In dimension four, the differences between continuous and differential topology are vast but fundamentally unstable, disappearing when manifolds are enlarged in various ways. Wall-type stabilization problems aim to quantify this instability. I will discuss an approach to these problems that uses Khovanov-type homology theories and relies on blending theoretical calculations with intensive machine computation. Time permitting, I will discuss how one might push these techniques further and use these Khovanov-type tools for knots and surfaces to calculate Floer-theoretic invariants for associated 3- and 4-manifolds.