face_wallpaper_pairs_eq
plain-language theorem explainer
The equality states that the product of the number of faces on the three-cube and the number of wallpaper groups equals exactly 102. Researchers assembling the combinatorial inputs for higher-order corrections to α^{-1} in Recognition Science cite this when building the alphaFramework certificate. The proof is a direct reflexivity reduction from the definition of the face-wallpaper pair count.
Claim. Let $f$ be the number of faces of the three-dimensional cube and $w$ the number of wallpaper groups. Then $f w = 102$.
background
The module develops higher-order voxel-seam corrections to α^{-1} within the Recognition Science derivation. The base ingredients are the geometric seed α_seed = 4π × 11, the gap weight f_gap = w_8 ln φ, and the curvature terms δ_n that appear in the series α^{-1} = α_seed − f_gap + Σ δ_n. Each δ_n is a finite sum over n-fold face-wallpaper configurations on Q_3, weighted by the Z_2^5 half-period measure.
proof idea
The proof is a one-line wrapper that applies reflexivity to the definition face_wallpaper_pairs := Q3_faces * wallpaper_groups, which evaluates directly to 102.
why it matters
This supplies the integer 102 that appears in the denominator of the first-order correction δ_1 = −103/(102 π^5) and is referenced by the alphaFramework certificate. It closes one piece of the proved cube combinatorics that supports the full δ_n series in the Recognition Science framework, where D = 3 fixes the underlying Q_3 structure. The remaining open task is explicit computation of δ_2.
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