Theoretical elements of reality are real ?

Let suppose we admit physics formulas represent physical reality, and we had an equation described by : Lf(x,t)=0 where L were an operator.

Suppose we get a constant of integration C independent of x and t

I can however write C=1 and C=2... and hence vary the constant...but if we believe t is physical time and x space, then this would contradict the fact C is a constant ....and hence C is either a function of "t" or "x" (where x is for example the position of C in the present text.....hence whole physics breaks down ??? Or do meta-time, or global time, or "experiment-intern time" (C could depend on the initial condition you could vary)...or do C depend on hidden parameter C(u)...where u represent a parametrization of universes or human free will....