N_colors_eq_dim
plain-language theorem explainer
The equality N_colors D = D holds definitionally for any natural number D because the color charge count is introduced as the face-pair count on the D-cube. Researchers tracing the Recognition Science route from spatial dimensions to SU(3) color would cite the result when equating ledger axes to quark charges. The proof is a one-line reflexivity that follows at once from the definition of N_colors.
Claim. For any natural number $D$, the number of color charges $N_c(D)$ equals $D$, where $N_c(D)$ is defined as the number of face pairs of the $D$-cube.
background
The module derives the number of quark colors from the spatial dimension count. N_colors D is introduced as face_pairs D, which tallies the independent charge directions carried by the axes of the D-cube ledger. The module documentation states that D equals 3 from DimensionForcing, so each spatial axis supplies one color charge and the cube's three opposite-face pairs fix N_c at 3.
proof idea
The proof is a direct reflexivity step. N_colors D expands by definition to face_pairs D, and the asserted equality to D is therefore definitionally true.
why it matters
The theorem completes the count identification inside P-007, confirming that the color charge number equals the spatial dimension. It supplies the numerical link between DimensionForcing and the color structure used in ParticleGenerations. No downstream theorems yet depend on it.
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