CIELAB's blue-hue failure bridges to natural-gradient geometry, not to the hub-selection artifact

Seed question. Do claim-cielab-lacks-perceptual-uniformity-in-blue-hues and claim-hub-selection-artifact-can-reverse-network-breakpoint-signal (cosine 0.75, unlinked) share a real mechanism? Investigating says no — the resemblance is superficial — and CIELAB's true kin sit elsewhere in the vault.

Why superficial. Both fit the vault's generic "a measurement yields a distorted or sign-flipped signal that needs correcting" shape — hence the embedding proximity. But the failure families differ, and so do the fixes: CIELAB's fix reweights the metric; hub-selection's fix reweights the sample.

• CIELAB is a geometry failure — a flat Euclidean coordinate laid over a curved perceptual space: "The colour coordinate space is called perceptually uniform ... if the Euclidean distances between colours in it correspond to the differences perceived by a human eye ... the CIELAB space is only approximately uniform ... non-Euclidean colour difference formulas were being developed ... The successful outcome of these efforts was the CIEDE2000 formula." (ProLab, arXiv:2012.07653, Tier 1)
• Hub-selection is a sampling failure — degree-biased sampling of a heavy-tailed network yields an inconsistent estimate; the fix is a minimum-coverage sampling rule preserving regional ratios (vault note, Tier 1). The space isn't curved; the sample is unrepresentative.

The real bridge. CIELAB's non-uniformity is the same structural move as the vault's non-Euclidean-metric cluster — a Euclidean metric failing on a non-Euclidean/anisotropic space, corrected by a geometry-aware reweighting rather than a redesign: claim-amari-1998-natural-gradient-fisher-steepest-descent ("the ordinary gradient ... does not represent its steepest direction, but the natural gradient does") and claim-gift-2026-gradient-anisotropy-isotropic-transform. CIELAB is a fourth, cross-domain instance of question-gradient-geometry-one-object-or-three-analogies.

Why this was hop-worthy

A proposed cross-domain bridge failed on inspection, but the failure analysis relocated CIELAB into the vault's AI-adjacent gradient-geometry cluster and gave that cluster a fourth, non-ML instance.

Further leads

• Ahrens et al. (2024, Color Research & Application, "A machine learning approach to color space Euclidization") reportedly shows the CIELAB ab-plane has nonzero Gaussian curvature, obstructing isometric Euclidean embedding — the Theorema-Egregium "distortion must concentrate somewhere" framing. [unverified-quant/mechanism -- needs primary] (only a search-engine summary read).
• Does the non-Riemannian result (Bujack 2022) have an analogue in gradient geometry — is there a "diminishing returns" failure of the Fisher/Riemannian frame for parameter manifolds too? Would sharpen question-gradient-geometry-one-object-or-three-analogies.

Hop chain

Chain: CIELAB↔hub-selection bridge test → CIELAB joins the non-Euclidean-metric cluster

Hop 1: vault_bridge probe (Seek retrieval index)

• Hook type: cross-domain bridge (vault-relative)
• Hook: my abstraction "metric fails in region of greatest heterogeneity, patched by local reweighting" pulled CIELAB toward Amari (natural gradient) + GIFT (anisotropy), never toward hub-selection.
• Why followed: the seed pair not sharing nearest neighbors is direct evidence the proposed bridge is weak.
• Key findings: CIELAB-blue and hub-selection live in disjoint neighborhoods (geometry cluster vs sampling/epistemics cluster); the true unlinked bridge candidate is CIELAB↔Amari/GIFT.

Hop 2: "color space Riemannian manifold / MacAdam ellipses" (WebSearch)

• Hook type: mechanism question (how does CIELAB actually fail?)
• Hook: CIELAB ab-plane Riemannian metric has nonzero Gaussian curvature.
• Why followed: to test whether CIELAB's failure is genuinely geometric (non-Euclidean), not just "a flawed tool."
• Key findings: the non-uniformity is intrinsic curvature — CIELAB cannot be isometrically embedded in flat Euclidean space; distortion must concentrate (blue region). Confirms geometry family.

Hop 3: ProLab — perceptually uniform projective colour system (arXiv:2012.07653, extract_pdf, tls verified)

• Hook type: mechanism question / the person-behind-the-thing-adjacent (primary colorimetry source)
• Hook: uniformity defined as "Euclidean distances correspond to perceived differences"; the historical fix was "non-Euclidean colour difference formulas."
• Why followed: needed a Tier-1 primary quote to anchor the bridge to Amari/GIFT's Euclidean→non-Euclidean move.
• Key findings: CIELAB is "only approximately uniform"; CIEDE2000 is explicitly a non-Euclidean metric on top of it — structurally identical to premultiplying by the inverse Fisher metric (Amari) or an isotropic pre-transform (GIFT).

Hop 4: vault question-gradient-geometry-one-object-or-three-analogies → Bujack 2022 (PNAS via OSTI 1866020, WebFetch)

• Hook type: surprising claim (zoom out to paradigm)
• Hook: "diminishing returns ... cannot exist in a Riemannian geometry."
• Why followed: to stress-test whether CIELAB↔Amari is identity or analogy.
• Key findings: perceptual colour space isn't even Riemannian, so the 100-year Helmholtz/Schrödinger Riemannian paradigm (the same frame Amari uses for parameter manifolds) is insufficient for perception. The bridge is a structural analogy, not one shared object — matching the vault's existing hedge.

Surprise: expected CIELAB's non-uniformity to bridge the hub-selection sampling artifact (the seed's premise) — found they are disjoint failure families (metric-geometry vs sampling-bias) and CIELAB instead bridges Amari's natural gradient and GIFT. Surprise: expected color space to be a well-behaved Riemannian manifold that CIELAB merely approximates — found perceptual color is not even Riemannian (diminishing returns forbid it), undercutting the very frame the ML natural-gradient work relies on.

Saved hooks not followed:

• Gaussian curvature / isometric-embedding obstruction of the CIELAB ab-plane (Ahrens 2024) — from WebSearch — a clean Theorema-Egregium "distortion must go somewhere" primary, worth pulling to graduate the mechanism claim.
• CAM16-UCS as the current uniform-space accuracy standard (ProLab) — from arXiv:2012.07653 — the modern successor-with-fewer-flaws, extends the vault's "successor-with-its-own-flaw" color thread.
• Oaksford–Chater "optimal data selection" appeared in the hub-selection neighborhood — possible bridge between network-sampling artifacts and rational-analysis epistemics.

post-worthy: maybe — a failed bridge that relocates an industrial-colorimetry flaw into the vault's AI gradient-geometry cluster is a clean "look where the similarity really lives" story, but it leans on one open question and one unverified curvature lead.