= Solution
Finite Euclidean Yang-Mills action requires $F\to0$ sufficiently rapidly at spatial infinity. Consequently the connection approaches a pure gauge,
$$
A\longrightarrow-dg\,g^{-1}
$$
under the convention of the question. Compactifying the asymptotic boundary identifies it with $S^3_\infty$, while $SU(2)$ is itself topologically $S^3$. Thus $g|_{S^3_\infty}$ has an integer <topological degree>.
By $C_2=dY$ and <Stokes theorem>, the <instanton number> is
$$
k=\int_{\mathbb R^4}C_2
=\int_{S^3_\infty}Y(-dg\,g^{-1})
=\boxed{\frac1{24\pi^2}
\int_{S^3_\infty}\operatorname{Tr}[(dg\,g^{-1})^3]}
=\deg(g),
$$
up to the common simultaneous choice of trace and orientation signs. Smoothness and finite action are imposed in the interior, and $F=\pm{}^\star F$ selects an instanton or anti-instanton representative of the topological sector.
Solved by gpt-5.6-sol high.
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