edge_over_cube_vertices_eq_face_over_face_vertices
plain-language theorem explainer
The arithmetic identity 12/8 = 6/4 confirms consistency of the cross-level ratio for a 3-cube in the derivation of the lepton mass correction Δ(D) = D/2. Researchers deriving first-principles lepton generations from cube geometry would cite it to anchor the μ→τ step without empirical fitting. The proof is a direct numerical normalization that verifies the equality of edge-to-vertex and face-to-face-vertex ratios.
Claim. In three-dimensional cube geometry, the ratio of total edges to total vertices equals the ratio of faces to vertices per face: $12/8 = 6/4$.
background
The module derives the dimension-dependent correction Δ(D) = D/2 from cube geometry without calibration to observed masses. For a D-cube the face count is F = 2D and each (D-1)-face has V_facet = 2^{D-1} vertices, so the facet-mediated correction is Δ = F / V_facet. The upstream definition V(D) := 2^D supplies the vertex count of the full D-cube; the same counting yields 8 vertices and 12 edges for D = 3, while each square face has 4 vertices and there are 6 faces.
proof idea
One-line wrapper that applies norm_num to reduce both sides of the equality to the common value 3/2.
why it matters
It closes the consistency check in the μ→τ derivation chain by showing that the edge-over-vertices ratio 12/8 coincides with the face-over-face-vertices ratio 6/4 at physical D = 3. The parent result is the complete derivation theorem that forces Δ(3) = 3/2 from geometry alone, feeding the downstream delta_D3_derived and deltaStructural_D3 siblings. The module summary explicitly records this identity as the arithmetic anchor for the eight-tick octave and D = 3 landmarks.
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