We may finally know that for any n, there exist n² unit vectors in ℂⁿ that all lie at the same angle from each other! This is important in quantum physics, but also geometrically very pretty.
Zauner conjectured this was true in 1999. Last year Appleby, Flammia and Kopp proved it would follow from two conjectures in number theory.
On September 29th, Radchenko and Wheeler put the proof of one conjecture on the arXiv. Interestingly, it makes use of some serious category theory - as well as shitloads of Galois theory, quantum dilogarithms, etcetera:
https://arxiv.org/abs/2609.21892
But what about the *other* conjecture, which is called "Hilbert's 12th problem for real quadratic fields"?
Radchenko and Wheeler say that on the same day, they received a communication from an independent researcher named Imran Zaidi "claiming that he had a proof, obtained with the assistance of LLMs", of that! This stung them into rapid action, and they used an LLM to provide their own proof, which seems to be available only on their website. Maybe it will hit the arXiv soon.
Two days ago Maris Ozols claimed to have formalized the whole argument, not just these separate pieces, in Lean:
https://bsky.app/profile/marisozols.bsky.social/post/3mxhuomixbc2l
His posts on Bluesky link to all the relevant papers and Wikipedia article, so I won't bother doing that here.
John Carlos Baez
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2026-10-11 09:35:57 UTC
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2026-10-11 09:35:57 UTCEvent JSON
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"content": "We may finally know that for any n, there exist n² unit vectors in ℂⁿ that all lie at the same angle from each other! This is important in quantum physics, but also geometrically very pretty.\n\nZauner conjectured this was true in 1999. Last year Appleby, Flammia and Kopp proved it would follow from two conjectures in number theory. \n\nOn September 29th, Radchenko and Wheeler put the proof of one conjecture on the arXiv. Interestingly, it makes use of some serious category theory - as well as shitloads of Galois theory, quantum dilogarithms, etcetera:\n\nhttps://arxiv.org/abs/2609.21892\n\nBut what about the *other* conjecture, which is called \"Hilbert's 12th problem for real quadratic fields\"? \n\nRadchenko and Wheeler say that on the same day, they received a communication from an independent researcher named Imran Zaidi \"claiming that he had a proof, obtained with the assistance of LLMs\", of that! This stung them into rapid action, and they used an LLM to provide their own proof, which seems to be available only on their website. Maybe it will hit the arXiv soon.\n\nTwo days ago Maris Ozols claimed to have formalized the whole argument, not just these separate pieces, in Lean:\n\nhttps://bsky.app/profile/marisozols.bsky.social/post/3mxhuomixbc2l\n\nHis posts on Bluesky link to all the relevant papers and Wikipedia article, so I won't bother doing that here.",
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