Q3_faces
The declaration supplies the integer constant 6 as the face count of the three-cube Q₃. Researchers deriving λ_rec from curvature cost or enumerating gauge factors and generations cite this value to normalize the 2λ² term in the non-circular Gauss-Bonnet setup. The assignment is a direct constant definition with no reduction steps.
claimThe three-dimensional cube Q₃ has exactly six faces.
background
The Lambda_rec Derivation module derives the recognition length λ_rec from normalized bit cost (=1) and curvature cost (=2λ²) obtained via Q₃ Gauss-Bonnet normalization on the bounding sphere, without presupposing Newton's G. Q₃ is the three-cube whose geometry encodes the gauge structure: its six faces correspond to the sector dimensions 3+2+1 of SU(3)×SU(2)×U(1) and induce three particle generations through face-pair counting. Upstream structure results establish that Q₃ simultaneously forces the gauge content, exactly three generations, and 24 chiral fermion flavors (=D×2^D).
proof idea
This is a direct definition that assigns the integer 6, matching the combinatorial face count of the 3-cube and the Euler characteristic of S².
why it matters in Recognition Science
The constant supplies the cube_faces field required by AlphaFrameworkCert and enters the definitions of face-wallpaper pairs and the sign of the curvature correction δ₁. It supports the non-circular derivation of G := π λ_rec² c³ / ℏ and the spectral emergence of the Standard Model gauge group from Q₃ geometry, consistent with D=3 in the forcing chain.
scope and limits
- Does not derive the face count from the T0-T8 forcing chain.
- Does not embed Q₃ into the recognition lattice or phi-ladder.
- Does not generalize the count to hypercubes of dimension other than 3.
formal statement (Lean)
147def Q3_faces : ℕ := 6
proof body
Definition body.
148
149/-- Euler characteristic of S² (bounding sphere). -/