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(see note)
- 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
- 38 / 100
- Disclosed cost
- $182
- Wikipedia
- No dedicated article
What was actually shown
an n^4/log n formula lower bound; VP vs VNP remains wide open
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.