Foundation Quark Colors N Colors

Color count

In particle physics, quarks carry a property called color charge, which comes in three values conventionally labeled red, green, and blue. The theory of the strong force, quantum chromodynamics (QCD), treats these three colors as the fundamental representation of the symmetry group SU(3). The number three is not arbitrary in the standard model; it is a measured feature of nature, confirmed by processes such as the decay rate of the neutral pion and the ratio of hadron production in electron-positron collisions.

The ledger, a discrete record of recognition events, is the central object in the Recognition Science framework. The framework's library, a machine-checked collection of formal theorems, defines a function N_colors that counts color charges from a spatial dimension D. The definition is simple: N_colors D equals the number of opposite face-pairs of a D-dimensional cube, which is just D itself. For the physical case of three spatial dimensions, the library proves N_colors 3 = 3, and it also proves that N_colors 3 is neither 2 nor 4.

In Recognition Science, the number of colors is not an independent input. The framework derives the three spatial dimensions from a chain of theorems about recognition cost and linking, and then identifies each spatial axis with one color charge. The cube's three pairs of opposite faces correspond to the three color values. This identification is a definitional choice within the framework, not a theorem about quantum chromodynamics.

The library's theorems are conditional on the framework's own derivation of three dimensions. The declaration N_colors itself only defines a counting function; the claim that the physical color count equals three depends entirely on the prior claim that physical space is three-dimensional. The framework does not derive the gauge group SU(3) or any dynamical property of the strong force; it only provides a geometric count that matches the known number of color charges.

What the declaration does not claim is just as important as what it proves. It does not claim that the framework explains confinement, asymptotic freedom, or any other dynamical feature of QCD. It does not claim that the colors red, green, and blue have any physical reality beyond being labels. It does not claim that the identification of spatial axes with color charges is unique or forced by the mathematics alone; that identification is a modeling step, not a theorem.

/-- For D = 3, there are exactly 3 color charges. -/
theorem three_colors_from_D3 : N_colors 3 = 3 := by
  unfold N_colors face_pairs
  rfl

/-- For D = 3, we cannot have 2 or 4 colors. -/
theorem not_two_colors : N_colors 3 ≠ 2 := by norm_num [N_colors, face_pairs]

theorem not_four_colors : N_colors 3 ≠ 4 := by norm_num [N_colors, face_pairs]

/-- Number of color charges = number of cube face-pairs = D.
    In the ledger, each independent "axis" of the D-cube carries one color. -/
def N_colors (D : ℕ) : ℕ := face_pairs D

/-- N_colors D = D (by definition of face_pairs). -/
theorem N_colors_eq_dim (D : ℕ) : N_colors D = D := rfl

/-- **P-007 Resolution**: Three colors follow from D = 3.

    In the RS framework:
    1. DimensionForcing proves D = 3 (linking, 8-tick, spinors).
    2. The D-cube has D pairs of opposite faces (face_pairs D = D).
    3. Ledger face identification assigns one color per face-pair.
    4. Thus N_c = 3.

    This matches SU(3) color in QCD. The gauge group rank is forced
    by the same dimension argument that gives 3 generations. -/
theorem three_colors_forced :
    N_colors DimensionForcing.D_physical = 3 := by
  unfold N_colors DimensionForcing.D_physical face_pairs
  rfl

What this page does not claim

The framework does not derive SU(3) as a gauge group or any dynamical property of the strong force. The color labels red, green, and blue are not claimed to have physical reality beyond being names. The identification of spatial axes with color charges is not claimed to be unique or mathematically forced.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/QuarkColors.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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