Square Functions and the Complete Crouzeix Conjecture in Dimension Three

What the math gives to ML

The paper gives a sharp operator-functional-calculus bound: polynomial functions of a potentially highly nonnormal matrix are controlled by the maximum polynomial magnitude over its numerical range, with constant 2 in the complete conjecture. This is relevant to neural dynamical systems because spectral radius alone can miss large transient amplification caused by nonnormal recurrent or state-space transition matrices. A practical transfer is to constrain or regularize the numerical range of learned transition or Jacobian operators, obtaining direct bounds on finite-horizon polynomial propagation. The most promising initial target is a linear state-space or recurrent layer, where numerical-range support values can be estimated using matrix-vector products.