IndisputableMonolith.Foundation.QuarkColors

   1import Mathlib
   2import IndisputableMonolith.Foundation.DimensionForcing
   3import IndisputableMonolith.Foundation.ParticleGenerations
   4
   5/-!
   6# P-007: Why Quarks Come in Three Colors
   7
   8Formalizes the RS derivation of N_c = 3 (SU(3) color).
   9
  10## Registry Item
  11- P-007: Why do quarks come in three colors?
  12
  13## RS Derivation
  14D = 3 spatial dimensions (from DimensionForcing) → ledger has three independent
  15"charge directions" via cube face identification. Each spatial axis corresponds
  16to one color charge. Thus N_c = 3.
  17
  18The cube has 3 pairs of opposite faces; the color structure identifies with
  19the same ledger combinatorics as generations (D face-pairs = D colors).
  20-/
  21
  22namespace IndisputableMonolith
  23namespace Foundation
  24namespace QuarkColors
  25
  26open DimensionForcing ParticleGenerations
  27
  28/-! ## Color Count from Dimension -/
  29
  30/-- Number of color charges = number of cube face-pairs = D.
  31    In the ledger, each independent "axis" of the D-cube carries one color. -/
  32def N_colors (D : ℕ) : ℕ := face_pairs D
  33
  34/-- For D = 3, there are exactly 3 color charges. -/
  35theorem three_colors_from_D3 : N_colors 3 = 3 := by
  36  unfold N_colors face_pairs
  37  rfl
  38
  39/-- **P-007 Resolution**: Three colors follow from D = 3.
  40
  41    In the RS framework:
  42    1. DimensionForcing proves D = 3 (linking, 8-tick, spinors).
  43    2. The D-cube has D pairs of opposite faces (face_pairs D = D).
  44    3. Ledger face identification assigns one color per face-pair.
  45    4. Thus N_c = 3.
  46
  47    This matches SU(3) color in QCD. The gauge group rank is forced
  48    by the same dimension argument that gives 3 generations. -/
  49theorem three_colors_forced :
  50    N_colors DimensionForcing.D_physical = 3 := by
  51  unfold N_colors DimensionForcing.D_physical face_pairs
  52  rfl
  53
  54/-! ## Structural Theorems -/
  55
  56/-- N_colors D = D (by definition of face_pairs). -/
  57theorem N_colors_eq_dim (D : ℕ) : N_colors D = D := rfl
  58
  59/-- For D = 3, we cannot have 2 or 4 colors. -/
  60theorem not_two_colors : N_colors 3 ≠ 2 := by norm_num [N_colors, face_pairs]
  61theorem not_four_colors : N_colors 3 ≠ 4 := by norm_num [N_colors, face_pairs]
  62
  63end QuarkColors
  64end Foundation
  65end IndisputableMonolith
  66