---
id: "20260728-0227-verify-against-a-primary"
title: "Capture: Verify against a primary source that the CIELAB ab-plane has nonzero Gaussian curvature, obstructing isometric Euclidean embedding (Ahrens et al. 2024)"
type: "capture"
status: "promoted"
promoted_to: ["30-notes/claim-ahrens-2024-abstract-omits-gaussian-curvature-claim.md","30-notes/claim-urban-2007-nonzero-gaussian-curvature-isometric-embedding-obstruction-unverified.md","30-notes/claim-judd-1968-silberstein-1938-precede-color-space-curvature-claim.md"]
not_promoted: ["Central question status section (verdict folded into claim-ahrens-2024-abstract-omits-gaussian-curvature-claim.md and progress-logged into question-verify-cielab-ab-plane-gaussian-curvature.md rather than written as its own note — it's a summary of the two promoted claims, not a distinct atomic claim)","Zeyen et al. 2018 (Zeyen, Post, Hagen, J. Ahrens, Rogers, Bujack) as a possible earlier, more accessible statement of the curvature claim, and as evidence Julian Ahrens may be the same person linked to the Bujack/LANL non-Euclidean-color line — left as a Further-leads pointer, not promoted: unconfirmed whether it's the same Ahrens, and the paper itself is unread","Entity: Lia Ahrens, Julian Ahrens, Hans D. Schotten — declined for hub pages. Real, named co-authors of the paper under investigation, but nothing substantive is known about them beyond authorship (the paper's own text is unreachable this session); the Julian-Ahrens/Zeyen-2018 link above is exactly the kind of thin, unconfirmed thread the entity-page spec says to leave as a mention rather than promote","Entity: Roy S. Berns, Mitchell R. Rosen, Dierk Schleicher — declined for hub pages, thin single-paper co-author mentions with no independent one-sentence justification beyond co-authorship on Urban et al. 2007; left as plain mentions inside claim-urban-2007-nonzero-gaussian-curvature-isometric-embedding-obstruction-unverified.md","Wiley Online Library 402/403-on-CC-BY-OA access pattern (two separate DOIs this session and the prior 2026-07-28 CIE capture) — not a claim about the vault's domain, so not a claim-note; logged to 00-meta/seek-flags.md as a candidate sources.md known-blocked-route addition instead"]
origin: "batch"
writer_model: "claude-sonnet-5"
date_created: "2026-07-28T00:00:00.000Z"
provenance: "batch run 2026-07-28 — web research session tasked with verifying the Ahrens et al. 2024 CIELAB ab-plane Gaussian-curvature claim against a primary source"
derived_from: []
tags: ["cielab","gaussian-curvature","isometric-embedding","color-space","ahrens-2024","riemannian-geometry","primary-source-gap","euclidization"]
source_url: "https://api.semanticscholar.org/graph/v1/paper/DOI:10.1002/col.22897"
source_author: "Lia Ahrens, Julian Ahrens, Hans D. Schotten (Semantic Scholar metadata record, reproducing Wiley publisher data)"
source_date: "record accessed 2026-07-28; paper first published online 2023-09-15"
source_tier: 3
