AlphaFrameworkCert
plain-language theorem explainer
AlphaFrameworkCert records the 3-cube counts (6 faces, 12 edges), 17 wallpaper groups, 102 face-wallpaper pairs, curvature numerator 103, measure dimension 5, the explicit ratio δ₁ = −103/(102 π^5), the n=2 configuration numbers 10404 and 217, and 32 Z₂ sectors. A researcher extending the fine-structure series to higher-order voxel-seam corrections would cite this certificate to lock the n=1 inputs before summing δ₂. It is realized as a structure definition that directly asserts the numerical identities and the curvature formula.
Claim. The structure asserts that the 3-cube has 6 faces and 12 edges, 11 passive edges, 17 wallpaper groups, 102 face-wallpaper pairs, curvature numerator 103, measure dimension 5, the first correction satisfies $δ_1 = -103/(102 π^5)$, there are 10404 two-fold configurations with 217 reduced, and 32 $Z_2$ sectors.
background
Q₃ is the 3-dimensional hypercube. Upstream definitions give cube_faces(d) = 2d so that Q3_faces = 6, cube_edges(d) = d·2^{d-1} so that Q3_edges = 12, and wallpaper_groups = 17 (the classical count of distinct 2D crystallographic groups). Face-wallpaper pairs are their product, 102. Curvature numerator is defined as face-wallpaper pairs plus active edges, producing 103. The first correction is then δ₁ = −(curvature numerator) / (face-wallpaper pairs · π^{measure dimension}) with measure dimension = 5. Z₂ sectors count the half-period sectors in the integration measure, fixed at 32.
proof idea
This is a structure definition that directly encodes the combinatorial identities and the explicit formula for δ₁. It populates each field from the upstream definitions Q3_faces, Q3_edges, wallpaper_groups, face_wallpaper_pairs, curvature_numerator, delta_1, and the n-fold configuration counts already established in the module.
why it matters
The certificate supplies the verified geometric and first-order data that alphaFramework packages for use in the higher-order α^{-1} series. It closes the framework for δ_n at arbitrary order while leaving the explicit δ₂ sum as the remaining open computation. The construction sits inside the Recognition Science derivation of the fine-structure constant, where the ~8 ppm residual after the first correction motivates the full alternating series Σ δ_n.
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