= Solution
At the threshold $P_*=3+2\sqrt2$, the <determinant> quadratic has a double zero at
$$
k_*^2=\frac{P_*-1}{2P_*}=P_*^{-1/2}=\sqrt2-1.
$$
Consequently the scaled disturbance length in $\cos(s/L)$ is
$$
\boxed{L_{\rm crit}=P_*^{1/4}=\sqrt{1+\sqrt2}.}
$$
The physical length is $\ell_uL_{\rm crit}=\sqrt{\ell_u\ell_v}$ at threshold; its full spatial <wavelength> is $2\pi$ times this length, not this length itself. For much shorter lengths, the large negative <diffusion> terms smooth both components. For much longer lengths, <diffusion> contributes little and the stable homogeneous reaction feedback, especially the <inhibitor> response, suppresses disturbances. The selected length lies between those two stabilization mechanisms.
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