze not seeing the whole picture.
Whether this is thinking is more philosophical. To me this demonstrates more that we are closer to bio computers than an LLM is to having some sort of divine soul.
Thanks for your input. The way I saw this and how it looks Knuth interpreted it is that there were some reasoning steps taken by Claude independently. Some internal decisions in the model that made it try different things, finally succeeding.
>Are not LLMs supposed to just find the most probable word that follows next like many people here have touted?
The base models are trained to do this. If a web page contains a problem, and then the word "Answer: ", it is statistically very likely that what follows on that web page is an answer. If the base model wants to be good at predicting text, at some point learning the answer to common question becomes a good strategy, so that it can complete text that contains these.
NN training tries to push models to generalize instead of memorizing the training set, so this creates an incentive for the model to learn a computation pattern that can answer many questions, instead of just memorizing. Whether they actually generalize in practice... it depends. Sometimes you still get copy-pasted input that was clearly pulled verbatim from the training set.
But that's only base models. The actual production LLMs you chat with don't predict the most probable word according to the raw statistical distribution. They output the words that RLHF has rewarded them to output, which includes acting as an assistant that answers questions instead of just predicting text. RLHF is also the reason there are so many AI SIGNS [1] like "you're absolutely right" and way more use of the word "delve" than is common in western English.
[1]: https://en.wikipedia.org/wiki/WP:AISIGNS
Yes, that is exactly what they do.
But that does not mean that the results cannot be dramatic. Just like stacking pixels can result in a beautiful image.
Those people still exist? I only know one guy who is still fighting those windmills
Yes, I am one.
Imagine training a chess bot to predict a valid sequence of moves or valid game using the standard algebraic notation for chess
Great! It will now correctly structure chess games, but we've created no incentive for it to create a game where white wins or to make the next move be "good"
Ok, so now you change the objective. Now let's say "we don't just want valid games, we want you to predict the next move that will help that color win"
And we train towards that objective and it starts picking better moves (note: the moves are still valid)
You might imagine more sophisticated ways to optimize picking good moves. You continue adjusting the objective function, you might train a pool of models all based off of the initial model and each of them gets a slightly different curriculum and then you have a tournament and pick the winningest model. Great!
Now you might have a skilled chess-playing-model.
It is no longer correct to say it just finds a valid chess program, because the objective function changed several times throughout this process.
This is exactly how you should think about LLMs except the ways the objective function has changed are significantly significantly more complicated than for our chess bot.
So to answer your first question: no, that is not what they do. That is a deep over simplification that was accurate for the first two generations of the models and sort of accurate for the "pretraining" step of modern llms (except not even that accurate, because pretraining does instill other objectives. Almost like swapping our first step "predict valid chess moves" with "predict stockfish outputs")
Are you feigning ignorance? The best way to answer a question, like completing a sentence, is through reasoning; an emergent behavior in complex models.
Thinking is a big word that sweeps up a lot of different human behavior, so I don't know if it's right to jump to that; HOWEVER, explanations of LLMs that depend heavily on next-token prediction are defunct. They stopped being fundamentally accurate with the rise of massive reinforcement learning and w/ 'reasoning' models the analogy falls apart when you try to do work with it.
Be on the lookout for folks who tell you these machines are limited because they are "just predicting the next word." They may not know what they're talking about.
Fascinating report by DEK himself.
Time to sit down, read, digest and understand it without the help of LLM.
I don't have time to do that myself yet so I just dug a quick TL;DR rabbit hole for fun:
https://ontouchstart.github.io/rabbit-holes/llm_rabbit_hole_...
time to use claude code to understand DEKs paper, in plain English. As someone who did a bit of formal verification in grad school. I feel like, there are a long tail of problems that can be solved by human-model collab like this one. The problems may not mean much but hopefully it can stack up understanding of intelligence.
I asked Claude to solve the pentominoes puzzle made famous by Arthur C. Clarke. It struggled mightily until I told it how I'd solved the problem using 64 bit unsigned integers to represent the board and pieces. Then, it created a C# program that solved the problem very quickly. However, in the 20x3 case it found four solutions when there are only two. Turns out it had incorrectly mapped one of the pentominoes. Sort of a silly mistake; the sort a human might make.
