Smale's mean value constant exceeds 1.1559

We prove that Smale's mean value constant exceeds 1.1559837, improving the bound 1.1018 of Jatar and Ng. Smale proved in 1981 that for every complex polynomial P of degree at least 2 and every point z with P′(z) ≠ 0, some critical point ζ of P satisfies |P(z) − P(ζ)| ≤ K |z − ζ| |P′(z)| with K = 4, and he conjectured that K = 1 suffices; K* is the best constant. A computer-checked proof that K* > 1 appeared in 2026. Jatar and Ng showed that every pair (W, B) of a normalized univalent map W of the unit disk and a finite Blaschke product B with B(0) > 0, such that (WB)′ has no zero in the disk, gives K* ≥ 1/B(0). We construct such a pair explicitly, with B of degree two, from a polynomial of degree 18 with Gaussian rational coefficients, and we verify the required properties by a computation in exact arithmetic. It follows that the best constant in degree d also exceeds 1.1559837 for all large d. Every result of this paper is also proved in the Lean 4 proof assistant, from the cited results of Jatar and Ng and of Dieudonné and from one fact that is checked by computer outside Lean.