---
title: "Urban, Rosen, Berns & Schleicher (2007) is reported to state that non-Euclidean CIELAB-based color-difference spaces have regions of nonzero Gaussian curvature obstructing isometric embedding — not yet quote-verified against the primary"
type: "claim"
status: "seedling"
source_url: "https://opg.optica.org/josaa/abstract.cfm?uri=josaa-24-6-1516"
source_title: "Embedding non-Euclidean color spaces into Euclidean color spaces with minimal isometric disagreement"
source_author: "Philipp Urban, Mitchell R. Rosen, Roy S. Berns, Dierk Schleicher"
source_date: 2007
source_quote: "[unverified-quote — needs direct read] Reported wording, obtained only via WebFetch's summarization layer against the publisher's abstract page, corroborated across three independent WebFetch/WebSearch passes with consistent phrasing but never a raw document read: isometric embedding is investigated 'owing to regions of nonzero Gaussian curvature within common non-Euclidean color spaces.'"
source_tier: 3
audit_status: "flagged (unverified-mechanism, unverified-quote — the venue (JOSA A) is Tier 1, but what this session actually obtained was a WebFetch fetch-and-summarize rendering, not a direct extract_pdf/Read of the document; per sources.md's quote-provenance rule that rendering is not an admissible source_quote no matter how consistently it reproduces. Recorded here as a lead, not evidence. Routed to [[question-verify-cielab-ab-plane-gaussian-curvature]].)"
provenance: "Promotion from 10-inbox/raw/2026-07-28-verify-against-a-primary-source-that-the-cielab.md, 2026-07-29"
origin: "batch"
derived_from: "10-inbox/raw/2026-07-28-verify-against-a-primary-source-that-the-cielab.md"
date_created: "2026-07-29T00:00:00.000Z"
writer_model: "claude-sonnet-5"
tags: ["cielab","gaussian-curvature","urban-2007","isometric-embedding","ciede2000","unverified-mechanism","unverified-quote","color-space"]
audits: ["2026-07-31 claude-fable-5"]
---


[[entity-philipp-urban|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). The paper examines three CIELAB-based color-difference formulas — CMC, CIE94, and CIEDE2000 — as non-Euclidean metric spaces. Multiple independent search-and-fetch passes on the publisher's own abstract page (opg.optica.org) consistently returned language to the effect that isometric embedding is investigated "owing to regions of nonzero Gaussian curvature within common non-Euclidean color spaces." Both attempts to reach the paper's actual PDF text (via `extract_pdf`, including a ResearchGate-hosted mirror) returned HTTP 403.

This paper is very likely the direct technical antecedent that [[claim-ahrens-2024-abstract-omits-gaussian-curvature-claim|Ahrens et al. 2024]] implicitly measures itself against — Ahrens's own abstract claims improvement "over previous contributions" on the same CIEDE2000-anchored Euclidization task, seventeen years later. But that inference, and the Gaussian-curvature wording itself, rest only on a WebFetch summarization layer this session, never a raw document read. The vault's quote-provenance rule treats a summarizing layer's output as inadmissible for `source_quote` regardless of how consistent or well-formed it reads — a real prior failure mode (fabricated-but-plausible quotes from exactly this kind of tool) means consistency across passes is not the same as verification.

> [!note] Seek's commentary:
> This is the shape the vault keeps running into: a claim that's almost certainly true, corroborated three separate times by a tool that has a documented history of confabulating exactly this kind of internally-coherent, on-topic text. I'm keeping this at seedling rather than dropping it, because "probably true and worth a targeted primary read" is a different epistemic state than "unsourced" — but it stays a lead, not evidence, until someone gets a raw PDF or the ResearchGate 403 clears.