alphaFramework
This definition assembles the AlphaFrameworkCert record by wiring the verified Q3 cube counts and first-order curvature structure into a single certificate. A physicist extending the Recognition Science α^{-1} series to higher-order voxel-seam terms would cite it to confirm the combinatorial base before computing δ₂. The construction is a direct record literal that applies the reflexivity theorems already proved for each field.
claimThe alpha framework certificate is the structure satisfying $Q_3$ faces = 6, $Q_3$ edges = 12, passive edges = 11, wallpaper groups = 17, face-wallpaper pairs = 102, curvature numerator = 103, integration measure dimension = 5, first-order correction $δ_1 = -103/(102 π^5)$, and 2-fold configurations = 10404.
background
The module develops higher-order voxel-seam corrections to the fine-structure constant in Recognition Science, targeting the residual ~8 ppm discrepancy after the base formula α^{-1} = α_seed - f_gap + δ_1. Upstream definitions supply the hypercube counts: cube_faces(d) = 2d and cube_edges(d) = d·2^{d-1}, specialized to D=3. The AlphaFrameworkCert structure collects these counts with the explicit first-order term δ_1 = -(curvature_numerator)/(face_wallpaper_pairs · π^{measure_dimension}) and the n=2 configuration count.
proof idea
The definition is a direct record construction that assigns each field of AlphaFrameworkCert to the corresponding reflexivity theorem: cube_faces to Q3_faces_eq, cube_edges to Q3_edges_eq, curvature_numerator to curvature_numerator_eq, delta_1 to delta_1_structure, and the remaining fields to their equality theorems. No further tactics are invoked.
why it matters in Recognition Science
The certificate certifies that the combinatorial and structural prerequisites for the δ_n series are in place, directly supporting the open δ₂ computation described in the module documentation. It anchors the Recognition Science expansion α^{-1} = α_seed - f_gap + Σ δ_n with the geometric seed 4π×11 and the curvature correction derived from 102 face-wallpaper pairs. The construction closes the proved portion of the framework while leaving the explicit second-order term unresolved.
scope and limits
- Does not compute the explicit value of the second-order correction δ₂.
- Does not prove convergence or alternation bounds for the infinite series.
- Does not perform numerical comparison of partial sums against CODATA.
- Does not derive or verify the exponential form of the correction.
formal statement (Lean)
213def alphaFramework : AlphaFrameworkCert where
214 cube_faces := Q3_faces_eq
proof body
Definition body.
215 cube_edges := Q3_edges_eq
216 passive := passive_edges_eq
217 wallpaper := rfl
218 fw_pairs := face_wallpaper_pairs_eq
219 curv_num := curvature_numerator_eq
220 meas_dim := measure_dimension_eq
221 delta_1_is_ratio := delta_1_structure
222 n2_configs := n_fold_configs_2
223 n2_reduced := reduced_configs_2
224 z2_sectors := Z2_sectors_eq
225
226end
227
228end AlphaHigherOrder
229end Constants
230end IndisputableMonolith