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Internal stabilization adds a trivial handle to an embedded surface in acoordinate chart. It is known that any pair of smoothly knotted surfaces in asimply-connected $4$-manifold become smoothly isotopic after sufficiently manyinternal stabilizations. In this paper, we show that there is no upper bound onthe number of internal stabilizations required. In fact, this behavior isfairly generic. The definition of a subtly smoothly knotted pair of surfaces isgiven and it is shown that many surfaces may be modified to obtain subtlyknotted surfaces with large internal stabilization distance. Furthermore, it isshown that after stabilizing any $4$-manifold with contain topologicallyisotopic, smoothly related, non-isotopic copies of any $3$-manifold having apositive first betti number.

Smoothly knotted surfaces that remain distinct after many internal stabilizations