= Solution
For homogeneous corrections at the square boundary, <integration by parts> gives $\langle\psi_0\Delta\psi_1\rangle=\langle\psi_1\Delta\psi_0\rangle$ and $\langle\theta_0\Delta\theta_1\rangle=\langle\theta_1\Delta\theta_0\rangle$. The first-derivative cross terms give $\langle\psi_0\theta_{1x}\rangle=-\langle\psi_{0x}\theta_1\rangle$ and $\langle\theta_0\psi_{1x}\rangle=-\langle\theta_{0x}\psi_1\rangle$. Using $\Delta\psi_0=q_c^2\theta_{0x}$ and $\Delta\theta_0=-\psi_{0x}$, these four contributions cancel in pairs. This proves exactly the supplied <solvability condition>, including the weight $q_c^2$ on its second row. Equivalently $(\psi_0,q_c^2\theta_0)$ is the relevant <adjoint eigenfunction> for this coupled system.
At first order in the detuning, the inhomogeneous equations are
$$
\Delta\psi_1-q_c^2\theta_{1x}=2q_cq_1\theta_{0x},\qquad \psi_{1x}+\Delta\theta_1=\lambda\theta_0.
$$
Insert these right-hand sides into the verified <solvability condition>. One obtains
$$
2q_cq_1\langle\psi_0\theta_{0x}\rangle+q_c^2\lambda\langle\theta_0^2\rangle=0.
$$
A further <integration by parts>, using the leading <heat equation>, gives
$$
\langle\psi_0\theta_{0x}\rangle=-\langle\psi_{0x}\theta_0\rangle=\langle\theta_0\Delta\theta_0\rangle=-\langle|\nabla\theta_0|^2\rangle.
$$
Hence the <growth-rate solvability for conducting-square Darcy convection> yields
$$
\boxed{\lambda=\frac{2q_1}{q_c}\frac{\langle|\nabla\theta_0|^2\rangle}{\langle\theta_0^2\rangle}.}
$$
\b[The PDF prints the reciprocal of the required integral ratio.] With its declared $\theta_t=\epsilon\lambda\theta$, the displayed reciprocal does not follow and is false for $q_1\neq0$. Both integrals are positive for these nonzero modes. The square's <Poincaré inequality> gives their ratio at least $2\pi^2$, so it cannot equal its reciprocal. For $q_1=1$, the two explicit parities give approximately $\lambda=11.34770$ for odd $\theta$ and $\lambda=7.30164$ for even $\theta$; the printed expression instead gives about $0.00446439$ and $0.00693825$. These values provide direct mode-based counterexamples.
The corrected growth rate is positive for $q_1>0$ and negative for $q_1<0$, as expected at a <convection> threshold. Because the critical <eigenspace> is two-dimensional, an arbitrary superposition generally splits into two different first-order growth rates. The specified parity modes remain independent under the detuning: the <temperature> <linear operator> preserves reflection parity, so the cross-parity projection vanishes. This justifies applying the scalar <solvability condition> to either listed <eigenfunction>; it should not be applied as a single common <eigenvalue> to an arbitrary mixture.
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