SieveColorSpace

Theory

From primes to color

The Sieve of Eratosthenes, starting from s = 1/2, produces a geometric distribution on prime gaps. This distribution has a unique fixed point at mu* = 15, where exactly 3 primes ({3, 5, 7}) are “active” — they dominate the sieve dynamics.

These 3 primes become the 3 chromatic channels. The prime p = 2 becomes the luminance channel.

Nothing is chosen. Everything is computed.

The simplex

A color’s chromaticity lives on the 2-simplex:

pi = (pi_3, pi_5, pi_7)    with    pi_3 + pi_5 + pi_7 = 1

The coordinates are the gamma-weighted LMS cone responses, normalized:

w_i = gamma_i * LMS_i
pi_i = w_i / (w_1 + w_2 + w_3)

where gamma_p are the effective dimensions derived from the sieve:

gamma_3 = 0.808    (L-cone channel)
gamma_5 = 0.696    (M-cone channel)
gamma_7 = 0.595    (S-cone channel)

The conservation law

On the simplex, two quantities are defined:

• Saturation S = D_KL(pi

  uniform) — how far from achromatic

• Luminance entropy L = H(pi) — Shannon entropy of the simplex point

The sum rule states:

S + L = log 3    (exactly, in nats)

This means: the total informational budget is fixed at log 3 ≈ 1.0986 nats. If saturation increases, chromatic entropy decreases by exactly the same amount. This is a generic mathematical identity on any three-outcome probability simplex with uniform reference, not a heuristic. What is SCS- specific is the choice of three active primes {3, 5, 7} from the sieve, and the physical reading of S as saturation and L as chromatic entropy.

The metric

The natural metric on the simplex is the Fisher information metric, weighted by gamma_p. For two colors with coordinates (ell_1, pi_1) and (ell_2, pi_2):

dE^2 = (3/4) * d_lum^2 + (1/4) * d_chrom^2

where:

• d_lum = 2|arcsin(sqrt(ell_1)) - arcsin(sqrt(ell_2))| (Fisher distance on the Bernoulli manifold, p=2 channel)

• d_chrom = 2 * arccos(sum_i sqrt(pi_1_i * pi_2_i)) (Bhattacharyya distance on the simplex)

• Weights 3/4 and 1/4 come from N/(N+1) with N = 3 active primes

The metric is unique (Cencov’s theorem): it is the only Riemannian metric on statistical manifolds that is invariant under sufficient statistics.

Derivation chain

s = 1/2                              (unique input, T0)
  -> geometric distribution           (Cauchy + max-entropy, L0)
  -> holonomy angles sin^2(theta_p)   (T6)
  -> fixed point mu* = 15             (T7)
  -> active primes {3, 5, 7}          (N = 3)
  -> effective dimensions gamma_p     (T7 at mu*)
  -> Fisher metric on simplex         (Cencov uniqueness theorem)
  -> Bhattacharyya geodesic on D^2    (geodesic of Fisher metric)
  -> 3/4 luminance + 1/4 chroma       (N/(N+1) balance)
  -> dE_SCS                          (color difference formula)

Zero fitted parameters at every step.

Reference

Y. Senez, “The Sieve Color Space: A First-Principles Color Space from the Sieve of Eratosthenes” (2026). See article/ directory.

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