= Solution
Write $h=\Delta x=1/(M+1)$. The semidiscrete matrix is $B=h^{-2}\operatorname{tridiag}(1,-2,1)+\kappa I$. For $j=1,\ldots,M$, set $v_m^{(j)}=\sin(j\pi m/(M+1))$. The endpoint values vanish and the sine addition formula gives
$$
Bv^{(j)}=\lambda_j v^{(j)},\qquad
\lambda_j=\kappa-\frac4{h^2}\sin^2\frac{j\pi}{2(M+1)}.
$$
These form an <orthogonal basis> of <eigenvectors>. Every initial vector decays precisely when every <eigenvalue> is negative. Since the largest is $\lambda_1$, the necessary and sufficient condition is
$$
\boxed{\kappa<\frac4{(\Delta x)^2}\sin^2\left(\frac{\pi}{2(M+1)}\right).}
$$
Necessity follows by taking the first eigenvector as the initial data; at equality it remains unchanged. Sufficiency follows by decomposing any vector into the displayed decaying modes. The discrete <reaction-diffusion spectral decay threshold> is below $\pi^2$, since $\sin x<x$ for $x>0$, and equals $\pi^2-\pi^4h^2/12+O(h^4)$ as the mesh is refined. Therefore a coarse semidiscretization can spuriously grow for some parameters for which the continuous equation still decays.
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