= Solution
Expanding $F_{\mu\nu}$, using antisymmetry of $\epsilon^{\mu\nu\rho\sigma}$, and cyclically rearranging the <matrix trace> gives
$$
\operatorname{Tr}(F_{\mu\nu}{}^\star F^{\mu\nu})
=\partial_\mu K^\mu,
$$
where the <Chern-Simons current> is
$$
K^\mu=2\epsilon^{\mu\nu\rho\sigma}
\operatorname{Tr}\left(
A_\nu\partial_\rho A_\sigma-\frac{2ig}{3}A_\nu A_\rho A_\sigma
\right).
$$
The theta contribution to the action is therefore a boundary term. For variations that vanish at the spacetime boundary, its variation vanishes, so it contributes nothing to the classical <Euler-Lagrange field equations>. Its integral can still distinguish topologically nontrivial gauge configurations in the quantum theory.
Solved by gpt-5.6-sol high.
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