n_fold_configs
plain-language theorem explainer
The definition specifies the cardinality of ordered n-fold face-wallpaper configurations on the 3-cube as the nth power of the base face-wallpaper pair count. Researchers computing higher-order voxel-seam corrections to the fine-structure constant in Recognition Science cite this when indexing the summation space for the nth term δ_n in the series for α^{-1}. It is realized as a direct exponentiation of the upstream combinatorial base.
Claim. Let $k$ denote the number of face-wallpaper pairs on the 3-cube. The number of ordered n-fold face-wallpaper configurations is $k^n$.
background
This module develops the framework for higher-order corrections to α^{-1} in Recognition Science, starting from the geometric seed α_seed = 4π × 11, gap weight f_gap = w_8 · ln φ, and curvature term δ_1. The cube combinatorics supply the base count via the product of Q3 faces and wallpaper groups. The module states that each δ_n is a finite combinatorial sum over n-fold face-wallpaper configurations on Q₃, weighted by the Z₂⁵ half-period integration measure. The upstream definition establishes the base as the product of the number of faces on Q3 and the number of wallpaper groups.
proof idea
The definition is a one-line wrapper that raises the face-wallpaper pair count to the power n.
why it matters
This supplies the configuration cardinality used to index the sum in δ_n and to populate the AlphaFrameworkCert structure that certifies all elements are in place for δ_2 computation. It advances the series α^{-1} = α_seed − f_gap + Σ_{n=1}^∞ δ_n toward closing the residual discrepancy with CODATA. The module notes that the series framework is proved while explicit δ_2 computation remains open.
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