From boundary geometry to counterexamples in Smale's mean value problem

Abstract

This paper provides a human digestion of the recent counterexample to Smale's mean value problem found by GPT-6 Astra. We give a geometric mechanism for constructing counterexamples. An outward normal that re-enters an analytic Jordan domain, subject to a curvature condition, yields a holomorphic model whose normalized divided differences exceed one in modulus on the boundary. Polynomial perturbations then concentrate every critical point near the boundary whilst preserving the strict inequality. This turns boundary geometry into counterexamples in every sufficiently large degree. We also locate these critical points to first order and show that their limiting distribution is the logarithmic equilibrium measure on the boundary. We end with an explicit example of degree $29\,984$, verified by interval arithmetic.