No I cannot, and I'd argue most people can't either. We rely on mathematicians, peer review, and letting the scientific process run its course.
Granted for the sake of argument. But for this process, there's no requirement anyone from Clay needs to be involved, nor that anything has to be published in a traditional journal.
Terrence Tao can download the pdf, think it over, and publish on his blog, and it would be just as valid. Then some other mathematician can write something on 4chan or whatever. (See https://en.wikipedia.org/wiki/Superpermutation#Lower_bounds,... for 4chan.)
Many people, see prior conversation on HN, have already decided that AI solved it. The standards of reasoning and rigor in academia are complex enough that we all argue over them and harumph as we epistemically trespass on each other's domains.
The public, really humans if care for Herbert Simon, are much more apt to evaluate knowledge emotionally and by other standards. We may see them as wrong but standards only matter in context. The NYT, HN, and Annals of Mathematics will always have different standards of truth.
My understanding is that the theorem statement is quite simple, so i guess the latter is not very likely, but the former is very much a possibility in a proof this large, and it will take some human eyeballs to go over it before convincing mathematicians.
They do a Comparator Challenge to validate that they actually solved the correct theorem from the result, which they copied from Google/DeepMind: https://github.com/openai/NavierStokesAndEuler/blob/f9e8bc5b... - this is valid for both Euler and NS. Also, they validated with an external kernel from the Lean Kernel Arena. That way bugs in the Lean kernels were found in the past already, iirc.
Having said that, I strongly believe a positive result with the challenge above is the reason why they published it. I highly doubt anybody at OpenAI (or anywhere else) fully gets the proof after such a short time since publishing. This is also what Terry Tao criticized the most in my opinion.
Independent of the remaining drama [0], from my point of view, the proof is correct and an achievement.
[0]: https://news.ycombinator.com/item?id=49661928 - I am pretty much on the critical side, however, one can not ignore that it is an achievement. Esp. the unforced result.
If that's the current burden of proof required in your world for maths then that's fine! 't'ain't in my world: I want to see peer reviewed and published. Surely that's not too much to ask. Its not perfect but generally works rather well for maths.
I'm not a sodding programmer so please don't assume everyone here is one. I'm not a mathematician either but I do have standards: Your counter argument is a poorly constructed and inappropriately deployed example of "proof by whataboutism".
Not in the world mathematicians have been living in for the past decades at least. Nearly all big theorems that have been formalized so far had been published beforehand, and it was usually regarded as a step up in rigor. Wrong results get published in peer reviewed journals all the time.
OK but this member of the general public has at least subscribed to New Scientist since 1987, nine O levels, two A levels, two AS levels and a HND in Civ Eng. All pretty mediocre but I have a fair idea on how sciencing is supposed to work and how it ... actually works. Obviously, I ended up in IT.
I should also point out that maths "peer reviewed" is a bit special. For example Mr Wiles went through quite a maelstrom before his proof of some dodgy marginalia was accepted as "true".
While it's a convenient to assume that mathematics deals with logical statements, any attempt to evaluate those statements relies on physical processes with both known and unknown failure modes. There cannot be a test that establishes it unambiguously whether a claim is true or false. In all nontrivial situations, mathematical truth is based on expert consensus. When a new claim is made, people will try to raise and resolve objections, until a consensus emerges one way or another.
As for C++, all compilers are different. For any given compiler, there are valid C++ programs the compiler fails to compile and invalid programs it compiles without any errors or warnings. And now that I think of it, a new version of a compiler crashing with valid code earlier versions used to handle is the only class of compiler bugs I see with any regularity.
Just because Lean can compile it, does not mean it is safely proven. It is the start of a process to check whether something actually holds, not the end.
The more familiar analogy was when I look at the code that Claude spews for my partner. They take it at face value and hope it works. I usually find it very problematic, but only because I knew what to look for.