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There is no localization for eigenspinors of the natural Dirac operator on $S^4$ as can be seen from explicit formulae for these.  However, this is not the case for solutions of Yang-Mills equation for a principal $G$-bundle.  Recall that these equations arise from minimizing the Yang-Mills functional $YM(A) = \int_M |F_A|^2 dA$ over gauge-equivalent connections on a fixed $G$-bundle.  The absolute minima of the Yang-Mills equation are $*F^+_A = \pm F_A$.   A nice thesis describing explicit solutions of the Yang-Mills equation is LindenhoviusInstantonsAndADHMConstruction
$A|_x = \im ( \frac{(x-b)d\bar{x}}{\rho^2+|x-b|^2} ),$
where $\rho^2 \in\mathbf{R}$ and $b \in mathbf{H}$.  These have a pseudo-localization falling off like $1/r$ from its maximum.