Smale’s K=1 mean-value conjecture
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Smale’s mean-value question asks whether every normalized complex polynomial p with p(0)=0 and p′(0)=1 has a critical point c satisfying |p(c)/c|≤1. A universal constant 4 is known.
Why it matters and what counts as progress
Why it matters: The constant-one case tests the proposed sharp form of the polynomial mean-value inequality.
What counts as progress: A constant-one theorem, an improved universal constant, or a normalized polynomial for which every critical point gives |p(c)/c|>1.
How the problem developed
- 1981 · Conjecture [1]
Stephen Smale Proved a universal bound with constant 4 and asked whether it could be reduced to 1 (or the degree-dependent value (d−1)/d).
- September 2026 · Counterexample claim [2]
Tom Adamczewski and GPT-6 Astra A Lean-formalized project repository reports a counterexample to the K=1 conjecture, with the construction found autonomously by GPT-6 Astra.
- 21 Oct 2026 · Quantitative result [3]
Aneesh Jatar and Tuen Wai Ng Reformulated the constant using univalent maps and finite Blaschke products, proved it exceeds 1.1018, and in version 2 answered Smale's separate Problems 1A and 1B negatively.
AI contribution
Astra’s Lean proof refutes Smale’s K=1 mean-value conjecture
Astra proved in Lean that a complex polynomial p exists with p(0)=0 and p′(0)=1 such that |p(c)/c|>1 at every critical point c. Thus the universal K=1 mean-value inequality fails at z=0.
Sources
- Stephen Smale, The Fundamental Theorem of Algebra and Complexity Theory
- Tom Adamczewski, Lean formalization and counterexample for Smale's mean-value problem
- Jatar and Ng, Smale's mean value constant and admissible univalent Blaschke pairs
- Adamczewski’s Smale counterexample record
- Astra’s Lean proof
- Jatar and Ng’s explanation and later bounds
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Astra’s Lean proof refutes Smale’s K=1 mean-value conjecture