Source authenticated

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 $1$ .

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 3, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-verified. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Smale’s Mean Value Conjecture ($K=1$) · Smale’s Mean Value

Confidence: Not scored

Registry verification: lean verified · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

GPT-6 Astra (pre-release)
model · ai model contributor · OpenAI

Tom Adamczewski
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Smale’s Mean Value Conjecture ( $K=1$ )

specialist registry · passed

Lean-verified on this site's ladder: kernel-checked, and the statement was written independently of the prover. Checked here on 5 September 2026 from a clone of the repository at f99ab98: 2,266 lines of Lean across the development, zero `sorry` outside the Challenge.lean stub, zero `axiom` declarations, no `native_decide`, `unsafe` or `implemented_by`; the mathlib revision is pinned; CI runs Comparator against the trusted statement allowing only propext, Quot.sound and Classical.choice. The compared statement is byte-identical to Formal Conjectures' `mean_value_problem` at commit 9cbe1d3c, which is Smale's K = 1 form exactly: the quantified K is unused there, and the division convention in Lean (x/0 = 0) only makes the conjecture easier to satisfy, so it cannot help a disproof. Candidate rather than resolved because, two days after the run, no mathematician outside it has read the proof; the witness is a nonconstructive limiting perturbation of large unspecified degree and violates the bound by a small margin, which is consistent with everything known. The proof account was machine-generated and is unaudited.

Source ↗

Solution.lean and the Lean development

lean artifact · passed

lean-proof link supplied by the VibeMathed registry.

Source ↗

X Announcement

witness · pending

discussion link supplied by the VibeMathed registry.

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Formal Conjectures source statement (Google DeepMind)

formal registration · pending

lean-statement link supplied by the VibeMathed registry.