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OpenAI's Astra produced proofs resolving 10 long-standing math problems (most…

2026-08-03 · 16:47 UTC ·@alex_prompter ·2 min read

Brief

An OpenAI model (Astra) produced proofs settling ten longstanding open problems—most frozen for at least a decade—collected in a 249-page paper and formalized in Lean after human editing. Highlights include an 80-year-old Erdős conjecture disproved in May 2026, the first non-sofic group (27-year question), Connes's rigidity conjecture refuted, improved sphere-packing and coding bounds, lattice hardness for post-quantum cryptography, and a reported token cost of ≈ $2,000.

Why it matters

OpenAI's Astra produced proofs resolving 10 long-standing math problems (most frozen ≥ a decade), compiled into a 249-page paper and fully formalized in Lean after human edits.

Key details

  • Claimed breakthroughs include: disproving an 80-year-old Erdős conjecture (May 2026); constructing the first 'non-sofic' group (question open 27 years); disproving Connes's rigidity conjecture; tighter high-dimensional sphere-packing bounds; sharper error-correcting-code limits; stronger circuit lower bounds for the permanent; and lattice hardness results relevant to post-quantum cryptography.
  • OpenAI reports the total token cost to find all 10 solutions was roughly $2,000; OpenAI and Google DeepMind reached International Math Olympiad gold-medal level in 2025, highlighting rapid progress from competition-level math to claimed original research.
Source evidence

An AI solved 10 math problems that stumped humans for decades. You don't need a PhD to get them.

In 2025, OpenAI and Google DeepMind reached gold-medal level at the International Math Olympiad. In May 2026, an OpenAI model disproved an Erdős conjecture that had stood for 80 years. Astra, OpenAI's next major model, went bigger and produced results on 10 open problems at once, each frozen for at least a decade and most for far longer.

The full paper runs 249 pages of dense mathematics. So I asked Claude's Fable 5 to weigh in and translate every result into plain English. Here's what it gave me.

  1. It proved tighter limits on how densely spheres can pack in high-dimensional space, a question tied to how data gets packed and transmitted.

  2. It sharpened the known limits of error-correcting codes, the math that lets WiFi and hard drives survive noise.

  3. It built the first example of a "non-sofic" group, a structure mathematicians spent 27 years unsure even existed.

  4. It disproved Connes's rigidity conjecture, which claimed certain groups are uniquely pinned down by their von Neumann algebras, the same algebra quantum theory runs on.

  5. It raised the proven floor on how much circuitry the permanent, a famously stubborn calculation, actually requires.

  6. It showed that repeating a quantum game in parallel crushes a cheater's odds exponentially, a building block for quantum security proofs.

  7. It proved the lattice problem behind post-quantum encryption stays hard even when you only need an approximate answer, which is good news for tomorrow's locks.

  8. It settled, in every dimension, how large a convex shape can be when its balance point is the only grid point inside it.

  9. It showed that guaranteed patterns in multicolored networks appear far later than expected, resolving one of Erdős's open problems.

  10. It resolved two more Erdős problems about when a network gets so connected that specific patterns become unavoidable.

The cost is the wildest part. OpenAI says finding all 10 solutions took roughly $2,000 in tokens. Humans edited the write-ups, then the model formalized every proof in Lean, software that machine-checks each logical step.

Mathematicians are still reviewing the claims, and that caution is fair. But 12 months separate winning a student competition from producing new mathematics. That gap keeps shrinking.

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