= Whole-plane reduction for continuous superposition
For $f\in C(\mathbb R^2)$, choose a positive <continuous function> $A$ with $A(t)\ge(1+t)\max_{x^2+y^2\le t}|f(x,y)|$ for $t\ge0$. Such an $A$ is obtained by piecewise linear interpolation of a sufficiently large increasing sequence. Then $b(x,y)=f(x,y)/A(x^2+y^2)$ tends uniformly to zero at infinity. Transport $b$ to $(0,1)^2$ by $x=\tan(\pi(s-1/2))$ and extend it by zero on the boundary; the extension is continuous. Apply the compact <Kolmogorov-Arnold representation theorem>, then multiply by $A(x^2+y^2)$. Multiplication introduces no additional primitive operation because $uv=[(u+v)^2-(u-v)^2]/4$ uses only addition and continuous one-variable squaring and scaling.
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