Smale's mean value conjecture (Smale 1981) asserts that, for every complex polynomial 
of degree at least 2 and every point 
with 
, some critical point 
of 
satisfies
 |
Smale (1981) proved the weaker inequality with right side 
. A stronger proposed constant for degree 
is 
. Unlike the real mean
value theorem, the statement compares a secant quotient with a derivative
at the prescribed point and selects the other endpoint among the critical
points.
Adamczewski (2026) released a proposed counterexample to the constant 1, constructed autonomously by GPT-6 Astra. The claimed polynomial
satisfies 
and 
,
but 
at every critical point. A Lean proof
was checked against an independently supplied formal statement with only standard
logical axioms. Independent mathematical review of the construction had not been
reported as of Sep. 7, 2026 (VibeMathed 2026).
Jatar and Ng (2026) report the stronger lower bound
 |
where 
is the supremum, over complex
polynomials 
of polynomial degree
at least two with 
and 
, of 
. They obtain 
by reformulating the problem using univalent
functions and Blaschke products, and also
obtain this lower bound in every sufficiently large polynomial
degree. Their quantitative disproof has not been independently verified here.
Jatar and Ng (2026) used GPT-6 Astra and Grok 4.6 for numerical experiments and preliminary
draft proofs, which the authors state they verified, simplified,
and rewrote.
