= Solution
{c}
A semicolon denotes the <covariant derivative> for the <Levi-Civita connection>. For example, the four lower indices of the <Riemann curvature tensor> contribute four connection terms:
$$
\begin{aligned}
R_{\lambda\mu\nu\kappa;\eta}
={}&\partial_\eta R_{\lambda\mu\nu\kappa}
-\Gamma^a_{\lambda\eta}R_{a\mu\nu\kappa}
-\Gamma^a_{\mu\eta}R_{\lambda a\nu\kappa}\\
&-\Gamma^a_{\nu\eta}R_{\lambda\mu a\kappa}
-\Gamma^a_{\kappa\eta}R_{\lambda\mu\nu a}.
\end{aligned}
$$
Here $\partial_\eta=\partial/\partial x^\eta$ is a <partial derivative>, $\Gamma^a_{b\eta}$ are <Christoffel symbols>, and repeated indices are summed from $0$ to $3$. Write $g_{\mu\nu}$ for the <metric tensor>, $g^{\mu\nu}$ for its <inverse metric>, $R_{\mu\nu}=g^{\lambda\nu'}R_{\lambda\mu\nu'\nu}$ for the <Ricci tensor>, and $R=g^{\mu\nu}R_{\mu\nu}$ for the <Ricci scalar>. This <tensor contraction> agrees with the paper's $R_{\mu\nu}=-R^a{}_{\mu\nu a}$.
By <metric compatibility>, the <inverse metric> may pass through each <covariant derivative>. Contracting the <second Bianchi identity> with $g^{\lambda\nu}$ gives
$$
\nabla_\eta R_{\mu\kappa}-\nabla_\kappa R_{\mu\eta}
+\nabla^\lambda R_{\lambda\mu\kappa\eta}=0,
$$
where $\nabla_\eta$ is the same <covariant derivative> as the semicolon. Next contract with $g^{\mu\kappa}$. Antisymmetry of the first pair gives $g^{\mu\kappa}R_{\lambda\mu\kappa\eta}=-R_{\lambda\eta}$, so
$$
\partial_\eta R-\nabla^\mu R_{\mu\eta}-\nabla^\lambda R_{\lambda\eta}=0.
$$
Thus the <contracted Bianchi identity> is $\nabla^\mu R_{\mu\eta}=\tfrac12\partial_\eta R$. Raising the remaining index and using <metric compatibility> proves
$$
\boxed{\nabla_\mu G^{\mu\nu}=0,\qquad
G^{\mu\nu}=R^{\mu\nu}-\tfrac12g^{\mu\nu}R.}
$$
The <Einstein tensor> $G^{\mu\nu}$ therefore has identically zero <covariant divergence>, without imposing any field equation. This makes the <Einstein field equations> compatible with <stress-energy conservation>, $\nabla_\mu T^{\mu\nu}=0$, where $T^{\mu\nu}$ is the <stress-energy tensor>. The connection terms describe how <energy> and <momentum> are compared at neighboring points; ordinary componentwise <partial derivatives> would not give a coordinate-independent conservation law in curved <spacetime>.
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