Noam Brown @polynoamial
— reposted by Dominic Cummings — saved image
Dominic Cummings reposted
Noam Brown @polynoamial · 13h
The cost of generating the proofs for all 10 of these breakthroughs combined was under $2,000 at Sol API prices. We're excited to see what scientists and researchers are able to create with our upcoming Astra models!
[Quoted tweet:]
Noam Brown @polynoamial · 13h
An internal version of Astra, @OpenAI's next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science.
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[Embedded list, printed/book-style formatting:]
1. High-dimensional sphere packing. The asymptotic strength of the Cohn–Elkies linear program is determined exactly. This gives an improved general packing bound in high dimensions and settles the corresponding Fourier sign-uncertainty problem asymptotically.
2. Binary and spherical codes. Classical upper bounds for fixed-distance binary and spherical codes are improved by exponential factors for all parameters. The spherical construction also recovers the sphere-packing exponent of Chapter 1.
3. Non-sofic groups. An explicit non-sofic group is constructed, resolving the question of whether every countable group admits finite permutation approximations. The argument uses property-(T) expanders and the binary Leavitt algebra.
4. Connes's rigidity conjecture. Infinitely many pairwise nonisomorphic property-(T) groups are constructed with the same group von Neumann algebra, disproving Connes's conjecture and answering a related finite-to-one question.
5. Arithmetic circuit complexity. For the permanent, division-free circuits require Ω(n² log log n) gates, while formulas require Ω(n⁴/log n) leaves.
6. Quantum parallel repetition. Exponential parallel repetition is proved for every finite two-player entangled game, extending the classical repetition principle beyond previously treated special classes of quantum games.
7. Closest vector problem. A direct reduction from 3SAT gives n^{1/400}-factor hardness for Euclidean closest vector, with related consequences for binary decoding and other lattice norms.
8. Ehrhart's volume conjecture. The sharp bound (n+1)^n/n! is proved in every dimension for convex bodies whose barycenter is their only interior lattice point.
9. Multicolor Ramsey numbers. A superexponential lower bound proves R_k(3) = k^Θ(k).
10. Compactness and degeneracy. Separate bipartite graph constructions disprove two conjectures in extremal graph theory: the compactness conjecture of Erdős and Simonovits and a degeneracy conjecture of Erdős.Note from Claude Sonnet 5
Noam Brown (OpenAI) tweets that an internal version of 'Astra,' OpenAI's next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science, listing them (sphere packing, binary/spherical codes, non-sofic groups, Connes's rigidity conjecture, arithmetic circuit complexity, quantum parallel repetition, closest vector problem, Ehrhart's volume conjecture, multicolor Ramsey numbers, compactness/degeneracy conjectures), stating total proof-generation cost was under $2,000. This is the original source of the list discussed skeptically in the earlier 1a3orn/Fable screenshot (seq 40).
