In Mr. Purcell's Electricity and Magnetism  ,page 103, it is stated,

An isolated conductor carrying a charge $Q$ has a certain potential $\phi _{0}$, with zero potential at infinity. $Q$ is proportional to $\phi _{0}$. The constant of proportionality depends only on the size and shape of the conductor. We call this factor the capacitance of that conductor and denote it by C. $$Q=C \phi _{0}$$

I understand that for a given charge $Q_{0}$ and its corresponding potential $\phi_{0}$ we could define a $C_{0}$ as a function of the shape and size of the conductor such that $Q _{0} =C_{0}\phi_{0}$.

When we change the charge to $Q_{1}$, the potential will become $\phi_{1}$. How can we prove that it is the same constant $C_{0}$ that will link $Q_{1}$ and $\phi _{1}$ ? In other words is charge being linearly proportional to potential an experimental result or can we prove it?

If one argues that it is the same constant because it depends only on the shape and size of the conductor, then they must also prove that this constant does satisfy $$Q=C _{0} \phi$$ for every given charge and its corresponding potential.

In Mr. Purcell's Electricity and Magnetism, page 103, it is stated,

An isolated conductor carrying a charge $Q$ has a certain potential $\phi _{0}$, with zero potential at infinity. $Q$ is proportional to $\phi _{0}$. The constant of proportionality depends only on the size and shape of the conductor. We call this factor the capacitance of that conductor and denote it by C. $$Q=C \phi _{0}$$

I understand that for a given charge $Q_{0}$ and its corresponding potential $\phi_{0}$ we could define a $C_{0}$ as a function of the shape and size of the conductor such that $Q _{0} =C_{0}\phi_{0}$.

When we change the charge to $Q_{1}$, the potential will become $\phi_{1}$. How can we prove that it is the same constant $C_{0}$ that will link $Q_{1}$ and $\phi _{1}$ ? In other words is charge being linearly proportional to potential an experimental result or can we prove it?

If one argues that it is the same constant because it depends only on the shape and size of the conductor, then they must also prove that this constant does satisfy $$Q=C _{0} \phi$$ for every given charge and its corresponding potential.