delta_1_structure
delta_1_structure asserts that the first-order curvature correction equals the negative ratio of the curvature numerator to the product of face-wallpaper pairs and pi to the measure dimension. Researchers assembling the alpha inverse series in Recognition Science cite this when adding the initial voxel-seam term to the geometric seed and gap weight. The proof is a single reflexivity step that matches the definition of delta_1 exactly.
claim$δ_1 = -κ / (f ⋅ π^d)$, where κ is the curvature numerator (face-wallpaper pairs plus active edges), f is the number of face-wallpaper pairs (cube faces times wallpaper groups), and d is the integration measure dimension (equal to 5).
background
The module develops higher-order voxel-seam corrections to α^{-1} inside the Recognition Science derivation. It begins from the geometric seed α_seed = 4π × 11, subtracts the gap weight f_gap = w_8 ln φ, and adds the series Σ δ_n. Each δ_n sums combinatorial face-wallpaper configurations on the Q_3 cube under the Z_2^5 half-period measure.
proof idea
The proof applies rfl. This one-line wrapper confirms the equality because the definition of delta_1 is written verbatim as the displayed ratio of curvature_numerator, face_wallpaper_pairs, and measure_dimension.
why it matters in Recognition Science
The result supplies the explicit first-order term δ_1 that feeds the alphaFramework certification of cube combinatorics. It closes the n=1 slot in the series α^{-1} = α_seed − f_gap + Σ δ_n, targeting the 8 ppm CODATA residual. The framework still leaves δ_2 computation open as the next required step.
scope and limits
- Does not evaluate the numerical value of δ_1.
- Does not construct or bound δ_n for n ≥ 2.
- Does not prove convergence of the full correction series.
- Does not include the seed or gap-weight terms in the equality.
formal statement (Lean)
104theorem delta_1_structure :
105 delta_1 = -(curvature_numerator : ℝ) / ((face_wallpaper_pairs : ℝ) * π ^ measure_dimension) :=
proof body
Decided by rfl or decide.
106 rfl
107