three_colors_from_D3
For three spatial dimensions the number of color charges equals three. Researchers deriving the SU(3) structure of QCD from Recognition Science would cite this result to link the dimension-forcing argument to the observed quark color multiplicity. The proof is a one-line wrapper that unfolds N_colors to the face-pair count and reduces by reflexivity.
claimIn three spatial dimensions the number of color charges equals three: $N_c(3)=3$.
background
N_colors(D) is defined as the number of cube face-pairs for dimension D; each pair of opposite faces supplies one independent color charge in the ledger. The module sets this inside the P-007 derivation: DimensionForcing supplies D=3 from the eight-tick octave and spinor structure, after which the D-cube combinatorics directly yields three colors. This ledger identification is the same mechanism used for particle generations, so the color count inherits the same D dependence.
proof idea
The tactic proof unfolds N_colors to face_pairs and then applies reflexivity, reducing the goal to the definitional equality face_pairs 3 = 3.
why it matters in Recognition Science
The declaration discharges the P-007 registry item, showing that the same dimension-forcing step (T8) that fixes D=3 also fixes the gauge-group rank of QCD at three. It therefore supplies the color multiplicity that matches the three-generation structure already obtained from the same ledger face-pair counting. No downstream theorems are recorded yet, leaving the result as a direct bridge between the forcing chain and the standard-model gauge content.
scope and limits
- Does not derive the full SU(3) Yang-Mills dynamics or coupling running.
- Does not address quark confinement or hadron spectroscopy.
- Does not compute the numerical value of the strong fine-structure constant.
- Does not extend the color count to non-cubic lattices or higher-dimensional embeddings.
formal statement (Lean)
35theorem three_colors_from_D3 : N_colors 3 = 3 := by
proof body
Tactic-mode proof.
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. -/