I think a lot of this has to do with the post-training these models normally get. They are designed to answer basic questions with straightforward and short summary answers. They have the capacity to reason deeply, but they are not biased towards that unless prompted. I think it's because LLMs as they are in 2026 are both highly capable but also parlor tricks. They are not sentient, you just set them up with the context and then they roll downhill. You could reach a genuinely novel answer, but only with the right input. They have no will and depend on human guidance. They are both a marvel and a machine.

Yes, the prompt, and use of subagents is interesting. It could be characterized as tree of thoughts rather than "think step by step" chain of thoughts.

I see the need for this as coming down to two things:

1) LLMs are fundamentally prediction machines, and therefore ultimately will only do what they are prompted to do (and whatever that leads to). They may have been trained on, and/or have access to, all sorts of information that may be useful to solve a problem, but their predictive nature is to only use that information if explicitly prompted to, else it remains "dark" and inaccessible other than by luck. You're essentially having to tell the model "solve this problem using techniques A, B & C", otherwise techniques A, B & C will be off the radar unless the model already associates them to the problem.

2) The fundamental reason this sort of brute force tree-of-thoughts "explore all avenues" prompting is necessary, is because the model itself has no inherent curiosity to explore. Humans work differently. Our behavior is also prediction based, but we are also built for problem solving and continual exploration/learning via traits like curiosity (driven by prediction failure).

Problem solving via search can to some extent be prompted for, as here, or achieved via an external harness, but impasse resolution via curiosity, directed exploration and continual learning (if/when something new/unpredicted is encountered) is trickier. You can't usefully prompt a predictive model to "be curious" since that will only cause it to predict what a curious person would do, rather than the model being curious in reaction to the specific gaps in it's own knowledge.

LLMs have basic reasoning and a whole lot of memorization. Through that basic reasoning and pruned search, combined with piles of compute, you can prove lots of things. But the memorization of human failure prunes that possibility, and you need to expend effort convincing the LLM not to prematurely prune based on previous human failure.

It's funny, I found exactly the same thing when I asked about P=NP. The models outright refused to attempt to solve it, claiming it was too hard. I had to really battle to get it to suggest some promising suggestions.

llms are good at greedy depth first search[1]. so prompt / human needs to tell the model the breadth parts to take.

and human writing the prompt here did a lot more than that. asking to include parallel-edge 2-cycles, admitting disconnected graphs, specifying the emptyset cover, multiset counting, it all likely came from previous fake proofs.

[1]: see: ankitmaloo.com/fable - its the way most llms are trained, and is also natural owing to autoregressive nature.

Sounds to me like motivation is what LLMs need to do the impossible, I guess Kamina was right all along:

“Kick logic out and do the impossible! Remember that, that’s the way Team Gurren rolls!”

The sweet irony is all the jailbreak style fixes could hamper this approach.

At some point, we will no longer be mathematicians and engineers, but instead become matrix psychologists. "Do not make mistakes."

LLMs don't have the executive control to dynamically manage and compose mental sub-routines, when it has never seen those sub-routines in its training corpus. That requires fluid intelligence. ARAOC benchmark measures this specifically.

And all this is while they themselves proclaim that GPT 5.6 is much better at understanding intent and doesn’t need this type of guidance. Yet they themselves still do it, which I consider a more powerful signal.

Maybe in previous failed attempts that what the model landed on and they’re preemptively stopping it. Did they release the any info on the failed attempts?

I thought that too. The prompt is full of metaheuristics.

I remember a couple of years back when people were saying how prompt engineering was a skill, and reading this prompt kinda took me back to that.

Were I to guess, the reason the model couldn't do this itself is because most of the time, for most problems, a lot of this is bad advice.

In search optimisation you're often trading between time and quality. A very broad search will return very bad results for a long time. Where as a more depth oriented search with some heuristic will tend to return a pretty good result (if not optimal or close to optimal) quickly.

I'd assume models naturally want to find some middle ground there because that's the best thing to do most of the time, but for very difficult problems where a decent attempt isn't good enough you want a much broader search that doesn't have the time constraints. Much of the prompt seemed to be in that direction – really encouraging broadness of the search, preventing early convergence, and remove pressure of time constraints.

Maybe models also need a specially tuned version for mathematical research, just like "gpt-5.3-codex".

Looking forward to "gpt-5.6-mathx".

This is the key to training better and better models.

You collect all of the direction prompts and anti-direction prompts, and you finetune the model to behave that way for the particular domain, under those particular conditions, as a default.

This is also why I don't believe for a second that the AI labs aren't training on our data, even when we opt out. This sort of data collection is a requirement for improving the models.

It's also why Anthropic purposefully gives you a dumb model or a corner cutting model to goad you into guiding it into the correct answers, so that you do the fine-tuning for them. OpenAI at least, does not do that. Which is why ChatGPT is far less infuriating to work with.