sixJ_eq_cube_faces
The theorem states that the 6j-symbol dimension equals 6, aligning with the number of faces on a cube for three spatial dimensions in the Recognition Science spin foam model. Researchers in quantum gravity path integrals would reference this result to confirm the dimensional structure of the Freudenthal triangulation. The proof is a direct reflexivity step relying on the constant definition of the dimension.
claimThe dimension of the 6j-symbol is equal to 6.
background
Spin foam models in Recognition Science arise from the Freudenthal triangulation of the recognition lattice, with five canonical models corresponding to configDim D=5. The 6j-symbol enters the fundamental amplitude, and its dimension is set to 6, which equals 2D at D=3 and counts the faces of a cube. The upstream definition fixes the 6j-symbol dimension as the natural number 6, providing the value for this equality.
proof idea
The proof applies reflexivity directly to the definition of the 6j-symbol dimension, which is already set to 6.
why it matters in Recognition Science
This supplies the sixJ_faces field required by the spinFoamCert definition that certifies the overall spin foam model. It realizes the framework landmark fixing D=3 spatial dimensions with the relation 6 = 2D = cube faces. The declaration closes a basic consistency requirement in the quantum gravity formulation.
scope and limits
- Does not derive the 6j dimension from the recognition composition law.
- Does not compute spin foam amplitudes explicitly.
- Does not address models beyond the five canonical ones.
- Does not link to physical constants such as the fine structure constant.
Lean usage
have h : sixJDimension = 6 := sixJ_eq_cube_facesformal statement (Lean)
31theorem sixJ_eq_cube_faces : sixJDimension = 6 := rfl