I don't know much about Gauge theory but from some bits and pieces I know about it, the following problem seems to be related to it.
Consider the classical Lagrangian $$\mathcal{L}=T-U$$ Where $T$ is kinetic energy and $U$ is potential energy.
Potential energy is defined via $$\vec F=-\vec \nabla U \tag{1}$$
This comes from the fact that for a consevative force $$\oint\vec F\cdot d\vec l=0\rightarrow \vec F=\vec\nabla V \tag{2}$$
Here the minus sign is conventional in equation $1$. In addition, we can introduce $U'=U+c$ for some constant $c$ and nothing will change about the physics in question.
Thus the freedom to choose the sign and value $c$ is gauge dependence for this problem.
Now consider total energy $$E=T+U$$ Adding this to Lagrangian we get $$\mathcal{L}+E=2T$$ which is gauge-independent.
Thus when we add two gauge dependents we are getting a gauge independent.
I was wondering if there is any mathematical significance in this result?