The recently proposed o(n log n) integer multiplication algorithm [pdf]

But perhaps mathematicians are no longer able to be the best judge if a proof is valid. Yet nobody but them is able to even understand what is claimed to have been resolved.

It is really strange to see mathematicians of the highest rank taking these results seriously[3], holding lectures explaining them, yet see it dismissed here as a mass psychosis. [1]https://smf.emath.fr/actualites-smf/20-10-26-communique-sur-... [2]https://www.mathsoc.jp/publicity/news20261016.html [3]https://terrytao.wordpress.com/2026/10/19/a-sketch-of-the-pr...

I think that would change the conclusion of the argument a little.

Math is about clear communication between people first and foremost. That's why Shinichi Mochizuki's claimed proof of the ABC conjecture isn't accepted - he's unable to communicate it to enough mathematicians to get a consensus on its validity. LLMs generating Lean proofs are basically following that same schema. Sure, something has been proven. But what? Where's the clear communication explaining it to the satisfaction of other mathematicians?

The author of those sentences is also the author of the main result that this generated improvement builds on, and prior to this improvement his work was thought to be unimprovable. He is, in all of the important ways, literally the single most qualified human on the planet to asses this result. And his assessment is 1) this improvement is real, 2) this improvement is exponentially bigger than his work and the work he improved on, 3) it is effectively impossible for him (and by implication, humanity) to understand the result as presented, 4) it is seemingly trivial for “frontier circa 2026”-class models to understand the result as presented, and 5) to the extent that he is able to understand it, it is only with significant frontier model assistance.

So the place we are at, at least for this highly intellectual field, is “intelligence is the property of ~2 corporations, you must pay them a fee and agree to their terms and conditions to access it”.

At risk of stating the obvious, this is not a healthy place to be because intelligence is now structurally subordinate to the profit motives of particular corporations and at the mercy of all the attendant economic/sociopolitical incentives that apply to them. If the landscape is such that it is in these corporations interests to withhold this intelligence from some and make it available to others, that is simply what will happen.

One trivial example, of which there are an arbitrary number: An elderly leader is convinced to issue a decree that all number-math-computation intelligence is to be made available only to the intelligence agencies of his nation, and failure to comply means no IPO and no further investment. From this point on there would be a permanent nation-state-shaped shadow over practically all modern cryptography, with no recourse possible.

If the Church had a Large Astronomy Model in the time of Copernicus, the earth would have stood still - regardless of what it may privately prove about heliocentrism, it would never declare e pur si muove.

None of them were okay. Those were small small tools with few (sometimes only one) uses.

Generations earlier, famines, diseases and lead-leeching plumbing made rulers insane and entire cities of people die without any help from the "best tools".

Scifi is always written about the best tools. The issues arrive in the tools and in the hands that wield them both.

The amount of unimaginable complexity contained in a seemingly very tiny amount of matter to us, is incredible, and it’s all around us in nature

We’ve never had the capacity to fully comprehend neither ourselves nor the environment we live in. But we do our best, and we do survive and sometimes even thrive

At one point intuition helps to build out usage of the math itself. See stuff like "oh I can do rotations through matrix multiplication"-style ideas.

It is helpful for us to be able to reach for those tools quickly. The more tools that require advanced prep to use, the less we'll be able to use the results in easier contexts. Everything will be hard

Of course nothing really prevents people from looking for the simpler tools and proofs. Lots of math stuff starts off complicated and gets simpler.

Just important that this is still a goal in mind

How can you tell?

1. These tools are wrong a significant % of the time

2. They have an upper complexity bound where they start to fall apart

Relying on AI to check AI papers is a terrible idea

If you don't understand the subject and relies on the understanding of the tool, how will you have a frigging chance of catching an error?

Anyway this reminds me that scene in Matrix 2, where the mayor (of sorts, don't remember his exact title) of Zion tells Neo about the machines that keeps everyone alive, but that there is nobody left that understands how they work or could repair them if they broke.

All this anthropomorphism of literal statistical text generators.

Yes they exhibit generated sentences of text that describe coherent responses about a topic when there's an inordinate amount of research material in the training data from which they are distilling sentences of sound and rational thought. That's the ONLY way it can possibly work. There are so many scenarios where the training data isn't setup like that and low and behold the generated responses are just literal garbage.

There's a huge lot of suicidal people who ideate their afterlife as "pursuing hobbies".

Fable is good, especially at conceptual stuff, don’t get me wrong, but it has limits. Opus 5.5 is a machine at getting correct code out once the concepts are in place. But I can’t trust either to not make mistakes.

The big O notation is very often used to denote exact complexity of an algorithm even if the notation is technically an upper bound.

The little o notation is usually used to denote a small asymptotic improvement over some known complexity.

For example if some sublinear algorithm would have n^(7/8) complexity - would be probably advertised as big O of that. But n^(1-eps) is shared as o(n) with exact constant provided as a detail.

f(n) = o(g(n)) means as n increases, f(n)/g(n) must shrink

O(2n+1) = O(2n) = O(n)

o(2n+1) > o(2n) > o(n)

Both define upper boundary, but big O can read like "the function is generally proportional to ___" and little o can read like "the function is never more than _____"

  Lim f/g -> C a constant

  Lim f/h -> 0

https://stackoverflow.com/questions/1364444/difference-betwe...

Which is exactly what the capitalists that own these systems want. Divide and conquer.

"Pay no attention to the men who are responsible for you situation, instead pointlessly fight amongst yourselves."