Smale’s Mean Value Conjecture ($K=1$)

The formal proof constructs a complex polynomial $p$ such that $$p(0)=0,\qquad p'(0)=1,$$ and for every critical point $c$ of $p$, $$\left|\frac{p(c)}{c}\right|>1.$$ Thus at $z=0$ there is no critical point satisfying $$\frac{|p(0)-p(c)|}{|c|}\le |p'(0)|=1,$$ which disproves Smale's conjectured universal constant $K=1$. The counterexample has very large unspecified degree and violates the bound only by a small margin, so it is consistent with Smale's original $K=4$ theorem, the known low-degree positive cases, and previous asymptotic improvements toward $1$.

The formal proof constructs a complex polynomial $p$ such that $$p(0)=0,\qquad p'(0)=1,$$ and for every critical point $c$ of $p$, $$\left|\frac{p(c)}{c}\right|>1.$$ Thus at $z=0$ there is no critical point satisfying $$\frac{|p(0)-p(c)|}{|c|}\le |p'(0)|=1,$$ which disproves Smale's conjectured universal constant $K=1$. The counterexample has very large unspecified degree and violates the bound only by a small margin, so it is consistent with Smale's original $K=4$ theorem, the known low-degree positive cases, and previous asymptotic improvements toward $1$.