Smale’s Mean Value Conjecture ($K=1$)

Lineage and corrections

Corrects Smale’s Mean Value Conjecture ($K=1$) →

Supersedes Smale’s Mean Value Conjecture ($K=1$) →

Smale’s Mean Value Conjecture ($K=1$) parent of this event

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 $1$. parent of this event

VibeMathed record: Smale’s Mean Value Conjecture ($K=1$) evidence for this event

Smale’s Mean Value Conjecture ($K=1$) evidence for this event

X Announcement evidence for this event

Challenge.lean: the compared statement, copied from Formal Conjectures evidence for this event

Solution.lean and the Lean development evidence for this event

Formal Conjectures source statement (Google DeepMind) evidence for this event

Epoch AI LeanOpenProblems, the evaluation the run was part of evidence for this event

This event attributed to GPT-6 Astra (pre-release)

This event attributed to Tom Adamczewski