Q3_faces
The three-dimensional cube Q₃ possesses exactly six 2-faces. Derivations extracting the SU(3) sector dimension and three-generation count from forced D=3 cite this identity when assembling the gauge group and fermion state space. The proof is a direct native_decide evaluation of the face-counting function on input 3.
claimThe number of 2-faces of the 3-cube satisfies $F_2(3)=6$.
background
The Spectral Emergence module starts from the forced dimension D=3 (T8) and defines the binary cube Q₃={0,1}³ with 8 vertices. The function F₂ counts the 2-dimensional faces of the n-cube; upstream definitions give equivalent counts such as 2*3 and explicit theorems confirming the value 6 for n=3. This supplies the combinatorial input for deriving SU(3)×SU(2)×U(1) content (dimensions 3+2+1) and three generations from face-pair counting, all inside the Recognition Science self-consistency loop that equates |Aut(Q₃)|=48 with the 48 chiral fermion states.
proof idea
The proof is a one-line wrapper that applies the native_decide tactic to evaluate F₂ 3 directly. It relies on the definitional reduction already established in the imported Constants and CubeSpectrum modules without expanding further lemmas.
why it matters in Recognition Science
This supplies the cube_faces field required by the AlphaFrameworkCert structure that certifies readiness for the δ₂ fine-structure computation. It fills the explicit combinatorial step from T8 (D=3) through the eight-tick octave and φ-ladder to the observed gauge dimensions and three particle generations. The result closes one link in the loop that identifies the cube symmetry group with the fermion state space.
scope and limits
- Does not derive the full gauge group Lie algebra structure.
- Does not compute the automorphism group order of Q₃.
- Does not address the φ-ladder mass formula or J-cost edges.
- Does not prove that D=3 is the unique dimension satisfying all requirements.
formal statement (Lean)
103theorem Q3_faces : F₂ 3 = 6 := by native_decide