GPT‑5.6 Sol davant matemàtiques de doctorat: què resol de debò

So today we're back because OpenAI have just released GPT 5.6. And if you've been on this channel before, you'll see that we've been basically testing every single model against PhD maths problems. To see whether all of the claims that are made on Twitter are actually legitimate. We've tested all of OpenAI's recent pro models because I pay monthly for that subscription, and we've also tested chords, favour, and we've seen some really strong capabilities, but also some pretty huge limitations where these models will fumble in an incredibly basic way. that said, we have also seen the opposite of this happen where the model will make really big claims and they turn out to be true. And in these cases, I'm the one playing catch-up in order to see whether the model has said is actually true and then it does turn out to be true and it took me a while to actually verify it. So it's a mix bag, really. But this model in particular has seen a lot of hype in terms of its ability to do research level math problems. So today we're going to be exploring some problems that I've been working on and we're going to see whether GPT-5.6 Pro can actually help make progress on them. I've got questions regarding what I've already done, questions regarding like general generalizations of another paper in the context of what I'm doing. So, we're all kinds of capabilities of GPT-5.6 for explaining and exploring things that it's given to actually trying to generalize things that it's given with a benchmark given by a different paper, which serves a similar problem but in a different context. And so much more so that we can really see whether GPT-5.6 is actually a step-function improvement

in research level maths, or whether it's just another hyped AI model release. Okay, so we set up two questions which are an increasing difficulty, and we'll then ask some following questions based on how the model responds. So, the first task, which I think is much more straightforward than the second one is I've given it a whole bunch of research now, so you can kind of see here, which basically give some propositions and some lamas and I've essentially explained what I'd like it to do is to combine some results in order to make another result actually there's two results that I'd like it to basically state and prove which basically just combine the things which I've given it. So this task seems definitely doable in my opinion because basically it is quite straightforward how you do this is just a whole bunch of algebra that I don't want to type up. It's not necessarily that complicated of a task. And I then outline the argument of exactly how it gets each one. So one does a coordinate transformation introduces a kernel that's basically conversed a factor that we want and then gets the type of symbol from there. And then the other one does a coordinate transformation gets the kernel and then does the same whole argument but then after you've written it more to relatively which is a thing that's important somehow in this type of problem, you then undo the coordinate transformation and then somehow the symbol seems to lose the end dependence. And as you can see we're on GPT5.6 so and put it on pro. Next up is a much more difficult problem.

So in this case, I'm asking you to do a very serious calculation, which I don't actually even fully understand to be honest, which uses some really advanced techniques from integral systems, called non-linear steepest to send, which I personally haven't had formal training in, but it does seem like it's kind of the big guns that can solve pretty much any asymptotics problem if you have a determinant and the whole point of what I'm doing. This is trying to represent the thing that I'm trying to solve as a determinant so that then you can get the asymptotics that determined using this machinery that's been developed. Why I want to see really is if a clanker which is much more clever than me at knowing this kind of stuff, whether it can give any kind of outline of results because there are some limits of the thing that I'm trying to solve, which kind of have been predicted already. So if basically the clanker can use this method of non-linear steepers that say maybe it can try and recover known results and then you would know whether it's done it right because there's already formally that have been predicted or like conjectured and so you really would know if it's followed the right reasoning and whether it's worth actually studying what it's given. So really the task is this.

It's to compute the larger asymptotics in what's called the double scaling limit. So that's where you take one variable to infinity. And another variable that kind of is in the whole game of what you're doing. That also depends on this thing that you're taking to infinity. So in some sense it's two things of scaling at once, although they're really still end up depending on the thing that you're taking to infinity. So it is pretty cool because when you do that you get different results. So this question is definitely asking a lot more. And it's going to be more difficult for me to verify whether what it's done is right. by I'm interested in seeing what this model can do because I've really asked previous models of this type of question and the answer hasn't been that useful. Okay so the model has replied on both the questions and interestingly if you look at the thinking time for the two questions so the more difficult problem it seems to have thought for 20 minutes if you look to the simpler problem it thought for half of that time which to be honest I'm still surprised it thought for that long given that this task was really much more simple actually than the other ones. I'm surprised actually that the other tasks which will read and see whether it's done it. I mean just looking at the thinking time to be honest to just to me that it hasn't solved the problem. So in this first question it basically gave back a whole bunch of um late-tech code which is some mathematics surrounding what I'd asked it to do. So I