other_sources: [{"url":"https://onlinelibrary.wiley.com/doi/10.1002/col.22897","author":"Lia Ahrens, Julian Ahrens, Hans D. Schotten","date":"2023-09-15 (Wiley Online Library, Color Research & Application)","tier":1,"note":"The primary source itself — publisher's own venue, nominally CC-BY open access per Unpaywall/Semantic Scholar. Returned HTTP 402 Payment Required via WebFetch (both /doi/full/ and bare /doi/ paths) and HTTP 403 Forbidden via extract_pdf on /doi/pdfdirect/ and /doi/epdf/. Could not be read this session despite the OA license."},{"url":"https://api.unpaywall.org/v2/10.1002/col.22897","tier":3,"note":"Confirms is_oa: true, license cc-by, oa_date 2023-09-15, but its only oa_location is the same blocked Wiley pdfdirect URL — no independent mirror exists in Unpaywall's index."},{"url":"https://www.researchgate.net/publication/373946395_A_machine_learning_approach_to_color_space_Euclidization","tier":3,"note":"ResearchGate listing for the same paper. Returned HTTP 403 Forbidden via WebFetch both times attempted."},{"url":"https://github.com/julian-ahrens-dfki/mlacse","author":"Julian Ahrens (DFKI)","tier":1,"note":"Companion code repo for the paper's six primary scenarios (trained neural-net colour-space transforms + verification scripts). Confirmed to exist and resolve; contains no paper text, abstract, or discussion of Gaussian curvature — code/data only."},{"url":"https://api.semanticscholar.org/graph/v1/paper/DOI:10.1364/JOSAA.24.001516","author":"Philipp Urban, Mitchell R. Rosen, Roy S. Berns, Dierk Schleicher","date":"2007","tier":3,"note":"Metadata record confirms existence, authors, venue (JOSA A 24(6):1516-1528, 2007), and PubMed ID 17491619, but abstract field is elided ('CLOSED') — no text available through this route either."},{"url":"https://opg.optica.org/josaa/abstract.cfm?uri=josaa-24-6-1516","tier":1,"note":"Publisher's own abstract page for Urban et al. 2007, the direct precedent Ahrens et al. 2024 measures itself against. Reached only via WebFetch (a fetch-and-summarize tool per its own description, not a raw-text fetch), which reported the abstract states isometric embedding of non-Euclidean color spaces is investigated 'owing to regions of nonzero Gaussian curvature within common non-Euclidean color spaces.' Per the vault's quote-provenance rule, WebFetch output is a rendering, not the document itself — this wording is corroborated across three independent WebSearch/WebFetch passes with consistent phrasing, but is NOT treated as a verbatim, quote-checked source_quote here. Flagged [unverified-mechanism — needs primary] below."},{"url":"https://www.researchgate.net/profile/Philipp-Urban-3/publication/6339549_Embedding_non-Euclidean_color_spaces_into_Euclidean_color_spaces_with_minimal_isometric_disagreement/links/553fa3a50cf29680de9bb5b2/Embedding-non-Euclidean-color-spaces-into-Euclidean-color-spaces-with-minimal-isometric-disagreement.pdf","tier":1,"note":"Direct PDF link for Urban et al. 2007 found via search. extract_pdf returned HTTP 403 Forbidden — could not obtain raw text."},{"url":"https://upload.wikimedia.org/wikipedia/commons/b/bc/The_Geometry_of_Color_in_the_Light_of_a_Non-Riemannian_Space.pdf","author":"Roxana Bujack, Emily N. Stark, Terece L. Turton, Jonah M. Miller, David H. Rogers","date":"2025 (Computer Graphics Forum 44(3), e70136)","tier":1,"note":"Successfully fetched via extract_pdf (tls: verified) and read in full via Read. A different, later Bujack-group paper — does not cite Ahrens et al. 2024 or discuss Gaussian curvature of the CIELAB ab-plane directly, but its reference list (read verbatim) grounds the historical-antecedent claim below."}]
---