Probably because a human made it before their code got harvested and then regurgitated to you in the form of slop...
Regurgitation is pretty rare, and very difficult to coax out, if not even impossible, for things that aren't massively overrepresented in the training set relative to the size of the training set. Even the famous regurgitation paper showed this: while they got most of the models to regurgitate the first book of the Harry Potter series, only Claude 3.7 Sonnet was able to regurgitate any significant portion of any of the other books that had a high nv-recall rate, and basically all of them dropped off precipitously for works like GoT, The Catcher in the Rye, Beloved, and remembered almost nothing about the Da Vinci Code or Catch-22[0]. So you really need huge amounts of examples to get any kind of meaningful regurgitation on any kind of reliable basis. Thus, you'd have to prove that hypothesis.
[0]: https://arxiv.org/pdf/2601.02671
TLDR (story, not math) - Knuth poses a problem, his friend uses Claude to conduct 30 some explorations, with careful human guidance, and Claude eventually writes a Python program that can find a solution for all odd values. Knuth then writes a proof of the approach and is very pleased by Claude's contribution. Even values remain an open question (Claude couldn't make much progress on them)
> with careful human guidance,
I think this is pretty clearly an overstatement of what was done. As Knuth says,
"Filip told me that the explorations reported above, though ultimately successful, weren’t really smooth. He had to do some restarts when Claude stopped on random errors; then some of the previous search results were lost. After every two or three test programs were run, he had to remind Claude again and again that it was supposed to document its progress carefully. "
That doesn't look like careful human guidance, especially not the kind that would actually guide the AI toward the solution at all, let alone implicitly give it the solution — that looks like a manager occasionally checking in to prod it to keep working.
Solves? It's a part of the training set. Nothing more, nothing less.
Opening sentences:
> Shock! Shock! I learned yesterday that an open problem I’d been working on for several weeks had just been solved by Claude Opus 4.6— Anthropic’s hybrid reasoning model that had been released three weeks earlier! It seems that I’ll have to revise my opinions about “generative AI” one of these days. What a joy it is to learn not only that my conjecture has a nice solution but also to celebrate this dramatic advance in automatic deduction and creative problem solving.
I encourage you to look at what the current models with a bit of harnessing are capable of, e.g. Opus 4.6 and Claude Code. Try to make it solve some mathematics-heavy problem you come up with. If only to get a more accurate picture of whats going on.
Unfortunately, these tools generalize way beyond regurgitating the training set. I would not assume they stay below human capabilities in the next few years.
Why any moral person would continue building these at this point I don't know. I guess in the best case the future will have a small privileged class of humans having total power, without need for human workers or soldiers. Picture a mechanical boot stomping on a human face forever.
If this was a joke, it certainly flew over most people's heads...
Prove it.
I would like to note that it would be trivial to definitively prove or disprove such things if we had a searchable public archive of the training data. Interestingly, the same people (and corporate entities) who loudly claim that LLMs are creating original work seem to be utterly disinterested in having actual, definitive proof of their claims.
This would be awesome. Even titles and shasums could be enough.
Did you read the article? It was an open problem.
Was it? It was an open problem to Knuth - who generally knows how to search literature. However there is enough literature to search that it wouldn't be a surprise at all to discover it was already solved but he just used slightly different terms and so didn't find it. Or maybe it was sovled because this is a specialization of something that looks unrelated and so he wouldn't have realized it when he read it. Or...
Overall I'm going with unsolved, because Knuth is a smart person who I'd expect to not miss the above. I'm also sure he falls for the above all the time even though the majority of the time he doesn't.
Agreed with all of that, but with the added point that Knuth has done a lot of work in this exact area in The Art of Computer Programming Volume 4. If he considers this conjecture open given his particular knowledge of the field, it likely is (although agreed, it's not guaranteed).
> If he considers this conjecture open given his particular knowledge of the field, it likely is (although agreed, it's not guaranteed).
It is as good as guaranteed. If Knuth says it doesn't know how to solve the problem, and if anyone knows, then they will inform Knuth about it. Knuth not just a very knowledgeable person, but a celebrity also.