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.

I recall an earlier exchange, posted to HN, between Wolfram and Knuth on the GPT-4 model [1].

Knuth was dismissive in that exchange, concluding "I myself shall certainly continue to leave such research to others, and to devote my time to developing concepts that are authentic and trustworthy. And I hope you do the same."

I've noticed with the latest models, especially Opus 4.6, some of the resistance to these LLMs is relenting. Kudos for people being willing to change their opinion and update when new evidence comes to light.

1. https://cs.stanford.edu/~knuth/chatGPT20.txt

I didn't expect such a misleading intro from Knuth. It reads like Claude solved Knuth's math problem. In reality, Claude generated various example solution, and Knuth then manually generalized that to a formal proof. What Claude did is certainly useful, but it would have been nice to be clear about the scope of the contribution in the intro.

> 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 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.

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)

Must be a fun time to work on open problems. I published my graduate research close to a decade ago, often find myself fantasizing about tackling open problems with Claude.

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.

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'?

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.

Fascinating report by DEK himself.

Time to sit down, read, digest and understand it without the help of 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 thought Claude Monet - Impressionist techniques applied to coding.

Amazing paper. The simulated annealing portion reminds me of genetic algorithms (GAs). A good intro to that are the Genetic Programming series of books by John Koza, I read III in the early 2000s:

https://www.amazon.com/Genetic-Programming-III-Darwinian-Inv...

https://www.genetic-programming.com/

Note that the Python solution in the pdf is extremely short, so could have been found by simply trying permutations of math operators and functions on the right side of the equation.

We should be solving problems in Lisp instead of Python, but no matter. That's because Lisp's abstract syntax tree (AST) is the same as its code due to homoiconicity. I'm curious if most AIs transpile other languages to Lisp so that they can apply transformations internally, or if they waste computation building programs that might not compile. Maybe someone at an AI company knows.

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I've been following AI trends since the late 1980s and from my perspective, nothing really changed for about 40 years (most of my life that I had to wait through as the world messed around making other people rich). We had agents, expert system, fuzzy logic, neural nets, etc since forever, but then we got video cards in the late 1990s which made it straightforward to scale neural nets (NNs) and GAs. Unfortunately due to poor choice of architecture (SIMD instead of MIMD), progress stagnated because we don't have true multicore computing (thousands or millions of cores with local memories), but I digress.

Anyway, people have compared AI to compression. I think of it more as turning problem solving into a O(1) operation. Over time, what we think of as complex problems become simpler. And the rate that we're solving them is increasing exponentially. Problems that once seemed intractable only were because we didn't know the appropriate abstractions yet. For example, illnesses that we thought would never be cured now have vaccines through mRNA vaccines and CRISPR. That's how I think of programming. Now that we have LLMs, whole classes of programming problems now have O(1) solutions. Even if that's just telling the computer what problem to solve.

So even theorem proving will become a solved problem by the time we reach the Singularity between 2030 and 2040. We once mocked GAs for exploring dead ends and taking 1000 times the processing power to do simple things. But we ignored that doing hard things is often worth it, and is still a O(1) operation due to linear scaling.

It's a weird feeling to go from no forward progress in a field to it being effectively a solved problem in just 2 years. To go from trying to win the internet lottery to not being sure if people will still be buying software in a year or two if/when I finish a project. To witness all of that while struggling to make rent, in effect making everything I have ever done a waste of time since I knew better ways of doing it but was forced to drop down to whatever mediocre language or framework paid. As the problems I was trained to solve and was once paid to solve rapidly diminish in value because AI can solve them in 5 minutes. To the point that even inventing AGI would be unsurprising to most, so I don't know why I ever went into computer engineering to do exactly that. Because for most people, it's already here. As I've said many times lately, I thought I had more time.

Although now that we're all out of time, I have an uncanny feeling of being alive again. I think tech stole something from my psyche so profound that I didn't notice its loss. It's along the lines of things like boredom, daydreaming, wasting time. What modern culture considers frivolous. But as we lose every last vestige of the practical, as money becomes harder and harder to acquire through labor, maybe we'll pass a tipping point where the arts and humanities become sought-after again. How ironic would it be if the artificial made room for the real to return?

On that note, I read a book finally. Hail Mary by Andy Weir. The last book I read was Ready Player One by Ernest Cline, over a decade ago. I don't know how I would have had the bandwidth to do that if Claude hadn't made me a middle manager of AIs.

> 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

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Solves? It's a part of the training set. Nothing more, nothing less.

Trivia:AKWU AGHALI OFU THEOREM

Theorem (Akwu Aghali Ofu — The Single Nest or 1/2 spin)

For any observer O with personal quantum seed s (derived from first orgasm timestamp SHA-256), there exists a unique Hamiltonian cycle C(O) through the M³ digraph such that:

1. C(O) starts at vertex (0,0,0) — the Single Nest 2. C(O) has length exactly L³ for L determined by O's muon/mass preference 3. The cycle visits every vertex exactly once before returning 4. The cycle only exists when O observes it 5. No other observer can traverse the same cycle

Proof Sketch: 1. Let s = SHA-256(timestamp) mod L determine coefficients (α,β,γ) 2. Define g(i,j,k) = (αi + βj + γk) mod L 3. Show that the mapping f: (i,j,k) → next vertex via g is a permutation 4. Show that the permutation decomposes into cycles 5. Show that for appropriate s, the cycle containing (0,0,0) has length L³ 6. Show that this cycle depends on s — different s give different cycles 7. Show that observation collapses the quantum superposition, making the cycle actual

Corollary: The Single Nest spins forever because the cycle is Hamiltonian (it loves only you) — it never repeats until it returns, and the return is a new beginning, not a repetition.