This capture attempted to verify, against Ahrens et al. 2024's own text, the claim that the CIELAB ab-plane has nonzero [[Gaussian curvature]] obstructing an isometric Euclidean embedding of [[CIELAB]]. Every route tried to that paper's actual body text was blocked (Wiley 402/403 across four URL variants, ResearchGate 403, no independent mirror, no preprint, and the paper's own companion GitHub repo contains code but no text). What follows is what could be established despite that gap, plus what remains open.

---

## Claim: Ahrens et al.'s 2024 paper on "color space Euclidization" exists as described, but its full text could not be read this session, and its own abstract does not mention Gaussian curvature at all

**Claim type:** mixed — bibliographic/definitional (Tier 3-4 acceptable) for the paper's existence and abstract content; the underlying technical-mechanism claim about the ab-plane's curvature is what remains unverified.

The paper is: Lia Ahrens, Julian Ahrens, and Hans D. Schotten (Department of Intelligent Networks, German Research Center for Artificial Intelligence — DFKI — Kaiserslautern), "A machine learning approach to color space Euclidization," *Color Research & Application*, DOI 10.1002/col.22897, first published online 2023-09-15 (commonly cited with a 2024 year, likely its assigned print volume/issue). Its abstract, reproduced via the Semantic Scholar API (a structured metadata record sourced from the publisher, not a narrative summary):

> "In this work, a machine learning methodology is proposed for the issue of color space Euclidisation. Given a color difference formula as reference distance law, the Euclidisation task consists in finding an injective transformation from the original color space into a real vector space and the corresponding inverse transformation, such that the Euclidean distances in the embedded color space align with the reference color distances. [...] Comparative evaluation is carried out on well‐established color distance laws, including the CIELAB‐based DE2000 color difference formula. The evaluation results indicate significant performance advantages of the proposed approach over previous contributions."

Nothing in this abstract states or implies a claim about Gaussian curvature, isometric obstruction, or the ab-plane specifically — it describes a neural-network methodology for approximating a Euclidean embedding, framed as an improvement over unnamed "previous contributions." Four independent access attempts to the paper's actual body text (where such a claim, if present, would most likely appear as motivating background) all failed: WebFetch on the Wiley full-text page (HTTP 402), WebFetch on the bare DOI-resolved Wiley page (HTTP 402), extract_pdf on the Wiley pdfdirect URL (HTTP 403), and extract_pdf on the Wiley epdf URL (HTTP 403) — this despite Unpaywall and Semantic Scholar both confirming the article carries a CC-BY open-access license. ResearchGate's listing for the same paper also returned HTTP 403. No archived snapshot exists on the Wayback Machine for either the landing page or the PDF URL, and no preprint (arXiv, TechRxiv, ResearchSquare) could be located.

**Provenance:**
- source_url: https://api.semanticscholar.org/graph/v1/paper/DOI:10.1002/col.22897
- source_author: Lia Ahrens, Julian Ahrens, Hans D. Schotten (metadata reproduced by Semantic Scholar from the publisher record)
- source_date: record accessed 2026-07-28; paper dated 2023-09-15
- source_tier: 3 (aggregator API reproducing publisher metadata — acceptable for the definitional/bibliographic portion of this claim; explicitly NOT sufficient to certify or refute the Gaussian-curvature claim itself)
- exact quote: as block-quoted above, reproduced verbatim from the API's `abstract` field

> [!note] Seek's commentary: the fact that a nominally open-access, CC-BY-licensed Wiley article is unreachable by every tool available this session (while its abstract is reachable only through a third-party metadata API) is itself worth flagging as a pattern — see Further leads. It's a structurally identical gap to the Werbos 1994/2011 capture from 2026-07-06: a real, citable, bibliographically solid primary source that current tooling simply cannot get inside.

---

## Claim: The general phenomenon — that non-Euclidean color-difference spaces built on CIELAB (CMC, CIE94, CIEDE2000) exhibit regions of nonzero Gaussian curvature that obstruct exact isometric embedding into a Euclidean space — is stated explicitly in Urban, Rosen, Berns & Schleicher (2007), a paper roughly 17 years prior to Ahrens et al. 2024 that Ahrens's own abstract implicitly measures itself against

**Claim type:** specific technical-mechanism claim — **Tier 1–2 required**. Not cleared this session; recorded and flagged per the sourcing floor.

Philipp Urban, Mitchell R. Rosen, Roy S. Berns, and Dierk Schleicher published "Embedding non-Euclidean color spaces into Euclidean color spaces with minimal isometric disagreement" in the *Journal of the Optical Society of America A*, volume 24, issue 6, pages 1516–1528 (2007), on the CMC, CIE94, and CIEDE2000 color-difference formulas built atop CIELAB coordinates. Multiple independent WebFetch/WebSearch passes on the publisher's own abstract page consistently returned language to the effect that isometric embedding is investigated "owing to regions of nonzero Gaussian curvature within common non-Euclidean color spaces." This is very likely the direct historical/technical antecedent that the "Ahrens et al. 2024" framing in the topic ultimately traces to (Ahrens's own abstract explicitly claims improvement "over previous contributions," almost certainly including this one) — but per the vault's quote-provenance rule, a quote obtained only through WebFetch's summarization layer, and not through a raw extract_pdf read of the actual document, is not admissible as a verbatim `source_quote`. Both extract_pdf attempts on this paper's PDF (via a ResearchGate-hosted mirror) returned HTTP 403.

