= Solution
{c}
To derive <fermionic Einstein-coefficient relations>, first consider an idealized emitter that can exchange one ultrarelativistic <fermion> with the bath, with zero <chemical potential>. Let $\mathcal D_\nu=d\,4\pi\nu^2/c^3$ be its <density of states> per volume and ordinary <frequency>, with $d$ populated helicity states; for one <neutrino> helicity $d=1$, whereas the corresponding <photon> count is two. The <Fermi-Dirac distribution> gives
$$
f_\nu=\frac1{e^{h\nu/(k_BT)}+1},\qquad u_\nu=h\nu\mathcal D_\nu f_\nu.
$$
<Pauli blocking> changes the final-state factor in downward emission from the bosonic $1+f$ to $1-f$. Define positive coefficients $B_{12},C_{21}$ and $A_{21}=C_{21}h\nu\mathcal D_\nu$. The upward rate per lower emitter is $B_{12}u_\nu$, and the downward rate per upper emitter is
$$
A_{21}(1-f_\nu)=A_{21}-C_{21}u_\nu.
$$
Thermal <detailed balance>, $n_1B_{12}u_\nu=n_2(A_{21}-C_{21}u_\nu)$ with $n_2/n_1=(g_2/g_1)e^{-h\nu/(k_BT)}$, yields
$$
\boxed{g_1B_{12}=g_2C_{21},\qquad
\frac{A_{21}}{C_{21}}=d\,\frac{4\pi h\nu^3}{c^3}.}
$$
For example, substituting $e^{h\nu/(k_BT)}=x$ reduces the balance equation to $B_{12}=(g_2/g_1)C_{21}$ because $x f_\nu=1-f_\nu$.
If one insists on the original additive notation $A_{21}+B_{21}u_\nu$, its formal coefficient is negative:
$$
\boxed{B_{21}=-C_{21},\qquad g_1B_{12}=-g_2B_{21},\qquad
A_{21}/B_{21}=-d\,4\pi h\nu^3/c^3.}
$$
A negative correction means inhibited emission into occupied modes, not a negative total decay rate; $0\leq f_\nu\leq1$ ensures positivity. The dimensions are those of part (a), and the factor of two relative to the <photon> density is a helicity count, independent of the statistical sign.
There is a physical qualification: ordinary electronic <energy levels> of the same <chemical atom> have the same <fermion parity>, and cannot interconvert by emission or absorption of a single <neutrino>. A realizable single-<fermion> emitter must change the relevant particle-number and parity sector, with matching <conservation laws>. For ordinary atomic levels, neutrino-pair processes would require a different calculation. A bath with nonzero <chemical potential> would also require the corresponding chemical-equilibrium population ratio. The calculation above is the intended single-mode statistical analogy under explicit assumptions.
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