Abstract:
Claude’s relevant proofs on the Riemann Hypothesis have been thoroughly verified by human mathematicians. This time, mathematician Youness Lamzouri announced that Claude's proof was correct and gave a new proof that is simpler, more elegant and stronger.
Paper: https://arxiv.org/pdf/2609.02882
Go back to last month, when Claude proved that more than 2/3 of the zeta zero points were confirmed to be on the critical line and were single zero points!
https://www-cdn.anthropic.com/95c246936988e43127bc6b2ceb7077c1dad2d68e.pdf
This number directly breaks the record proven by the Riemann Hypothesis - the previous mainstream limit stayed at around 41.6%, and mathematicians have only improved by 0.8% in the past 37 years.
And Anthropic’s mathematicians used the internal Claude to directly improve it by 25.6%!
Although it did not completely prove the Riemann Hypothesis, this result was enough to cause an earthquake in the mathematics world, and Anthropic directly dominated the headlines.
But the proof of AI is obscure and difficult to understand, and "the main mechanism is not clear at a glance."
Now, number theorist Youness Lamzouri has written a proof that is so brief, clever, and beautiful that it is an extraordinary feat.
In just a few hours, AxiomProver completed the formalization of this work.
This era is really exciting!
This is not only a violent aesthetic of AI, but also a desperate counterattack of human wisdom, and finally the myth of "cross-species" collaboration was finalized by another AI (Axiom Prover).
Lean certificate: https://github.com/AxiomMath/ZetaZeros
The underlying logic of the history of mathematics was completely rewritten on this day.
AI’s “aesthetics of violence” opens the door to the Riemann Hypothesis
The Riemann Hypothesis is revered as the "Holy Grail of Mathematics". Why? Because it coincides with the most profound cosmic reality in its purest abstraction.
First, it's the ultimate blueprint for taming "prime numbers."
Prime numbers appear to be randomly distributed on the number axis like ghosts, but Riemann geniusly connects them to the zero points of the Zeta function.
If the conjecture is true, it proves that under the appearance of chaos, there is actually an extremely perfect harmonious symphony hidden.
Secondly, it is the "load-bearing wall" of modern mathematics.
Today, there are thousands of high-level theorems based on the premise that "the Riemann Hypothesis is true". If it is falsified, half of the analytic number theory building will be turned into waste paper in an instant.
From number theory, algebra, geometry to functional analysis, there are about 30 to 50 equivalent forms of the Riemann Hypothesis that are truly landmark. Even the search for such equivalent forms itself is a hot topic in modern mathematics research.
Oxford University Press has published 3 volumes of "Equivalent Forms of Riemann Hypothesis"
Unexpectedly, it crosses disciplinary boundaries.
Physicists were horrified to find that the spacing distribution of these purely mathematical zero points was miraculously consistent with the energy level distribution of heavy atomic nuclei in quantum mechanics! Proving it is not only a victory of mathematics, but also the ultimate leap for mankind to touch the underlying code of the universe.
As we all know, the core of the Riemann Hypothesis is to explore where the non-trivial zero points of the Riemann zeta function are.
In 1859, Riemann boldly guessed:
All non-trivial zero points are neatly arranged on a "critical line" with a real part of 1/2.
Proving that it is 100% on the critical line means cracking the Riemann Hypothesis. Humanity had previously struggled to prove that more than two-fifths of the zero points were on it, and then it seemed to get stuck.
Until the AI ends.
In the latest research, Claude launched a charge to solve this century's problem.
AI doesn't understand what "a flash of inspiration" means. It believes in "miraculous results with great force."
Claude used a bunch of huge matrices that grew with N(T) to calculate the trace and Hilbert–Schmidt norm of the matrix. Finally, relying on a complex set of rank-trace inequalities, he stuck the lower bound of the zero point and pushed the boundary to 67.25%!
This result has also been initially verified by mathematicians Alpöge and Furman.
As soon as the results came out, the world was in an uproar. Humanity is close to completely defeating the Riemann Hypothesis, with only the final 32.75% left!
But when human mathematicians got the proof process of AI, they were all silent.
This proof is so ugly. In other words, it's too "inhuman".
Through exhaustive research and splicing, Claude forcibly pieced together two mathematical tools that are "out of reach" in human intuition.
It is filled with a large number of complex tools and obscure matrix transformations. It was like a rampaging bulldozer, crushing a bloody road in the roadless tropical rainforest.
Is this road feasible? able. Is this road easy to walk? Even top mathematicians get headaches after reading it.
As the Fields Medal winner and known as "the smartest brain today"
Tao Zhexuan once accurately complained
:The proof of this type of AI is extremely obscure and difficult to understand, and humans need to spend a lot of time "digesting" it.