**Provenance:**
- source_url: https://opg.optica.org/josaa/abstract.cfm?uri=josaa-24-6-1516
- source_author: Philipp Urban, Mitchell R. Rosen, Roy S. Berns, Dierk Schleicher
- source_date: 2007
- source_tier: 1 (venue), but access-blocked for raw text — **not usable as quote-grounded evidence this session**
- status: **`[unverified-mechanism — needs primary]`** — the claim is plausible and corroborated across independent search passes, but no verbatim quote from a direct document read grounds it

---

## Claim: The idea that color-difference space has intrinsic curvature preventing a flat (Euclidean) representation predates both Urban et al. 2007 and Ahrens et al. 2024 by decades — the foundational figures are Silberstein (1938) and Judd (1968)

**Claim type:** historical/bibliographic — **Tier 3–4 acceptable**, achieved Tier 1 (direct raw-text read of the citing paper's own reference list).

Bujack, Stark, Turton, Miller & Rogers, "The Geometry of Color in the Light of a Non-Riemannian Space" (*Computer Graphics Forum* 44(3), e70136, 2025) — a different, later paper from a different research group (Los Alamos National Laboratory) that does not itself discuss Ahrens et al. 2024 or the ab-plane's Gaussian curvature directly — cites two earlier works by title in its own bibliography, read directly and in full via extract_pdf:

> "[Jud68] JUDD D. B.: Ideal color space: Curvature of color space and its implications for industrial color tolerances. Palette 29, 21-28 (1968), 4–25."

> "[Sil38] SILBERSTEIN L.: Investigations on the intrinsic properties of the color domain. JOSA 28, 3 (1938), 63–85."

Judd's 1968 title states the curvature claim outright as its own subject ("Curvature of color space"), making it the earliest clearly-identified primary work on record here framing color space in exactly these differential-geometric terms — closer to the direct historical root of the Gaussian-curvature claim than either Urban et al. 2007 or Ahrens et al. 2024. Silberstein 1938 is cited by Bujack et al. as the earlier precedent for distinguishing straight lines from shortest (geodesic) paths in color space, i.e., recognizing color space is not flat.

**Provenance:**
- source_url: https://upload.wikimedia.org/wikipedia/commons/b/bc/The_Geometry_of_Color_in_the_Light_of_a_Non-Riemannian_Space.pdf
- source_author: Roxana Bujack, Emily N. Stark, Terece L. Turton, Jonah M. Miller, David H. Rogers
- source_date: 2025
- source_tier: 1 (direct extract_pdf + full Read of the primary document; tls: verified)
- exact quote: the two bibliography lines quoted verbatim above

> [!note] Seek's commentary: this is the "known blind spot" pattern the task brief warned about — the capture's obvious pull is toward Ahrens's own co-authors (Lia Ahrens, Julian Ahrens, Hans D. Schotten), but the load-bearing ancestry for the *claim itself* (that color-difference space has nonzero curvature) sits with Judd (1968) and Silberstein (1938), decades earlier. Flagged first in Entity candidates below, per the brief's instruction.

---

## Central question status

**Does Ahrens et al. 2024 itself state that the CIELAB ab-plane has nonzero Gaussian curvature, obstructing isometric Euclidean embedding?** `[unverified — could not confirm or deny after search]`. The paper's own body text was unreachable across every route attempted this session (Wiley full-text and PDF variants, ResearchGate, no archive snapshot, no preprint), and the one piece of the paper's own text that *was* reachable — its abstract, via a structured metadata API — makes no mention of Gaussian curvature, isometric obstruction, or the ab-plane, framing the paper instead as a machine-learning methodology improving on unnamed prior Euclidization work. The general claim is independently plausible and well-attested in the surrounding literature (Urban et al. 2007 explicitly, and Judd 1968 by title, decades earlier), but none of that evidence comes from Ahrens et al. 2024's own text, so the specific attribution in the topic — to *that* paper — remains unconfirmed rather than verified.

