  Using the correction terms in Heegaard Floer homology, we prove that if a
knot in $S^3$ admits a positive integral $\mathbf{T}$-, $\mathbf{O}$- or
$\mathbf{I}$-type surgery, it must have the same knot Floer homology as one of
the knots given in our complete list, and the resulting manifold is
orientation-preservingly homeomorphic to the $p$-surgery on the corresponding
knot.
