Lower Bounds for the Permanent in Arithmetic Circuits
How large must arithmetic circuits and formulas computing the permanent be? New lower bounds include an arithmetic-formula bound of order , far beyond the quadratic barrier that stood for decades.
- Result
- Proved (an n^4/log n formula lower bound; VP vs VNP remains wide open)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Algebraic complexity
- Posed by
- Leslie Valiant
- Year posed
- 1979
- Years open
- 47y
- Solved
- 2026-08-01
- Model
- Astra (internal preview)
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-verified
- Publication
- Announced
- Significance
- 40 / 100
- Disclosed cost
- $182
- Wikipedia
- No dedicated article
What the AI did
Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.
Verification
Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending.
Sources
OpenAI: Ten advances in mathematics and theoretical computer science