= Solution
For fixed mode numbers $k,l$, expansion of the <sines> gives
$$
\lambda_{5,kl}=-\pi^2(k^2+l^2)+\frac{\pi^4h^2}{12}(k^4+l^4)+O(h^4),
$$
whereas the mixed <Kronecker product> correction gives
$$
\lambda_{9,kl}=-\pi^2(k^2+l^2)+\frac{\pi^4h^2}{12}(k^2+l^2)^2+O(h^4).
$$
Thus \b[both <eigenvalue> approximations are second order for fixed modes]. The leading nine-point error includes an additional positive term $\pi^4h^2k^2l^2/6$.
There is also an exact comparison, not just an asymptotic one. Since $\sin x<x$ for $0<x<\pi/2$, and both sine factors in the correction are positive,
$$
-\pi^2(k^2+l^2)<\lambda_{5,kl}<\lambda_{9,kl}.
$$
Both estimates lie on the less-negative side of the continuum <Laplacian> <eigenvalue>, so \b[the five-point formula is the better <eigenvalue> estimate for every corresponding interior mode]. The <five-point versus nine-point Laplacian eigenvalue accuracy> comparison does not contradict the useful isotropy or special Poisson accuracy properties of a nine-point stencil. Those properties concern a different error criterion. The fixed-mode expansion is not uniform for modes with $k$ or $l$ comparable to $h^{-1}$; wavelengths close to the mesh scale need not have small relative spectral error.
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