n_fold_configs_2
plain-language theorem explainer
The theorem states that the number of ordered 2-fold face-wallpaper configurations on Q₃ equals 10404. Researchers computing the n=2 term in the δ_n series for higher-order corrections to α^{-1} would cite this count when assembling the combinatorial sum. The proof is a one-line native evaluation of the power expression that defines n_fold_configs.
Claim. Let $N(n)$ denote the number of ordered $n$-fold face-wallpaper configurations on the cube $Q_3$. Then $N(2) = 10404$.
background
In the Recognition Science framework the inverse fine-structure constant receives additive corrections δ_n from voxel-seam combinatorics on the Q₃ cube. Each δ_n is obtained by summing over n-fold face-wallpaper configurations, where the base enumeration of face-wallpaper pairs is raised to the n-th power to count ordered n-tuples. The module AlphaHigherOrder supplies the cube combinatorics (faces, edges, wallpaper groups) and the series structure α^{-1} = α_seed − f_gap + Σ_{n=1}^∞ δ_n, with the present result giving the explicit n=2 configuration count. Upstream, n_fold_configs is defined directly as face_wallpaper_pairs ^ n, while the module imports the self-reference structure from UniversalForcingSelfReference to certify the overall geometric setting.
proof idea
The proof is a one-line term that applies native_decide to evaluate the defining expression face_wallpaper_pairs ^ 2, returning the integer 10404.
why it matters
This theorem supplies the n=2 configuration count required by the δ_n series for the fine-structure constant correction and feeds the alphaFramework certificate that assembles the full combinatorial data for the module. It advances the open δ_2 computation by providing the exact multiplicity needed for the weighted sum over Q₃ configurations. The result sits inside the higher-order voxel-seam framework whose target is the ~8 ppm residual between the RS seed value and CODATA.
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