A solution to Crouzeix's conjecture

What the math gives to ML

The paper proves Crouzeix's functional-calculus inequality: evaluating any polynomial on a possibly highly non-normal operator is controlled, up to a factor of 2, by the polynomial's maximum over the operator's numerical range. This is directly relevant to recurrent, state-space, and deep residual dynamics, where powers and polynomial filters of a transition operator can exhibit large transient amplification despite a seemingly safe eigenvalue spectrum. The transferable asset is a computable geometric stability certificate based on the numerical range rather than eigenvalues alone. A practical adaptation is to regularize or constrain the numerical range of learned transition operators and use the bound to monitor worst-case amplification of multi-step dynamics.