Kůrková's 1991 rebuttal to Girosi and Poggio resolved the neural-Kolmogorov dispute by substituting an approximate representation for the exact one, not by waiting for new tooling

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Two independent Tier-1 sources — Braun & Griebel (2009) and Ismayilova & Ismailov (2023) — describe the same resolution to Girosi and Poggio's 1989 objection: Věra Kůrková's 1991 paper "Kolmogorov's theorem is relevant," published in the same journal (Neural Computation) two years after Girosi and Poggio's "Kolmogorov's theorem is irrelevant," followed by "Kolmogorov's theorem and multilayer neural networks" (Neural Networks, 1992). Braun and Griebel summarize the mechanism: Kůrková "partly eliminated these difficulties by substituting the exact representation... with an approximation of the function f," replacing the theorem's one-variable inner functions with finite linear combinations of affine transformations of a single sigmoidal function, an approach that also let her estimate the number of hidden units needed for a given accuracy. Ismayilova and Ismailov's independent account matches: Kůrková showed "the relevance of Kolmogorov's superposition theorem to approximation by neural networks is different," substituting the precise representation with an approximation. Neither account attributes the resolution to a new training algorithm or hardware; the pivot is a change in what the papers are proving — trade the exact, unlearnable Kolmogorov network for a learnable approximation of it.