The regular expansion threshold for every higher-order zero

For an integer $m\ge3$ let $\lambda_{n,1}\le\cdots\le\lambda_{n,n}$ be the eigenvalues of the Toeplitz matrix $T_n(g)$ generated by $g(\theta)=(2-2\cos\theta)^m$, and put $h=(n+2)^{-1}$. There are real $C^\infty$ functions $d_0=g,d_1,\ldots,d_{2m}$ on $[0,\pi]$, given by an explicit phase built from the stable roots of the characteristic equation, such that $\lambda_{n,j}-\sum_{k=0}^p d_k(j\pi h)h^k=O(h^{p+1})$ uniformly in $j$ for every $p\le2m-1$, and for $p=2m$ when $j\ge\lceil(\log(n+2))^2\rceil$, while the order-$2m$ estimate cannot hold uniformly for all $j$. The expansion comes from a normalized exact boundary determinant and a uniform indexing of the eigenangles. The obstruction compares two values of $\lim(n+2)^{-2m}\operatorname{tr}(T_n(g)^{-1})$: a rational one from the classical inverse-kernel limit of Böttcher and Widom, and an irrational one that a uniform expansion would force.