Claude Fable produced a counterexample to the Jacobian Conjecture - https://news.ycombinator.com/item?id=48973869 - July 2026 (508 comments)
I can use AI for coding after decades of coding. I can't use it for theoretical physics because I can't evaluate the responses.
> I’m also surprised to see that even Terrence Tao seems to use it in a way that resembles, in progression, how I use llms in my area of expertise
I've mostly seen people trying to oneshot a result, while I'll quickly experienced that going through steps/discovery was more effective and more satisfying, since you can always steer it back in the right direction; while oneshotting is hit (and it kind feel like magic) or miss (and you'll have to rework your prompt).
I think it's not even about the ability to steer the AI. Just the ability to ask the right questions
The first one was someone proving another conjecture false by just repeatedly saying "keep going" to ChatGPT: https://x.com/DmitryRybin1/status/2079904005652893709
What a world we live in.
1. The counter example wasn't just a brute force selection, the polynomial is structured in a very specific way that ends up getting the result.
2. Terry Tao's questions are very specific and prompts the AI in a useful way, that without high math training you are not going to get the same information out of it. Terry seems to see some aspects of the problem and counter example and uses AI to brute force some parts of it.
Just awesome to see new knowledge hit an incredible mind like this. Having these "what if" discussions is what I miss most from JPL and academia.
Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same unfamiliar sense of total loss like when encountering hard mathematics.
Yes, Tao is guiding it to where he wants to go. But also, Tao is actively learning from it and relying on its explaining, analysis, and inference abilities. You can easily imagine this conversation having taken place between Tao and a PhD thesis student, or even another professor, explaining their results.
What can we imagine and predict about the future anymore? Maybe a year - or two model releases - from now, the AI assistant will be undeniably stronger than Tao, and not an equal anymore.
a) The model thinks on some questions while straight answers on others. (I wish I'd knew from the questions if this is somehow correlated to hard tasks or "inventive" tasks, but that's way out of my league).
b) The model sometimes pushes back. Again, I'd wish I knew if it was warranted, but I counted 2 instances where it said "yes, but with caveats", one where it said "mostly yes but with this correction" and one where it said "careful here, because x y z".
c) The model did q&a + pdf ingestion + code writing + more q&a + thinking + more q&a, for a looong while, while seemingly staying on topic (at least Terrence Tao seems to think they're still productive, so I'll trust that).
This is what model progress is, not number goes up on xBency or yBencher. Damn.
Where will we be in another 4 years? What a time to be alive!
Another satisfied customer!
https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the...
The fascinating this is that the LLM is not acting as a tool here AFAIk, but very much like a colleague.
I have no knowledge of the domain and have only PhD EE level math knowledge, so maybe my bar is too low.
I wonder how many ppl can actually follow what's happening, I mean the math.
Expand the entire expression, then change the representation to find the core axis. You can't see the axis from just one perspective, so you change the representation. In programming terms, it's like applying multiple domain models. Then break it down into small contract units. Why is it a Jacobian monomial? Why does x satisfy a cubic equation? And so on.
Then swap out the modeling under a hypothesis, assemble it all back together, and verify it through the equation.
This feels similar to modeling in programming.
Observe the whole -> explore better modeling -> decompose local problem -> verify independently -> reason about the highre level structure -> integrate back into the original problem.
This feels similar to when I receive work from a client and write a programming proposal
Two, at some point AIs will be able to use other context like the fact that this is Terrence Tao and not your average Joe and change how it answers, either in tone or structure.
One thing I notice is many models say statements along the lines of “okay we have exhausted this thread it’s diminishing returns from here and we should stop and move on”
It’s funny because I’ve been building a tiny neural network maze solver (23 bytes solves 92.75% of unseen 2D mazes)
When I asked ChatGPT/Fable if we had anymore threads to pull to increase capability and decrease byte size, they both basically said no way - back when I was at ~166 byte models with a ~85% solve rate.
Throughout the experiment I just kept trying different approaches and eventually had 3 mini “breakthroughs” in this particular niche. But if I had listened to the models…
Anyway, these models are amazing to experiment with quickly, but they are dumb as hell and so absolute
Yes—for a continuous-time autonomous system
x ˙ =f(x),f(x ∗ )=0,
this is the standard linearization criterion, with J=Df(x ∗ ):
If every eigenvalue of J has strictly negative real part, then x ∗ is locally exponentially asymptotically stable. If at least one eigenvalue has strictly positive real part, then x ∗ is unstable. If no eigenvalue has positive real part but at least one has real part 0, linearization is generally inconclusive. Nonlinear terms or a center-manifold analysis are needed.
The last case really can go either way. For example, all three scalar equations below have Jacobian J=0 at x=0:
x ˙ =−x 3 , x ˙ =x 3 , x ˙ =0.
Yet 0 is respectively asymptotically stable, unstable, and neutrally stable.
A slightly more precise wording is therefore:
If the spectral abscissa
α(J)= λ∈σ(J) max
Reλ
is negative, the equilibrium is locally exponentially stable. If α(J)>0, it is unstable. If α(J)=0, the Jacobian test is inconclusive.
This criterion concerns the Jacobian matrix of a dynamical system at an equilibrium; it is unrelated to the “constant Jacobian determinant” condition in the Jacobian conjecture.
Is there any way to tell a conversation's model and thinking level?
Physics, they describe general relativity as a two way street. Space tells mass how to move, and mass tells space how to bend.
Here, Terence tells AI what question to explore, AI tells him what questions to ask next.
The exchange that ensues is just magic to watch.
```
A question is salient to the degree that its answer changes what we do next. Operationally, saliency = the product of four things:
- Decision-leverage — would resolving it one way vs another force a different design or invalidate a stated decision? (No leverage → drop, however interesting.)
- Residual uncertainty given current evidence — is it still genuinely open after reading the docs and the code? (Already settled → drop, however deep.)
- Load-bearing-ness — how much rests on the premise.
- Cost of finding out late — architecture-deciding / expensive-to-unwind raises priority; cheap-to-fix-later lowers it.
```
There are a few things to note about this prompt
1. There is no reason from looking at it that it should work, it even has the word load-bearing which people loathe, but it remarkably produces a stable design with questions from claude (atleast from claude Opus 4.8 and even better from Fable5). Otherwise the design document claude likes to really write are implementation level(code or otherwise). I usually pair this with matt pocock's grilling skill to make claude behave.
2. From design -> implementation, its is generally about understanding when claude is trying to trick you into making something sound like a good/easy solution but has tons of untested assumptions. Here you have to read and patiently spot if a how you would get to the solution is not clear. A common error here are when claude makes a big deal based on what it read and interpreted too seriously without questioning the assumptions. There are several more.
But it also comes down to your experience as a SWE, much like a mathematician's. The frustrating thing about it is, it feels tha a skilled mathematician working with AI can make them productive in ways that are more reliable as compared to a SWE (e.g. lean is deterministic and can provide very strong feedback and LLMs are very good at using that feedback). Maybe a mathematician can chime in on that?
Modern AI feels like a godsend to mathematicians. It helps them break down boundaries and connect concepts in ways a mere mortal couldn't imagine.
Almost all of Tao's questions begin with what, or why. That forces an open ended response, which is a great way to reduce or eliminate sycophancy and severe hallucinations. The less you steer, the more accurate it gets.
Classic Gabe-bot
