= Kolmogorov-Arnold representation theorem
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= Kolmogorov superposition theorem
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{synonym}
Every real <continuous function> on a compact square can be represented as $\sum_{q=1}^5G_q(a_q(x)+b_q(y))$, with continuous one-variable functions and fixed inner functions $a_q,b_q$ independent of the represented function. <Grid-separated additive coordinates> give the inner functions. For a residual $r$ of norm $M$, assign $r$ at a sample point divided by three to each separated rectangle image, and continuously interpolate between images. Each point lies in at least three rectangles, so if $k\in\{3,4,5\}$ summands are accurate, the error is at most $[|1-k/3|+(5-k)/3]M+5\eta/3=2M/3+5\eta/3$. Choose oscillation $\eta\le M/10$ to obtain contraction $5/6$. Iteration gives outer functions by <uniform convergence> of corrections bounded by $M(5/6)^j/3$. <Whole-plane reduction for continuous superposition> removes compactness of the original domain without assuming boundedness of the original function.
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