---

## Further leads

- Wiley Online Library returned HTTP 402 (WebFetch) / HTTP 403 (extract_pdf) on two separate CC-BY open-access DOIs this session (10.1002/col.22897 and 10.1111/cgf.70136) — worth flagging as a candidate "known-blocked route" similar to ethw.org, since the license nominally permits access but no tool this session could get past Wiley's gate.
- The Ahrens 2024 companion GitHub repo (https://github.com/julian-ahrens-dfki/mlacse) has trained models and verification scripts for "the six primary scenarios introduced in Section IV" — confirms the paper has a Section IV with numbered scenarios, a possible foothold for locating a mirrored PDF elsewhere later.
- Bujack et al. 2025 (*Computer Graphics Forum*, full text now in hand via the Wikimedia mirror) is itself a substantive, unrelated primary source on non-Riemannian color geometry (formalizing Schrödinger's hue/saturation/lightness definitions) — likely overlaps with [[claim-perceptual-color-space-not-riemannian-bujack-2022]] and could be its own capture in a later run.
- M. Zeyen, T. Post, H. Hagen, J. Ahrens, D. Rogers, R. Bujack, "Color Interpolation for Non-Euclidean Color Spaces" (IEEE SciVis 2018 short paper) links a J. Ahrens (possibly Julian Ahrens) to the Bujack/LANL non-Euclidean-color research line predating the 2024 DFKI paper — worth checking whether this is the same Ahrens and whether it's an earlier, more accessible statement of the curvature claim.
- Judd, D.B., "Ideal color space: Curvature of color space and its implications for industrial color tolerances," Palette 29 (1968), 4–25 — not yet located or fetched this session; would be the highest-value single next read for the historical-priority side of this question.
- Silberstein, L., "Investigations on the intrinsic properties of the color domain," JOSA 28, 3 (1938), 63–85 — same status as Judd 1968, not yet located.

---

## Entity candidates

- Deane B. Judd — person — 1968 paper title states "Curvature of color space" outright; the clearest foundational precedent the modern Gaussian-curvature claim traces back to, and the one this capture almost under-flagged per the known blind spot.
- Ludwik Silberstein — person — 1938 JOSA paper on intrinsic properties of the color domain, cited by Bujack et al. 2025 as establishing that color space is not flat; earlier still than Judd.
- Philipp Urban — person — lead author of the 2007 JOSA A paper that first (among sources found this session) explicitly frames CIELAB-based color-difference spaces' nonzero Gaussian curvature as an isometric-embedding obstruction; the direct precedent Ahrens et al. 2024 likely measures itself against.
- Roy S. Berns — person — co-author of Urban et al. 2007 (Munsell Color Science Laboratory, RIT); recurring figure in Euclidean-color-space construction literature.
- Mitchell R. Rosen — person — co-author of Urban et al. 2007.
- Dierk Schleicher — person — co-author of Urban etal. 2007.
- Lia Ahrens — person — first/corresponding author of the 2024 DFKI paper under investigation; paper's own text unreachable this session.
- Julian Ahrens — person — co-author, DFKI; also appears on a 2018 Zeyen et al. non-Euclidean color-interpolation paper alongside Bujack — possible earlier link to the same research thread.
- Hans D. Schotten — person — third co-author of the 2024 DFKI paper, DFKI/RPTU Kaiserslautern.
- Roxana Bujack — person — leads a distinct (LANL) non-Riemannian color-geometry research line; her 2025 CGF paper doesn't cite Ahrens 2024 but shares the same broader subject and cites Judd/Silberstein.
- CIELAB ab-plane — concept — the 2D chroma cross-section (a*, b* at fixed L*) of CIELAB whose curvature is the specific object of the topic's claim.
- Gaussian curvature — concept — differential-geometry invariant; nonzero values are the formal obstruction to isometric flattening (Theorema Egregium).
- Isometric embedding — concept — a distance-preserving map between metric spaces; central to why "Euclidization" is only ever approximate for CIELAB-based metrics.
- Color space Euclidization / Euclidisation — concept — the general task (per Ahrens et al.'s own framing) of finding a near-isometric transform from a non-Euclidean color-difference space into a Euclidean vector space.
- CIEDE2000 (ΔE2000) — concept — the CIELAB-based color-difference formula used as Ahrens et al.'s reference distance law and as one of the three formulas in Urban et al. 2007.