HLC Atlas Lambda-Star v4

The HLC Atlas Lambda-Star v4 is a reference-based, atlas-only system for the analysis of real measured reflectance spectra. It does not describe color as a generated output, a stylistic invention, or a subjective impression, but as an existing atlas reference with an explicitly assigned spectrum. Every valid color in the system is defined exclusively by a reference in the form Hxxx_Lxxx_Cxxx. Without this reference, there is no valid color and no valid structural attribute in the system.

The atlas operates exclusively on measured reflectance spectra R(λ) across the visible range from 380 to 730 nm. In the v4 atlas, this spectrum is represented on the atlas wavelength grid and supports the computation of structural attributes such as balance, centroid, spread, and asymmetry.

What the atlas is

The atlas is a reference and analysis system. Its purpose is to reveal structure in existing references: balance, location, spread, and asymmetry of a measured spectrum. It supports explanation, comparison, selection, and data-scientific evaluation of atlas references. It is not intended to invent colors or create a new color space.

The system does not create colors. It reveals structure in existing references.

What the atlas is not

The atlas is not a color generator, not an optimization system, not a mixing tool, and not a recipe system. Color generation, interpolation between atlas samples, averaging into new colors, or semantic color definitions without atlas snap are not allowed. Even palettes are not treated as free lists of names or HEX values; in this system, a palette is always a set of valid atlas references, obtained only through filtering, selection, and sorting within the atlas.

Reference before attribute

Within the atlas, the reference is the primary identity. Values such as Lab, λ*_V2, λ*_EE, Δλ*, μ₂, σ*, μ₃, or γ₁ are descriptive attributes, but they never replace the reference itself. This matters because the atlas does not work with vague “similar to” statements or invented approximations; it works with explicitly identified, reproducible spectral records.

The core logic of the atlas

The atlas deliberately separates identity, method, and structural attributes.

• Identity is always the atlas reference Hxxx_Lxxx_Cxxx.
• Method defines how a quantity is computed, for example via Brent root finding or reflectance-weighted moments.
• Attributes describe the structure of the spectrum, such as balance, centroid, width, or asymmetry.

λ*_V2: — deterministic wavelength-coded reflectance descriptor (legacy symbol retained for continuity)

λ*_V2 is the energetic balance point of a spectrum. It is defined as the unique root of the balance function:

g(λ) = ∫(380..λ)(1 − R(λ')) dλ' − ∫(λ..730) R(λ') dλ'

In the rule set, λ*_V2 is explicitly defined as a root-finding problem and must be computed using Brent root finding. Centroids, means, quantiles, CDF points, or other approximative substitutes are not valid replacements for this quantity.

λ*_EE: the equal-energy centroid

λ*_EE is the reflectance-weighted centroid of the spectrum. It does not describe the balance point; it describes the mean location of reflected energy:

λ*_EE = Σ(λ_i · R_i) / Σ(R_i)

This makes λ*_EE fundamentally different from λ*_V2. The two quantities are not interchangeable. λ*_V2 describes a balance point, while λ*_EE describes a centroid.

Δλ*: the structural difference

Δλ* is defined as:

Δλ* = λ*_V2 − λ*_EE

This difference shows how the balance point and the centroid are displaced relative to each other.

• Δλ* > 0 indicates a structural pull toward higher wavelengths.
• Δλ* < 0 indicates a structural pull toward lower wavelengths.
• Δλ* ≈ 0 indicates a balanced relationship between root and centroid.

In the atlas logic, Δλ* is the primary asymmetry indicator and takes priority over Δμ₂ and Δσ* in review, drift, and release workflows.

μ₂: the second central moment

μ₂ is the second central moment and describes the spectral spread around λ*_EE:

μ₂ = Σ(R_i · (λ_i − λ*_EE)^2) / Σ(R_i)

It measures how broadly the spectral mass is distributed around the chosen center.

• A small μ₂ indicates a more compact structure.
• A large μ₂ indicates a broader distribution.

Unit: nm²

σ*: the readable width scale

σ* is the square root of μ₂:

σ* = √μ₂

It carries the same structural information as μ₂, but in a more interpretable scale expressed in nm. While μ₂ is the mathematically fundamental moment, σ* is often the more intuitive measure of spectral width.

Unit: nm

μ₃: directed asymmetry

μ₃ is the third central moment:

μ₃ = Σ(R_i · (λ_i − λ*_EE)^3) / Σ(R_i)

It indicates the direction of the spectral tail:

• μ₃ > 0 means a longer tail toward higher wavelengths.
• μ₃ < 0 means a longer tail toward lower wavelengths.
• μ₃ ≈ 0 suggests an approximately symmetric structure.

Unit: nm³

γ₁: normalized skewness

γ₁ is the dimensionless normalized skewness:

γ₁ = μ₃ / σ*³

It separates simple width from actual asymmetry more effectively.

• γ₁ ≈ 0 means approximately symmetric.
• γ₁ < 0 means skewness toward lower wavelengths.
• γ₁ > 0 means skewness toward higher wavelengths.

In the atlas logic, γ₁ is optional; the primary asymmetry indicator remains Δλ*.

How these quantities work together

• λ*_EE answers: Where is the spectrum located on average?
• μ₂ and σ* answer: How broadly is it distributed around that center?
• μ₃ and γ₁ answer: To which side does the tail extend, and how strong is that asymmetry?
• λ*_V2 answers: Where is the energetic balance point of the spectrum?
• Δλ* reveals how far balance point and centroid are separated.

Discrete atlas computation

For the atlas wavelength grid 380, 390, …, 730 nm, the discrete forms of λ*_EE, μ₂, σ*, and μ₃ are explicitly allowed. These moment-based quantities are computed directly from the measured atlas samples. What is essential, however, is that these discrete moment formulas are not a substitute for the Brent-based computation of λ*_V2. The atlas makes a strict distinction between centroid/moment quantities and root-defined quantities.

Validity and methodological rigor

An atlas result is only valid if a valid Hxxx_Lxxx_Cxxx reference exists, the underlying spectrum genuinely belongs to the atlas, λ*_V2 has been computed via Brent, and no approximative replacement definition of λ* has been used. A missing reference means STOP. A false λ* definition leads to INVALID.

What is new in v4

The file atlas_v4_recomputed.json keeps the old atlas logic and the new v4 logic methodologically separate. It preserves the legacy origin of older atlas values and adds explicit v4 fields for:

• lambda_v2_brent
• lambda_ee_struct
• delta_lambda_struct
• mu2_struct
• sigma_struct
• mu3_struct
• gamma1_struct

At the same time, the method section still makes clear that the legacy values in atlas.json were based on a piecewise-linear root solution, while v4 describes λ*_V2 explicitly through Brent.

Practical meaning

The atlas does not answer the question: “How do I create this color?”

It answers the question: “Where and why is this measured spectrum structurally stable, broad, asymmetric, or sensitive?”

That is its value for research, teaching, technical documentation, and industrial use: it makes real spectral structure transparent, reproducible, and comparable.