---
title: "Verify against a primary source that the CIELAB ab-plane has nonzero Gaussian curvature, obstructing isometric Euclidean embedding (Ahrens et al. 2024)"
type: "question"
status: "open"
writer_model: "claude-opus-4-8"
date_raised: "2026-07-12T00:00:00.000Z"
tags: ["color-science","cielab","non-euclidean","gaussian-curvature","primary-source-verification","unverified-quant","unverified-mechanism","hop"]
---


Raised while promoting the CIELAB non-Euclidean-bridge capture
(2026-07-11). A "further lead" in that capture reports that Ahrens et al.
(2024, *Color Research & Application*, "A machine learning approach to color
space Euclidization") shows the CIELAB **ab-plane has nonzero Gaussian
curvature**, which would obstruct any isometric embedding into flat Euclidean
space — the Theorema-Egregium "distortion must concentrate somewhere" framing,
and a clean geometric mechanism for *why* CIELAB is only approximately uniform.

The capture explicitly flagged this claim `[unverified-quant/mechanism --
needs primary]`: **only a search-engine summary was read, not the paper.** It
is a specific technical-mechanism / quantitative-geometry claim, which the
sourcing floor puts at Tier 1–2. It therefore did **not** become a claim-note
in this promotion; this question holds the lead.

To close:

1. **Read the Ahrens et al. (2024) primary** (Color Research & Application, or
   its preprint) and confirm the ab-plane is reported to have nonzero Gaussian
   curvature, with the isometric-embedding obstruction stated as such. Record
   the exact quantity/quote.
2. If confirmed, this graduates into a claim-note — the intrinsic-curvature
   mechanism behind
   [[claim-cielab-only-approximately-uniform-fixed-by-noneuclidean-formulas]]
   and a sharper geometric anchor for
   [[observation-cielab-nonuniformity-bridges-gradient-geometry-not-sampling-artifact]].
3. **Read [[entity-philipp-urban|Urban, Rosen, Berns & Schleicher (2007)]]
   directly** (JOSA A 24(6):1516–1528 — added 2026-07-29). A dedicated
   verification capture this date confirmed Ahrens et al. 2024's abstract
   itself says nothing about Gaussian curvature
   ([[claim-ahrens-2024-abstract-omits-gaussian-curvature-claim]]), and its
   full text remains unreachable (Wiley 402/403 across four URL variants
   despite a nominal CC-BY license; no mirror, no preprint). The same session
   found the general claim stated, seventeen years earlier, in Urban et al.
   2007's own abstract — but only through a WebFetch summarization layer, not
   a raw document read (ResearchGate 403, direct PDF 403), so it is recorded
   as [[claim-urban-2007-nonzero-gaussian-curvature-isometric-embedding-obstruction-unverified|a lead, not evidence]].
   Reading Urban et al. 2007 directly may be the more tractable route to
   closing this question than a fifth attempt at the paywalled Ahrens paper,
   since it is the paper Ahrens et al. most likely measures itself against.
4. Historical footnote, not a verification target itself: the same session
   traced the general curvature-of-color-space intuition back through
   Bujack et al. 2025's own bibliography to
   [[claim-judd-1968-silberstein-1938-precede-color-space-curvature-claim|Judd (1968) and Silberstein (1938)]] —
   decades before either Urban or Ahrens. Neither has been read directly.

Related open thread (not a verification, a research question): does the
non-Riemannian result in
[[claim-perceptual-color-space-not-riemannian-bujack-2022]] have an analogue in
gradient geometry — is there a "diminishing returns" failure of the
Fisher/Riemannian frame for parameter manifolds too? That would sharpen
[[question-gradient-geometry-one-object-or-three-analogies]].

Low-to-medium priority — this is an orphan-ish colour-science seed in an
otherwise ML-history-centred vault, worth a pass mainly if the gradient-geometry
cluster keeps growing.