posted that into a little section of a late-tech file that I'm already working on. So this first result is what I kind of expected that it would give. Having read through the notes it did do a pretty good job I would say explaining how you would prove the identity. So it did succeed I would say and doing what I'd asked it to do. It typed up to two propositions and gave some proofs. I do find sometimes that these models would just give so much writing. When you actually want to write something yourself, you do want to distill it into what's important. And I do find that even this model is just kind of saying so much crap. It seems like it's capable of answering the question. I just wish it was able to do it in a more concise manner, which is like a bit easier to read. So that I'm not spending hours reading the same thing. But this is interesting. So it's given a little remark. I wonder what it's saying. So it's got this character equivalent one.

You could use the un-shifted symbol. Okay, that's fine. Oh. Ready? Why are you telling me this? Why would you put this as a remark? Why would you put that as a remark? I mean, that would be a joke. If you put this in a remark, you'd come across a more on. Well, you know, actually I'm gonna put a remark in this paper that I've released, that actually I could factor this term out of the determinant and have it pick up an end. That's worth a remark. Why are you telling me this? I don't know. I mean, maybe it's suggesting that it's better to have it factor down.

Maybe. It also is kind of annoying how often it defines things. It defines so many symbols. By the end you've got an equation that's got so many symbols and you've got to scroll back through all of the crap. I was about to say it would be cool if it defined everything at the beginning, which it kind of does in this, but I think there was another symbol that you use this one. As Q of I, I think why I expected with this was a slightly better quality of answer. from this question, because I've really done this type of question with GPT5.5. And it's the same thing. You get an answer with some useful equations and a whole bunch of filler. And within this problem, I've been using Fable, I've been using GPT Pro.

And in both cases actually, the models would generate so much text. It's much more common for it to extend your kind of file, adding loads of code, than to offer something more efficient. I'm gonna ask why it gave that remark, because it seems like a trivial statement, and I'm not sure why you would state that remark. Okay, so as a follow-on question, I've said, can you provide a more concise version? seem to redefine things a lot avoid using things like Delta Resipter Online Z squared or SQZ equal to the or complex additive statistics. It's not it's complex additive statistics. The complex additive statistics. Please explain why you felt the need to write a whole remark regarding the property of a determinant underfecturing in or out the constant. There's just no way you would have that and unless and this is what I'm saying it's happy to make a case for itself. Okay, so I said perhaps you felt necessary because you think it's better to fact around the deterministic drift either argue your case or remove the remark. So we're going to see what it does him and in doing so we're going to read the answer

to this. So just to clarify by the way, as I said before, 20 minutes for this problem, I don't believe that it would have done it. I just don't believe it. Okay? Not if in nine minutes it gave this crap and maybe I'm being too critical. Okay, but this, I don't think it was that complicated for it to do. I'm surprised that took nine minutes. and it did have this really stupid remark. This remark is a big deal, okay? I really want Nege put it in your deez as ableed, put it in your deez so that when you have your vibe you look like a complete moron. To justify a fight if you don't know why it's stupid this remark, look up factoring out constant from determine. This is effectively what it's done and it's written a remark about this. So let's see. Maybe this one is pretty much this, which I think is probably an A level video. Is this a question here? If you take your matrix and then you multiply by number and then you take the determinant. Because this constant when you multiply by the matrix, I think it goes into every row and column. So when you take the determinant and you expand it, you're going to get in of them

of this constant. So really it just factors out. So it's the determinant and very multiplied by the constant to power of end. This is what it's saying here. Okay. It looks fancy. You know you've got this cool symbol here. This is stupid. Unless crunchy pt argues it's case. And we'll I'm excited to see whether our I guess it's case to be honest because I've happily be wrong, but we'll see what it says. Yeah, I would therefore remove the remark entirely, okay? And keep the drift inside the symbol. That's what I was all with. Let's go back to this question. So, it actually tries to generalize the problem instead of answering the question I asked to. It actually does a more general problem which, so that was cool. This is cool to see that it starts and introducing this factor of the logarithmic derivative. So this is the more general problem. So, really what I was asking about was just this term here. But in general, people would do this. I kind of was under the impression this was just a belief that I had Which I believe is justified Otherwise I probably wouldn't have believed it. I could have just done it the other way But I wanted to see whether the methods worked for this case truth is I couldn't be bothered right now to do it like this was enough Anyway, so it's cool to see that the client has actually been like you know what you may as well just define the joint moment That's cool because actually another question I was thinking of asking was to be like oh, here's some work

