Verify against a primary source that the CIELAB ab-plane has nonzero Gaussian curvature, obstructing isometric Euclidean embedding (Ahrens et al. 2024)

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 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 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 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.