n_fold_configs_1
plain-language theorem explainer
The theorem states that the number of ordered one-fold face-wallpaper configurations on the cube equals 102. Researchers computing the series expansion for the inverse fine-structure constant cite this base case when assembling the first correction term. The proof is a direct reflexivity evaluation of the power definition at exponent one.
Claim. Let $N(n)$ be the number of ordered $n$-fold face-wallpaper configurations on the cube $Q_3$. Then $N(1)=102$.
background
The module develops higher-order voxel-seam corrections to the fine-structure constant in Recognition Science. The inverse fine-structure constant is written as the series α^{-1} = α_seed − f_gap + Σ_{n=1}^∞ δ_n, where each δ_n is a weighted sum over n-fold face-wallpaper configurations on Q3. The count function is defined by raising the number of face-wallpaper pairs to the nth power, so the n=1 case supplies the combinatorial factor for the first correction δ1 = −103/(102π^5).
proof idea
The proof is a one-line reflexivity that substitutes the definition of the count function as the power of the face-wallpaper pair count and evaluates the expression at n=1.
why it matters
This base case anchors the combinatorial prefactor for the first-order term δ1 in the alternating series for α^{-1} that targets the CODATA value 137.035999206. It completes the n=1 entry in the framework of cube combinatorics and Z2^5-weighted corrections. The open deliverable remains the explicit evaluation of the n=2 term.
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