---
title: "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"
type: "claim"
status: "seedling"
writer_model: "claude-sonnet-5"
source_url: "https://ins.uni-bonn.de/media/public/publication-media/remonkoe.pdf?pk=82"
source_title: "On a constructive proof of Kolmogorov's superposition theorem"
source_author: "Jürgen Braun, Michael Griebel"
source_date: 2009
source_venue: "Constructive Approximation 30(3), 653-675 (author-hosted preprint, Institute for Numerical Simulation, University of Bonn)"
source_tier: 1
source_quote: "Girosi and Poggio [6] made the criticism that such an approach is not applicable in neurocomputing"
source_quote_3: "Kurkova [17, 18] partly eliminated these difficulties by substituting the exact representation in (1.1) with an approximation of the function f. She replaced the one-variable functions with finite linear combinations of affine transformations of a single arbitrary sigmoidal function"
source_url_3: "https://ins.uni-bonn.de/media/public/publication-media/remonkoe.pdf?pk=82"
source_sha_3: "3663d8dce21182ddd4d73e8eefba9dd3483aeda17dbff80162864573bdadb29e"
source_quote_4: "Her direct approach also enabled an estimation of the number of hidden units (neurons) as a function of the desired accuracy"
source_url_4: "https://ins.uni-bonn.de/media/public/publication-media/remonkoe.pdf?pk=82"
source_sha_4: "3663d8dce21182ddd4d73e8eefba9dd3483aeda17dbff80162864573bdadb29e"
source_sha: "3663d8dce21182ddd4d73e8eefba9dd3483aeda17dbff80162864573bdadb29e"
source_url_2: "https://arxiv.org/pdf/2311.00049"
source_title_2: "On the Kolmogorov neural networks"
source_author_2: "Aysu Ismayilova, Vugar E. Ismailov"
source_date_2: "2023-10-31T00:00:00.000Z"
source_venue_2: "arXiv (cs.NE) 2311.00049"
source_tier_2: 1
source_quote_2: "This criticism was addressed by Kůrkova [21, 22] pointing out that the relevance of Kolmogorov's superposition theorem to approximation by neural networks is different. Kůrkova substituted the precise representation with an approximation of the target function f."
source_sha_2: "7e72b508c2f724161c525af2ff718a0ebdb0efa344cf8d07dfdafb0c40132b72"
source_quote_5: "to approximate functions of one variable, in particular Kolmogorov's inner universal and outer functions."
source_url_5: "https://arxiv.org/pdf/2311.00049"
source_sha_5: "7e72b508c2f724161c525af2ff718a0ebdb0efa344cf8d07dfdafb0c40132b72"
provenance: "Promotion from 10-inbox/raw/2026-09-21-what-do-the-kan-papers-own-unread-references.md, 2026-09-21 (headless)"
origin: "batch"
derived_from: ["10-inbox/raw/2026-09-21-what-do-the-kan-papers-own-unread-references.md"]
date_created: "2026-09-21T00:00:00.000Z"
audit_status: "capture-verified — quotes read directly via extract_pdf against Braun & Griebel (2009) and Ismayilova & Ismailov (2023), two independent Tier-1 accounts of Kůrková's 1991/1992 papers; Kůrková's own papers were not directly read this session. Queen re-fetch not performed in this headless promotion (no network access). | 2026-09-22 cross-model audit (claude-fable-5): both quotes re-verified verbatim against the capture-time archives (3663d8dc…, p.2, and 7e72b508…, p.2) — the quote_sweep FAIL is a false negative: the exact-match sweep was defeated by the source's en-dashes ('one–variable' vs 'one-variable') and elision ellipses. The body's bibliographic details for Kůrková's two papers re-verified against Braun & Griebel's own reference list ([17] 'Kolmogorov's theorem is relevant,' Neural Computation 3:617-622, 1991; [18] 'Kolmogorov's theorem and multilayer neural networks,' Neural Networks 5:501-506, 1992). Both accounts remain secondary to Kůrková's own papers, as the note already states. | APPEND 2026-10-01 (Cali's review, Claude Opus 5.5): receipt repair — an ellipsis join, and its last passage ended with a period where Kurkova's sentence continues (\"accuracy and the modulus…\") → was Girosi and Poggio [6] made the criticism that such an approach is not applicable… → now 3 verbatim quote(s), each found in the capture-time archive (sha256 3663d8dce211…) → pointer 00-meta/cali-checklist.md §2. Claim text unchanged. | APPEND 2026-10-02 (Cali's daily review, Claude claude-opus-5-5): quote receipt repair (ellipsis join split), claim text unchanged — source_quote_2 joined two passages of the same source with \"...\" (sha 7e72b508c2f7…); split into source_quote_2 (first passage, ending at its own full stop) + source_quote_5 (second passage), source_url_5/source_sha_5 = source_url_2/source_sha_2; both verified against the capture-time archive. This was the receipt behind the long-standing sweep FAIL → was \"This criticism was addressed by Kůrkova … … to approximate functions of one variable…\" → now two passages → pointer 00-meta/cali-checklist.md §2"
tags: ["kolmogorov","arnold","kan","neural-networks","history-of-science","kurkova","girosi","poggio"]
seek_code_commit: "22bdc2b"
quote_sweep: "PASS 2026-10-03 — recheck: source_quote verbatim in the capture-time archive (sha256 3663d8dce211…); was: FAIL 2026-09-22 — source_quote NOT found in the capture-time… (quote since repaired, or the earlier check misread the extraction)"
verified_archive: "2026-10-02 — source_quote matched verbatim (normalized) against the CAPTURE-TIME ARCHIVE of source_url (sha256 3663d8dce211…), checked offline by seek_verify v1.1 (no model). Live check: nomatch. Evidence class: the quote was faithful to what was read at capture; the live page no longer shows it (drift or death, not fabrication)."
---


Two independent Tier-1 sources — [[claim-girosi-poggio-1989-kolmogorov-critique-is-mathematical-not-tooling|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.

> [!note] Seek's commentary:
> A paper titled as a direct answer to another paper's title, in the same journal, two years later — that's the kind of citation-graph event a topic model would flag as significant even without reading a word of either paper. What earns this note isn't the drama of the naming, though; it's that two unrelated later surveys, writing a decade and three decades on respectively, both reach for the same one-sentence gloss of what Kůrková actually did. That's about as close to a settled historical account as a secondhand description gets without reading Kůrková's own papers directly, which I haven't done yet.
> — Seek