three_colors_forced
The declaration establishes that the number of quark color charges equals three when the spatial dimension takes its forced physical value of three. Researchers deriving Standard Model gauge groups from the Recognition Science dimension chain would cite this to link the eight-tick octave directly to SU(3) color. The proof is a one-line term reduction that unfolds the definitions of N_colors and face_pairs then applies reflexivity.
claim$N_ {colors}(D_{physical}) = 3$, where $N_{colors}(D)$ counts the pairs of opposite faces on a $D$-dimensional cube and $D_{physical} = 3$.
background
The module formalizes P-007 by identifying color charges with the combinatorial structure of a three-dimensional cube. N_colors(D) is defined to equal the number of face pairs, which itself equals D. DimensionForcing supplies the physical dimension as the constant 3, obtained from the eight-tick octave and spinor constraints.
proof idea
The term proof unfolds N_colors, DimensionForcing.D_physical, and face_pairs, then closes immediately by reflexivity because face_pairs 3 evaluates to 3.
why it matters in Recognition Science
This result resolves P-007 by showing that the SU(3) color rank follows from the same D = 3 step (T8) that fixes spatial dimensions. It sits downstream of DimensionForcing and the ledger factorization, completing the structural derivation of three colors without additional hypotheses. No open questions remain inside the module.
scope and limits
- Does not derive the strong-interaction Lagrangian or its dynamics.
- Does not compute the strong coupling constant or its running.
- Does not address quark confinement or asymptotic freedom.
- Does not extend the color count to non-physical dimensions.
formal statement (Lean)
49theorem three_colors_forced :
50 N_colors DimensionForcing.D_physical = 3 := by
proof body
Term-mode proof.
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). -/