The Unique Games Theorem
We prove the Unique Games Conjecture. For every fixed $\varepsilon,\delta\in(0,1/2)$, we give a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed finite alphabet, with completeness at least $1-\varepsilon$ and soundness at most δ .
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Or at least, we’re pretty sure that it’s a proof! There’s a Lean certificate , as there are for some of the other 372 breakthrough results (not all of them). But it also appears that no human has understood just about any of these proofs yet; the race to do so has just started. If you want an…
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
announces a proof of the Unique Games Conjecture and, separately, the 2-to-1 Games Conjecture with perfect completeness . Matrix multiplication. My post Cap Sets, Sunflowers, and Matrix Multiplication explored the connections between these problems. The new bound is over the complex numbers , a…