Proposition: Smooth interpolation non-determination

Suppose a finite code table specifies distinct readings $s_1,\dots ,s_m$ and the values $F(s_i,x)$ of a smooth uniformly spacelike graph family on a compact regular spatial patch. Those samples need not determine its intermediate normal deformation component.

Proof. Choose $s_c$ different from all sample readings, a smooth time bump $b$ supported in an unsampled interval with $b(s_c)=0$ and $b^{\prime }(s_c)\ne 0$, and a nonzero smooth spatial bump $h$. Put

All sampled graphs are unchanged. By compactness and uniform spacelikeness, sufficiently small $ϵ$ preserves spacelikeness. If ordering is required and ${\partial }_{s}F$ has a positive lower bound on the perturbation support, small $ϵ$ preserves it as well.

At $s_c$, the spatial graph gradient is unchanged, but

Equation (eq:normal) therefore gives different normal components wherever $h\ne 0$.