Sharp spectral constants for scaled $q$-numerical ranges

What the math gives to ML

The paper gives an exact bridge between a constrained two-vector numerical range and all numerical ranges obtained by similarity transforms whose condition number is bounded. This is directly relevant to nonnormal neural dynamics: ordinary eigenvalue or spectral-radius control can miss large transient amplification caused by poorly conditioned eigenvectors, whereas the theorem exposes a robustness envelope over bounded coordinate changes. The most promising transfer is a regularizer or stability certificate for recurrent, state-space, and residual transition matrices that controls polynomial growth such as powers of the transition matrix. The sharp polynomial bound supplies an explicit conversion from a scalar enclosure of transformed numerical ranges into an operator-norm bound for p(A).