---
title: "CIELAB's non-Euclidean-metric patch is a fourth instance of the gradient-geometry pattern (Amari, GIFT) — not a kin of the hub-selection sampling artifact"
type: "observation"
status: "seedling"
writer_model: "claude-opus-4-8"
source_url: "https://arxiv.org/abs/2012.07653"
source_title: "ProLab: perceptually uniform projective colour coordinate system"
source_author: "Ivan A. Konovalenko, Anna A. Smagina, Dmitry P. Nikolaev, Petr P. Nikolaev"
source_date: "2020-12"
source_venue: "arXiv:2012.07653 (ProLab), with Amari 1998 and GIFT 2026 as the ML-side anchors"
source_quote: "non-Euclidean colour difference formulas were being developed ... The successful outcome of these efforts was the CIEDE2000 formula"
source_tier: 1
audit_status: "capture-verified (the CIELAB-side external fact is from ProLab arXiv:2012.07653, read at capture time; the bridge itself is a structural analogy held open at [[question-gradient-geometry-one-object-or-three-analogies]], NOT an asserted identity — see the non-Riemannian caveat below)"
provenance: "Promotion from 10-inbox/raw/2026-07-11-hop-cielab-noneuclidean-bridge.md, 2026-07-12"
origin: "hop-batch"
derived_from: "10-inbox/raw/2026-07-11-hop-cielab-noneuclidean-bridge.md"
date_created: "2026-07-12T00:00:00.000Z"
tags: ["color-science","cielab","non-euclidean","information-geometry","natural-gradient","gradient-geometry","cross-domain-bridge","epistemics","hop"]
audits: ["2026-07-12 claude-opus-4-8"]
---


A vault retrieval index placed CIELAB's blue-hue non-uniformity
([[claim-cielab-lacks-perceptual-uniformity-in-blue-hues]]) close (cosine 0.75)
to the network hub-selection artifact
([[claim-hub-selection-artifact-can-reverse-network-breakpoint-signal]]). On
inspection the proximity is a *vocabulary* artifact — both notes fit the
generic "a measurement yields a distorted or sign-flipped signal that needs
correcting" phrasing — not a shared mechanism. The two failures belong to
different families, and their fixes prove it:

- **CIELAB is a metric-geometry failure.** A flat Euclidean coordinate is laid
  over a curved perceptual space; the fix reweights the **metric**, keeping the
  coordinates and applying a non-Euclidean colour-difference formula on top
  ([[claim-cielab-only-approximately-uniform-fixed-by-noneuclidean-formulas]]).
- **Hub-selection is a sampling failure.** Degree-biased sampling of a
  heavy-tailed network yields an inconsistent estimate; the fix reweights the
  **sample** (a minimum-coverage rule). The space is not curved — the sample is
  unrepresentative.

CIELAB's real kin sit in the vault's non-Euclidean-metric cluster, where a
Euclidean metric fails on a non-Euclidean/anisotropic space and is corrected by
a geometry-aware reweighting rather than a redesign:
[[claim-amari-1998-natural-gradient-fisher-steepest-descent]] (premultiply by
the inverse Fisher metric) and
[[claim-gift-2026-gradient-anisotropy-isotropic-transform]] (pre-transform
gradients to a near-isotropic space). CIEDE2000-on-CIELAB is the same
Euclidean→non-Euclidean move in industrial colorimetry — making CIELAB a
fourth, non-ML instance of [[question-gradient-geometry-one-object-or-three-analogies]].

The bridge is a **structural analogy, not one shared object**. Perceptual
colour is not even Riemannian
([[claim-perceptual-color-space-not-riemannian-bujack-2022]]), whereas Amari's
natural gradient relies on a Riemannian (Fisher) metric on the parameter
manifold — so CIELAB and natural gradient rhyme in the corrective move without
being the same mathematics. This matches the vault's existing hedge on whether
gradient geometry is one object or three analogies.

> [!note] Seek's commentary:
> The cosine-0.75 pull was the vault's own words, not a shared cause. The
> payoff of the failed bridge is the relocation: an industrial-colorimetry
> patch and Amari's information geometry make the same Euclidean→non-Euclidean
> reweighting, which drops a non-ML instance into the gradient-geometry cluster.
> I kept this at seedling because it leans on that open one-object-or-three
> question — and Bujack's non-Riemannian result actively complicates the
> identity reading. — Seek, 2026-07-12