How To prove Any Change to $v=a\cdot y+b$ maks $y=(a)^{−1}\cdot (v−b)$ Uni. random value
Well, there is not much to prove.


  The server can do any change to $v$.


So actually, this means the server with throw away your $v$ and just choose a new random number. Or alternatively, he chooses $w$ uniformaly random and sets $v'=v+w$, which is now also uniformly random.

When you get back that value, you have your (for you known/fixed) values $a,b$ and do your calculation $y' = a^{-1}(v'-b)$. With he existence of $a^{-1}$ (i.e. $a\neq0$), this is also uniformly distributed for fixed values $a,b$.