Q3_faces
The declaration supplies the face count of the three-dimensional cube as six. Researchers assembling combinatorial weights for the voxel-seam corrections in the Recognition Science derivation of the fine-structure constant would cite this count. The definition is a direct arithmetic evaluation of two times three.
claimLet $Q_3$ denote the three-dimensional cube. The number of faces of $Q_3$ equals 6.
background
This module develops higher-order voxel-seam corrections to the fine-structure constant inverse. The cube combinatorics supply the weights for the series terms in the expansion of alpha inverse as the seed term minus the gap weight plus the sum of corrections. The face count enters the definition of face-wallpaper pairs as their product with the number of wallpaper groups. Upstream results establish the same count via the Euler characteristic of the bounding sphere, the hypercube vertex formula for dimension three, and direct computation in the spectral emergence setting.
proof idea
The definition evaluates the arithmetic expression two multiplied by three.
why it matters in Recognition Science
This definition supplies the face count required by the AlphaFrameworkCert structure, which certifies that all structural elements are in place for the second-order correction computation. It fills the cube combinatorics step in the higher-order voxel-seam corrections to alpha inverse. The open question remains the explicit computation of the second-order term.
scope and limits
- Does not derive the count from the Euler characteristic of the sphere.
- Does not extend the count to higher-dimensional cubes.
- Does not incorporate wallpaper group weighting or integration measures.
formal statement (Lean)
67def Q3_faces : ℕ := 2 * 3