= Solution
The <normalized stationary fluctuation-dissipation relation for a reaction network> is
$$
\boxed{M\eta+\eta M^{\mathsf T}=D,\qquad
\eta_{ij}=\frac{\operatorname{Cov}(X_i,X_j)}{\mu_i\mu_j}.}
$$
Here $M$ is the negative drift <Jacobian matrix> expressed in normalized population coordinates, and $D$ is the normalized noise matrix. More explicitly, with $S=\operatorname{diag}(\mu_i)$, drift Jacobian $A$, <stoichiometric vectors> $\nu_r$, and stationary <reaction propensity functions> $a_r$, these matrices are $M=-S^{-1}AS$ and $D=S^{-1}BS^{-1}$, where $B=\sum_ra_r\nu_r\nu_r^{\mathsf T}$. The <linear noise approximation> gives a covariance evolution $\dot\eta=-M\eta-\eta M^{\mathsf T}+D$; stationarity is the displayed <Continuous Lyapunov equation>.
In one dimension, the normalized restoring coefficient is $M=H/\tau=1/(2\tau)$. Squared chemical jumps determine the noise:
$$
B=9\lambda+\beta\sqrt\mu=12\lambda=\frac{4\mu}{\tau},\qquad
D=\frac{B}{\mu^2}=\frac4{\tau\mu}.
$$
The factor nine from a birth burst is essential; its three molecules arise in one correlated reaction event. The scalar <Continuous Lyapunov equation> is $2M\eta=D$, giving
$$
\boxed{\eta\approx\frac4\mu,\qquad \operatorname{Var}(X)\approx4\mu.}
$$
Equivalently, the normalized variance is $\langle s\rangle/(H\mu)=2/((1/2)\mu)$. The <Fano factor> is approximately four. This calculation uses the prescribed mean closure and the <linear noise approximation>, so its small-relative-fluctuation regime is $\mu\gg1$; it is not an exact assertion for arbitrarily small molecular copy numbers.
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