= Solution
Set $\sigma=0$ under the assumed stationary-onset hypothesis, and let $\kappa=h\kappa'$ with fixed nonzero $\kappa'$. Substitute $W=hW_0+h^2W_1+\cdots$, $N=N_0+hN_1+h^2N_2+\cdots$, and $R_c=h^{-1}(R_0+hR_1+\cdots)$ into the modal equations and plate <boundary conditions>.
At leading order the concentration equation is $N_0''=0$, with $N_0'=0$ at both plates. Normalize its nonzero constant to $N_0=1$. The first nonzero momentum equation is
$$
W_0''''=-\kappa'^2R_0,\qquad W_0=W_0'=0\quad(z=-1,0).
$$
Integrating four times and imposing all four clamped <boundary conditions> gives
$$
\boxed{N_0=1,\qquad W_0=-\frac{\kappa'^2R_0}{24}z^2(z+1)^2.}
$$
At the next concentration order, $\bar n=1+O(h)$ and $\bar n'=h+O(h^2)$, so the vertical advection of the basic concentration does not yet contribute. Thus
$$
N_1''=-GW_0'',\qquad N_1'=N_0=1\quad(z=-1,0).
$$
Integration gives $N_1=-GW_0+z+C_1$. The constant $C_1$ is an amplitude-normalization freedom; set it to zero, obtaining
$$
\boxed{N_1=-GW_0+z.}
$$
There is no need to require zero depth average of this concentration <eigenfunction>: for a nonzero horizontal <wavenumber> the horizontal average is already zero.
Use $\bar n=1+h(z+1/2)+O(h^2)$ and $\bar n'=h+O(h^2)$. At order $h^2$, the concentration equation is
$$
N_2''-N_1'-\kappa'^2=W_0-GW_1''-G\left(z+\frac12\right)W_0''.
$$
The corresponding cell-flux <boundary conditions> are $N_2'=N_1$ at both plates. Integrate the equation from $-1$ to zero. Its left derivative term satisfies
$$
N_2'(0)-N_2'(-1)=N_1(0)-N_1(-1)=\int_{-1}^0N_1'dz.
$$
The $N_1'$ terms therefore cancel. The no-slip conditions imply $\int W_1''dz=0$. Also <integration by parts> gives
$$
\int_{-1}^0\left(z+\frac12\right)W_0''dz
=\left[\left(z+\frac12\right)W_0'-W_0\right]_{-1}^0=0.
$$
Consequently the <solvability condition> is simply
$$
0=\kappa'^2+\int_{-1}^0W_0dz
=\kappa'^2\left(1-\frac{R_0}{720}\right),
$$
since $\int_{-1}^0z^2(z+1)^2dz=1/30$. For every fixed nonzero $\kappa'$, the <long-wave bioconvection threshold in weak stratification> is therefore
$$
\boxed{R_0=720,\qquad R_c\sim\frac{720}{h}.}
$$
The cancellation shows why $G$ does not enter this leading coefficient; neither does $S$ at stationary onset. In dimensional terms, the leading critical mean number density and volume fraction are
$$
\boxed{C_{0c}\sim\frac{720\rho\nu D^2}{gv\Delta\rho V_sH^4},\qquad \varphi_{0c}\sim\frac{720\rho\nu D^2}{g\Delta\rho V_sH^4}.}
$$
These assume heavier-than-fluid cells, $\Delta\rho>0$. Exactly $\kappa'=0$ makes the integrated equation identically zero and corresponds to a spatially uniform change of conserved cell number, not a convective mode; the statement independent of $\kappa'$ is interpreted for nonzero long waves or their limit. This is the leading threshold within the specified long-wave, real-<eigenvalue> assumptions. It does not by itself rule out a different finite-wavelength or oscillatory instability outside those assumptions.
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