delta_1_numerator
plain-language theorem explainer
The declaration establishes that the numerator counting face-wallpaper pairs plus active edges on the three-dimensional cube equals 103. Researchers assembling the series for the fine-structure constant inverse in Recognition Science would cite this integer when evaluating the first-order voxel-seam correction. The proof is a direct one-line wrapper that invokes the reflexivity equality for the curvature numerator definition.
Claim. The numerator of the first-order curvature correction term satisfies $N = 103$, where $N$ is defined as the sum of face-wallpaper pairs and active edges on the 3-cube.
background
The module develops higher-order voxel-seam corrections to the fine-structure constant inverse. The RS derivation combines a geometric seed $4π × 11$, a gap weight $w_8 ln φ$, and the series $α^{-1} = α_{seed} - f_{gap} + Σ δ_n$, with the leading correction $δ_1 = -103/(102 π^5)$. The curvature numerator counts the total combinatorial contributions from face-wallpaper pairs and active edges under Euler closure on the 3-cube $Q_3$, using the $Z_2^5$ half-period integration measure of dimension $D + 2$.
proof idea
The proof is a one-line wrapper that applies the upstream theorem curvature_numerator_eq.
why it matters
This supplies the explicit numerator for the $δ_1$ term in the alternating series that targets the CODATA value 137.035999206. It completes the first-order combinatorial count inside the higher-order corrections module, leaving the $δ_2$ computation as the principal open deliverable. The result sits within the framework landmarks of the eight-tick octave and the phi-ladder, though it addresses only the constant correction rather than mass or dimension derivations.
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