Q3_vertices
The declaration sets the vertex count of the three-dimensional cube Q₃ to eight. Workers on the Recognition Science derivation of α⁻¹ cite it when counting lattice sites for voxel-seam corrections and curvature packets. The definition is a direct power computation that matches the general D-cube vertex formula specialized to D=3.
claimThe number of vertices of the three-cube $Q_3$ is defined to be $2^3$.
background
The AlphaHigherOrder module develops higher-order corrections to the fine-structure constant α⁻¹, starting from the geometric seed 4π×11, the gap weight w₈ ln φ, and curvature terms δ_n summed over n-fold face-wallpaper pairs on Q₃. Q₃ supplies the underlying 3-dimensional lattice whose vertices carry the curvature packets in the Gauss-Bonnet application. Upstream results in LambdaRecDerivation and PlanckScaleMatching already record that the same cube has eight vertices, identified with the eight ticks of the Gray cycle and the order of the hyperoctahedral group B₃.
proof idea
The definition is a direct one-line computation that instantiates the standard vertex count 2^D for a D-cube at D=3.
why it matters in Recognition Science
This supplies the base combinatorial datum for the GDerivationChain structure, which links Q₃ geometry through Gauss-Bonnet total curvature 4π to the unique λ_rec that yields G = λ_rec² c³/(π ℏ). It anchors the cube combinatorics block inside the α⁻¹ series framework, where each δ_n sums over configurations weighted by the Z₂⁵ measure. The count 8 directly realizes the eight-tick octave (T7) and the spatial dimension D=3 (T8) of the forcing chain. The open δ₂ computation remains the next required step downstream.
scope and limits
- Does not compute any δ_n for n>1.
- Does not evaluate the full α⁻¹ series or its convergence bounds.
- Does not encode the curvature cost per vertex or the λ_rec balance.
- Does not reference the CODATA target value.
formal statement (Lean)
59def Q3_vertices : ℕ := 2^3