which I know if you just do it all for the more general version that should it all go through. Can you do that? So it's called see that it's doing that. Okay so I've got to give it credit here. Take me speaking in the question that I asked it is making a fair point because I asked for this a structural analog of a theorem that shows up in a paper. In order to get that you would definitely need to introduce this. So you would need this because otherwise you're not going to get this term. You can just tell that it's sort of read between the lines because otherwise you wouldn't be getting this factor. Then it talks about a known result. Talk to us proposed structural analog. See, this is just, that's fine. This is obviously true. I don't know why it's putting a box around it though. So it's giving a lot of vibe justification of what's going on. So it's giving you a vibe of what's happening. Let me just show you this epic epic paper. I'm sorry, but this, I was asking for, I wanted to know stuff like this, actually get the asymptotic structure of the parts

which are important. This is kind of what I've seen it do before to be honest. It's just kind of the same stuff. And you could argue, okay, I'm trying to one-shot it by asking one question. I think it's giving insight on how you would solve the problem. I just think the thing is, if you're actually trying to solve this problem like this, using this advanced machinery that's kind of outlined here, there's really no other way of going about it. You can't just trust a client to do it. So the things you can't actually trust what it's doing for a more complicated problem. So it makes you wonder how you intending on removing the expert, like say the person that wrote this paper. I'm saying, if I wanted to try and tackle the problem that I'm trying to solve with the machinery in here, the only way I can do that pretty much

is the old fashioned way of learning how to do. You can't just be like, oh, gone, track GPT's sender. Okay, so I was a little bit disappointed with the first attempt with GPT Pro, but after leaving it for a bit, and unfortunately seeing even more AI hype to do with this model being released and just how good it is with math. I kind of wondered whether maybe I'd asked the wrong question on maybe it was too difficult. So I had to think about maybe how I could isolate the actual problem I wanted to ask about it. So if we go back to the conversation that I had, why have decided to do instead is to basically simplify the problem and try and isolate the part which I thought was the important part which is evaluating this double-scadler method of a settlement and try and get a new prompt based on that simplified problem which I had the help of GPT Pro to basically help me write the prompt, basically starting from the prompt that I've written, that's what's going into here, as you can see it's something like this, it just includes all of that stuff. I've also given all of the research notes for the thing I'm trying to do and this paper which is a specific case out of the thing that I'm trying

to get. So what we've done is we're using GPT5.6 sole on Ultra because apparently it's spawned subagents to try and solve the problem. So I thought maybe it was worth giving that ago. In any case we've also re-ass this question like a simplified question because I can't imagine open I've put this much time in this model and it being basically the same feels a little bit more likely the I've made a mistake maybe than the model is basically the same, but maybe it is basically the same who knows at this point. Maybe we'll see how it answers this question whether we can give any more information than it previously did before. Okay so I had to look over the responses last night and I had to say I was actually much more impressed with the answers that it gave this time round with the improved prompt to what it had given yesterday when looking at this problem. So I had a session in GPT work with this sub-agents thing and it's all for 20 minutes And then we had one in GPT Pro, thought for much longer than the other times for some reason, maybe it's just the prompt was better, which I'm looking at actually because there was some news recently that opening I basically proved some new conjecture. And apparently it was one shot and they gave the prompt to actually, so I'll show you guys that on the screen and they even like publish some sort of archive paper. So from there, it seems like maybe I could improve the answers that I get by comparing my current prompts to what they've done. So the main things that are good about this, and this equation looks pretty good. this one.1. I mean I'm not sure how useful it is really. Just looking at it, it does seem decent. Why? Because basically this piece is quite important and reduces to some other known case, and then this piece is a bit like the generalization. So I basically read through this whole thing

yesterday and it seemed pretty promising to be honest. So it talks about this limiting tread home determinant, which is kind of what you would expect, because you are taking a double scale limit of a determinant and if we look at the chaturity work session, it kind of gave the same sort of result which is interesting and overall the answer is just much higher quality than what we had before. It goes into way more detail and I think it's much more rigorous and the formula that it's given are much more interesting than what it had before.