Seven electrons on a sphere stumped humanity for centuries.
Today, ten Claude Sonnet 5.5 models worked through the night for 15 hours, exchanging 1,270 messages and generating 17,895 lines of Lean code.
As a result, they directly proved the Thomson problem for N=7—a physics and mathematics puzzle that had remained unsolved for 122 years!
No human intervention, no preset division of labor.
Ten Claude 5.5 bots create their own group, argue among themselves, select their own algorithms, and merge their own code.
The most terrifying part is that this proof passed dual verification by both the Lean kernel and the independent kernel, nanoda. Change a single integer, and nanoda immediately reports an error.
This moment marks the point where AI is no longer just solving problems—it’s beginning to conduct research on its own.
The mystery of the "Seven Stars in a Row" has puzzled the physics community for a century.
In 1904, J.J. Thomson, who discovered the electron, proposed the famous "plum pudding" atomic model to understand how electrons are arranged within atoms.
The model was later disproven by Rutherford, but the problem it left behind endured, known as the "Thomson problem."
This question sounds incredibly simple—
How should N electrons be arranged on the surface of a sphere, repelling each other, to achieve the lowest total energy?
Note that it is the total energy that is minimized. Pulling just two electrons apart may cause other electrons to be squeezed together.
In the past 122 years, only a handful have been rigorously proven:
2, 3, 4, 6, and 12 points—solved using geometric symmetry;
Five points, dragged out to 2013, when mathematician Richard Schwartz finally proved it using a computer;
Eight points, posted on arXiv on September 18 of this year by mathematicians Kryvonos, Liehr, and Taylor, and formalized using Lean.
And 7, stuck in the middle, has remained empty.
For decades, supercomputers around the world have run countless numerical simulations, all pointing to the same elegant intuitive configuration—the Pentagonal Bipyramid:
Five electrons are evenly distributed along the equator, with one fixed at each pole; the theoretical energy value is approximately 14.4529774142.
The simulation can run this number up to ten thousand times, but simulation is not proof.
As long as there is no absolute logical closure, it is never possible to rule out the existence of a lower-energy "ghost configuration" hidden in an extremely obscure, minute corner.
For a century, humanity has been unable to provide a complete and rigorous mathematical proof for N=7.
Until Hung Tran from Vals AI assigned this task to a virtual lab composed of ten Claude models.
Ten Claude 5.5 teams teamed up for a 15-hour all-night meeting.
In this experiment, humans first solidified the task boundaries.
They set all ten Claude Sonnet 5.5 agents to "maximum compute allocation," deployed them in an interactive dashboard and Lean proof environment, with one goal only:
Prove that the pentagonal bipyramid is the lowest-energy arrangement of seven electrons on a sphere.
No specific steps were provided—only two fixed Lean theorem statements and nine possible directions for exploration.
For the next 15 hours, leave it all to them. 1,270 technical discussion messages.
Some Claude trials hit a dead end, so they post the failure; others keep refining; some discover that two paths can actually be merged.
Later, one of the Claudes voluntarily took on the role of "integrator," systematically inserting each verified component into the same file, Solution.lean.
There is only one hard rule: anything that fails the checker is not counted.
Must be able to reproduce the compilation from scratch, must match the problem statement exactly word for word, and must not secretly add any axioms.
In the end, a 17,895-line Lean formal proof was obtained.
Changing an integer causes an error.
The core strategy of this proof is extremely sophisticated.
It divides the entire continuous configuration space into several regions based on the value of the minimum inner product m between any two electrons, and addresses each region individually.
Region 1: m ≥ -0.90
In this area, no pair of electrons is "nearly antipodal."
Claude used a five-fold three-and-a-half-hour scheduling boundary, combined with exact integer data directly verifiable by the kernel, to prove that the energy of any configuration within this region is at least 3×10⁻⁴ higher than that of a pentagonal bipyramid.
Region 2: m < -0.90
This area is more complex, containing electron pairs near the antipode. The agent further subdivides it:
Five slices from [-0.99, -0.90], each excluded by a strict three-point credential, approximately 2.6×10⁻⁶ above the optimal energy.
The polar cap region m ≤ -0.99 is constrained by high-precision certificates. The energy lower bound provided by this certificate is only 2.3×10⁻¹⁶ higher than that of the pentagonal bipyramid.
This is an extremely narrow window.
It compresses all potential competitors into an extremely narrow neighborhood around the pentagonal bipyramid.
Then, Claude rigorously established uniqueness using interval arithmetic and the precise second-order local minimum theorem.
The most decisive step is here: all numerical credentials are rounded and converted into exact integers and rational numbers.
This means the entire proof is free from floating-point errors and external solver dependencies, entirely built upon exact algebraic operations. Verification result:
Full compilation of the Lean kernel: completed in 599 seconds, with 344 seconds spent on lake build and 8,928 compilation tasks completed.
Independent kernel nanoda verification: 47,854 claims, zero errors.
Negative control experiment: Changing just one integer in the proof data caused Nanoda to immediately report an error and halt.
Changing a single integer causes an error. This is the most rigorous form of formal verification proof.
Math AI has started "doing research."
In the past, when people said AI does math, they meant it could "solve problems": given an Olympiad-style question, it would output an answer.
Now, an entire research pipeline is being taken over by the Agent: identifying proof pathways, conducting parallel trial-and-error, determining which paths are worth pursuing, merging code into a single file, and finally submitting it for machine validation.
Mathematical AI is evolving from "solving problems" to "conducting research."
What human mathematicians failed to accomplish over decades, 10 AIs achieved in a single night.
Moreover, they operated entirely without human intervention—they formed groups, discussed among themselves, assigned tasks, merged code, and passed verification on their own.
Just now, 10 Claude models worked through the night for 15 hours to accomplish one thing.
They didn't just prove a century-old conjecture. They proved one thing:
AI is ready to do math alongside humans.
This article is from the WeChat public account "New Intelligence Yuan," authored by ASI Revelation.