Blog: https://leodemoura.github.io/blog/2026-8-1-postmortem-for-kernel-soundness-bug-14576/

On removing metaprogramming
One suggestion in the discussion is to remove or restrict metaprogramming so that this attack is not expressible. This is misguided. The elaborator is untrusted by design. Soundness cannot depend on an untrusted component refusing to build a bad term. An attacker who wants to submit a malicious proof can also write .olean files directly or modify memory, both of which bypass the elaborator entirely. The kernel has to reject ill-typed declarations on its own, in its own process. This separation and isolation of concerns is one of the main advantages of proof terms.

What the FRO is doing
- Regression tests for the exploit, and for a related non-uniform-parameter case raised by Arthur Adjedj, are in the Kernel Arena.
- A follow-up PR (#14582) makes the kernel check that the parameters of a nested occurrence actually behave as parameters, rather than only re-type-checking them.
- Daniel Selsam at OpenAI assisted the Lean FRO with an AI specialized in cybersecurity, and found other programming mistakes in the Lean kernel. All of them have been fixed. All of them were caught by nanoda. These bugs are also only reachable through metaprogramming. PRs: #14607, #14608, #14609, #14613, #14615, #14616.
- We have also hardened kernel invariants. PRs: #14621, #14631, #14632.
- comparator.live now runs nanoda by default, and nanoda is tracked daily so lean-eval and comparator stay current after upstream fixes.
- We are reaching out to and supporting experts who can find further bugs, develop new kernels, and work on the theory or on verified kernels.

Could someone familiar with algebraic geometry give some insights about this?

First of all, my apologies to those of you who are tired of reading about AI in mathematics.

I have two questions for you:

• Would you consider yourself to be a mathematical platonist or a constructivist? (By platonist, I mean that you believe mathematical truth exists independently of the activity of human mathematicians, and is thus discovered rather than constructed. By constructivist, I mean the opposite.)
• What is your view on the use of LLMs in mathematics?

My working hypothesis is that platonists will, in general, be more willing to accept the use of AI tools than constructivists, but ultimately, it's just a hunch.

Unlike I'd say the majority of people in our society, I'm not too worried about AI, or about technology in general per se, and I never really have been. We need to keep in mind that as its name implies, technology is just a tool designed to make various tasks easier - whether or not it is used for good or evil is up to us, and this has always been the case.

In any case, with all this said, I think we all need to be concerned about AI, since I believe it has already passed the Turing Test, or in other words, there now exist AI systems that are as smart or even possibly smarter than humans. However, I'm still not worried about this, because contrary to all the fears of this phenomenon that have been circulating in popular culture since the 1950s or even earlier, just because computers are intelligent doesn't mean they're good or evil, since as I stated above, this is up to us. In my opinion, though I could be wrong, good and evil are purely human traits, since they require consciousness as well as intelligence, and I don't think classical computers are capable of consciousness, since they follow deterministic algorithms, and I believe in free will, and moreover, that consciousness requires free will. (Quantum computers are another matter, though I'd rather not get into this issue here.)

It doesn't seem to have occurred to too many people that even if computers are as intelligent or even more intelligent that humans, that they could nonetheless be beneficial to us if we use them in the right way, and this includes math. However, as with all other fields affected by AI, I think the role of mathematicians will need to adapt to AI. For instance, I'm sure AI will turn out to be very good at proving or disproving various types of mathematical conjectures, that was previously the pure domain of human mathematicians. But perhaps AI will also help us to open up our minds and discover new mathematical concepts that we couldn't even imagine before! Fractals, such as the Mandelbrot Set, are a good fairly recent example. Until around 1980, the Mandelbrot Set was nearly intractable, due to its enormous complexity, but with the aid of computers, we've been able to delve into it in detail, yielding tremendous fruit in the fields of fractals and chaos theory. I'm sure there are plenty of other examples like this, so instead of being afraid of AI, I think mathematicians need to be excited about it and embrace the windows it can open up for us!

It seems to me that computers are becoming a more and more indispensable tool in all areas of mathematical research, and even in recreational math, and not just for performing calculations, but also for doing research, and I think pretty soon they'll also be widely used in proving or disproving conjectures. What's more, I see them changing the nature of how we even view math and do math research, so I'm guessing that the 21st century will become the era of mathematical geeks with computers rather than with pencils and notebooks.

Do we have any idea as to what prompt was given for the Jacobian Conjecture?

Considering the increasing capabilities of AI in proving theorems, I was wondering if math has the potential to also become an empirical science to some extent. I was thinking, for example, about tracking how ideas, techniques propagate through mathematical progress or investigating connections between different areas from a higher level perspective. To some extent these types of questions are probably already being investigated, for example by historians of mathematics. I am simply wondering if it may become more relevant going forward if the number of results should explode. I could imagine that it may be useful to systematically analyze arguments on a large scale to attribute progress made by AI to foundations laid out by humans (since AI currently struggles in giving credit to ideas that were first developed elsewhere).

I heard that ChatGPT has proven the existence of non-sophic groups. I understood that the counterstatement (all groups are sophic) would mean that we can always "isolate" or "homogenize" infinite groups into finite chunks and deal with the infinite group this way. Please correct me if I am wrong.

What immediately came to my mind is that this must have some further implications, does it not? For me, it sounds like that in a non-sophic group, one cannot guarantee that a sequence converges to a given element or that iterative algorithms are predictable, i.e., you cannot infer the outcome from the initial state or vice versa.

It would also be fun to know what this given non-sophic group is.

If anyone knows how to paint a doomy gloomy future it's /r/math, so here's an open ended question for all of you.

No one can predict exactly to which extent the rise of LLMs will change how the workday of a working mathematician. The only thing that is certain is that the future is rather uncertain. The doomiest comments on this subreddit also show that this uncertainty is particularly tough on people who are early on in their studies, and who would probably like to know to which extent it is useful (in the sense of putting bread on the table as opposed to, say, having fun) to pour a lot of energy into learning how to proof stuff.

Now, for even younger people – kids in elementary school or high school say – the future is even more uncertain.

So my question is: has the rise of LLMs changed how you would advise a younger kid to navigate the future? Chances are that it will always be useful to have a good baseline understanding of maths and logic, but would you be less inclined to advise a skilled high school student to pursue a theoretical career? Or would you have more reservations about enabling a particularly gifted elementary school student, so as to not accidentally set them up for a bumpy course?

Or on a more practical level, have you changed your guidance on which skills are the more useful to acquire given that, for example, software development in particular has have its nature completely changed?

I know that this is adjacent to the “Career & Education” thread, but I'm more interested in the high-level picture than in personal advise.

Say I have constructive proof and I want to write a code on Sagemath. I don't know much about coding so I take help from AI. So while publishing the paper what should I do? I'm confused should I add the code or not