N_colors
N_colors sets the color charge count to the number of opposite face pairs on the D-cube. Gauge theorists tracing SM structure to cube geometry cite this for the N_c = 3 result at D = 3. The implementation is a direct alias to face_pairs.
claimThe number of color charges equals the number of opposite face pairs on a $D$-dimensional cube, hence $N_c(D) = D$.
background
Recognition Science forces D = 3 spatial dimensions via the unified forcing chain. The face_pairs combinator counts the D pairs of opposite faces on the D-cube; each axis corresponds to one color charge in the ledger. Module P-007 uses this to derive three colors from the same geometry that yields three generations.
proof idea
One-line definition delegating directly to face_pairs D.
why it matters in Recognition Science
This supplies N_colors 3 = 3 for the gauge_group_certificate theorem, which extracts SU(3) from the S3 factor of the 3-cube automorphism group. It completes the color sector of the gauge-generation unification. The framework landmark is the identification of color with axis permutations in D = 3.
scope and limits
- Does not establish the full Lie algebra structure of SU(3).
- Does not connect to quark confinement mechanisms.
- Does not derive the dimension D from first principles.
formal statement (Lean)
32def N_colors (D : ℕ) : ℕ := face_pairs D
proof body
Definition body.
33
34/-- For D = 3, there are exactly 3 color charges. -/