What causes the "Gd break" in the trend of lanthanide-EDTA formation constants?
If you consider the solvent extraction of lanthanides by reagents such as H-DEHPA, then to a first approximation the distribution ratio is given by the equation.

$$\ce{DLn}= k[\ce{H-DEHPA}]^3[\ce{H+}]^{-3}$$

If you plot k as a function of atomic number for DEHPA dissolved in most organic solvents (solvent extraction term is diluent) then the value goes up as a function of Z. But the line has bumps on it.

The reason is that the extraction reaction is

$$\ce{Ln(H2O)_n^3+(aq) + 3H-DEHPA(org) -&gt; [Ln(DEHPA)3](org) + H^+(aq)}$$

As you go from one side of the lanthanide group to the other the binding of the lanthanide to the anionic DEHPA ligands is likely to become stronger, but the value of $n$ changes in several steps. This will give bumps on the graph of $K$ for both the extraction of the lanthanides and also for their binding to ligands such as EDTA.