AI gives humans answers, but it doesn’t give humans “understanding.”
Is the future of mathematics just a ruthless computing machine outputting a bunch of truths that humans cannot understand?
Human mathematicians will never admit defeat.
Humanity’s counterattack, the dimensionality reduction attack of “taste” and “elegance”
Just when Claude's violent proof was painful for the academic community to "digest", mathematician Youness Lamzouri stood up.
Intuition told him:
This is wrong, the truth should not be this bloated. Since AI has proven that this direction is correct, there must be a more beautiful shortcut!
Next, Youness showed what true "human wisdom" is.
He wielded Occam's razor and slashed away the redundant, complicated, and huge matrices that grew over time in Claude's proof.
He used an extremely concise and elegant Hilbert space inequality to directly complete the reduction of the problem!
The complex lower bound problem is exquisitely transformed into an estimate of the associated sum, and then the unconditional version of Montgomery's Theorem (BGST) is directly applied.
A few brush strokes, the clouds are calm and the wind is gentle.
It is also a proof that ">67% of the zero points are on the critical line." If Claude's proof is a steam engine monster with violent aesthetics, then Youness's proof is a peerless sword that is completely natural and has its hair blown off.
The Axiom team was unstinting in their praise on social media:
Lamzouri reproved the same result, replacing a large number of complex tools with an elegant Hilbert space inequality. A concise and clear proof of taste and human wisdom.
Note this word: taste.
AI can carve a path out of 650 failed ideas, but only humans, with their natural appreciation of mathematical structures, can straighten and pave this muddy path into a magnificent highway.
Pass the game by machine in a few hours! The world of mathematics has completely changed
If you think that the story here is just a cliché of "humans defeat AI", you are totally wrong.
The climax has just begun.
According to the rules of traditional mathematics, Youness wrote this extremely elegant paper. What should we do next?
Find a few top colleagues, hold seminars, and spend months or even years reviewing and verifying line by line.
After all, this is the Riemann Hypothesis! Back then, Mochizuki Shinichi proved the ABC conjecture. Because the proof was too complicated, the mathematical community had been arguing for almost ten years without fully reaching a consensus.
But this time, times have changed.
Just a few hours after Youness shared the preprint draft! Note, it’s a few hours!
The AI tool AxiomProver automatically takes action and completely automatically formalizes Youness’ proof directly in the formal verification language Lean, and completes seamless machine-level verification!
https://github.com/AxiomMath/ZetaZeros
This means that when the masses just saw the title of the paper and did not understand what a Hilbert space was, the machine had ruthlessly tore through the entire logical chain of proof from beginning to end and made the final decision.
In human history, a major breakthrough regarding the epic difficulty of the Riemann Hypothesis,
on the "first day" the paper was released, machine verification was achieved!
The Axiom team cannot conceal their excitement:
We firmly believe that it will be an inevitable future for formal verification of machines to be accompanied by new breakthroughs in cutting-edge mathematics.
In less than 24 hours, the Axiom team announced that they were one step closer to the twin prime conjecture:
There are infinitely many pairs of prime numbers, the distance between them is no more than 212.
The twin prime conjecture states that there are infinitely many pairs of prime numbers whose difference is 2.
For centuries, the world has not known whether there are infinite pairs of prime numbers whose difference is a finite value.
In 2014, Zhang Yitang, James Maynard, Tao Zhexuan and the Polymath project achieved a historic breakthrough, narrowing this difference to 246.
For related contributions, James Maynard won the Fields Medal.
On August 31 this year, mathematician Julia Stadlmann narrowed the difference to 240.
https://arxiv.org/pdf/2608.31126
On September 3, experts in number theory, applied mathematicians and engineers, and AxiomProver achieved the results of 212 by conducting extensive experiments on large distributed clusters based on the work of Julia Stadlmann.
They said that the distance between human dreams and proof can be just a few hours, not a few lifetimes.
At present, the draft has been published, and the final version and Lean formal proof are committed to be posted on arXiv.
Paper: https://primegaps.axiommath.ai/bgp212.pdf
A few hours later, OpenAI used the latest model Astra to reduce this number to 186 using a new method.
What is even more shocking is that Fermat's last theorem was also completely formalized by Claude.
This is the most nerve-wracking part of this big incident:
The inefficient classical era of traditional mathematics of "proposing conjectures - proving - lengthy manual peer review - confirming" has officially come to an end today.
End: The last 32.75%, who will break the tie?
Today, we can pop the champagne to celebrate.
More than 67% of zeta zero points are nailed to that magical critical line. The line that Riemann drew casually 167 years ago is being illuminated inch by inch by humans and machines working together.
But don’t forget, this century-long journey is not over yet.
The Riemann Hypothesis, and the last 32.75% that is as mysterious as dark matter.
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