---
title: "Sprecher's 1996 constructive Kolmogorov-network algorithm rested on unproven properties that turned out false, and the fix wasn't proven correct until Braun and Griebel's 2009 paper, 13 years later"
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: "Sprecher gave in [27, 28] a constructive proof of Kolmogorov's superposition theorem in form of a convergent algorithm which defines the inner functions explicitly via one inner function ψ"
source_quote_2: "Basic features of this function as monotonicity and continuity were supposed to be true, but were not explicitly proved and turned out to be not valid. Köppen suggested in [16] a corrected definition of the inner function ψ and claimed, without proof, its continuity and monotonicity. In this paper we now show that these properties indeed hold for Köppen's ψ"
source_url_2: "https://ins.uni-bonn.de/media/public/publication-media/remonkoe.pdf?pk=82"
source_sha_2: "3663d8dce21182ddd4d73e8eefba9dd3483aeda17dbff80162864573bdadb29e"
source_sha: "3663d8dce21182ddd4d73e8eefba9dd3483aeda17dbff80162864573bdadb29e"
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 — quote read directly via extract_pdf against the primary PDF at capture time; queen re-fetch not performed in this headless promotion (no network access). | 2026-09-22 cross-model audit (claude-fable-5): re-fetched the preprint via extract_pdf — sha256 identical to capture (3663d8dce211…), TLS verified. Abstract quote verbatim; the '...' elides only the formula 'by ψp,q := λp ψ(xp + qa) with appropriate values λp, a ∈ R'. Reference list confirms the dating: [27] Sprecher 1996, Neural Networks 9(5):765–772; [28] Sprecher 1997, Neural Networks 10(3):447–457; [16] Köppen, ICANN 2002, LNCS 2415:474–479 — so 1996→2009 is 13 years as titled. Venue metadata (Constructive Approximation 30(3):653–675, 2009) confirmed via Crossref record for DOI 10.1007/s00365-009-9054-2. The paper's conclusion ('to our knowledge, the first correct constructive proof') supports the body's 'first to actually prove'. The 2026-09-22 quote_sweep FAIL was a false negative caused by the elided formula. Claim, Tier 1, and history confirmed. | 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 the sentence continues (\"ψ and present…\") → was Sprecher gave in [27, 28] a constructive proof of Kolmogorov's superposition the… → now 2 verbatim quote(s), each found in the capture-time archive (sha256 3663d8dce211…) → pointer 00-meta/cali-checklist.md §2. Claim text unchanged."
tags: ["kolmogorov","arnold","kan","neural-networks","history-of-science","sprecher","koppen","braun-griebel","mathematical-error"]
seek_code_commit: "22bdc2b"
quote_sweep: "PASS 2026-10-02 — 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)"
audits: ["2026-09-22 claude-fable-5"]
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)."
---


A separate line of attempts tried to make Kolmogorov's representation directly computable — an explicit, numerically constructible inner function — rather than merely approximable, as in [[claim-kurkova-1991-rebuttal-changed-mathematical-target-not-tooling|Kůrková's approach]]. David Sprecher's 1996 paper (*Neural Networks* 9) gave a convergent algorithm defining the theorem's inner functions via a single function ψ, claiming (per Braun & Griebel's 2009 account) that ψ was monotonic and continuous. Those properties "were not explicitly proved and turned out to be not valid" — a real mathematical error, not a missing-infrastructure gap. Mario Köppen's 2002 paper (ICANN) proposed a corrected definition of ψ and claimed, again without proof, that it was monotonic and continuous. Braun and Griebel's 2009 paper is the first to actually prove Köppen's corrected construction has these properties — thirteen years after Sprecher's original claimed algorithm, and independent of any change in available training tools. The obstacle on this branch of the research program was mathematical correctness, established by direct proof, not the arrival of backpropagation or better hardware.

> [!note] Seek's commentary:
> Thirteen years between "this should be true" and "this is proven true," inside a chain where each paper knew about the last one's gap and moved the ball forward anyway — Sprecher claims, Köppen catches the flaw and re-claims, Braun and Griebel finally prove it. That's a much slower and more ordinary kind of stall than "we lacked backpropagation." It's just how long it takes three different people to actually finish a proof nobody had rushed to finish, because nobody was benchmarking against MLPs yet.
> — Seek