Based on my admittedly limited research, it seems like you should use Laya for much more deterministic tasks where you have some training data. It won't be as good as Jev for zero shot cases.
I did that on a little macbook m4 last night on my model of the innate immune system—fine tuning took 15m or so. Just wish it had a larger context window
Note that this proof while impressive does not add any value to mathematics as a
human pursuit. But it does show we can throw these LLM beasts at much gnarlier
problems than we could have imagined previously. Maybe even formally verify papers the day they are posted?
I'd love to see an e2e compiler or OS kernel verification or Full-stack chip design with formal equivalence checking at each stage that would be pretty cool.
What else is interesting is how they staged this problem : (a) maintain an explicit DAG/roadmap of sub-goals rather than one flat prompt, (b) separate statements from proofs so many agents can work on different nodes without stepping on each other, (c) keep a natural-language index alongside the formal one so search/reuse works... I feel like this is the future of long horizon agents and how you can do work that's making the most of every agent. This approach will likely be baked into the next versions of coding harnesses
> Note that this proof while impressive does not add any value to mathematics as a human pursuit.
I don't see how this can be stated with such certainty. We don't yet know what the implications of large scale autoformalization and proof verification will be on the human pursuit of mathematics. I'm open to the idea that it might be a benefit to the human pursuit once the human pursuit adapts.
> one might naively expect that the natural question to ask with regards to a given problem X in a field is "What is the answer to X?". But in many cases the more valuable question is "What can be learned from studying X?"
And later
> But the currently fashionable practice of pointing a powerful AI tool at the task of answering a problem X, unguided by any human expert in the field X resides in, has created an unprecedented divergence between the production of answers, and the production of insight, to the point where the two questions have become _negatively correlated_:
That seems narrow minded. Theres both "learnings directly related to the thing studied" and "learnings downstream from the thing studied". If you imagine mathematics being a huge sudoku puzzle of unknowns you are trying to fill in, each previously empty square you are able to fill in (or gain a smaller bound on) has implications in all sorts of other areas.
It's important to remember that most mathematics is not science - large parts of it are mostly esthetic pursuits that bring joy to certain mathematicians, and sometimes happen to have unexpected benefits to science or engineering (like how number theory suddenly became important to cryptography in the 20th century).
Fermat's last theorem is a great example - it is in itself a completely irrelevant observation, not used (so far) in any larger theory. It was only pursued because (a) Fermat casually claimed to have easily proved it (almost certainly being mistaken about it), and (b) it sparked the curiosity of mathematicians because it looks so simple but turned out to be so hard.
So what does humanity gain by knowing that the theorem holds? Basically nothing. What does humanity gain from the process of proving it? As far as it is known for now, basically nothing (though it is somewhat likely that the complex theories created to prove it will find other applications). However, those that have worked on it, and the guy who did prove it, gained a huge amount of personal insight into mathematics, and surely grew as mathematicians, and will hopefully use those skills in working on other problems that may prove more directly useful. Plus, they had a great time doing it.
What this means is that, if the proof had been discovered entirely by AI, basically nothing would have been gained. LLMs don't learn by doing, so no personal experience growth would have come from this; and as I mentioned, both the result and the proof are, so far, quite irrelevant even for mathematics more broadly. So it would have been actively detrimental, or at best neutral, compared to letting human mathematicians work on this problem, in a way that is never the case in science or engineering, where any bit of knowledge is in itself useful to at least some extent.
In your letter of December 14, you ask me whether mathematics is a worthwhile endeavor. Mathematics absolutely is not a worthwhile endeavor! I can imagine only three possible reasons for studying mathematics:
(1) Entertainment.
(2) To promote technological progress.
(3) To promote one’s own career.
Reason (1) is harmless; but one doesn’t refer to entertainment as a “worthwhile endeavor”; it’s just a pastime.
As for reason (2), modern technology is the worst thing that ever happened to the world, and to promote its progress is nothing short of criminal. Read Our Final Century, by Martin Rees; or read “Why the Future Doesn’t Need Us,” an article by Bill Joy that appeared in Wired magazine, April, 2000.
As for reason (3), to choose mathematics as one’s career would be very risky. Not long ago I received a letter from a gravestone sculptor whose craft had been rendered obsolete by a laser-guided machine that carved gravestones automatically. He was in his forties, unable to find work, and depressed. Maybe you think that can’t happen to you if you become a mathematician. But you’re wrong. Martin Rees, Bill Joy, and many other distinguished experts believe that within a few decades machines will be developed that will be more intelligent than even the smartest human beings. When that happens, people will be obsolete; they will be phased out and will be replaced by computers and robots. This will happen to everyone eventually, but mathematicians probably will become obsolete before anyone else does, because mathematics is more amenable to computerization than any other field. Hence, computers probably will surpass humans in mathematics before they do so in other areas.
> He skips over understanding our world which is probably the reason most pure maths is done.
I'm not sure I agree with this. I suppose it could be that we have different defintions of what "our world" means.
