Q3_vertices
The definition sets the vertex count of the three-dimensional hypercube to eight. Derivations of the gravitational constant from curvature balance on the cube graph cite this number when applying the Gauss-Bonnet theorem to normalize total curvature. The assignment follows directly from the combinatorial structure of the hypercube as the unit cell of the three-dimensional integer lattice.
claimThe three-cube graph $Q_3$ has vertex set cardinality eight: $|V(Q_3)| = 8$.
background
The CubeSpectrum module introduces the combinatorial and spectral properties of the three-cube Q3 as the unit cell of the integer lattice Z^3. This graph possesses eight vertices, twelve edges, and six faces, together with Laplacian eigenvalues 0, 2, 2, 2, 4, 4, 4, 6. Upstream results establish that the hyperoctahedral group order for dimension three is 48 and confirm the vertex count via the formula 2^D for D equal to three. LambdaRecDerivation applies the vertex count to distribute curvature packets and normalize the magnitude kappa.
proof idea
The definition consists of a direct constant assignment to the integer eight, matching the explicit count 2 cubed from the hypercube construction.
why it matters in Recognition Science
This vertex count supplies the initial combinatorial step in the GDerivationChain that derives the gravitational constant from Q3 geometry, Gauss-Bonnet total curvature equal to 4 pi, and cost balance. It supports the forcing chain step that fixes three spatial dimensions and appears in Planck scale matching as the number of ticks in the Gray cycle. The value closes the geometric input to the curvature cost calculations that yield the fine structure constant band.
scope and limits
- Does not establish the eigenvalue multiplicities of the graph Laplacian.
- Does not extend the count to hypercubes in dimensions other than three.
- Does not incorporate symmetry group actions on the vertex set.
- Does not derive any physical constants or coupling strengths.
Lean usage
theorem total_curvature_gauss_bonnet : Q3_vertices * angular_deficit_per_vertex = 2 * Real.pi * euler_S2 := by simp [Q3_vertices]formal statement (Lean)
20def Q3_vertices : ℕ := 8