What is the correct domain of integration for the index of instantons? - $\mathbb{R}^4$ or $S^4$?

Question

I posted the original question on Math SE but it seems like a more appropriate question for Physics SE:

https://math.stackexchange.com/q/4417225/

In calculating the instanton solutions for $SU(2)$ gauge theory on $\mathbb{R}^4$ or its compactification $S^4$, what is the correct domain on which we integrate the Yang-Mills Lagrangian in order to obtain the instanton number? Is it $\mathbb{R}^4$ or $S^4$?

The details are in the above link. This is extremely confusing for me. Could anyone explain?

Answer 1

Your formula for $F_{\mu\nu}$ is not correct $$ F_{\mu\nu} = \frac{4\lambda^2\sigma_{\mu\nu}}{(r^2 +\lambda^2)^2} $$ and the $R^4$ integral is $$ \int tr {F^2} \propto \int \frac{r^3 dr}{(r^2+\lambda^2)^4} $$ which is convergent.