Finite measurement interpolation freedom ($g(x)=f(x)+\lambda {\prod }_i(x-x_i)$)

If $f(x_i)=y_i$ at finitely many distinct measured inputs, then $g(x)=f(x)+\lambda {\prod }_i(x-x_i)$ agrees at every measured point. Away from the roots it can disagree. This proves that the finite record alone does not determine an unrestricted function; it does not imply that all such functions satisfy a physical theory's independent constraints.