vertex_edge_slots
The definition sets the total vertex-edge incidence slots in a 3-cube to twice the hypercube edge count. CKM and PMNS mixing derivations cite this integer to obtain the factor 24 that fixes V_cb as 1/24 and links V_ub to fine-structure leakage. The definition is a direct one-line application of the upstream cube_edges formula at dimension 3.
claimLet $S$ denote the total number of vertex-edge incidence slots in the 3-cube. Then $S = 2d2^{d-1}$ evaluated at $d=3$, which equals 24.
background
The module supplies the geometric foundation for mixing matrices by imposing cubic voxel topology constraints on the CKM and PMNS parameters. The upstream cube_edges(d) counts the edges of a d-dimensional hypercube via the formula d * 2^(d-1). For d=3 this produces 12 edges; doubling yields the incidence slots because each edge meets two vertices.
proof idea
One-line definition that applies cube_edges to dimension 3 and multiplies the result by 2.
why it matters in Recognition Science
The count supplies the denominator 24 used in downstream theorems that set V_cb_pred = 1/vertex_edge_slots and V_ub_pred = fine_structure_leakage. It thereby anchors the Recognition Science claim that cubic topology forces the observed mixing angles and the Cabibbo correction coefficient. The construction sits inside the D=3 step of the forcing chain and the RCL-derived geometry for particle generations.
scope and limits
- Does not prove that vertex-edge slots correspond to any specific quantum field operator.
- Does not derive the cube model from the Recognition Composition Law.
- Does not address higher-dimensional or non-cubic topologies for mixing matrices.
Lean usage
have hslots : (vertex_edge_slots : ℝ) = 24 := by unfold vertex_edge_slots cube_edges; norm_numformal statement (Lean)
20def vertex_edge_slots : ℕ := 2 * cube_edges 3
proof body
Definition body.
21