= Solution
Take $M$ interior points in each direction and spacing $h=1/(M+1)$. The one-dimensional <Dirichlet discrete Laplacian> has sine eigenvectors $v_i^{(j)}=\sin(j\pi i h)$, because
$$
\frac{v_{i-1}^{(j)}-2v_i^{(j)}+v_{i+1}^{(j)}}{h^2}=-\frac4{h^2}\sin^2\left(\frac{j\pi h}{2}\right)v_i^{(j)}.
$$
The <five-point Dirichlet Laplacian as a Kronecker sum> therefore has product eigenvectors $v_i^{(j)}v_r^{(\ell)}$. Adding the reaction term gives modal growth rates
$$
\lambda_{j\ell}=\kappa-\frac4{h^2}\left[\sin^2\left(\frac{j\pi h}{2}\right)+\sin^2\left(\frac{\ell\pi h}{2}\right)\right],\qquad1\leq j,\ell\leq M.
$$
The semidiscrete matrix is real symmetric, so these modes are orthogonal in the mesh-weighted <discrete L2 norm>. Non-growth for all times is equivalent to every rate being nonpositive. The largest is $\lambda_{11}$, giving the <five-point heat-reaction stability threshold>
$$
\boxed{\kappa\leq\frac8{h^2}\sin^2\left(\frac{\pi h}{2}\right)=\frac4{h^2}(1-\cos\pi h).}
$$
At equality the first grid mode is neutral; above it that mode grows exponentially. This criterion concerns the spatially semidiscrete, continuous-time evolution. Any subsequent time integrator has an additional stability requirement.
The threshold expands as $2\pi^2-\pi^4h^2/6+O(h^4)$ and is below the continuum threshold for every finite mesh. Thus at the continuum neutral value the grid first mode has a small positive $O(h^2)$ rate. This is compatible with finite-time convergence, although it prevents an all-time contraction on that fixed mesh. In the weaker finite-time mesh-uniform sense, all fixed $\kappa$ have the bound $\|U(t)\|_h\leq e^{\kappa t}\|U(0)\|_h$ because the discrete Laplacian is nonpositive. The boxed range is the natural non-growing stability range, matching the threshold comparison in the preceding part; <finite-time stability versus power boundedness> is a distinct question.
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