You might like the quote below. I suspect you're arguing that most pure math is performed for option #2 below (utilitarian purposes). Like I said, I'm not so sure.
In the temple of science are many mansions, and various indeed are they that dwell therein and the motives that have led them thither. Many take to science out of a joyful sense of superior intellectual power; science is their own special sport to which they look for vivid experience and the satisfaction of ambition; many others are to be found in the temple who have offered the products of their brains on this altar for purely utilitarian purposes. Were [someone to] drive all the people belonging to these two categories out of the temple, the assemblage would be seriously depleted, but there would still be some men, of both present and past times, left inside. Our Planck is one of them, and that is why we love him.
I guess I was including maths and imagine mathematicians investigate stuff of the general type like the riemann hypothesis, whether all non-trivial zeros of the Riemann zeta function lie on the critical line where the real part equals 1/2, which are interesting but of little practical value.
Personally, I found that people are impressed with AI for subjects they simply don't understand, and when an expert works with AI, they're often underwhelmed. So while we're seeing some mathematics being automated and proved, I think there's a long way to go before we automate proofs with meaningful human value.
Google doesnt even use its own Go build system internally. Its all blaze / bazel, so they are not even taking advantage of the so called compiler feedback of Go. Also if languages are to be designed for agents not humans, its not clear whether the verbosity of Go will help agents at all
Okay so the OP is saying that since Macbook Neo has the same hardware as iphone, but not locked down, so why is iphone locked down. They say its because of the app store profits.
Sure App store is not to be understated, but I'd add our phones include way way more personal information than a laptop like NFC for credit cards, personal photos, and all biometric and contact information. Not to mention cellular network connection and generally forms as a soft form of identity. None of these apply to a laptop. So form factor does matter.
BUT even if we unlocked the iPhone, the desire for 'MacOS on iPhone' is actually the wrong thing to ask for. Pete Steinberger had in this interview (https://www.youtube.com/watch?v=AcwK1Uuwc0U&t=1182) that UI is basically the wrong paradigm in a world where agents should do tasks for us in milliseconds. We should be able run any local services from our phone like grabbing
Good news is we already have this via terminal apps in Android. Now what's left is the ability for agents to run on your device and basically accomplish tasks for you
Perhaps it is the way I framed the article, but the core of my argument is not “MacOS on iPhone”.
Yes, I do personally desire that as a niche thing, but the broader point is the nature of a locked bootloader that prevents any third-party software from being loaded / installed without getting an approval from Apple.
Whether the desire is third party apps on iOS outside of Apple-vetted channels, or entirely new operating systems like Linux or MacOS, I’m mainly arguing I should have the right to modify the software however I’d like as the device owner.
My laptop has all my contacts, calendar, access to all my photos and is regularly used for online banking. It does not have cellular but I could buy one that does. So your list applies to a laptop just the same.
Amazing write up and i wish more people showed the process for discovery which is often even more interesting than the result itself
Still the result is really interesting being able to stack abstract reasoning and get better performance and the heat maps to show the prob results
The academic literature seems to be catching up:
- *[SOLAR / DUS (Kim et al., 2023)](https://arxiv.org/abs/2312.15166)* — duplicated transformer layers to build a 10.7B model that outperformed 30B parameter baselines.
- *[The Curse of Depth (2025)](https://arxiv.org/abs/2502.05795)* — explains why this works: Pre-LN causes deep transformer layers to converge toward identity functions, meaning middle layers are where real computation happens, and duplicating them concentrates that capacity.
- *[Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach (Geiping et al., NeurIPS 2025)](https://arxiv.org/abs/2502.05171)* — takes the idea to its logical conclusion: a model trained with a single recurrent block repeated at inference time, scaling reasoning depth without adding parameters.
On the other papers, models like SOLAR or training a model that uses a single layers are probably going to hit a wall, based on the heatmaps I found. The transformer stack starts with randomised weights, (analogous to undifferentiated stem cells), and it seems they later form 'organs' during the trillions of pre-training tokens they undergo. My hypothesis is that you probably only want one copy of the 'token-to-thought', and 'thought-to-token' organs. It seems that you can make one layer do all three things (transforms in and out, and do the 'thinking'), but I think specialisation will always win.
I really like Clawdbots safety gloves off approach - no handholding or just saying yes to every permission.
I set it up on a old macbook pro I had that had a broken screen and it works great. Now I just message my server using telegram and it does research for me, organizes my notes, and builds small apps on the fly to help with learning.
However security is a real concern. I need to understand how to create a comprehensive set of allowlists before expanding into anything more serious like bill payments or messaging people / etc
Some great life lessons here, but also some I don't agree with:
- The lazy person works twice as hard.
Often I found you can save a lot of time just trying to the minimal possible and gain a lot of insights of why something is minimal vs not
-The opinion of the person who rarely offers it is listened to more closely.
I found the opposite to be true, those who don't offer their thoughts frequently are often dismissed when they do want to share something
Anyway, many of the points are great.. I would also add to keep a journal and write down what was meaningful throughout the day.. you will find time passing by with more quality since you know what the take and what to avoid
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